![34 Topology
that arbitrary continuous transformations can distort distances in a very flexible way; it
is not possible to define topological properties based on transformations that preserve
the distance function. One wants, in a sense, to preserve “infinitesimal” distances, i.e.
points which are infinitely close together should remain infinitely close together. Of
course, there is no notion of infinitesimal closeness for pairs of points. However, in a
metric space one can speak of a point being infinitesimally close to a set. For (X, d) a
metric space, x ∈ X, and U ⊆ X, we say that x is infinitesimally close to U if, for every
0, there is a point in u ∈ U such that d(x, u) ≤ . The collection of all points which
are infinitesimally close to a set U is called the closure of U and is denoted by U. Note
that if a point is infinitesimally close to a set U then any continuous transformation will
preserve that property, even if it does not preserve distances.
Example 3.11 The closure of the open interval (0, 1) is the closed interval [0, 1].
Example 3.12 The closure of the set Rn − {0} in Rn is the entire set Rn.
Example 3.13 The closure of the set of all points of the form n
2k , where 0 ≤ n ≤ 2k
and k ≥ 0, is the entire closed interval [0, 1].
Closure can be viewed as an operator on the subsets of Rn. It turns out to be idempotent
in the sense that U = U. One can develop the notion of a topological space using only an
abstract closure operator on the family of subsets, with certain properties. However, it is
more conventional to develop the notion using the ideas of closed and open sets. A closed
set V in Rn is a subset for which V = V. A subset U of Rn is open if its complement in X
is closed, i.e. if U = X V for some closed set V. Another characterization is that a set is
open if and only if it is an arbitrary union of open balls Br (x) = {y ∈ Rn | d(x, y) r}.
The open sets in a metric space Rn satisfy the following three properties.
• ∅ and X are both open sets in X.
• Arbitrary unions of open sets are open.
• Finite intersections of open sets are open.
The identification of these properties leads us to the notion of a topological space.
Definition 3.14 A topological space is a pair (X, U), where X is a set and U is a
family of subsets (called the open sets) of X satisfying the following three properties.
1. ∅ ∈ U and X ∈ U.
2. For any family {Uα}α∈A of open sets, where A is any parameter set,
S
α∈A Uα is also
in U.
3. For any family {Uα}α∈A of open sets, where A is any finite parameter set,
T
α∈A Uα
is also in U.
The family U is called a topology on X. A subset C ⊆ X is closed if X C is open.
Here are some examples of open and closed sets in Rn.
Example 3.15 Open (respectively closed) intervals are open (respectively closed) sets
in R.](https://image.slidesharecdn.com/22333107-250622052047-1eb352b4/85/Topological-Data-Analysis-With-Applications-Carlsson-Gunnar-Vejdemojohansson-51-320.jpg)
![3.2 Qualitative and Quantitative Properties 37
Example 3.32 Let X be the torus in three-dimensional space defined by the equation
(c
q
x2 + y2)2
+ z2
= a2
,
where 0 a c. Let Y ⊆ R4 be the subspace defined by the equations
(
x2 + y2 = 1,
z2 + w2 = 1.
We define a map f : X → Y via the formula
f (x, y, z) =
x
k(x, y)k2
,
y
k(x, y)k2
,
c k(x, y)k2
a
,
z
a
!
,
where k(x, y)k2 denotes
p
x2 + y2. It is easy to verify that this map has its image in
Y, and it is continuous since it is defined by simple formulas. It turns out that f is a
homeomorphism.
One way to prove that the map f in the above example is a homeomorphism is to
exhibit a formula for the inverse. This is somewhat involved even in this case and can
become much more so in more complicated examples. There is a simple statement that
allows one to bypass that process.
Proposition 3.33 Let f : X → Y be a continuous map, where X and Y are subsets
of Rm and Rn respectively. We say a subset of Rn is bounded if it is contained in a ball
BR (0) = {v | kvk R} for some R 0. Suppose that X and Y are closed and bounded.
If the map f is bijective as a map of sets, then f is a homeomorphism.
In Example 3.32 above, it is easy to check that both sets are closed and bounded and
that f , regarded as a map of sets, is bijective. We are therefore able to conclude that it is
a homeomorphism.
Remark 3.34 Note that the two descriptions of a topological space have different
strengths. The first description sits in three-dimensional space and can therefore be
readily visualized. The second is much simpler, in that the equations that define it are
much simpler. Depending on the kind of analysis one is doing, different coordinate
systems may be useful.
We should also ask how we can show that two spaces are not homeomorphic. Let
X = [0, 1] ⊆ R and let Y ⊆ R2 denote the closed unit ball {(x, y) ∈ R2 | x2 + y2 ≤ 1}.
These two spaces appear to be very different and we therefore suspect that they are
not homeomorphic, but in order to be convinced of this we must try to find some
topological property that applies to one but not to the other. Reasoning intuitively, we
observe that by removing a single point from X, such as the point 1
2 , we can form a
space which is disconnected in the sense that is breaks into two disjoint open pieces,
namely [0, 1
2 ) and (1
2, 1], while removing any single point from Y will always leave it in
a single connected piece. We have not formally defined connectedness yet (we will do so
in Section 3.2.8), but this informal argument suggests how we can develop topological
properties that enable discrimination between spaces, even when we allow ourselves to
modify the spaces by arbitrary coordinate changes.](https://image.slidesharecdn.com/22333107-250622052047-1eb352b4/85/Topological-Data-Analysis-With-Applications-Carlsson-Gunnar-Vejdemojohansson-54-320.jpg)
![North Sea and the English Channel but also for the Mediterranean
Sea. They cultivated what is known as the sea sense, and
developed shipbuilding and the art of navigation in accordance with
local needs. When Julius Cæsar came into conflict with the Veneti of
Brittany he tells that their vessels were greatly superior to those of
the Romans. The bodies of the ships, he says, were built entirely
of oak, stout enough to withstand any shock or violence.... Instead
of cables for their anchors they used iron chains.... The encounter of
our fleet with these ships was of such a nature that our fleet
excelled in speed alone, and the plying of oars; for neither could our
ships injure theirs with their rams, so great was their strength, nor
was a weapon easily cast up to them owing to their height.... About
220 of their ships ... sailed forth from the harbour. In this great
allied fleet were vessels from our own country.[55]
It must not be imagined that the sea sense was cultivated because
man took pleasure in risking the perils of the deep. It was stern
necessity that at the beginning compelled him to venture on long
voyages. After England was cut off from France the peoples who had
adopted the Neolithic industry must have either found it absolutely
necessary to seek refuge in Britain, or were attracted towards it by
reports of prospectors who found it to be suitable for residence and
trade.](https://image.slidesharecdn.com/22333107-250622052047-1eb352b4/85/Topological-Data-Analysis-With-Applications-Carlsson-Gunnar-Vejdemojohansson-58-320.jpg)
![CHAPTER VIII
Neolithic Trade and Industries
Attractions of Ancient Britain—Romans search for Gold,
Silver, Pearls, c.—The Lure of Precious Stones and
Metals—Distribution of Ancient British Population—
Neolithic Settlements in Flint-yielding Areas—Trade in Flint
—Settlements on Lias Formation—Implements from Basic
Rocks—Trade in Body-painting Materials—Search for
Pearls—Gold in Britain and Ireland—Agriculture—The
Story of Barley—Neolithic Settlers in Ireland—Scottish
Neolithic Traders—Neolithic Peoples not Wanderers—
Trained Neolithic Craftsmen.
The drift of peoples into Britain which began in Aurignacian times
continued until the Roman period. There were definite reasons for
early intrusions as there were for the Roman invasion. Britain
contains to reward the conqueror, Tacitus wrote,[56]
mines of gold
and silver and other metals. The sea produces pearls. According to
Suetonius, who at the end of the first century of our era wrote the
Lives of the Cæsars, Julius Cæsar invaded Britain with the desire to
enrich himself with the pearls found on different parts of the coast.
On his return to Rome he presented a corselet of British pearls to
the goddess Venus. He was in need of money to further his political
ambitions. He found what he required elsewhere, however. After the
death of Queen Cleopatra sufficient gold and silver flowed to Rome
from Egypt to reduce the loan rate of interest from 12 to 4 per cent.
Spain likewise contributed its share to enrich the great predatory
state of Rome.[57]](https://image.slidesharecdn.com/22333107-250622052047-1eb352b4/85/Topological-Data-Analysis-With-Applications-Carlsson-Gunnar-Vejdemojohansson-59-320.jpg)
![Long ages before the Roman period the early peoples entered Britain
in search of pearls, precious stones, and precious metals because
these had a religious value. The Celts of Gaul offered great
quantities of gold to their deities, depositing the precious metals in
their temples and in their sacred lakes. Poseidonius of Apamea tells
that after conquering Gaul the Romans put up these sacred lakes to
public sale, and many of the purchasers found quantities of solid
silver in them. He also says that gold was similarly placed in these
lakes.[58]
Apparently the Celts believed, as did the Aryo-Indians, that
gold was a form of the gods and fire, light, and immortality, and
that it was a life giver.[59]
Personal ornaments continued to have a
religious value until Christian times.
FLINT LANCE-HEADS FROM IRELAND (British Museum)](https://image.slidesharecdn.com/22333107-250622052047-1eb352b4/85/Topological-Data-Analysis-With-Applications-Carlsson-Gunnar-Vejdemojohansson-60-320.jpg)
![Mr. W. J. Perry, of Manchester University, who has devoted special
attention to the study of the distribution of megalithic monuments,
has been drawing attention to the interesting association of these
monuments with geological formations.[60]
In the Avebury district](https://image.slidesharecdn.com/22333107-250622052047-1eb352b4/85/Topological-Data-Analysis-With-Applications-Carlsson-Gunnar-Vejdemojohansson-64-320.jpg)
![thence went northward to Scotland. As Irish flints and ground axe-
heads occur chiefly in Ulster, it may be that the drift of early
Neolithic settlers into County Antrim, in which gold was also found,
was from south-western Scotland. The Neolithic settlement at
Whitepark Bay, five miles from the Giant's Causeway, was embedded
at a considerable depth, showing that there has been a sinking of
the land in this area since the Neolithic industry was introduced.
Neolithic remains are widely distributed over Scotland, but these
have not received the intensive study devoted to similar relics in
England. Mr. Ludovic Mann, the Glasgow archæologist, has, however,
compiled interesting data regarding one of the local industries that
bring out the resource and activities of early man. On the island of
Arran is a workable variety of the natural volcanic glass, called pitch-
stone, that of other parts of Scotland and of Ireland being too much
cracked into small pieces to be of use. It was used by the Neolithic
settlers in Arran for manufacturing arrowheads, and as it was
imported into Bute, Ayrshire, and Wigtownshire, a trade in this
material must have existed. If, writes Mr. Mann, the stone was
not locally worked up into implements in Bute, it was so manipulated
on the mainland, where workshops of the Neolithic period and the
immediately succeeding overlap period yielded long fine flakes,
testifying to greater expertness in manufacturing there than is
shown by the remains in the domestic sites yet awaiting adequate
exploration in Arran. The explanation may be that the Wigtownshire
flint knappers, accustomed to handle an abundance of flint, were
more proficient than in most other places, and that the pitch-stone
was brought to them as experts, because the material required even
more skilful handling than flint.[61]
In like manner obsidian, as has
been noted, was imported into Crete from the island of Melos by
seafarers, long before the introduction of metal working.[62]
It will be seen that the Neolithic peoples were no mere wandering
hunters, as some have represented them to have been, but they had
their social organization, their industries, and their system of trading
by land and sea. They settled not only in those areas where they](https://image.slidesharecdn.com/22333107-250622052047-1eb352b4/85/Topological-Data-Analysis-With-Applications-Carlsson-Gunnar-Vejdemojohansson-66-320.jpg)
![Broad-head (Brachycephalic) Skull
Both these specimens were found in Round Barrows in the East Riding
of Yorkshire
The discovery of metals in Britain and Ireland was, no doubt, first
made by prospectors who had obtained experience in working them
elsewhere. They may have simply come to exploit the country. How
these men conducted their investigations is indicated by the report
found in a British Museum manuscript, dating from about 1603, in
which the prospector gives his reason for believing that gold was to
be found on Crawford Moor in Lanarkshire. He tells that he saw
among the rocks what Scottish miners call mothers and English
miners leaders or metalline fumes. It was believed that the
fumes arose from veins of metal and coloured the rocks as smoke
passing upward through a tunnel blackens it, and leaves traces on
the outside. He professed to be able to distinguish between the
colours left by fumes of iron, lead, tin, copper, or silver. On
Crawford Moor he found sparr, keel, and brimstone between rocks,
and regarded this discovery as a sure indication that gold was in
situ. The mothers or leaders were more pronounced than any he
had ever seen in Cornwall, Somersetshire, about Keswick, or any
other mineral parts wheresoever I have travelled.[63]
Gold was found](https://image.slidesharecdn.com/22333107-250622052047-1eb352b4/85/Topological-Data-Analysis-With-Applications-Carlsson-Gunnar-Vejdemojohansson-70-320.jpg)
![in this area of Lanarkshire in considerable quantities, and was no
doubt worked in ancient times. Of special interest in this connection
is the fact that it was part of the territory occupied by Damnonians,
[64]
who appear to have been a metal-working people. Besides
occupying the richest metal-yielding area in Scotland, the
Damnonians were located in Devon and Cornwall, and in the east-
midland and western parts of Ireland, in which gold, copper, and tin-
stone were found as in south-western England. The Welsh Dyfneint
(Devon) is supposed by some to be connected with a form of this
tribal name. Another form in a Yarrow inscription is Dumnogeni. In
Ireland Inber Domnann is the old name of Malahide Bay north of
Dublin. Domnu, the genitive of which is Domnann, was the name of
an ancient goddess. In the Irish manuscripts these people are
referred to as Fir-domnann,[65]
and associated with the Fir-bolg (the
men with sacks). A sack-carrying people are represented in Spanish
rock paintings that date from the Azilian till early Bronze Age
times. In an Irish manuscript which praises the fair and tall people,
the Fir-bolg and Fir-domnann are included among the black-eyed
and black-haired people, the descendants of slaves and churls, and
the promoters of discord among the people.
The reference to slaves is of special interest because the lot of the
working miners was in ancient days an extremely arduous one. In
one of his collected records which describes the method of the
greatest antiquity Diodorus Siculus (a.d. first century) tells how
gold-miners, with lights bound on their foreheads, drove galleries
into the rocks, the fragments of which were carried out by frail old
men and boys. These were broken small by men in the prime of life.
The pounded stone was then ground in handmills by women: three
women to a mill and to each of those who bear this lot, death is
better than life. Afterwards the milled quartz was spread out on an
inclined table. Men threw water on it, work it through their fingers,
and dabbed it with sponges until the lighter matter was removed
and the gold was left behind. The precious metal was placed in a
clay crucible, which was kept heated for five days and five nights. It
may be that the Scandinavian references to the nine maidens who](https://image.slidesharecdn.com/22333107-250622052047-1eb352b4/85/Topological-Data-Analysis-With-Applications-Carlsson-Gunnar-Vejdemojohansson-71-320.jpg)
![turn the handle of the world mill which grinds out metal and soil,
and the Celtic references to the nine maidens who are associated
with the Celtic cauldron, survive from beliefs that reflected the habits
and methods of the ancient metal workers.
It is difficult now to trace the various areas in which gold was
anciently found in our islands. But this is not to be wondered at. In
Egypt there were once rich goldfields, especially in the Eastern
Desert, where about 100 square miles were so thoroughly worked in
ancient times that only the merest traces of gold remain.[66]
Gold,
as has been stated, was formerly found in south-western England,
North Wales, and, as historical records, archæological data, and
place names indicate, in various parts of Scotland and Ireland.
