UNIT 09 Alan_Morris_EMI_Chap_08 Display, Recording, and Presentation.pdf

CHAPTER 8
Display, Recording, and Presentation
ofMeasurement Data
8.1 Introduction 183
8.2 Display of Measurement Signals 184
8.2.1 Electronic Output Displays 184
8.2.2 Computer Monitor Displays 185
8.3 Recording of Measurement Data 185
8.3.1 Chart Recorders 185
Pen strip chart recorder 181
Multipoint strip chart recorder 188
Heated-stylus chart recorder 188
Circular chart recorder 189
Paperless chart recorder 189
Videographic recorder 190
8.3.2 Ink-Jet and Laser Printers 190
8.3.3 Other Recording Instruments 190
8.3.4 Digital Data Recorders 190
8.4 Presentation of Data 191
8.4.1 Tabular Data Presentation 191
8.4.2 Graphical Presentation of Data 192
Fitting cutves to data points on a graph 194
Regression techniques 194
Linear least-squares regression 196
Quadratic least-squares regression 199
Polynomial least-squares regression 199
Confidence tests in cutve fitting by least-squares regression 200
Correlation tests 201
8.5 Summary 202
8.6 Problems 203
8.1 Introduction
Earlier chapters in this book have been essentially concerned with describing ways of producing
high-quality, error-free data at the output of a measurement system. Having gotten that data,
the next consideration is how to present it in a form where it can be readily used and analyzed.
This chapter therefore starts by covering the techniques available to either display measurement
Measurement and Instrumentation: Thoory and Application
© 2012 Elsevier Inc. All rights reserved.
183

184 Chapter 8
data for current use or record it for future use. Following this, standards of good practice for
presenting data in either graphical or tabular form are covered, using either paper or a computer
monitor screen as the display medium. This leads to a discussion of mathematical regression
techniques for fitting best lines through data points on a graph. Confidence tests to assess the
correctness of the line fitted are also described. Finally, correlation tests are described that
determine the degree of association between two sets of data when both are subject to random
fluctuations.
8.2 Display of Measurement Signals
Measurement signals in the form of a varying electrical voltage can be displayed either by an
oscilloscope or by any of the electrical meters described earlier in Chapter 7. However, if
signals are converted to digital form, other display options apart from meters become possible,
such as electronic output displays or use of a computer monitor.
8.2. 1 Electronic Output Displays
Electronic displays enable a parameter value to be read immediately, thus allowing for any
necessary response to be made immediately. The main requirement for displays is that they
should be clear and unambiguous. Two common types of character formats used in displays,
seven-segment and 7 x 5 dot matrix, are shown in Figure 8.1. Both types of displays have the
advantage of being able to display alphabetic as well as numeric information, although the
seven-segment format can only display a limited 9-letter subset of the full 26-letter alphabet.
This allows added meaning to be given to the number displayed by including a word or letter
code. It also allows a single display unit to send information about several parameter values,
cycling through each in turn and including alphabetic information to indicate the nature ofthe
variable currently displayed.
Electronic output units usually consist of a number of side-by-side cells, where each cell
displays one character. Generally, these accept either serial or parallel digital input signals, and
1-0 •••••
•DODD
••••o
0-1 DODD•
ODDO•
•ODD•
- D•••D
(a) (b)
Figure 8.1
Character formats used in electronic displays: (a) seven segment and (b) 7 x 5 dot matrix.