During the period of the Great Thaw a great deal of alluvial gold
must have distributed throughout the country. Silver was found in
various parts. In Sutherland it is mixed with gold as it is elsewhere
with lead. Copper was worked in a number of districts where the
veins cannot in modern times be economically worked, and tin was
found in Ireland and Scotland as well as in south-western England,
where mining operations do not seem to have been begun, as
Principal Sir John Rhys has shown,[67]
until after the supplies of
surface tin were exhausted. Of special interest in connection with
this problem is the association of megalithic monuments with ancient
mine workings. An interesting fact to be borne in mind in connection
with these relics of the activities and beliefs of the early peoples is
that they represent a distinct culture of complex character. Mr. T. Eric
Peet[68]
shows that the megalithic buildings occupy a very
remarkable position along a vast seaboard which includes the
Mediterranean coast of Africa and the Atlantic coast of Europe. In
other words, they lie entirely along a natural sea route. He gives
forcible reasons for arriving at the conclusion that it is impossible to
consider megalithic building as a mere phase through which many
nations passed, and it must therefore have been a system
originating with one race, and spreading far and wide, owing either
to trade influence or migration. He adds:](https://image.slidesharecdn.com/22333107-250622052047-1eb352b4/85/Topological-Data-Analysis-With-Applications-Carlsson-Gunnar-Vejdemojohansson-72-320.jpg)
![Derbyshire is precisely that of old mines; in Cornwall the megalithic
structures are mainly grouped west of Falmouth, precisely in that
district where mining has always been most active.
Pearls, amber, coral, jet, c., were searched for as well as metals.
The megalithic monuments near pearling rivers, in the vicinity of
Whitby, the main source of jet, and in Denmark and the Baltic area
where amber was found were, in all likelihood, erected by people
who had come under the spell of the same ancient culture.
When, therefore, we come to deal with groups of monuments in
areas which were unsuitable for agriculture and unable to sustain
large populations, a reasonable conclusion to draw is that precious
metals, precious stones, or pearls were once found near them. The
pearling beds may have been destroyed or greatly reduced in value,
[69]
or the metals may have been worked out, leaving but slight if any
indication that they were ever in situ. Reference has been made to
the traces left by ancient miners in Egypt where no gold is now
found. In our own day rich gold fields in Australia and North America
have been exhausted. It would be unreasonable for us to suppose
that the same thing did not happen in our country, even although
but slight traces of the precious metal can now be obtained in areas
which were thoroughly explored by ancient miners.
When early man reached Scotland in search of suitable districts in
which to settle, he was not likely to be attracted by the barren or
semi-barren areas in which nature grudged soil for cultivation, where
pasture lands were poor and the coasts were lashed by great billows
for the greater part of the year, and the tempests of winter and
spring were particularly severe. Yet in such places as Carloway,
fronting the Atlantic on the west coast of Lewis, and at Stennis in
Orkney, across the dangerous Pentland Firth, are found the most
imposing stone circles north of Stonehenge and Avebury. Traces of
tin have been found in Lewis, and Orkney has yielded traces of lead,
including silver-lead, copper and zinc, and has flint in glacial drift.
Traces of tin have likewise been found on the mainlands of Ross-
shire and Argyllshire, in various islands of the Hebrides and in](https://image.slidesharecdn.com/22333107-250622052047-1eb352b4/85/Topological-Data-Analysis-With-Applications-Carlsson-Gunnar-Vejdemojohansson-74-320.jpg)
![Stirlingshire. The great Stonehenge circle is like the Callernish and
Stennis circles situated in a semi-barren area, but it is an area where
surface tin and gold were anciently obtained. One cannot help
concluding that the early people, who populated the wastes of
ancient Britain and erected megalithic monuments, were attracted
by something more tangible than the charms of solitude and wild
scenery. They searched for and found the things they required. If
they found gold, it must be recognized that there was a
psychological motive for the search for this precious metal. They
valued gold, or whatever other metal they worked in bleak and
isolated places, because they had learned to value it elsewhere.
Who were the people that first searched for, found, and used metals
in Western Europe? Some have assumed that the natives themselves
did so as a matter of course. Such a theory is, however, difficult to
maintain. Gold is a useless metal for all practical purposes. It is too
soft for implements. Besides, it cannot be found or worked except by
those who have acquired a great deal of knowledge and skill. The
men who first washed it from the soil in Britain must have
obtained the necessary knowledge and skill in a country where it
was more plentiful and much easier to work, and where—and this
point is a most important one—the magical and religious beliefs
connected with gold have a very definite history. Copper, tin, and
silver were even more difficult to find and work in Britain. The
ancient people who reached Britain and first worked metals or
collected ores were not the people who were accustomed to use
implements of bone, horn, and flint, and had been attracted to its
shores merely because fish, fowl, deer, and cows, were numerous.
The searchers for metals must have come from centres of Eastern
civilization, or from colonies of highly skilled peoples that had been
established in Western Europe. They did not necessarily come to
settle permanently in Britain, but rather to exploit its natural riches.
This conclusion is no mere hypothesis. Siret,[70]
the Belgian
archæologist, has discovered in southern Spain and Portugal traces
of numerous settlements of Easterners who searched for minerals,](https://image.slidesharecdn.com/22333107-250622052047-1eb352b4/85/Topological-Data-Analysis-With-Applications-Carlsson-Gunnar-Vejdemojohansson-75-320.jpg)
![Valentine
THE RING OF STENNIS, ORKNEY (see page 94)
Among the archæological finds, which prove that the Easterners
settled in Iberia before bronze working was introduced among the
natives, are idol-like objects made of hippopotamus ivory from
Egypt, a shell (Dentalium elephantum) from the Red Sea, objects
made from ostrich eggs which must have been carried to Spain from
Africa, alabaster perfume flasks, cups of marble and alabaster of
Egyptian character which had been shaped with copper implements,
Oriental painted vases with decorations in red, black, blue, and
green,[71]
mural paintings on layers of plaster, feminine statuettes in
alabaster which Siret considers to be of Babylonian type, for they
differ from Ægean and Egyptian statuettes, a cult object (found in
graves) resembling the Egyptian ded amulet, c. The Iberian burial
places of these Eastern colonists have arched cupolas and entrance
corridors of Egyptian-Mycenæan character.
Of special interest are the beautifully worked flints associated with
these Eastern remains in Spain and Portugal. Siret draws attention to](https://image.slidesharecdn.com/22333107-250622052047-1eb352b4/85/Topological-Data-Analysis-With-Applications-Carlsson-Gunnar-Vejdemojohansson-77-320.jpg)
![the exploitation for its exclusive and personal profit of the virgin
riches of the latter. These early Westerners had no idea of the use
and value of the metals lying on the surface of their native land,
while the Orientals valued them, were in need of them, and were
anxious to obtain them. As Siret puts it:
The West was a cow to be milked, a sheep to be fleeced,
a field to be cultivated, a mine to be exploited.
In the traditions preserved by classical writers, there are references
to the skill and cunning of the Phœnicians in commerce, and in the
exploitation of colonies founded among the ignorant Iberians. They
did not inform rival traders where they found metals. Formerly, as
Strabo says, the Phœnicians monopolized the trade from Gades
(Cadiz) with the islanders (of the Cassiterides); and they kept the
route a close secret. A vague ancient tradition is preserved by Pliny,
who tells that tin was first fetched from Cassiteris (the tin island) by
Midacritus.[72]
We owe it to the secretive Phœnicians that the
problem of the Cassiterides still remains a difficult one to solve.
To keep the native people ignorant the Easterners, Siret believes,
forbade the use of metals in their own colonies. A direct result of
this policy was the great development which took place in the
manufacture of the beautiful flint implements already referred to.
These the natives imitated, never dreaming that they were imitating
some forms that had been developed by a people who used copper
in their own country. When, therefore, we pick up beautiful Neolithic
flints, we cannot be too sure that the skill displayed belongs entirely
to the Stone Age, or that the flints evolved from earlier native
forms in those areas in which they are found.
The Easterners do not appear to have extracted the metals from
their ores either in Iberia or in Northern Europe. Tin-stone and
silver-bearing lead were used for ballast for their ships, and they
made anchors of lead. Gold washed from river beds could be easily
packed in small bulk. A people who lived by hunting and fishing were
not likely to be greatly interested in the laborious process of gold-](https://image.slidesharecdn.com/22333107-250622052047-1eb352b4/85/Topological-Data-Analysis-With-Applications-Carlsson-Gunnar-Vejdemojohansson-79-320.jpg)
![settlements of Easterners in Brittany were invaded at one and the
same time by the enemy and the ocean. Other refugees from the
colonies may have settled in Etruria, and founded the Etruscan
civilization. Etruscan menhirs resemble those of the south of France,
while the Etruscan crozier or wand, used in the art of augury,
resembles the croziers of the megaliths, c., of France, Spain, and
Portugal. There are references in Scottish Gaelic stories to magic
wands possessed by wise women, and by the mothers of
Cyclopean one-eyed giants. Ammianus Marcellinus, quoting
Timagenes,[73]
attributes to the Druids the statement that part of the
inhabitants of Gaul were indigenous, but that some had come from
the farthest shores and districts across the Rhine, having been
expelled from their own lands by frequent wars and the
encroachments of the ocean.
The bronze-using peoples who established overland trade routes in
Europe, displacing in some localities the colonies of Easterners and
isolating others, must have instructed the natives of Western Europe
how to mine and use metals. Bronze appears to have been
introduced into Britain by traders. That the ancient Britons did not
begin quite spontaneously to work copper and tin and manufacture
bronze is quite evident, because the earliest specimens of British
bronze which have been found are made of ninety per cent of
copper and ten per cent of tin as on the Continent. Now, since a
knowledge of the compound, wrote Dr. Robert Munro, implies a
previous acquaintance with its component elements, it follows that
progress in metallurgy had already reached the stage of knowing the
best combination of these metals for the manufacture of cutting
tools before bronze was practically known in Britain.[74]
The furnaces used were not invented in Britain. Professor Gowland
has shown that in Europe and Asia the system of working mines and
melting metals was identical in ancient times. Summarizing Professor
Gowland's articles in Archæologia and the Journal of the Royal
Anthropological Institute, Mr. W. J. Perry writes in this connection:[75]
The furnaces employed were similar; the crucibles were of the](https://image.slidesharecdn.com/22333107-250622052047-1eb352b4/85/Topological-Data-Analysis-With-Applications-Carlsson-Gunnar-Vejdemojohansson-82-320.jpg)
![man is restored in his original form.[76]
A similar belief, which is
widespread, is that bees have their origin as maggots placed in
trees. One form of the story was taken over by the early Christians,
which tells that Jesus was travelling with Peter and Paul and asked
hospitality from an old woman. The woman refused it and struck
Paul on the head. When the wound putrified maggots were
produced. Jesus took the maggots from the wound and placed them
in the hollow of a tree. When next they passed that way, Jesus
directed Paul to look in the tree hollow where, to his surprise, he
found bees and honey sprung from his own head.[77]
The custom of
placing crape on hives and telling the bees when a death takes
place, which still survives in the south of England and in the north of
Scotland, appears to be connected with the ancient belief that the
maggot, bee, and tree were connected with the sacred animal and
the sacred stone in which was the spirit of a deity. Sacred trees and
sacred stones were intimately connected. Tacitus tells us that the
Romans invaded Mona (Anglesea), they destroyed the sacred groves
in which the Druids and black-robed priestesses covered the altars
with the blood of captives.[78]
There are a number of dolmens on this
island and traces of ancient mine-workings, indicating that it had
been occupied by the early seafarers who colonized Britain and
Ireland and worked metals. A connection between the tree cult of
the Druids and the cult of the builders of megaliths is thus suggested
by Tacitus, as well as by the Irish evidence regarding the Ulster idol
Crom Chonnaill, referred to above (see also Chapter XII).
Who were the people that followed the earliest Easterners and
visited our shores to search like them for metals and erect megalithic
monuments? It is impossible to answer that question with certainty.
There were after the introduction of bronze working, as has been
indicated, intrusions of aliens. These included the introducers of the
short-barrow method of burial and the later introducers of burial by
cremation. It does not follow that all intrusions were those of
conquerors. Traders and artisans may have come with their families
in large numbers and mingled with the earlier peoples. Some
intruders appear to have come by overland routes from southern](https://image.slidesharecdn.com/22333107-250622052047-1eb352b4/85/Topological-Data-Analysis-With-Applications-Carlsson-Gunnar-Vejdemojohansson-84-320.jpg)
Topological Data Analysis With Applications Carlsson Gunnar Vejdemojohansson
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- 6. Topological Data Analysis with Applications The continued and dramatic rise in the size of data sets has meant that new methods are required to model and analyze them. This timely account introduces topological data analysis (TDA), a method for modeling data by geometric objects, namely graphs and their higher-dimensional versions, simplicial complexes. The authors outline the necessary background material on topology and data philosophy for newcomers, while more complex concepts are highlighted for advanced learners. The book covers all the main TDA techniques, including persistent homology, cohomology, and Mapper. The final section focuses on the diverse applications of TDA, examining a number of case studies ranging from monitoring the progression of infectious diseases to the study of motion capture data. Mathematicians moving into data science, as well as data scientists or computer scientists seeking to understand this new area, will appreciate this self-contained resource which explains the underlying technology and how it can be used. Gunnar Carlsson is Professor Emeritus at Stanford University. He received his doctoral degree from Stanford in 1976, and has taught at the University of Chicago, at the University of California, San Diego, at Princeton University, and, since 1991, at Stanford University. His work within mathematics has been concentrated in algebraic topology, and he has spent the last 20 years on the development of topological data analysis. He is also passionate about the transfer of scientific findings to real-world applications, leading him to found the topological data analysis-based company Ayasdi in 2008. Mikael Vejdemo-Johansson is Assistant Professor in the Department of Mathematics at City University of New York, College of Staten Island. He received his doctoral degree from Friedrich-Schiller-Universität Jena in 2008, and has worked in topolog- ical data analysis since his first postdoc with Gunnar Carlsson at Stanford 2008– 2011. He is the chair of the steering committee for the Algebraic Topology: Methods, Computation, and Science (ATMCS) conference series and runs the community web resource appliedtopology.org.
- 8. Topological Data Analysis with Applications GUNNAR CARLSSON Stanford University, California MIKAEL VEJDEMO-JOHANSSON City University of New York, College of Staten Island and the Graduate Center
- 9. University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 103 Penang Road, #05–06/07, Visioncrest Commercial, Singapore 238467 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108838658 DOI: 10.1017/9781108975704 © Gunnar Carlsson and Mikael Vejdemo-Johansson 2022 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2022 Printed in the United Kingdom by TJ Books Limited, Padstow Cornwall A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Carlsson, G (Gunnar), 1952– author | Vejdemo-Johansson, Mikael, 1980– author Title: Topological data analysis with applications / Gunnar Carlsson, Mikael Vejdemo-Johansson. Description: New York : Cambridge University Press, 2021. | Includes bibliographical references and index. Identifiers: LCCN 2021024970 | ISBN 9781108838658 (hardback) Subjects: LCSH: Topology. | Mathematical analysis. | BISAC: MATHEMATICS / Topology Classification: LCC QA611 .C29 2021 | DDC 514/.23–dc23 LC record available at https://lccn.loc.gov/2021024970 ISBN 978-1-108-83865-8 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
- 10. Contents Preface page vii Part I Background 1 1 Introduction 3 1.1 Overview 6 1.2 Examples of Qualitative Properties in Applications 6 2 Data 11 2.1 Data Matrices and Spreadsheets 11 2.2 Dissimilarity Matrices and Metrics 15 2.3 Categorical Data and Sequences 18 2.4 Text 20 2.5 Graph Data 21 2.6 Images 22 2.7 Time Series 23 2.8 Density Estimation in Point Cloud Data 24 Part II Theory 27 3 Topology 29 3.1 History 29 3.2 Qualitative and Quantitative Properties 30 3.3 Chain Complexes and Homology 71 4 Shape of Data 94 4.1 Zero-Dimensional Topology: Single-Linkage Clustering 94 4.2 The Nerve Construction and Soft Clustering 98 4.3 Complexes for Point Cloud Data 102 4.4 Persistence 113 4.5 The Algebra of Persistence Vector Spaces 117 4.6 Persistence and Localization of Features 128 4.7 Non-Homotopy-Invariant Shape Recognition 130
- 11. vi Contents 4.8 Zig-Zag Persistence 139 4.9 Multidimensional Persistence 142 5 Structures on Spaces of Barcodes 145 5.1 Metrics on Barcode Spaces 145 5.2 Coordinatizing Barcode Space and Feature Generation 147 5.3 Distributions on B∞ 153 Part III Practice 157 6 Case Studies 159 6.1 Mumford Natural Image Data 159 6.2 Databases of Compounds 169 6.3 Viral Evolution 171 6.4 Time Series 174 6.5 Sensor Coverage and Evasion 180 6.6 Vectorization Methods and Machine Learning 187 6.7 Caging Grasps 193 6.8 Structure of the Cosmic Web 195 6.9 Politics 199 6.10 Amorphous Solids 201 6.11 Infectious Diseases 204 References 207 Index 216
- 14. Preface ix • Cluster analysis seeks to divide data into disjoint groups to create a taxonomy of the data set. In many situations where cluster analysis is applied, however, one finds that the natural output is not a partition of the data into disjoint groups but, rather, a “soft clustering” in which the data is broken into groups that may overlap. This kind of information is very naturally modeled with a simplicial complex, which is able to describe the relationships between groups implied by the overlaps, using an appropriate shape or space. An ordinary cluster decomposition, by a partition, is in this situation modeled by a zero-dimensional simplicial complex, i.e. a finite set of points. • Data science is often confronted with the problem of deciding the appropriate shape for a data set, so as to be able to model it effectively. The theory of simplicial complexes is equipped with a method for describing the shape structure of the output of a model; this is based on an extension of the homology construction from the algebraic topology of spaces. The extension is called persistent homology and will be the subject of many of the ideas that we present. • Feature selection and feature engineering is a major task in data science. It is particularly challenging for data sets which are unstructured in the sense that they are not well represented by a data matrix or a spreadsheet with numerical entries. For example, a database of large molecules is regarded as unstructured because it consists of an unordered set of atoms and an unordered set of bonds and, further, because the spatial coordinates of the atoms can be varied via a rigid motion of space while the structure of the molecule remains unchanged. This means that a representation by the coordinates of the atoms is not meaningful. It turns out, though, that the molecules themselves have a geometry expressed through inter- atomic distances, which allows us to apply the homology tools described above to generate meaningful numerical quantities that can be used for analysis. Images form another class of data which can be viewed as unstructured and which can be studied with homological methods. • Another way in which topological methods can be used for feature engineering is the notion of topological signal processing (Robinson 2014). The idea here is that, when given a data matrix, it is also useful to develop a topological model for the columns of the data matrix (i.e. the features) rather than for the points or samples of the data (i.e. the rows). In this way, each of the original data points can be viewed as a function on the set of features and ultimately as a function on the topological model. Incorporating various methods, including graph Laplacians, one can impose structure on data points using this approach, and obtain topologically informed dimensionality reductions. We believe that the use of TDA in data science will motivate interesting and useful developments within topology. It is therefore useful to see where TDA methods fit within standard algebraic topology and homotopy theory. Here are some important points about this fit. • Persistent homology can be described as the study of diagrams whose shape is defined by the partially ordered set R. A number of other diagrams have been studied,
- 15. x Preface including those used in zig-zag persistence and multidimensional persistence. As the work in TDA broadens and deepens, it is likely that increasingly sophisticated diagrams will be useful for extracting more detailed information from data sets. It follows that the construction of invariants for diagrams of various shapes will be a useful endeavor. • Because TDA operates by studying samples of discrete sets of points, the dimensions of spaces that can be analyzed solely by TDA methods are fairly low, for the most part ≤ 5. A data set which would faithfully represent a space of dimension 10 would be expected to require at least 1010 points, if one assumes a resolution of 10 points for each dimension. This is already a very large number, and demonstrates the point that, for example, 50-dimensional homology is not likely to occur in a useful way in data sets. This suggests that studying more sophisticated unstable homotopy invariants (such as cup products, Massey products, etc.) would be a good direction to pursue. For example, the use of cup products is a key part of Carlsson & Filippenko (2020). • Within algebraic topology and homotopy theory, a very interesting aspect is the topology of spaces equipped with a reference map to a base space B; this is referred to as parametrized topology. All maps are then required to respect the reference map. The category of spaces over a base contains a much richer set of invariants than in the absolute case (i.e. ordinary topology, without a reference map), where B is a single point. This idea comes up in the study of evasion problems (Carlsson & Filippenko 2020) and can be used to define the idea of data science over a base, or parametrized topological data analysis (Nelson 2020), which appears to be a useful framework for an iterative method of data analysis. The study of unstable invariants in this case is particularly rich, and warrants further attention. • One is often interested in studying the invariants of a space X which are not necessarily topological in nature but which nevertheless can be thought of as qualitative, for example, the notion of the corners or ends of spaces are examples of this kind of situation. One way to approach such problems is to perform constructions on X so as to produce an associated space which reflects the property one wants to study, and then to use topological methods such as homology to perform the analysis. A powerful example of this philosophy is the work of Simon Donaldson on the topology of smooth 4-manifolds, where he showed that certain moduli spaces attached to smooth 4-manifolds allow one to study the topology of the manifold itself (Donaldson 1984). This kind of approach can be used to investigate various shape distinction problems that are not directly topological in nature. The goal of this book is to introduce the ideas of topological data analysis to both data scientists and topologists. We have omitted much of the technical material about topology in general, as well as for homology in particular, with the expectation that the reader who has studied the book will be able to go further in the subject as needed. We hope that it will encourage both groups to participate in this exciting intellectual development.
- 16. Preface xi As a matter of convention, we choose to use the terms injective, surjective, and bijective instead of one-to-one, onto, and “one-to-one and onto”. The authors are very grateful for helpful conversations with many people, including R. Adler, A. Bak, E. Carlsson, J. Carlsson, F. Chazal, J. Curry, V. de Silva, P. Diaconis, H. Edelsbrunner, R. Ghrist, L. Guibas, J. Harer, S. Holmes, M. Lesnick, A. Levine, P. Lum, B. Mann, F. Mémoli, K. Mischaikow, D. Morozov, S. Mukherjee, J. Perea, R. Rabadan, H. Sexton, P. Skraba, G. Singh, R. van de Weijgaert, S. Weinberger, and A. Zomorodian. We particularly thank A. Blumberg, whose collaboration on early drafts of this book has helped immensely. Deep thanks also go to the editorial staff at Cambridge University Press, for their patience and all their help.
- 20. 1 Introduction In the last two or three decades, the need for machine learning and artificial intelligence has grown dramatically. As the tasks we undertake become ever more ambitious, both in terms of size and complexity, it is imperative that the available methods keep pace with these demands. A critical component of any such method is the ability to model very large and complex data sets. There is a large suite of powerful modeling methods based on linear algebra and cluster analysis that can often provide solutions for the problems that arise. Although they are often successful, these methods suffer from some weaknesses. In the case of algebraic methods, it is the fact that they are not always flexible enough to model complex data, such as data sets of financial transactions or of surveys. Clustering methods by definition cannot model continuous phenomena. Additionally, they often require choosing thresholds for which there are no good theoretical justifications. What we will discuss in this volume is a modeling methodology called topological data analysis, or TDA, in which data is instead modeled by geometric objects, namely graphs and their higher-dimensional versions, simplicial complexes. Topological data analysis has been under development during the last 20 or so years, and has been applied in many diverse situations. Its starting point is a set equipped with a metric, typically given as a dissimilarity measure on the data points, which can be regarded as endowing the data with a shape. This shape is very informative, in that it describes the overall organization of the data set and therefore enables interrogation of various kinds to take place; TDA provides methods for measuring the shape, in a suitable sense. This is useful in as much as it allows one to access information about the overall organization. In addition, TDA can be used to study data sets of complex description, which might be thought of as unstructured data, where the data points themselves are sets equipped with a dissimilarity measure. For example, one might consider data sets of molecules, where each data point consists of a set of atoms and a set of bonds between those atoms, and use the bonds to construct a metric on the set of atoms. This idea leads to powerful methods for the vectorization of complex unstructured data. The methodology uses and is inspired by the methods of topology, the mathematical study of shape, and we now give a more detailed description of how it works. Much of mathematics can be characterized as the construction of methods for organizing infinite sets into understandable representations. Euclidean spaces are organized using the notions of vector spaces and affine spaces, which allows one to arrange the (infinite) underlying sets into understandable objects which can readily be manipulated and which can be used to construct new objects from old in systematic ways.
- 23. 6 Introduction homology, which attach a list of non-negative integers (called Betti numbers) to any topological space, and we also discuss the adaptation of homology to a tool for the study of point clouds. This adaptation is called persistent homology. • To introduce the mathematics surrounding the collection of persistence barcodes or persistence diagrams, which are the values taken by persistent homology construc- tions. Unlike Betti numbers, which are integer valued, persistent homology takes its values in multisets of intervals on the real line. As such, persistence barcodes have a mix of continuous and discrete structure. The study of these spaces from various points of view, so as to be able to make them maximally useful in various problem domains, is one of the most important research directions within applied topology. • To describe various examples of applications of topological methods to various problem domains. 1.1 Overview The purpose of this book is to develop topological techniques for the study of the qualitative properties of geometric objects, particularly those objects which arise in real- world situations such as sets of experimental data, scanned images of various geometric objects, and arrays of points arising in engineering applications. The mathematical formalism called algebraic topology, and more specifically homology theory, turns out to be a useful tool in making precise various informal, intuitive, geometric notions such as holes, tunnels, voids, connected components, and cycles. This precision has been quite useful in mathematics proper, in situations where we are given geometric objects in closed form and where calculations are carried out by hand. In recent years, there has been a movement toward improving the formalism so that it becomes capable of dealing with geometric objects from real world situations. This has meant that the formalism must be able to deal with geometric objects given via incomplete information (i.e. as a finite but large sample, perhaps with noise, from a geometric object) and that automatic techniques for computing the homology are needed. We refer to this extension of standard topological techniques as computational topology, and it is the subject of this volume. We will assume that the reader is familiar with basic algebra, groups, and vector spaces. In this introductory chapter, we will sketch all the main ideas of computational topology, without going into technical detail. The remaining chapters will then include a precise technical development of the ideas as well as some applications of the theory to actual situations. 1.2 Examples of Qualitative Properties in Applications 1.2.1 Diabetes Data and Clustering Diabetes is a metabolic disorder which is characterized by elevated blood glucose levels. Its symptoms include excessive thirst and frequent urination. In order to understand the
- 24. 1.2 Examples of Qualitative Properties in Applications 7 disease more precisely, it is important to understand the possible configurations of values that various metabolic variables can exhibit. The kind of understanding we hope for is geometric in nature. An analysis of this type was carried out in the 1970s by Reaven & Miller (1979). In this study, a collection of five parameters (four metabolic quantities and the relative weight) were measured for each patient. Each patient then corresponds to a single data point in five-dimensional space. In Reaven & Miller (1979), the projection pursuit method was used to produce a three-dimensional projection of the data set, which looks like the situation on the left in Figure 1.1. Figure 1.1 Diabetes patient distribution. See the main text for a description of the figure. From Reaven & Miller (1979), reproduced with permission of Springer–Nature, © 1979. Each patient was classified as being normal, chemical diabetic, or overt diabetic. This is a classification which physicians devised using their observation of the patients. It was observed that the normal patients occupied the central rounded object, and the chemical diabetics and overt diabetics corresponded to the two “ears” in the picture. Another very interesting method for visualizing the set was introduced in Diaconis & Friedman (1980). These visualizations suggest that the two forms of diabetes are actually fundamentally different ailments. In fact, physicians have understood that diabetes occurs in two forms, “Type I” and “Type II”. Type I patients often have juvenile onset, and the disease may be independent of the patient’s life style choices. Type II diabetes more often occurs later in life and appears to depend on life style choices. The chemical diabetics are likely to be thought of as Type II diabetics who eventually may arrive at the overt diabetic stage, while the overt diabetics might arrive at overt status directly. In this case, the qualitative property of the figure that is relevant is that it has the two distinct ears, coming out of a central core. Although human beings can recognize this fact in this projection, it is important to formalize mathematically what this means, so that one can hope to automate the recognition of this qualitative property. For example, there may be data sets for which no two- or three-dimensional projection gives a full picture of the nature of the set. In this case, the mathematical version of this statement would be as follows. We suppose that the three categories of patients (normal, chemical, and overt) correspond to three different regions A, B, and C in five-dimensional Euclidean space.
- 25. 8 Introduction What this experiment suggests is that if we consider the union X = A ∪ B ∪ C (which corresponds to all patients) and then remove A, the region corresponding to the normal patients, the region we are left with breaks up into two distinct connected pieces, which do not overlap and in fact are substantially removed from each other. Clustering techniques from statistics were used in Symons (1981) to find methods to differentiate between these two components. The qualitative question above about the nature of the disease can now be stated as asking how many connected components are present in the space of all patients having some form of diabetes. Finding the number of connected components of a geometric object is a topological question. 1.2.2 Periodic Motion Imagine that we are tracking a moving object in space, and that the information is given in terms of a three-dimensional coordinate system, so that we are given coordinates (x(t), y(t), z(t)). If we want to know whether the object is moving periodically, say, because it is orbiting around a planet, we can simply check whether the values of the coordinates repeat after some fixed period of time. Suppose, however, that we are not given the time values corresponding to the points but just a set of positions, and want to determine whether the object is undergoing periodic motion. We would thus like to know whether the set of positions forms a closed loop in space. If the object is orbiting around a single planet or star, and we therefore know by Kepler’s laws that the geometric shape of the orbit must be that of an ellipse, we can determine that the object is orbiting by simply curve-fitting an ellipse to the data set of positions. Suppose, however, that the object is being acted on gravitationally by several other objects, so that the path is not a familiar kind of closed curve. We would then still want to know whether the space of positions is a closed loop, but perhaps not one for which we have a familiar set of coordinatizations. The qualitative property in which we are interested is whether the space is a closed loop, and we would like to develop techniques which allow us to determine this without necessarily asking for a particular coordinatization of the curve. In other words, we are asking whether our space is a closed loop of some kind, not exactly what type of loop it is. A more difficult situation is where we are not actually given the values of the position of the object but, rather, a family of images taken from a digital camera. In this case, the set of these images actually lies in a very high-dimensional space, namely the space of all p-vectors, where p is the number of pixels. Each pixel of each image is given a value, the gray-scale intensity at that pixel, and so each image corresponds to a vector, with a coordinate for each pixel. If we take many images sequentially, we will obtain a family of points in the p-dimensional space, which lies along a subset which should be identified topologically with the set of positions of the object, i.e. a circle. So, although this set is not identified with a circle through any simple set of equations in p variables, the qualitative information that it is a circle is contained in this data. This is an example of an exotic coordinatization of a space (namely the circle) and shows that, in order to analyze this kind of data, it would be very useful to have tools which can tell whether a space is a closed loop, without its having to be any particular loop. In other words, coordinate-free tools are very useful.
- 27. 10 Introduction We will see later that cues involving “corners”, “curved arcs”, “vertices”, and “edges” are not directly topological. We will develop methods for recognizing these cues topologically on new spaces that we have constructed from the old ones using tangential information.