Display, Recording, and Presentation ofMeasurement Data 185
the input format can be either binary-coded decimal or ACSIL Technologies used for the
individual elements in the display are either light-emitting diodes or liquid-crystal elements.
8.2.2 Computer Monitor Displays
Now that computers are part of the furniture in most homes, the ability of computers to display
information is widely understood and appreciated. Computers are now both inexpensive and
highly reliable and provide an excellent mechanism for both displaying and storing information.
As well as alphanumeric displays of industrial plant variable and status data, for which the plant
operator can vary the size of font used to display the information at will, it is also relatively
easy to display other information, such as plant layout diagrams and process flow layouts. This
allows not only the value ofparameters that go outside control limits to be displayed, but also their
location on a schematic map of the plant. Graphical displays of the behavior of a measured
variable are also possible. However, this poses difficulty when there is a requirement to display
the variable's behavior over a long period of time, as the length of the time axis is constrained
by the size of the monitor's screen. To overcome this, the display resolution has to decrease
as the time period of the display increases.
Touch screens have the ability to display the same soli of information as a conventional
computer monitor, but also provide a command-input facility in which the operator simply has
to touch the screen at points where images of keys or boxes are displayed. A full "qweliy"
keyboard is often provided as part of the display. The sensing elements behind the screen are
protected by glass and continue to function even if the glass gets scratched. Touch screens are
usually totally sealed, thus providing intrinsically safe operation in hazardous environments.
8.3 Recording of Measurement Data
As well as displaying the current values ofmeasured parameters, there is often a need to make
continuous recordings of measurements for later analysis. Such records are paliicularly useful
when faults develop in systems, as analysis of the changes in measured parameters in the time
before the fault is discovered can often quickly indicate the reason for the fault. Options
for recording data include chart recorders, digital oscilloscopes, digital data recorders, and
hard-copy devices such as inkjet and laser printers. The various types of recorders used are
discussed here.
8.3.1 Chart Recorders
Chart recorders have paliicular advantages in providing a non-corruptible record that has the
merit of instant "viewability." This means that all but paperless forms of chart recorders
satisfy regulations set for many industries that require variables to be monitored and recorded
continuously with hard-copy output. ISO 9000 quality assurance procedures and ISO 14000

186 Chapter 8
environmental protection systems set similar requirements, and special regulations in the
defense industry go even further by requiring hard-copy output to be kept for 10 years. Hence,
while many people have been predicting the demise of chart recorders, the reality of the
situation is that they are likely to be needed in many industries for many years to come.
Originally, all chart recorders were electromechanical in operation and worked on the same
principle as a galvanometric moving coil meter (see analogue meters in Chapter 7) except that
the moving coil to which the measured signal was applied carried a pen, as shown in Figure 8.2,
rather than carrying a pointer moving against a scale as it would do in a meter. The pen drew an
ink trace on a strip of ruled chart paper that was moved past the pen at constant speed by an
electrical motor. The resultant trace on chart paper showed variations with time in the magnitude
ofthe measured signal. Even early recorders commonly had two or more pens ofdifferent colors
so that several measured parameters could be recorded simultaneously.
The first improvement to this basic recording arrangement was to replace the galvanometric
mechanism with a servo system, as shown in Figure 8.3, in which the pen is driven by a
servomotor, and a sensor on the pen feeds back a signal proportional to pen position. In this
form, the instrument is known as a potentiometric recorder. The servo system reduces the
typical inaccuracy of the recorded signal to ±0.1%, compared to ±2% in a galvanometeric
mechanism recorder. Typically, the measurement resolution is around 0.2% of the full-scale
reading. Originally, the servo motor was a standard d.c. motor, but brushless servo motors
are now invariably used to avoid the commutator problems that occur with d.c. motors.
The position signal is measured by a potentiometer in less expensive models, but more
expensive models achieve better performance and reliability using a noncontacting ultrasonic
sensor to provide feedback on pen position. The difference between the pen position and
Rotating
coil
Figure 8.2
paper
Original form of galvanometric chart recorder.