- 28. 2 Data In this chapter, we will discuss the properties and methods associated with various important data types, in order to indicate the wide range of applications for all methods of data science. Since the purpose of this volume is to leverage geometric structures which are applicable to data, we will point out the relevant geometric structures in each case. We will also discuss the conventional methods that are applicable in each situation. In particular, we will see that in many of these situations, it is important to use structures on the set of features attached to data sets, and we will point them out. In particular, geometric structures are often relevant for the features as well as for the data points themselves. We are certainly not claiming to be exhaustive in what we present; rather, we are attempting to give a reasonable sample of what is possible. 2.1 Data Matrices and Spreadsheets Perhaps the most common representation of data sets is as tidy data: a spreadsheet with real numerical entries. Each data point corresponds to a row in the spreadsheet, and each column corresponds to a feature in the data set. In more mathematical terms, the data set is represented as a matrix of real numbers, where the number of rows is the number of data points and the number of columns is the number of features in the spreadsheet. This interpretation suggests that linear algebra should be useful in the analysis of data sets, and this is indeed the case. An important method for the study of data sets given in matrix form is principal component analysis (PCA); see Hastie et al. (2009) for a comprehensive description. It is a method for modeling data by finite (but possibly very large) subsets of inner product spaces. Of course, if we are given a data matrix, the data is exactly identified with a finite subset of an inner product space, namely V = RN , where N is the number of columns of the matrix. The inner product on V is the standard inner product, in which the unit basis vectors form an orthonormal basis. If N is small, PCA permits some useful kinds of analysis. If N = 1, 2, or 3, then one can actually visualize the data set as a scatter plot. Even if N is greater than 3, but is of relatively small size, PCA makes many kinds of calculations possible and can give insight. This is the case in the text analysis of corpora of documents, where the data sets might have nearly a million features (one for every word in a dictionary), but where one can often reduce the model to a few hundred
- 30. 2.1 Data Matrices and Spreadsheets 13 X into two classes, X+1 and X 1, and whose goal is to produce a simple mathematical procedure (called a linear classifier) which predicts the subset to which a new data point should belong. The formula is produced by considering the decomposition of V using a hyperplane, where one class lies on one side of the hyperplane and the other on the other side. To sketch the idea, we suppose first that the sets X+1 and X−1 are in fact linearly separable, i.e. that there is a hyperplane such that X+1 and X−1 are entirely on opposite sides of it, with no point of X actually lying on it. We define a set H to be the collection of all pairs of parallel hyperplanes (H+1, H−1) such that X+1 lies on the opposite side of H+1 from H−1, and such that X−1 lies on the opposite side of H−1 from H+1. Each such pair of hyperplanes determines a classifier, as follows. We consider the plane H0 which is “halfway between” H+1 and H 1, and describe it by an equation of the form ϕ(~ v) = ~ w · ~ v b = 0. The parallel planes H+1 and H 1 are then given by equations ~ w · ~ v b+ = 0 and ~ w · ~ v b = 0 and, after possibly multiplying the equation by 1, we may assume that b+ > b > b and that b+ b = b b . The classification is now achieved by asserting that a vector ~ v belongs to (a) X+1 if ~ w · ~ v b > 0 and (b) X 1 if ~ w · ~ v b < 0. Of course, this can works for many pairs of hyperplanes. However, for each pair of parallel hyperplanes (H+, H ) ∈ H, there is a well-defined distance between the hyperplanes, and the choice that is made for the classifier is to select the pair (H+, H ) ∈ H which maximizes this distance (it is unique), and then to use the classifier attached to this pair. This would arguably be the best way to choose a linear classifier for new data points for which a predicted class is desired. Not all X = X+1 ∪ X 1 are linearly separable, in which case the above construction is not applicable. However, it gives the motivation for a different optimization problem which operates in non-separable situations. The idea is to define a loss function attached to a hyperplane and a set of points divided into two classes within a finite-dimensional inner product space. We note first that we will be optimizing over pairs (~ w, b), where ~ w ∈ V and b is a real number. This data represents the hyperplane ~ w · ~ v − b = 0. For each data point xi ∈ X, we assign yi ∈ {±1} by declaring that yi = 1 if xi ∈ X+1 and yi = 1 if xi ∈ X−1. To this configuration we now associate a loss function L = (i, ~ w, b) = max(0, 1 yi (~ w · xi b)) with each data point xi. We note that L = 0 if either yi = +1 and ~ w · xi b ≥ 1 or yi = 1 and ~ w · xi b ≤ 1. This means that the loss function is zero for points xi ∈ X+1 which are correctly correctly classified by the classifier ~ w · xi b ≥ 1, and similarly for points in X 1. Points which are not classified by either of these classifiers lie between the hyperplanes given by ~ w · ~ v − b = 1 and ~ w · ~ v − b = −1 and are assigned a positive loss value depending on how close they are to one or the other hyperplane. The hard loss function for the entire configuration is given by the sum N X i=1 L(i, ~ w, b), which one can attempt to minimize. If the subsets X+1 and X 1 are linearly separable then the value 0 can be achieved, as one can easily see from the linearly separable analysis above. One could now decide simply to use this loss function even in the
- 31. 14 Data linearly inseparable case, but it turns out to be useful to introduce a parameter λ and consider instead the modified loss function 1 N N X i=1 L(i, ~ w, b) + λk~ wk. The aim here is that there should be a penalty for k~ wk being too large. The reasoning is as follows. A hyperplane can be represented by many equations of the form ~ w ·~ v −b = 0, since we may multiply the equation by any non-zero real number C and obtain an equally valid equation. What does change with C are the hyperplanes ~ w · ~ v − b = ±1, which become closer to each other as C → ∞. In the infinite limit, this means that the “band” between them shrinks until the two hyperplanes become one. Thus the hard loss function is essentially just an evaluation of the fraction of points that are misclassified. This is in general too rigid, since one wants the loss function not to change too much with small changes in the positions of the data points. The parameter λ allows one to tune the classifier, to arrive at a model which is more robust to such small changes. This method can be extended to classification problems with more than two classes. In addition, one might hope to extend it to non-linear situations. This is sometimes done by embedding the problem non linearly into a much higher-dimensional inner product space and performing the linear classification there. See Vapnik (1998) for a more detailed discussion. Another method for addressing classification problems with linear algebra is logistic regression. In the simplest case, this method requires a collection of data points which consist of a number of continuous variables, called the independent variables, and one {0, 1}-valued outcome variable. The aim is to design a procedure that estimates the probability that the outcome variable equals 1, given the values of the independent variables. It is assumed that the probability has the form σ( P i ci xi + b), where the xi are the independent variables, the ci and b are parameters to be estimated, and σ is the logistic function, given for a real number t by σ(t) = 1 1 + e−t . In order to fit a model, we must choose a measure of the model fit that we can optimize. The standard choice is the maximum likelihood function. We suppose that the data points are {~ x1, . . ., ~ xN }, and that the outcome for the point ~ xi is yi such that yi ∈ {0, 1}. Given the model determined by the coefficient vector ~ c and the number b, the likelihood that the outcome variable equals 1 at the data point ~ xi is given by h~ c,b (~ xi) = σ(~ c · ~ xi + b) and the likelihood that it equals 0 is 1 h~ c,b (~ xi). The likelihood of the values being correct for all values of i is therefore Y i|yi =0 (1 h~ c,b (~ xi)) Y i|yi =1 h~ c,b (~ xi), or, equivalently, Y i (1 − h~ c,b (~ xi))(1−yi ) h~ c,b (~ xi)yi .
- 32. 2.2 Dissimilarity Matrices and Metrics 15 This function is now maximized using gradient descent methods. Logistic regression can also be extended to classification problems with more than two classes. A good discussion can be found in Hastie et al. (2009). 2.2 Dissimilarity Matrices and Metrics There is a straightforward extension of the notion of distance in R2 and in R3 to Rn, via the formula d(~ v, ~ w) = p (~ v − ~ w) · (~ v − ~ w). Therefore, when we have a data set of points S = {~ v1, . . .,~ vN } ∈ Rn, we create a symmetric matrix of distances, D(S), given by d(~ v1,~ v1) · · · d(~ v1,~ vN ) . . . . . . d(~ vN,~ v1) · · · d(~ vN,~ vN ). This matrix can be thought of as a dissimilarity matrix, because large distances can be thought of as indicating dissimilarity and small distances as indicating similarity. Definition 2.1 A dissimilarity matrix is a non-negative symmetric matrix which has zeros along the diagonal. It is also useful to think of it as a structure on a finite set X, defining a function D: X × X → [0, +∞) which (a) vanishes on the diagonal and (b) has the symmetry property D(x, x0) = D(x0, x). A dissimilarity space is a pair (X, D), where X is a set and D is a non-negative real-valued function on X × X satisfying conditions (a) and (b) above. A dissimilarity matrix is of metric type if additionally (a) all its off-diagonal entries are non-zero and (b) it satisfies the triangle inequality Dik ≤ Dij + Djk . The corresponding functions on the set X × X, where X is the set of columns of the matrix D, is then called a metric, and the pair consisting of X and the function D is called a metric space. There will be situations where the data we are working with are not necessarily embedded in Euclidean space, but where we can nevertheless construct a dissimilarity matrix. Here are some examples of when this occurs. 1. There are a number of abstract distance functions that can be defined on various kinds of categorical data, such as Hamming distances, which do not obviously arise from subsets of Euclidean space. 2. If we have points that we think of as sampled from a Riemannian manifold M embedded in Euclidean space, their dissimilarities may be better modeled by geodesic distances in M rather than the Euclidean distances. For data embedded in Rn, there are versions of the geodesic distance which are closely related to the graph distances between data points, where points sufficiently close are connected to give
- 33. 16 Data a graph structure. This distinction is the key to the utility of the ISOMAP algorithm (Tenenbaum et al. 2000). 3. There are situations where human beings have constructed intuitive but quantitative notions of similarity, and it is desirable to work with them geometrically. See for example Sneath & Sokal (1973) for numerous biological examples. It is therefore useful to create methods and algorithms which operate directly on dissimilarity matrices and which do not require a vector representation. The first such algorithm that we will discuss is multidimensional scaling (MDS) (Borg & Groenen 1997). Given an n × n dissimilarity matrix, interpreted as a dissimilarity measure on a set of points S = {x1, . . ., x2}, MDS produces an embedding of S in Euclidean space Ed, for some d, such that the loss function X i<j (kx̃i x̃j k d(xi, xj ))2 is minimized, where x̃i denotes the vector in Ed corresponding to xi ∈ S. In other words, the dissimilarity matrix is approximated as well as possible by the distances between the embedded points in Euclidean space. It is possible to choose other loss functions, depending on the situation, but this one has the attractive property that it is predictable in the sense that it produces a predictable minimum in the dissimilarity matrices which come from Euclidean space. This is demonstrated by the following analysis. Let D be a matrix of Euclidean distances, dij = kx̃i x̃j k. We form the associated matrix K: K = 1 2 HDH, where H = I 1 n 11T and 1 denotes the vector of all ones). When D is a Euclidean distance matrix, K is positive semi-definite and so standard minimization arguments imply that the solution is given by Λ1/2 ZT , where Λ is diagonal with the top k eigenvalues of K along the diagonal and Z contains the corresponding top k eigenvectors of K. This also demonstrates that the result is the same as that obtained by using PCA on the Euclidean data. Multidimensional scaling can be used for dimensionality reduction by choosing the dimension of the Euclidean space into which to embed. However, a significant advantage of MDS is the fact that the dissimilarity matrix used as input need not come from a distance function but can simply arise from any symmetric dissimilarity matrix (Borg & Groenen 1997). It is apparent that MDS is an unsupervised method for data analysis, and the discussion above shows that it generalizes PCA. Another class of methods for unsupervised data analysis is cluster analysis. In this case the output of the method is a partition of the data set rather than an embedding in a Euclidean space. There is a very rich family of methods that perform this task, and we will not attempt to be comprehensive here but, rather, will give a few simple examples. For a more complete treatment, see Everitt et al. (2011). By a clustering algorithm, we mean an algorithm which takes as its input a dissimilarity space (X, D), and from it defines a partition of the set X. The blocks of the partition are called clusters. Here are some examples. Example 2.2 Single-linkage clustering with scale parameter R is defined on a dissimi larity space (X, D) by declaring that two points x and x0 are in the same cluster if and only
- 35. 18 Data point, we have a clustering of X. Each cluster is a subset of X and, for each pair (ξ, ξ0) of clusters, we can evaluate L(ξ, ξ0 ) = min x∈ξ,x0 ∈ξ0 D(x, x0 ). We can now merge all pairs of clusters where the minimum positive value of L is achieved, and this process can be iterated to yield a family of clusterings of increasing coarseness. By keeping track of the values L computed at each stage, one can recover single-linkage hierarchical clustering in this way. This point of view permits a useful generalization, on using other choices for the function L. For example, one can replace L by Lave, defined by setting Lave(ξ, ξ0) equal to the average of the set of values {D(x, x0)} over pairs x ∈ ξ and x0 ∈ ξ0, where ξ , ξ0. One can also replace L by the function Lmax, defined by setting Lmax(ξ, ξ0) equal to the maximum of the set of values {D(x, x0)} over pairs x ∈ ξ and x0 ∈ ξ0, for any pairs of distinct clusters (ξ, ξ0). Each choice of L gives a different hierarchical clustering scheme, with L, Lave, and Lmax corresponding to hierarchical schemes referred to as single linkage, average-linkage, and complete linkage hierarchical clustering respectively. Average linkage and complete-linkage are often useful since they address problems arising with single-linkage problems, one being the chaining problem. This refers to the fact that single linkage often produces long sequences of clusters which consist of very small clusters adjoining one large one. See Everitt et al. (2011) for a discussion of this problem. Cluster analysis is a highly developed area in data science, and we refer the reader to Everitt et al. (2011) for a thorough treatment. 2.3 Categorical Data and Sequences Many data sets are not in numerical matrix format, and also they may not be naturally equipped with dissimilarity measures. Finding methods for putting them into matrix or dissimilarity format is of great value. Consider the following simple example. Suppose that we are given a data set whose entries are “shopping baskets” for a store. We are interested in understanding the set of these baskets in some way. Each entry in the data set is a list of products carried by the store, perhaps with multiplicities for purchases of multiple units, but the order in the entries is not meaningful for us because the order in which the items are rung up is not of significance. The data could be encoded as list of product identifiers. These identifiers might be numeric or alphanumeric, but a numerical quantity carries no meaning beyond the purpose of identification. As it stands, the methods in Sections 2.1 and 2.2 are of no use, because we do not have a numerical matrix representation, either as vectors or as a dissimilarity measure. Moreover, the data is given to us in a form which introduces extraneous information, namely the ordering of the items in the basket. We introduce one-hot encoding as a method for overcoming both of these problems. Let S be a finite set, and let X be a data set whose elements are subsets of S. We construct the vector space R#(S), with its standard basis {es}s∈S, and assign to each subset S0 ⊆ S the vector v(S0) = P s cses, where cs = 1 if s ∈ S0 and cs = 0 if
- 36. 2.3 Categorical Data and Sequences 19 s < S0. We have thus assigned a vector to each subset of S and consequently we have arrived at a data matrix, to which the matrix methods given above can be applied. The collection of subsets of S can now be viewed as a dissimilarity space, in which distances between subsets of S are determined by their symmetric differences. We note that the construction extends to multisets of elements, where elements can occur with multiplici- ties greater than one (see Hein 2003, for a discussion of these), in a straightforward way. Let us consider another situation, where a company is studying its sales representatives (we will call them reps), and the reps have collected data consisting of pairs (rep, productid), where the two variables take values in the discrete sets of reps and company products, respectively. Each data point represents one transaction generated by the rep. The company would like to have information about the performance of its reps. The given list is not informative about reps directly, since each data point corresponds to only a single transaction. We would instead like to create a data set where each data point corresponds only to a rep, and carries information about the distribution of products the rep has sold. A method for doing this is via pivot tables. For each rep, we have a multiset of products. Let us say the ith rep is associated with a set Si of products. The pivot table transform associated with a variable salesrep begins with a data set of transactions which are of the form (rep, p), where p is a product. Note that a particular value rep can occur in many data points, one for each transaction. The pivot table transform now associates with the data set of transactions a data set where the entries are of the form (rep, {p1, p2, . . ., pn}) and where each value rep of salesrep can occur only once and the second coordinate, within braces, is a multiset of products rather than a single product. The second coordinate is the multiset of all products that have been sold by the sales representative rep. To produce a matrix representation, we can apply one-hot encoding and obtain a data set consisting of elements (rep, vrep). It corresponds to an m × n matrix, where m is the number of sales reps and n is the number of all possible products. This transform is extremely useful, and can also be used in situations where some variables are actually numerical instead of belonging to a discrete set. Suppose, for example, that our data set of transactions instead consisted of pairs (rep, x), where x is the price of the item in a transaction. Then we could form the pivot table transform associated with the variable salesrep, and obtain a pair (rep, {x1, . . ., xn}), where {x1, . . ., xn} is the set of all prices for all transactions involving the sales representative rep. Since we might be interested in the total value of the transactions involving rep, we might here apply a different method to obtain a vector, in this case assigning to the set {x1, . . . xn} the sum x1 +· · ·+ xn. If we were interested in the average value of the transactions for a given rep, we would instead associate the average value of the xi. One could also consider the maximum or minimum values of the xi.
- 37. 20 Data The pivot table transform is in general defined in two steps. We suppose that we are given a data set X consisting of coordinates xs, where s ∈ S for some finite set S. The quantities s may correspond to variables of different types, namely one might consist of elements of a discrete set, another might consist of real numbers, another of positive integers, yet another of subsets of a fixed discrete set, etc. Suppose that S is the disjoint union of two sets S0 and S1. An Si-vector is a tuple {xs}s∈Si , where xs has the type of the variable s. For each data point x ∈ X, we have two projections π0(x) and π1(x), where πi (x) is an Si-vector for i = 0, 1. The first step in the pivot table transform associates with X a set P(X) consisting of pairs (ξ, S), where ξ is an S0-vector and S is a set of S1-vectors. The set P(X) has exactly one element for each unique value taken by π0(x) as x ranges over X, and, for a fixed ξ, the multiset S consists of all the S1-vectors that occur as π1(x) for data points x ∈ X for which π0(x) = ξ. The second step is the vectorization step, which applies a method for assigning vectors in a vector space to sets of S1-vectors. The use of one-hot encoding above, as well as the sum, average, max and min methods for sets of real numbers, are all such methods but one can imagine many others. In the examples above, we have examined situations where we have collections and are not interested in the ordering of the elements in the collection. There are many situations where one is interested in the ordering. A notable example is in genomics, where one considers sequences of elements belonging to an alphabet, such as the 20-element alphabet of amino acids. In this case, there is a simple vectorization step. Suppose that we are considering sequences of length N in an alphabet A of cardinality a. The we may use one-hot encoding to assign a vector h(a) of length a to members a of the alphabet A. We may then assign to the sequence (a1, . . . aN ) the vector (h(a1), . . . h(aN )), which is a vector of length aN. This is a simple vectorization method, but in this case it is often more useful to construct dissimilarity measures directly. For two sequences σ and σ0 of length N in the alphabet A, we define the Hamming distance between σ and σ0 to be the number of elements i ∈ {1, . . ., N} such that σi , σ0 i. This is clearly a dissimilarity measure, and it is in fact of metric type. This dissimilarity measure, and variants of it, are very useful in many situations, particularly in genomics. See Gusfield (1997) for a comprehensive treatment. 2.4 Text A very interesting data type arises in natural language processing, namely that of a corpus of documents. Each document is a sequence of words, and they may be of varying length. It is very useful in for example in analyzing sentiment, or detecting trends in collection of newspaper articles. The general setup is that a corpus is a collection of documents and each document is a sequence of words, perhaps with some punctuation symbols included. Since the documents could be regarded as sets of words (losing the information about their order in the sequence), it is possible to use one-hot encoding directly, where a basis element in a vector space would be associated with each word. What this effectively does is to assign to each document the collection of word counts within the document
- 38. 2.5 Graph Data 21 for every word within a dictionary. This is certainly a possible vectorization for the data set, but it is not satisfactory because very common words, such as the word “the”, will dominate the occurrences of other meaningful words in the document. A solution to this problem is via tf–idf (term-frequency–inverse-document-frequency). Let D be a collection of documents d, referred to as the corpus. In tf-idf methods, we assign to each word–document pair (w, d) the number of occurrences of the word w in the document d, and denote it by tf(w, d), the term frequency of w in d. In order to deal with the problem of frequently occurring words, we will want to weight this count, and the idea is to do that on the basis of the number of documents in D that include w. The most standard choice is to multiply this count by the inverse document frequency, which is defined to be idf(w, d) = log N |{d ∈ D such that w ∈ d}| . Assigning to each document d the vector {tf(w, d) idf(w, D)}w gives a data matrix with D rows and W columns, where D is the number of documents in the corpus and W is the number of words in the dictionary. Note that, with this vectorization, any word which occurs in every document will have an identically zero value, so that every column in the data matrix corresponding to such words will be identically zero and can therefore be ignored. The tf–idf approach has worked well in many situations; principal component analysis has been applied extensively to these data matrices with interesting results. Nevertheless, because the tf-idf approach treats each document as just a multiset of words, it does not take full advantage of all the structure that is present. One could, for example, begin to take advantage of the additional structure of the ordering by dealing with k-grams, i.e. sequences of k consecutive words. Another direction in natural language processing is the use of word embeddings. A word embedding consists of an inner product space with an assignment w in the dictionary. A canonical choice given above would be one-hot encoding, where the vector space V is RW and where the vector vw would be the standard basis vector ew in the vector space RW . The point of word embeddings is to embed the words in a vector space of much smaller dimension, so that the distances between the associated vectors in some way reflect correlations between words. It is possible to use idf reweighting in conjunction with any such word embedding. One standard method is to apply PCA and choose the top few hundred coordinates. Other popular examples include word2vec and Glove. Useful references for natural language processing are Manning & Schütze (1999) and Eisenstein (2018). 2.5 Graph Data There is a large class of data where either (a) the data is equipped with a graph structure or (b) the data set consists of elements which are graphs. The notion of a graph has several variants. Graphs may be directed or undirected, and they may carry weights or labels attached to edges and/or vertices. We present a number of different examples to show what is possible.
- 39. 22 Data 1. The internet is an example of a data set W of type (a) above, where the graph is directed and whose vertices are web pages and edges are hyperlinks. The web pages are considered as a data set, and the presence of a graph structure on W allows for feature generation. For example, page rank and the hubs and authorities construction (see Easley & Kleinberg 2011) create numerical features on the vertex set. One can also use simple graph-theoretic properties to create additional numerical features, such as total degree, inbound degree, and outbound degree. One can also create graph valued features by constructing a neighborhood of radius k, or other graph- valued constructions of a local nature. Using this idea, one can view the collection W as a data set of type (b) from above. It is also possible to assign weights to edges and vertices on the basis of the traffic at a webpage or across a particular hyperlink. 2. Any molecule can be regarded as an undirected graph whose vertices are atoms and whose edges are bonds. One can also attach weights or labels to the vertices using atomic weight, atomic number, or element label. Databases of molecules are therefore an example of type (b) above. Of course, simple measures, such as the number of atoms or various aggregate quantities concerning the bonds, are available but they do not reflect the full content of the data. In addition, there are standard collections of features called SMILES (see Weininger 1988). We will discuss how to apply topological methods to generate new useful features systematically in Section 6.2. 3. Crystals are regularly arranged collections of atoms or molecules. The theory of crystals is a highly developed area within physics, chemistry, and mathematics. The high degree of regularity makes a sophisticated theory, involving complex symmetry groups, possible. However, many materials of interest are amorphous, in that the atoms are not arranged with precise regularity but nevertheless exhibit geometric structure, which has been studied in various ways. One can encode information about an amorphous solid via a graph structure where the vertices correspond to the constitutent atoms or molecules and where the edges are assigned on the basis of chemical or physical properties. Often one can actually recognize bonds between the atoms and assign edges to them. In other situations, one assigns edges on the basis of distance thresholds. In Section 6.10, we will see how such methods can be used to generate features attached to various amorphous solid classes and to demonstrate that they carry chemical and physical information. 4. In genomics, the concept of a phylogenetic tree is a very important one. It describes the pattern of evolution within various classes of organisms and is certainly of a great deal of value in the study of viruses. The notion of moduli spaces of trees is a useful one, and it is studied in detail in Rabadan & Blumberg (2019). 2.6 Images Images present another very interesting data type. Here the data typically consists of rectangular arrays of pixels, with each pixel assigned either a gray-scale value or a collection of color intensities using one of a number of possible ways of encoding color. This means that image data sets can be expressed as matrices with MN columns, where
- 40. 2.7 Time Series 23 the image consists of an M × N rectangular array of pixels. What we note, though, is that the matrix is equipped with a kind of geometric structure, whether as a grid graph where every node is connected to its four nearest neighbors or with a distance function as a subset of the plane. What this means is that not only do we have the matrix entries, but we have knowledge about which nodes are close to a given node. Moreover, the regular grid structure supports translations in the image. Methods for working with images use the grid structure in a number of ways, as follows. 1. It is often useful to smooth images by replacing a pixel value at p by a sum of other pixel values, weighted by the inverse of their distance from p. 2. Convolutional neural networks (CNNs) (see Aggarwal 2018) constitute a computa tional method for computing with image data sets. They use the grid structure to devise a relatively sparse neural network architecture that performs strikingly well for a number of image analysis tasks, particularly for classification. 3. CNNs also use the translation property to insure that the object recognition methods that they construct have the property that an object (say a cat) in the lower left-hand corner of an image is recognized in exactly the same way as a cat in the upper right-hand corner. 4. The grid structure also allows for feature generation, one example being the so-called Gabor filters (see Feichtinger & Strohmer 1998). 2.7 Time Series Time series are data sets that consist of sequences of observations of some kind, typically made at regular intervals. The most common version involves real-valued observations, but vector-valued time series are also of great interest. One can also think of time series with observations in other data types, such as regularly occurring text documents. Just as images are data where the features are arranged geometrically in a rectangular grid, so the variables in a time series are arranged in a one-dimensional linear array. There is a large body of literature concerning the modeling of time series (see for example Kirchgässner & Wolters 2007 or Kantz & Schreiber 2004). A commonly used method for stationary times series, where the means and variances of the variables do not change over time, is the autoregressive moving average (ARMA) model, which we now describe. The assumption is that we generate a sequence of observations {Xi}i, say for i ≥ 0, and that Xi is the observation obtained at a time ti. It is also assumed that the ti are regularly spaced in time. The model further assumes that the observations at times ti depend linearly on a fixed number (say p) of the preceding observations and also that there are models for noise in the system. Formally, one writes Xt = c + εt + p X i=1 ϕi Xt−i + q X i=1 θiεt i.
- 41. 24 Data Here the variables εi are typically assumed to be independent identically distributed (i.i.d.) random variables sampled from a normal distribution with zero mean. Models other than normal ones are possible, but it is required that the variables should still be i.i.d. Because of the presence of the random variables εt, the above model is not amenable a standard linear regression model but, rather, describes a distribution for each variable Xt. The fitting of the model is then carried out using a method for fitting distributions, a common choice being the maximum likelihood method (see Hastie et al. 2009). There are simple extensions to vector-valued time series. Another extension is the so-called autoregressive integrative moving average (ARIMA) model, which does not require that the time series be stationary. There are of course situations where such linear models are not appropriate, but one nevertheless wants to predict the value of an observation using the preceding k observations for a fixed positive integer k. There is a construction that is available for time series, called delay embedding or Takens embedding. To introduce this, we first consider how to construct dissimilarity spaces out of sequences from a dissimilarity space. Let (X, D) be a dissimilarity space. We can form a dissimilarity space structure on the space of k-tuples X(k) = (Xk, Dk ) where Dk can be chosen in a number of different ways. The most popular ones are in direct analogy with Lp-metrics on sequences: Dk ((x1, . . ., xk ), (y1, . . ., yk )) = * , k X i=1 D(xi, yi)p+ - 1/p . Choosing p = 1 gives a particularly simple-to-use dissimilarity just sum up the pairwise dissimilarities of the two sequences while p = 2 is a very popular choice since in combination with the Euclidean metric it reduces to the Euclidean metric on concatenation of the observations. Now, for a single time series t = {xi}N i=1 we can define a delay embedding with delay and dimension k as the new time series Tk, t = {(xi (k 1), . . ., xi , xi)}N i=1+(k−1) ⊆ X(k). If T ⊆ X N is a collection of time series samples, where each t = (x1, . . ., xN ) ∈ T is a single time series taken over a family of times (t1, . . ., tN ) we may generate a new collection of time series by applying delay embedding separately on each time series in the collection, producing a delay-embedded collection Tk, T = {Tk, t | t ∈ T}. We will meet the Takens embedding again in Section 6.4, where the point cloud obtained by forgetting the time-ordering structure on Tk, T can be analyzed using persis- tent cohomology to generate information about quasiperiodicity in the time series itself. 2.8 Density Estimation in Point Cloud Data The theory of density estimation is a highly developed area of statistics. We will not attempt to survey this area but will instead refer the reader to Scott (2015) or Duvroye
- 43. 26 Data ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Figure 2.3 Points as in Figure 2.2. Left, codensity δ5. Right, codensity δ200. The 25% densest values are marked in red, indicating the sets P(5, 25) and P(200, 25) respectively. For a subset P0 ⊂ P, with a slight abuse of notation we shall write P0(k,T) for the T 100 |P0| points of P0 on which the codensity – as calculated on all of P – takes on the lowest values.
- 44. Part II Theory
- 46. 3 Topology 3.1 History Euler (1741) is usually cited as the first paper in topology. In this paper, Euler studies the so-called “Bridges of Königsberg” problem. The question that was asked about the bridges was whether it is possible to traverse all the bridges exactly once and return to one’s starting point. Euler answered this by recognizing that it is a question about paths in an associated network; see below. In fact, the question involves only certain properties of the paths which are independent of the rates at which the paths are traversed. His result concerned the properties of an infinite class of paths, i.e. a certain type of pattern in the network. Euler also derived a polyhedral formula relating the number of vertices, edges, and faces in polyhedra (Euler 1752a, 1752b). The subject developed in a sporadic fashion over the following century and a half; developments included the work of Vandermonde on knot theory (Vandermonde 1771), the proof of the Gauss–Bonnet theorem (never published by Gauss, but with a special case proved in Bonnet 1848), the first book in the subject, by Listing (1848), and the work of Riemann identifying the notion of a manifold (Riemann 1851). The paper of Poincaré (1895) was a seminal work in which the notions of homology and the fundamental group were introduced, with motivation from celestial mechanics. The subject then developed at a greatly accelerated pace throughout the twentieth century. The first paper in persistent homology was Robins (1999), and the subject of applying topological methodologies to finite metric spaces has been developing rapidly since that time.
- 47. 30 Topology 3.2 Qualitative and Quantitative Properties 3.2.1 Topological Properties The notion of a topological property is an old one in mathematics. We first discuss what is meant by topological properties, and why they are important in data science. In many problems in science, one can impose different coordinate systems on a single physical problem. For example, one can apply rigid motions (rotations and translations) to data in Euclidean space. This kind of coordinate change is, for example, ubiquitous in physics. It has the useful property that it preserves distances between points. Often, however, one needs to consider coordinate changes that are not rigid motions but are more complicated, in the sense that they distort the geometric properties (such as distance) of the data obtained from observation or experimentation. Example 3.1 Temperature can be measured using many different scales, including the Celsius, Fahrenheit, and Kelvin scales. The transition from degrees Celsius to degrees Fahrenheit is given by the formula F = 9 5 C + 32. This transformation does not preserve distances; it dilates them by a constant factor. The transformation law from degrees Celsius to kelvins is given by K = C + 273.15. This transformation does preserve distances, i.e. the size of a degree. Example 3.2 A change of coordinates which is frequently used to make apparent some properties of data is the so-called log–log coordinate system, in which every point (x, y) is replaced by (log(x), log(y)). This transformation carries curves given by power laws to straight lines of varying slopes. It does not preserve distances. Example 3.3 Polar coordinates provide a convenient way to study a number of problems. The transition law from polar coordinates to rectangular coordinates does not preserve distances. Example 3.4 Coordinate changes which apply multiplication by non-orthogonal matrices will distort the geometric properties of sets in the plane or in space. For example, dilation (multiplication by a positive multiple of the identity matrix) carries a circle centered at the origin to another circle centered at the origin with a different radius. Multiplication by a general diagonal matrix (perhaps with distinct eigenvalues) carries circles centered at the origin to ellipses centered at the origin. Example 3.5 It is often said that topology is the subject in which a coffee cup is regarded as being the same as a doughnut. This means that there is a coordinate change such that a set which is a coffee cup in one coordinate system is a doughnut in the new coordinates. See the illustration below.