Measured
signal
Pen
position
signal
Error Servomotor
and
gearbox
Potentiometer
Figure 8.3
Servo system of a potentiometric chart recorder.
Pen
position
the measured signal is applied as an error signal that drives the motor. One consequence ofthis
electromechanical balancing mechanism is that the instrument has a slow response time, in
the range of 0.2-2.0 seconds, which means that electromechanical potentiometric recorders
are only suitable for measuring d.c. and slowly time-varying signals.
All current potentiometric chart recorders contain a microprocessor controller, where the
functions vary according to the particular chart recorder. Common functions are selection
of range and chart speed, along with specification of alarm modes and levels to detect
when measured variables go outside acceptable limits. Basic recorders can record up to
three different signals using three different colored pens. However, multipoint recorders
can have 24 or more inputs and plot six or more different colored traces simultaneously.
As an alternative to pens, which can run out of ink at inconvenient times, recorders
using a heated stylus recording signals on heat-sensitive paper are available. Another
variation is the circular chart recorder, in which the chart paper is circular in shape and
is rotated rather than moving translationally. Finally, paperless forms of recorder exist
where the output display is generated entirely electronically. These various forms are
discussed in more detail later.
Pen strip chart recorder
A pen strip chart recorder refers to the basic form of the electromechanical potentiometric
chart recorder mentioned earlier. It is also called a hybrid chart recorder by some manufacturers.
The word "hybrid" was used originally to differentiate chart recorders that had a microprocessor
controller from those that did not. However, because all chart recorders now contain a micro-
processor, the term hybrid has become superfluous.
Strip chart recorders typically have up to three inputs and up to three pens in different colors,
allowing up to three different signals to be recorded. A typical commercially available model
is shown in Figure 8.4. Chart paper comes in either roll or fan-fold form. The drive mechanism

188 Chapter 8
Figure 8.4
Honeywel I DPR100 strip chart recorder (reproduced by permission ofHoneywell International, Inc.).
can be adjusted to move the chart paper at different speeds. The fastest speed is typically
6000 mm/hour and the slowest is typically 1 mm/hour.
As well as recording signals as a continuous trace, many models also allow for the printing of
alphanumeric data on the chart to record date, time, and other process information. Some
models also have a digital numeric display to provide information on the current values of
recorded variables.
Multipoint strip chart recorder
A multipoint strip chart recorder is a modification of the pen strip chart recorder that uses a
dot matrix print head striking against an ink ribbon instead ofpens. A typical model might allow
up to 24 different signal inputs to be recorded simultaneously using a six-color ink ribbon.
Certain models of such recorders also have the same enhancements as pen strip chart recorders
in terms of printing alphanumeric information on the chart and providing a digital numeric
output display.
Heated-stylus chart recorder
A heated-stylus chart recorder is another variant that records the input signal by applying
a heated stylus to heat-sensitive chart paper. The main purpose of this alternative printing
mechanism is to avoid the problem experienced in other forms of paper-based chart recorders
of pen cartridges or printer ribbons running out of ink at inconvenient times.

Circular chart recorder
A circular chart recorder consists of a servo-driven pen assembly that records the measured
signal on a rotating circular paper chart, as shown in Figure 8.5. The rotational speed of the
chart can be typically adjusted between one revolution in 1 hour to one revolution in 31 days.
Recorded charts are replaced and stored after each revolution, which means replacement
intervals that vary between hourly and monthly according to the chart speed. The major
advantage ofa circular chart recorder over other forms is compactness. Some models have up to
four different colored pen assemblies, allowing up to four different parameters to be recorded
simultaneously.
Paperless chart recorder
A paperless chart recorder, sometimes alternatively called a virtual chart recorder or a digital
chart recorder, displays the time history ofmeasured signals electronically using a color-matrix
liquid crystal display. This avoids the chore of periodically replacing chart paper and ink
cartridges associated with other forms of chart recorders. Reliability is also enhanced compared
with electromechanical recorders. As well as displaying the most recent time history of
measured signals on its screen, the instrument also stores a much larger past history. This stored
data can be recalled in batches and redisplayed on the screen as required. The only downside
compared with other forms of chart recorders is this limitation ofonly displaying one screen full
of information at a time. Of course, conventional recorders allow the whole past history of
signals to be viewed at the same time on hard-copy, paper recordings. Otherwise, specifications
are very similar to other forms of chart recorders, with vertical motion of the screen display
Record of
measured signal
Figure 8.5
Circular chart recorder.