- 51. 34 Topology that arbitrary continuous transformations can distort distances in a very flexible way; it is not possible to define topological properties based on transformations that preserve the distance function. One wants, in a sense, to preserve “infinitesimal” distances, i.e. points which are infinitely close together should remain infinitely close together. Of course, there is no notion of infinitesimal closeness for pairs of points. However, in a metric space one can speak of a point being infinitesimally close to a set. For (X, d) a metric space, x ∈ X, and U ⊆ X, we say that x is infinitesimally close to U if, for every 0, there is a point in u ∈ U such that d(x, u) ≤ . The collection of all points which are infinitesimally close to a set U is called the closure of U and is denoted by U. Note that if a point is infinitesimally close to a set U then any continuous transformation will preserve that property, even if it does not preserve distances. Example 3.11 The closure of the open interval (0, 1) is the closed interval [0, 1]. Example 3.12 The closure of the set Rn − {0} in Rn is the entire set Rn. Example 3.13 The closure of the set of all points of the form n 2k , where 0 ≤ n ≤ 2k and k ≥ 0, is the entire closed interval [0, 1]. Closure can be viewed as an operator on the subsets of Rn. It turns out to be idempotent in the sense that U = U. One can develop the notion of a topological space using only an abstract closure operator on the family of subsets, with certain properties. However, it is more conventional to develop the notion using the ideas of closed and open sets. A closed set V in Rn is a subset for which V = V. A subset U of Rn is open if its complement in X is closed, i.e. if U = X V for some closed set V. Another characterization is that a set is open if and only if it is an arbitrary union of open balls Br (x) = {y ∈ Rn | d(x, y) r}. The open sets in a metric space Rn satisfy the following three properties. • ∅ and X are both open sets in X. • Arbitrary unions of open sets are open. • Finite intersections of open sets are open. The identification of these properties leads us to the notion of a topological space. Definition 3.14 A topological space is a pair (X, U), where X is a set and U is a family of subsets (called the open sets) of X satisfying the following three properties. 1. ∅ ∈ U and X ∈ U. 2. For any family {Uα}α∈A of open sets, where A is any parameter set, S α∈A Uα is also in U. 3. For any family {Uα}α∈A of open sets, where A is any finite parameter set, T α∈A Uα is also in U. The family U is called a topology on X. A subset C ⊆ X is closed if X C is open. Here are some examples of open and closed sets in Rn. Example 3.15 Open (respectively closed) intervals are open (respectively closed) sets in R.
- 52. 3.2 Qualitative and Quantitative Properties 35 Example 3.16 Open balls (respectively closed balls) are open (respectively closed) sets in Rn. Example 3.17 Finite sets of points in Rn are always closed. Complements of finite sets are open. Example 3.18 For continuous function f : Rn → R, the set of points {v ∈ Rn | f (v) 0} is always an open set. Similarly for . The set of points {v ∈ Rn | f (v) ≤ 0} is always a closed set. Again, similarly for ≥. The terminology is consistent with the notions of open and closed conditions used in analysis. Example 3.19 For any family { f1, f2, . . ., fk } of continuous functions from Rn to R, the set {v ∈ Rn | fi (v) = 0 for all i} is closed. This means that algebraic varieties (sets of points defined by algebraic equations) are closed sets in Rn. As we progress, we will give numerous examples of topological spaces. Here we introduce the most elementary example. Definition 3.20 For any subset X ⊆ Rn, we define a topology on X by declaring that U ⊆ X is open if and only if U = V ∩ X for an open subset V ⊆ Rn. This defines familiar spaces such as spheres and tori as topological spaces. We are now able to be precise about what is meant by a topological property. Definition 3.21 A property of subsets of Rn is said to be topological if and only if it depends only on the topology defined above. Of course, topological properties can be defined on other topological spaces as well. The definition of a topological space is motivated by the case of Rn, and so clearly applies to it. The power of the definition lies in the fact that there are a number of systems far removed from the case of Rn but which nevertheless satisfy the axioms for a topological space. It turns out that this enables the use of geometric and topological intuition in cases where it was not anticipated. Example 3.22 For any set X, we can let the family of all subsets of X with finite complement be the open sets. This does give a topological space. Example 3.23 Let p be a prime number. A topology U on Z, the set of integers, is given by defining U to be the collection of sets U such that for any n ∈ U there is a positive integer r such that the set {n + kpr | k ∈ Z} is contained in U. This is called the p-adic topology, and it is of use in number theory. 3.2.2 Continuous Maps and Homeomorphisms In the study of multivariable calculus, we identify the notion of continuous maps from Rm to Rn using the –δ definition of continuity. We can characterize continuity directly using the topology on Rn as follows.
- 53. 36 Topology Proposition 3.24 A map f : Rm → Rn is continuous if and only if f 1(U) is open for all open sets U ⊆ Rn. This leads us to the notion of a continuous map between topological spaces X and Y. Definition 3.25 Let X and Y be topological spaces and f : X → Y a map of sets. Then f is said to be continuous if and only if f −1V is open for all open sets V ⊆ Y. Remark 3.26 One can formulate the notion of a closure operator in an arbitrary topological space and then interpret this definition as requiring that if x ∈ X is in the closure of a subset U ⊆ X then f (x) is in the closure of f (U). Using this notion of maps, we will now identify the key notion that will allow us to speak precisely about topological properties. Definition 3.27 Let f : X → Y be a continuous map of topological spaces. If there is a continuous function g: Y → X such that these functions are inverses of each other, i.e. f (g(y)) = y and g( f (x)) = x, then f is a homeomorphism, and we say that X and Y are homeomorphic. Remark 3.28 This notion is the exact analogue of the notion of the isomorphisms of groups or the bijections of sets. It means that the map f can be inverted and that the inverse is continuous. It also gives precise meaning to the idea that Y can simply be regarded as a reparametrization of X. Another point of view is that it makes precise the notion of Y being obtained from X by stretching or deforming, without tearing or “crushing”. If there is a homeomorphism from X to Y, we say that X and Y are homeomorphic. Here are examples of pairs of homeomorphic spaces. Example 3.29 Let X ⊆ R2 denote the unit circle, and let Y denote the ellipse given by the equation x2 a2 + y2 b2 = 1. Both sets are equipped with the subspace topology, defined in Definition 3.20. Then the map (x, y) → (ax, by) is a homeomorphism from the circle to the ellipse, so the circle and the ellipse are homeomorphic. Example 3.30 The parabola given by y = x2 and the real line are homeomorphic, the homeomorphism being given by the map (x, y) → x. Example 3.31 Let X ⊆ R2 denote the set X = {(x, y) ∈ R2 | x2 + y2 1}, and let Y ⊆ R2 denote the subset R2 − ~ 0. Polar coordinates are a good choice for describing a map g: X → Y, which we define by g(r, θ) = (r − 1, θ). The inverse map is given by g 1 (r, θ) = (r + 1, θ).
- 54. 3.2 Qualitative and Quantitative Properties 37 Example 3.32 Let X be the torus in three-dimensional space defined by the equation (c q x2 + y2)2 + z2 = a2 , where 0 a c. Let Y ⊆ R4 be the subspace defined by the equations ( x2 + y2 = 1, z2 + w2 = 1. We define a map f : X → Y via the formula f (x, y, z) = x k(x, y)k2 , y k(x, y)k2 , c k(x, y)k2 a , z a ! , where k(x, y)k2 denotes p x2 + y2. It is easy to verify that this map has its image in Y, and it is continuous since it is defined by simple formulas. It turns out that f is a homeomorphism. One way to prove that the map f in the above example is a homeomorphism is to exhibit a formula for the inverse. This is somewhat involved even in this case and can become much more so in more complicated examples. There is a simple statement that allows one to bypass that process. Proposition 3.33 Let f : X → Y be a continuous map, where X and Y are subsets of Rm and Rn respectively. We say a subset of Rn is bounded if it is contained in a ball BR (0) = {v | kvk R} for some R 0. Suppose that X and Y are closed and bounded. If the map f is bijective as a map of sets, then f is a homeomorphism. In Example 3.32 above, it is easy to check that both sets are closed and bounded and that f , regarded as a map of sets, is bijective. We are therefore able to conclude that it is a homeomorphism. Remark 3.34 Note that the two descriptions of a topological space have different strengths. The first description sits in three-dimensional space and can therefore be readily visualized. The second is much simpler, in that the equations that define it are much simpler. Depending on the kind of analysis one is doing, different coordinate systems may be useful. We should also ask how we can show that two spaces are not homeomorphic. Let X = [0, 1] ⊆ R and let Y ⊆ R2 denote the closed unit ball {(x, y) ∈ R2 | x2 + y2 ≤ 1}. These two spaces appear to be very different and we therefore suspect that they are not homeomorphic, but in order to be convinced of this we must try to find some topological property that applies to one but not to the other. Reasoning intuitively, we observe that by removing a single point from X, such as the point 1 2 , we can form a space which is disconnected in the sense that is breaks into two disjoint open pieces, namely [0, 1 2 ) and (1 2, 1], while removing any single point from Y will always leave it in a single connected piece. We have not formally defined connectedness yet (we will do so in Section 3.2.8), but this informal argument suggests how we can develop topological properties that enable discrimination between spaces, even when we allow ourselves to modify the spaces by arbitrary coordinate changes.
- 55. Another Random Document on Scribd Without Any Related Topics
- 56. towards Northern Europe is obtained from some of the rock paintings and carvings of Sweden. Among the canoes depicted are some with distinct Mediterranean characteristics. One at Tegneby in Bohuslän bears a striking resemblance to a boat seen by Sir Henry Stanley on Lake Victoria Nyanza. It seems undoubted that the designs are of common origin, although separated not only by centuries but by barriers of mountain, desert, and sea extending many hundreds of miles. From the Maglemosian boat the Viking ship was ultimately developed; the unprogressive Victoria Nyanza boatbuilders continued through the Ages repeating the design adopted by their remote ancestors. In both vessels the keel projects forward, and the figure-head is that of a goat or ram. The northern vessel has the characteristic inward curving stern of ancient Egyptian ships. As the rock on which it was carved is situated in a metal- yielding area, the probability is that this type of vessel is a relic of the visits paid by searchers for metals in ancient times, who established colonies of dark miners among the fair Northerners and introduced the elements of southern culture. The ancient boats found in Scotland are of a variety of types. One of those at Glasgow lay, when discovered, nearly vertical, with prow uppermost as if it had foundered; it had been built of several pieces of oak, though without ribs. Another had the remains of an outrigger attached to it: beside another, which had been partly hollowed by fire, lay two planks that appear to have been wash- boards like those on a Sussex dug-out. A Clyde clinker-built boat, eighteen feet long, had a keel and a base of oak to which ribs had been attached. An interesting find at Kinaven in Aberdeenshire, several miles distant from the Ythan, a famous pearling river, was a dug-out eleven feet long, and about four feet broad. It lay embedded at the head of a small ravine in five feet of peat which appears to have been the bed of an ancient lake. Near it were the stumps of big oaks, apparently of the Upper Forestian period. Among the longest of the ancient boats that have been discovered are one forty-two feet long, with an animal head on the prow, from
- 57. Loch Arthur, near Dumfries, one thirty-five long from near the River Arun in Sussex, one sixty-three feet long excavated near the Rother in Kent, one forty-eight feet six inches long, found at Brigg, Lincolnshire, with wooden patches where she had sprung a leak, and signs of the caulking of cracks and small holes with moss. These vessels do not all belong to the same period. The date of the Brigg boat is, judging from the geological strata, between 1100 and 700 b.c. It would appear that some of the Clyde vessels found at twenty-five feet above the present sea-level are even older. Beside one Clyde boat was found an axe of polished green-stone similar to the axes used by Polynesians and others in shaping dug-outs. This axe may, however, have been a religious object. To the low bases of some vessels were fixed ribs on which skins were stretched. These boats were eminently suitable for rough seas, being more buoyant than dug-outs. According to Himilco the inhabitants of the Œstrymnides, the islands rich in tin and lead, had most sea-worthy skiffs. These people do not make pine keels, nor, he says, do they know how to fashion them; nor do they make fir barks, but, with wonderful skill, fashion skiffs with sewn skins. In these hide-bound vessels, they skim across the ocean. Apparently they were as daring mariners as the Oregon Islanders of whom Washington Irving has written: It is surprising to see with what fearless unconcern these savages venture in their light barks upon the roughest and most tempestuous seas. They seem to ride upon the wave like sea-fowl. Should a surge throw the canoe upon its side, and endanger its over turn, those to the windward lean over the upper gunwale, thrust their paddles deep into the wave, and by this action not merely regain an equilibrium, but give their bark a vigorous impulse forward. The ancient mariners whose rude vessels have been excavated around our coasts were the forerunners of the Celtic sea-traders, who, as the Gaelic evidence shows, had names not only for the
- 58. North Sea and the English Channel but also for the Mediterranean Sea. They cultivated what is known as the sea sense, and developed shipbuilding and the art of navigation in accordance with local needs. When Julius Cæsar came into conflict with the Veneti of Brittany he tells that their vessels were greatly superior to those of the Romans. The bodies of the ships, he says, were built entirely of oak, stout enough to withstand any shock or violence.... Instead of cables for their anchors they used iron chains.... The encounter of our fleet with these ships was of such a nature that our fleet excelled in speed alone, and the plying of oars; for neither could our ships injure theirs with their rams, so great was their strength, nor was a weapon easily cast up to them owing to their height.... About 220 of their ships ... sailed forth from the harbour. In this great allied fleet were vessels from our own country.[55] It must not be imagined that the sea sense was cultivated because man took pleasure in risking the perils of the deep. It was stern necessity that at the beginning compelled him to venture on long voyages. After England was cut off from France the peoples who had adopted the Neolithic industry must have either found it absolutely necessary to seek refuge in Britain, or were attracted towards it by reports of prospectors who found it to be suitable for residence and trade.
- 59. CHAPTER VIII Neolithic Trade and Industries Attractions of Ancient Britain—Romans search for Gold, Silver, Pearls, c.—The Lure of Precious Stones and Metals—Distribution of Ancient British Population— Neolithic Settlements in Flint-yielding Areas—Trade in Flint —Settlements on Lias Formation—Implements from Basic Rocks—Trade in Body-painting Materials—Search for Pearls—Gold in Britain and Ireland—Agriculture—The Story of Barley—Neolithic Settlers in Ireland—Scottish Neolithic Traders—Neolithic Peoples not Wanderers— Trained Neolithic Craftsmen. The drift of peoples into Britain which began in Aurignacian times continued until the Roman period. There were definite reasons for early intrusions as there were for the Roman invasion. Britain contains to reward the conqueror, Tacitus wrote,[56] mines of gold and silver and other metals. The sea produces pearls. According to Suetonius, who at the end of the first century of our era wrote the Lives of the Cæsars, Julius Cæsar invaded Britain with the desire to enrich himself with the pearls found on different parts of the coast. On his return to Rome he presented a corselet of British pearls to the goddess Venus. He was in need of money to further his political ambitions. He found what he required elsewhere, however. After the death of Queen Cleopatra sufficient gold and silver flowed to Rome from Egypt to reduce the loan rate of interest from 12 to 4 per cent. Spain likewise contributed its share to enrich the great predatory state of Rome.[57]
- 60. Long ages before the Roman period the early peoples entered Britain in search of pearls, precious stones, and precious metals because these had a religious value. The Celts of Gaul offered great quantities of gold to their deities, depositing the precious metals in their temples and in their sacred lakes. Poseidonius of Apamea tells that after conquering Gaul the Romans put up these sacred lakes to public sale, and many of the purchasers found quantities of solid silver in them. He also says that gold was similarly placed in these lakes.[58] Apparently the Celts believed, as did the Aryo-Indians, that gold was a form of the gods and fire, light, and immortality, and that it was a life giver.[59] Personal ornaments continued to have a religious value until Christian times. FLINT LANCE-HEADS FROM IRELAND (British Museum)
- 61. Photo Oxford University Press CHIPPED AND POLISHED ARTIFACTS FROM SOUTHERN ENGLAND (British Museum) As we have seen when dealing with the Red Man of Paviland, the earliest ornaments were shells, teeth of wild animals, coloured stones, ivory, c. Shells were carried great distances. Then arose the habit of producing substitutes which were regarded as of great potency as the originals. The ancient Egyptians made use of gold to manufacture imitation shells, and before they worked copper they wore charms of malachite, which is an ore of copper. They probably used copper first for magical purposes just as they used gold. Pearls found in shells were regarded as depositories of supernatural influence, and so were coral and amber (see Chapter XIII). Like the Aryo-Indians, the Egyptians, Phœnicians, Greeks, and others connected precious metals, stones, pearls, c., with their deities, and believed that these contained the influence of their deities, and were therefore lucky. These and similar beliefs are of great antiquity in Europe and Asia and North Africa. It would be rash to assume that they were not known to the ancient mariners who reached our shores in vessels of Mediterranean type.
- 62. The colonists who were attracted to Britain at various periods settled in those districts most suitable for their modes of life. It was necessary that they should obtain an adequate supply of the materials from which their implements and weapons were manufactured. The distribution of the population must have been determined by the resources of the various districts. At the present day the population of Britain is most dense in those areas in which coal and iron are found and where commerce is concentrated. In ancient times, before metals were used, it must have been densest in those areas where flint was found—that is, on the upper chalk formations. If worked flints are discovered in areas which do not have deposits of flint, the only conclusion that can be drawn is that the flint was obtained by means of trade, just as Mediterranean shells were in Aurignacian and Magdalenian times obtained by hunters who settled in Central Europe. In Devon and Cornwall, for instance, large numbers of flint implements have been found, yet in these counties suitable flint was exceedingly scarce in ancient times, except in East Devon, where, however, the surface flint is of inferior character. In Wilts and Dorset, however, the finest quality of flint was found, and it was no doubt from these areas that the early settlers in Cornwall and Devon received their chief supplies of the raw material, if not of the manufactured articles. In England, as on the Continent, the most abundant finds of the earliest flint implements have been made in those areas where the early hunters and fishermen could obtain their raw materials. River drift implements are discovered in largest numbers on the chalk formations of south-eastern England between the Wash and the estuary of the Thames. The Neolithic peoples, who made less use of horn and bone than did the Azilians and Maglemosians, had many village settlements on the upper chalk in Dorset and Wiltshire, and especially at Avebury where there were veritable flint factories, and near the famous flint mines at Grimes Graves in the vicinity of Weeting in Norfolk and at Cissbury Camp not far from Worthing in Sussex. Implements were likewise
- 63. made of basic rocks, including quartzite, ironstone, green-stone, hornblende schist, granite, mica-schist, c.; while ornaments were made of jet, a hydrocarbon compound allied to cannel coal, which takes on a fine polish, Kimeridge shale and ivory. Withal, like the Aurignacians and Magdalenians, the Neolithic-industry people used body paint, which was made with pigments of ochre, hæmatite, an ore of iron, and ruddle, an earthy variety of iron ore. In those districts, where the raw materials for stone implements, ornaments, and body paint were found, traces survive of the activities of the Neolithic peoples. Their graves of long-barrow type are found not only in the chalk areas but on the margins of the lias formations. Hæmatite is found in large quantities in West Cumberland and north Lancashire and in south-western England, while the chief source of jet is Whitby in Yorkshire, where it occurs in large quantities in beds of the Upper Lias shale.
- 64. Mr. W. J. Perry, of Manchester University, who has devoted special attention to the study of the distribution of megalithic monuments, has been drawing attention to the interesting association of these monuments with geological formations.[60] In the Avebury district
- 65. stone circles, dolmens, chambered barrows, long barrows, and Neolithic settlements are numerous; another group of megalithic monuments occurs in Oxford on the margin of the lias formation, and at the south-end of the great iron field extending as far as the Clevelands. According to the memoir of the geological survey, there are traces of ancient surface iron-workings in the Middle Lias formation of Oxfordshire, where red and brown hæmatite were found. Mr. Perry notes that there are megalithic monuments in the vicinity of all these surface workings, as at Fawler, Adderbury, Hook Norton, Woodstock, Steeple Aston, and Hanbury. Apparently the Neolithic peoples were attracted to the lias formation because it contains hæmatite, ochre, shale, c. There are significant megaliths in the Whitby region where the jet is so plentiful. Amber was obtained from the east coast of England and from the Baltic. The Neolithic peoples appear to have searched for pearls, which are found in a number of English, Welsh, Scottish, and Irish rivers, and in the vicinity of most, if not all, of these megaliths occur. Gold was the first metal worked by man, and it appears to have attracted some of the early peoples who settled in Britain. The ancient seafarers who found their way northward may have included searchers for gold and silver. The latter metal was at one time found in great abundance in Spain, while gold was at one time fairly plentiful in south-western England, in North Wales, in various parts of Scotland and especially in Lanarkshire, and in north-eastern, eastern, and western Ireland. That there was a drift of civilized peoples into Britain and Ireland during the period of the Neolithic industry is made evident by the fact that the agricultural mode of life was introduced. Barley does not grow wild in Europe. The nearest area in which it grew wild and was earliest cultivated was the delta area of Egypt, the region from which the earliest vessels set out to explore the shores of the Mediterranean. It may be that the barley seeds were carried to Britain not by the overland routes alone to Channel ports, but also by the seafarers whose boats, like the Glasgow one with the cork plug, coasted round by Spain and Brittany, and crossed the Channel to south-western England and
- 66. thence went northward to Scotland. As Irish flints and ground axe- heads occur chiefly in Ulster, it may be that the drift of early Neolithic settlers into County Antrim, in which gold was also found, was from south-western Scotland. The Neolithic settlement at Whitepark Bay, five miles from the Giant's Causeway, was embedded at a considerable depth, showing that there has been a sinking of the land in this area since the Neolithic industry was introduced. Neolithic remains are widely distributed over Scotland, but these have not received the intensive study devoted to similar relics in England. Mr. Ludovic Mann, the Glasgow archæologist, has, however, compiled interesting data regarding one of the local industries that bring out the resource and activities of early man. On the island of Arran is a workable variety of the natural volcanic glass, called pitch- stone, that of other parts of Scotland and of Ireland being too much cracked into small pieces to be of use. It was used by the Neolithic settlers in Arran for manufacturing arrowheads, and as it was imported into Bute, Ayrshire, and Wigtownshire, a trade in this material must have existed. If, writes Mr. Mann, the stone was not locally worked up into implements in Bute, it was so manipulated on the mainland, where workshops of the Neolithic period and the immediately succeeding overlap period yielded long fine flakes, testifying to greater expertness in manufacturing there than is shown by the remains in the domestic sites yet awaiting adequate exploration in Arran. The explanation may be that the Wigtownshire flint knappers, accustomed to handle an abundance of flint, were more proficient than in most other places, and that the pitch-stone was brought to them as experts, because the material required even more skilful handling than flint.[61] In like manner obsidian, as has been noted, was imported into Crete from the island of Melos by seafarers, long before the introduction of metal working.[62] It will be seen that the Neolithic peoples were no mere wandering hunters, as some have represented them to have been, but they had their social organization, their industries, and their system of trading by land and sea. They settled not only in those areas where they
- 67. could procure a regular food supply, but those also in which they obtained the raw materials for implements, weapons, and the colouring material which they used for religious purposes. They made pottery for grave offerings and domestic use, and wooden implements regarding which, however, little is known. Withal, they had their spinners and weavers. The conditions prevailing in Neolithic settlements must have been similar to those of later times. There must have been systems of laws to make trade and peaceful social intercourse possible, and no doubt these had, as elsewhere, a religious basis. Burial customs indicate a uniformity of beliefs over wide areas. The skill displayed in working stone was so great that it cannot now be emulated. Ripple-flaking has long been a lost art. Craftsmen must have undergone a prolonged period of training which was intelligently controlled under settled conditions of life. It is possible that the so-called Neolithic folk were chiefly foreigners who exploited the riches of the country. The evidence in this connection will be found in the next chapter.
- 68. CHAPTER IX Metal Workers and Megalithic Monuments Broad-heads of Bronze Age—The Irish Evidence— Bronze Introduced by Traders—How Metals were Traced —A Metal Working Tribe—Damnonii in England, Scotland, and Ireland—Miners as Slaves—The Lot of Women Workers—Megalithic Monuments in English Metal-yielding Areas—Stone Circles in Barren Localities—Early Colonies of Easterners in Spain—Egyptian and Babylonian Relics associated with British Jet and Baltic Amber—A New Flint Industry of Eastern Origin—British Bronze identical with Continental—Ancient Furnaces of Common Origin —Stones of Worship adorned with Metals—The Maggot God of Stone Circles—Ancient Egyptian Beads at Stonehenge—Earliest Authentic Date in British History— The Aim of Conquests. It used to be thought that the introduction of metal working into Britain was the result of an invasion of alien peoples, who partly exterminated and partly enslaved the long-headed Neolithic inhabitants. This view was based on the evidence afforded by a new type of grave known as the Round Barrow. In graves of this class have been found Bronze Age relics, a distinctive kind of pottery, and skulls of broad-heads. The invasion of broad-heads undoubtedly took place, and their burial customs suggest that their religious beliefs were not identical with those of the long-heads. But it
- 69. remains to be proved that they were the actual introducers of the bronze industry. They do not appear to have reached Ireland, where bronze relics are associated with a long-headed people of comparatively low stature. The early Irish bronze forms were obviously obtained from Spain, while early English bronze forms resemble those of France and Italy. Cutting implements were the first to be introduced. This fact does not suggest that a conquest took place. The implements may have been obtained by traders. Britain apparently had in those ancient times its trading colonies, and was visited by active and enterprising seafarers. Long-head (Dolichocephalic) Skull
- 70. Broad-head (Brachycephalic) Skull Both these specimens were found in Round Barrows in the East Riding of Yorkshire The discovery of metals in Britain and Ireland was, no doubt, first made by prospectors who had obtained experience in working them elsewhere. They may have simply come to exploit the country. How these men conducted their investigations is indicated by the report found in a British Museum manuscript, dating from about 1603, in which the prospector gives his reason for believing that gold was to be found on Crawford Moor in Lanarkshire. He tells that he saw among the rocks what Scottish miners call mothers and English miners leaders or metalline fumes. It was believed that the fumes arose from veins of metal and coloured the rocks as smoke passing upward through a tunnel blackens it, and leaves traces on the outside. He professed to be able to distinguish between the colours left by fumes of iron, lead, tin, copper, or silver. On Crawford Moor he found sparr, keel, and brimstone between rocks, and regarded this discovery as a sure indication that gold was in situ. The mothers or leaders were more pronounced than any he had ever seen in Cornwall, Somersetshire, about Keswick, or any other mineral parts wheresoever I have travelled.[63] Gold was found
- 71. in this area of Lanarkshire in considerable quantities, and was no doubt worked in ancient times. Of special interest in this connection is the fact that it was part of the territory occupied by Damnonians, [64] who appear to have been a metal-working people. Besides occupying the richest metal-yielding area in Scotland, the Damnonians were located in Devon and Cornwall, and in the east- midland and western parts of Ireland, in which gold, copper, and tin- stone were found as in south-western England. The Welsh Dyfneint (Devon) is supposed by some to be connected with a form of this tribal name. Another form in a Yarrow inscription is Dumnogeni. In Ireland Inber Domnann is the old name of Malahide Bay north of Dublin. Domnu, the genitive of which is Domnann, was the name of an ancient goddess. In the Irish manuscripts these people are referred to as Fir-domnann,[65] and associated with the Fir-bolg (the men with sacks). A sack-carrying people are represented in Spanish rock paintings that date from the Azilian till early Bronze Age times. In an Irish manuscript which praises the fair and tall people, the Fir-bolg and Fir-domnann are included among the black-eyed and black-haired people, the descendants of slaves and churls, and the promoters of discord among the people. The reference to slaves is of special interest because the lot of the working miners was in ancient days an extremely arduous one. In one of his collected records which describes the method of the greatest antiquity Diodorus Siculus (a.d. first century) tells how gold-miners, with lights bound on their foreheads, drove galleries into the rocks, the fragments of which were carried out by frail old men and boys. These were broken small by men in the prime of life. The pounded stone was then ground in handmills by women: three women to a mill and to each of those who bear this lot, death is better than life. Afterwards the milled quartz was spread out on an inclined table. Men threw water on it, work it through their fingers, and dabbed it with sponges until the lighter matter was removed and the gold was left behind. The precious metal was placed in a clay crucible, which was kept heated for five days and five nights. It may be that the Scandinavian references to the nine maidens who
- 72. turn the handle of the world mill which grinds out metal and soil, and the Celtic references to the nine maidens who are associated with the Celtic cauldron, survive from beliefs that reflected the habits and methods of the ancient metal workers. It is difficult now to trace the various areas in which gold was anciently found in our islands. But this is not to be wondered at. In Egypt there were once rich goldfields, especially in the Eastern Desert, where about 100 square miles were so thoroughly worked in ancient times that only the merest traces of gold remain.[66] Gold, as has been stated, was formerly found in south-western England, North Wales, and, as historical records, archæological data, and place names indicate, in various parts of Scotland and Ireland. During the period of the Great Thaw a great deal of alluvial gold must have distributed throughout the country. Silver was found in various parts. In Sutherland it is mixed with gold as it is elsewhere with lead. Copper was worked in a number of districts where the veins cannot in modern times be economically worked, and tin was found in Ireland and Scotland as well as in south-western England, where mining operations do not seem to have been begun, as Principal Sir John Rhys has shown,[67] until after the supplies of surface tin were exhausted. Of special interest in connection with this problem is the association of megalithic monuments with ancient mine workings. An interesting fact to be borne in mind in connection with these relics of the activities and beliefs of the early peoples is that they represent a distinct culture of complex character. Mr. T. Eric Peet[68] shows that the megalithic buildings occupy a very remarkable position along a vast seaboard which includes the Mediterranean coast of Africa and the Atlantic coast of Europe. In other words, they lie entirely along a natural sea route. He gives forcible reasons for arriving at the conclusion that it is impossible to consider megalithic building as a mere phase through which many nations passed, and it must therefore have been a system originating with one race, and spreading far and wide, owing either to trade influence or migration. He adds:
- 73. Great movements of races by sea were not by any means unusual in primitive days. In fact, the sea has always been less of an obstacle to early man than the land with its deserts, mountains, and unfordable rivers. There is nothing inherently impossible or even improbable in the suggestion that a great immigration brought the megalithic monuments from Sweden to India or vice versa. History is full of instances of such migrations. But there must have been a definite reason for these race movements. It cannot be that in all cases they were forced merely by natural causes, such as changes of climate, invasions of the sea, and the drying up of once fertile districts, or by the propelling influences of stronger races in every country from the British Isles to Japan—that is, in all countries in which megalithic monuments of similar type are found. The fact that the megalithic monuments are distributed along a vast seaboard suggests that they were the work of people who had acquired a culture of common origin, and were attracted to different countries for the same reason. What that attraction was is indicated by studying the elements of the megalithic culture. In a lecture delivered before the British Association in Manchester in 1915, Mr. W. J. Perry threw much light on the problem by showing that the carriers of the culture practised weaving linen, and in some cases the use of Tyrian purple, pearls, precious stones, metals, and conch-shell trumpets, as well as curious beliefs and superstitions attached to the latter, while they adopted certain definite metallurgical methods, as well as mining. Mr. Perry's paper was subsequently published by the Manchester Literary and Philosophical Society. It shows that in Western Europe the megalithic monuments are distributed in those areas in which ancient pre- Roman and pre-Greek mine workings and metal washings have been traced. The same correspondence, he writes, seems to hold in the case of England and Wales. In the latter country the counties where megalithic structures abound are precisely those where mineral deposits and ancient mine-workings occur. In England the grouping in Cumberland, Westmorland, Northumberland, Durham, and
- 74. Derbyshire is precisely that of old mines; in Cornwall the megalithic structures are mainly grouped west of Falmouth, precisely in that district where mining has always been most active. Pearls, amber, coral, jet, c., were searched for as well as metals. The megalithic monuments near pearling rivers, in the vicinity of Whitby, the main source of jet, and in Denmark and the Baltic area where amber was found were, in all likelihood, erected by people who had come under the spell of the same ancient culture. When, therefore, we come to deal with groups of monuments in areas which were unsuitable for agriculture and unable to sustain large populations, a reasonable conclusion to draw is that precious metals, precious stones, or pearls were once found near them. The pearling beds may have been destroyed or greatly reduced in value, [69] or the metals may have been worked out, leaving but slight if any indication that they were ever in situ. Reference has been made to the traces left by ancient miners in Egypt where no gold is now found. In our own day rich gold fields in Australia and North America have been exhausted. It would be unreasonable for us to suppose that the same thing did not happen in our country, even although but slight traces of the precious metal can now be obtained in areas which were thoroughly explored by ancient miners. When early man reached Scotland in search of suitable districts in which to settle, he was not likely to be attracted by the barren or semi-barren areas in which nature grudged soil for cultivation, where pasture lands were poor and the coasts were lashed by great billows for the greater part of the year, and the tempests of winter and spring were particularly severe. Yet in such places as Carloway, fronting the Atlantic on the west coast of Lewis, and at Stennis in Orkney, across the dangerous Pentland Firth, are found the most imposing stone circles north of Stonehenge and Avebury. Traces of tin have been found in Lewis, and Orkney has yielded traces of lead, including silver-lead, copper and zinc, and has flint in glacial drift. Traces of tin have likewise been found on the mainlands of Ross- shire and Argyllshire, in various islands of the Hebrides and in
- 75. Stirlingshire. The great Stonehenge circle is like the Callernish and Stennis circles situated in a semi-barren area, but it is an area where surface tin and gold were anciently obtained. One cannot help concluding that the early people, who populated the wastes of ancient Britain and erected megalithic monuments, were attracted by something more tangible than the charms of solitude and wild scenery. They searched for and found the things they required. If they found gold, it must be recognized that there was a psychological motive for the search for this precious metal. They valued gold, or whatever other metal they worked in bleak and isolated places, because they had learned to value it elsewhere. Who were the people that first searched for, found, and used metals in Western Europe? Some have assumed that the natives themselves did so as a matter of course. Such a theory is, however, difficult to maintain. Gold is a useless metal for all practical purposes. It is too soft for implements. Besides, it cannot be found or worked except by those who have acquired a great deal of knowledge and skill. The men who first washed it from the soil in Britain must have obtained the necessary knowledge and skill in a country where it was more plentiful and much easier to work, and where—and this point is a most important one—the magical and religious beliefs connected with gold have a very definite history. Copper, tin, and silver were even more difficult to find and work in Britain. The ancient people who reached Britain and first worked metals or collected ores were not the people who were accustomed to use implements of bone, horn, and flint, and had been attracted to its shores merely because fish, fowl, deer, and cows, were numerous. The searchers for metals must have come from centres of Eastern civilization, or from colonies of highly skilled peoples that had been established in Western Europe. They did not necessarily come to settle permanently in Britain, but rather to exploit its natural riches. This conclusion is no mere hypothesis. Siret,[70] the Belgian archæologist, has discovered in southern Spain and Portugal traces of numerous settlements of Easterners who searched for minerals,
- 76. c., long before the introduction of bronze working in Western Europe. They came during the archæological Stone Age; they even introduced some of the flint implements classed as Neolithic by the archæologists of a past generation. These Eastern colonists do not appear to have been an organized people. Siret considers that they were merely groups of people from Asia—probably the Syrian coast—who were in contact with Egypt. During the Empire period of Egypt, the Egyptian sphere of influence extended to the borders of Asia Minor. At an earlier period Babylonian influence permeated the Syrian coast and part of Asia Minor. The religious beliefs of seafarers from Syria were likely therefore to bear traces of the Egyptian and Babylonian religious systems. Evidence that this was the case has been forthcoming in Spain. These Eastern colonists not only operated in Spain and Portugal, but established contact with Northern Europe. They exported what they had searched for and found to their Eastern markets. No doubt, they employed native labour, but they do not appear to have instructed the natives how to make use of the ores they themselves valued so highly. In time they were expelled from Spain and Portugal by the people or mixed peoples who introduced the working of bronze and made use of bronze weapons. These bronze carriers and workers came from Central Europe, where colonies of peoples skilled in the arts of mining and metal working had been established. In the Central European colonies Ægean and Danubian influences have been detected.
- 77. Valentine THE RING OF STENNIS, ORKNEY (see page 94) Among the archæological finds, which prove that the Easterners settled in Iberia before bronze working was introduced among the natives, are idol-like objects made of hippopotamus ivory from Egypt, a shell (Dentalium elephantum) from the Red Sea, objects made from ostrich eggs which must have been carried to Spain from Africa, alabaster perfume flasks, cups of marble and alabaster of Egyptian character which had been shaped with copper implements, Oriental painted vases with decorations in red, black, blue, and green,[71] mural paintings on layers of plaster, feminine statuettes in alabaster which Siret considers to be of Babylonian type, for they differ from Ægean and Egyptian statuettes, a cult object (found in graves) resembling the Egyptian ded amulet, c. The Iberian burial places of these Eastern colonists have arched cupolas and entrance corridors of Egyptian-Mycenæan character. Of special interest are the beautifully worked flints associated with these Eastern remains in Spain and Portugal. Siret draws attention to
- 78. the fact that no trace has been found of flint factories. This particular flint industry was an entirely new one. It was not a development of earlier flint-working in Iberia. Apparently the new industry, which suddenly appears in full perfection, was introduced by the Eastern colonists. It afterwards spread over the whole maritime west, including Scandinavia where the metal implements of more advanced countries were imitated in flint. This important fact emphasizes the need for caution in making use of such a term as Neolithic Age. Siret's view in this connection is that the Easterners, who established trading colonies in Spain and elsewhere, prevented the local use of metals which they had come to search for and export. It was part of their policy to keep the natives in ignorance of the uses to which metals could be put. Evidence has been forthcoming that the operations of the Eastern colonies in Spain and Portugal were extended towards the maritime north. Associated with the Oriential relics already referred to, Siret has discovered amber from the Baltic, jet from Britain (apparently from Whitby in Yorkshire) and the green-stone called callais usually found in beds of tin. The Eastern seafarers must have visited Northern Europe to exploit its virgin riches. A green-stone axe was found, as has been stated, near the boat with the cork plug, which lay embedded in Clyde silt at Glasgow. Artifacts of callais have been discovered in Brittany, in the south of France, in Portugal, and in south-eastern Spain. In the latter area, as Siret has proved, the Easterners worked silver-bearing lead and copper. The colonists appear to have likewise searched for and found gold. A diadem of gold was discovered in a necropolis in the south of Spain, where some eminent ancient had been interred. This find is, however, an exception. Precious metals do not as a rule appear in the graves of the period under consideration. As has been suggested, the Easterners who exploited the wealth of ancient Iberia kept the natives in ignorance. This ignorance, Siret says, was the guarantee of the prosperity of the commerce carried on by the strangers.... The first action of the East on the West was
- 79. the exploitation for its exclusive and personal profit of the virgin riches of the latter. These early Westerners had no idea of the use and value of the metals lying on the surface of their native land, while the Orientals valued them, were in need of them, and were anxious to obtain them. As Siret puts it: The West was a cow to be milked, a sheep to be fleeced, a field to be cultivated, a mine to be exploited. In the traditions preserved by classical writers, there are references to the skill and cunning of the Phœnicians in commerce, and in the exploitation of colonies founded among the ignorant Iberians. They did not inform rival traders where they found metals. Formerly, as Strabo says, the Phœnicians monopolized the trade from Gades (Cadiz) with the islanders (of the Cassiterides); and they kept the route a close secret. A vague ancient tradition is preserved by Pliny, who tells that tin was first fetched from Cassiteris (the tin island) by Midacritus.[72] We owe it to the secretive Phœnicians that the problem of the Cassiterides still remains a difficult one to solve. To keep the native people ignorant the Easterners, Siret believes, forbade the use of metals in their own colonies. A direct result of this policy was the great development which took place in the manufacture of the beautiful flint implements already referred to. These the natives imitated, never dreaming that they were imitating some forms that had been developed by a people who used copper in their own country. When, therefore, we pick up beautiful Neolithic flints, we cannot be too sure that the skill displayed belongs entirely to the Stone Age, or that the flints evolved from earlier native forms in those areas in which they are found. The Easterners do not appear to have extracted the metals from their ores either in Iberia or in Northern Europe. Tin-stone and silver-bearing lead were used for ballast for their ships, and they made anchors of lead. Gold washed from river beds could be easily packed in small bulk. A people who lived by hunting and fishing were not likely to be greatly interested in the laborious process of gold-
- 80. washing. Nor were they likely to attach to gold a magical and religious value as did the ancient Egyptians and Sumerians. So far as can be gathered from the Iberian evidence, the period of exploitation by the colonists from the East was a somewhat prolonged one. How many centuries it covered we can only guess. It is of interest to find, in this connection, however, that something was known in Mesopotamia before 2000 b.c. regarding the natural riches of Western Europe. Tablets have recently been found on the site of Asshur, the ancient capital of Assyria, which was originally a Sumerian settlement. These make reference to the Empire of Sargon of Akkad (c. 2600 b.c.), which, according to tradition, extended from the Persian Gulf to the Syrian coast. Sargon was a great conqueror. He poured out his glory over the world, declares a tablet found a good many years ago. It was believed, too, that Sargon embarked on the Mediterranean and occupied Cyprus. The fresh evidence from the site of Asshur is to the effect that he conquered Kaptara (?Crete) and the Tin Land beyond the Upper Sea (the Mediterranean). The explanation may be that he obtained control of the markets to which the Easterners carried from Spain and the coasts of Northern Europe the ores, pearls, c., they had searched for and found. It may be, therefore, that Britain was visited by Easterners even before Sargon's time, and that the Glasgow boat with the plug of cork was manned by dark Orientals who were prospecting the Scottish coast before the last land movement had ceased—that is, some time after 3000 b.c.
- 81. MEGALITHS Upper: Kit's Coty House, Kent. Lower: Trethevy Stone, Cornwall. When the Easterners were expelled from Spain by a people from Central Europe who used weapons of bronze, some of them appear to have found refuge in Gaul. Siret is of opinion that others withdrew from Brittany, where subsidences were taking place along the coast, leaving their megalithic monuments below high-water mark, and even under several feet of water as at Morbraz. He thinks that the
- 82. settlements of Easterners in Brittany were invaded at one and the same time by the enemy and the ocean. Other refugees from the colonies may have settled in Etruria, and founded the Etruscan civilization. Etruscan menhirs resemble those of the south of France, while the Etruscan crozier or wand, used in the art of augury, resembles the croziers of the megaliths, c., of France, Spain, and Portugal. There are references in Scottish Gaelic stories to magic wands possessed by wise women, and by the mothers of Cyclopean one-eyed giants. Ammianus Marcellinus, quoting Timagenes,[73] attributes to the Druids the statement that part of the inhabitants of Gaul were indigenous, but that some had come from the farthest shores and districts across the Rhine, having been expelled from their own lands by frequent wars and the encroachments of the ocean. The bronze-using peoples who established overland trade routes in Europe, displacing in some localities the colonies of Easterners and isolating others, must have instructed the natives of Western Europe how to mine and use metals. Bronze appears to have been introduced into Britain by traders. That the ancient Britons did not begin quite spontaneously to work copper and tin and manufacture bronze is quite evident, because the earliest specimens of British bronze which have been found are made of ninety per cent of copper and ten per cent of tin as on the Continent. Now, since a knowledge of the compound, wrote Dr. Robert Munro, implies a previous acquaintance with its component elements, it follows that progress in metallurgy had already reached the stage of knowing the best combination of these metals for the manufacture of cutting tools before bronze was practically known in Britain.[74] The furnaces used were not invented in Britain. Professor Gowland has shown that in Europe and Asia the system of working mines and melting metals was identical in ancient times. Summarizing Professor Gowland's articles in Archæologia and the Journal of the Royal Anthropological Institute, Mr. W. J. Perry writes in this connection:[75] The furnaces employed were similar; the crucibles were of the
- 83. same material, and generally of the same form; the process of smelting, first on the surface and then in the crucibles was found everywhere, even persisting down to present times in the absence of any fresh cultural influence. The study of the technique of mining and smelting has served to consolidate the floating mass of facts which we have accumulated, and to add support for the contention that one cultural influence is responsible for the earliest mining and smelting and washing of metals and the getting of precious stones and metals. The cause of the distribution of the megalithic culture was the search for certain forms of material wealth. That certain of the megalithic monuments were intimately connected with the people who attached a religious value to metals is brought out very forcibly in the references to pagan customs and beliefs in early Christian Gaelic literature. There are statements in the Lives of St. Patrick regarding a pagan god called Cenn Cruach and Crom Cruach whose stone statue was adorned with gold and silver, and surrounded by twelve other statues with bronze ornaments. The statue is called the king idol of Erin, and it is stated that the twelve idols were made of stone, but he ('Crom Cruach') was of gold. To this god of a stone circle were offered up the firstlings of every issue and the chief scions of every clan. Another idol was called Crom Dubh (Black Crom), and his name is still connected, O'Curry has written, with the first Sunday of August in Munster and Connaught. An Ulster idol was called Crom Chonnaill, which was either a living animal or a tree, or was believed to have been such, O'Curry says. De Jubainville translates Cenn Cruach as Bloody Head and Crom Cruach as Bloody Curb or Bloody Crescent. O'Curry, on the other hand, translates Crom Cruach as Bloody Maggot and Crom Dubh as Black Maggot. In Gaelic legends maggots or worms are referred to as forms of supernatural beings. The maggot which appeared on the flesh of a slain animal was apparently regarded as a new form assumed by the indestructible soul, just as in the Egyptian story of Bata the germ of life passes from his bull form in a drop of blood from which two trees spring up, and then in a chip from one of the trees from which the
- 84. man is restored in his original form.[76] A similar belief, which is widespread, is that bees have their origin as maggots placed in trees. One form of the story was taken over by the early Christians, which tells that Jesus was travelling with Peter and Paul and asked hospitality from an old woman. The woman refused it and struck Paul on the head. When the wound putrified maggots were produced. Jesus took the maggots from the wound and placed them in the hollow of a tree. When next they passed that way, Jesus directed Paul to look in the tree hollow where, to his surprise, he found bees and honey sprung from his own head.[77] The custom of placing crape on hives and telling the bees when a death takes place, which still survives in the south of England and in the north of Scotland, appears to be connected with the ancient belief that the maggot, bee, and tree were connected with the sacred animal and the sacred stone in which was the spirit of a deity. Sacred trees and sacred stones were intimately connected. Tacitus tells us that the Romans invaded Mona (Anglesea), they destroyed the sacred groves in which the Druids and black-robed priestesses covered the altars with the blood of captives.[78] There are a number of dolmens on this island and traces of ancient mine-workings, indicating that it had been occupied by the early seafarers who colonized Britain and Ireland and worked metals. A connection between the tree cult of the Druids and the cult of the builders of megaliths is thus suggested by Tacitus, as well as by the Irish evidence regarding the Ulster idol Crom Chonnaill, referred to above (see also Chapter XII). Who were the people that followed the earliest Easterners and visited our shores to search like them for metals and erect megalithic monuments? It is impossible to answer that question with certainty. There were after the introduction of bronze working, as has been indicated, intrusions of aliens. These included the introducers of the short-barrow method of burial and the later introducers of burial by cremation. It does not follow that all intrusions were those of conquerors. Traders and artisans may have come with their families in large numbers and mingled with the earlier peoples. Some intruders appear to have come by overland routes from southern
- 85. and central France and from Central Europe and the Danube valley, while others came across the sea from Spain. That a regular over- seas trade-route was in existence is indicated by the references made by classical writers to the Cassiterides (Tin Islands). Strabo tells that the natives bartered tin and hides with merchants for pottery, salt, and articles of bronze. The Phœnicians, as has been noted, monopolized the trade from Gades (Cadiz) with the islanders and kept the route a close secret. It was probably along this sea- route that Egyptian blue beads reached Britain. Professor Sayce has identified a number of these in Devizes Museum, and writes: They are met with plentifully in the Early Bronze Age tumuli of Wiltshire in association with amber beads and barrel-shaped beads of jet or lignite. Three of them come from Stonehenge itself. Similar beads of ivory have been found in a Bronze Age cist near Warminster: if the material is really ivory it must have been derived from the East. The cylindrical faience beads, it may be added, have been discovered in Dorsetshire as well as in Wiltshire. Professor Sayce emphasizes that these blue beads belong to one particular period in Egyptian history, the latter part of the Eighteenth Dynasty and the earlier part of the Nineteenth Dynasty.... The period to which they belong may be dated 1450-1250 b.c., and as we must allow some time for their passage across the trade routes to Wiltshire an approximate date for their presence in the British barrows will be 1300 b.c.
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