Pen carrier

190 Chapter 8
varying between 1and6000 mm/hour, typical inaccuracy less than ±0.1%, and capability of
recording multiple signals simultaneously in different colors.
Videographic recorder
A videographic recorder provides exactly the same facilities as a paperless chart recorder but
has additional display modes, such as bar graphs (histograms) and digital numbers. However,
it should be noted that the distinction is becoming blurred between the various forms of
paperless recorders described earlier and videographic recorders as manufacturers enhance
the facilities of their instruments. For historical reasons, many manufacturers retain the
names that they have traditionally used for their recording instruments but there is now much
overlap between their respective capabilities as the functions provided are extended.
8.3.2 Ink-Jet and Laser Printers
Standard computer output devices in the form of ink-jet and laser printers are now widely used
as an alternative means of storing measurement system output in paper form. Because a
computer is a routine pali of many data acquisition and processing operations, it often makes
sense to output data in a suitable form to a computer printer rather than a chali recorder.
This saves the cost of a separate recorder and is facilitated by the ready availability ofsoftware
that can output measurement data in a graphical format.
8.3.3 Other Recording Instruments
Many ofthe devices mentioned in Chapters 5 and 7 have facilities for storing measurement data
digitally. These include data logging acquisition devices and digital storage oscilloscopes.
These data can then be convelied into hard-copy form as required by transferring it to either a
chart recorder or a computer and printer.
8.3.4 Digital Data Recorders
Digital data recorders, also known as data loggers, have already been introduced in
Chapter 5 in the context of data acquisition. They provide a further alternative way of
recording measurement data in a digital format. Data so recorded can then be transferred at
a future time to a computer for further analysis, to any of the forms of measurement display
devices discussed in Section 8.2, or to one of the hard-copy output devices described in
Section 8.3.
Features contained within a data recorder/data logger obviously vary according to the particular
manufacturer/model under discussion. However, most recorders have facilities to handle
measurements in the form ofboth analogue and digital signals. Common analogue input signals
allowed include d.c. voltages, d.c. currents, a.c. voltages, and a.c. currents. Digital inputs can

usually be either in the form of data from digital measuring instruments or discrete data
representing events such as switch closures or relay operations. Some models also provide
alarm facilities to ale11 operators to abnormal conditions during data recording operations.
Many data recorders provide special input facilities optimized for particular kinds of
measurement sensors, such as accelerometers, thermocouples, thermistors, resistance
the1mometers, strain gauges (including strain gauge bridges), linear variable differential
transformers, and rotational differential transformers. Some instruments also have special
facilities for dealing with inputs from less common devices such as encoders, counters, timers,
tachometers, and clocks. A few recorders also incorporate integral sensors when they are
designed to measure a particular type of physical variable.
The quality of data recorded by a digital recorder is a function of the cost of the instrument.
Paying more usually means getting more memory to provide a greater data storage capacity,
greater resolution in the analogue-to-digital converter to give better recording accuracy,
and faster data processing to allow greater data sampling frequency.
8.4 Presentation of Data
The two formats available for presenting data on paper are tabular and graphical, and the
relative merits of these are compared later. In some circumstances, it is clearly best to use only
one or the other of these two alternatives alone. However, in many data collection exercises,
part of the measurements and calculations are expressed in tabular form and part graphically,
making best use of the merits of each technique. Very similar arguments apply to the relative
merits of graphical and tabular presentations if a computer screen is used for presentation
instead of paper.
8.4.1 Tabular Data Presentation
A tabular presentation allows data values to be recorded in a precise way that exactly maintains
the accuracy to which the data values were measured. In other words, the data values are
written down exactly as measured. In addition to recording raw data values as measured, tables
often also contain further values calculated from raw data. An example of a tabular data
presentation is given in Table 8.1. This records results of an experiment to determine the strain
induced in a bar ofmaterial subjected to a range of stresses. Data were obtained by applying a
sequence of forces to the end of the bar and using an extensometer to measure the change in
length. Values ofthe stress and strain in the bar are calculated from these measurements and are
also included in Table 8.1. The final row, which is of crucial importance in any tabular
presentation, is the estimate of possible error in each calculated result.

192 Chapter 8
Table 8.1 Table of Measured Applied Forces and Extensometer Readings
and Calculations of Stress and Strain
Extensometer
Force Applied (l<N) Reading (Divisions) Stress (N/m
2
) Strain
0 0 0 0
2 4.0 15.5 19.8 x 10
4 5.8 31.0 28.6 x 10
5
5
6 7.4 46.5 36.6 x 10 5
8 9.0 62.0 44.4 x 10 5
10 10.6 77.5 52.4 x 10 5
12 12.2 93.0 60.2 x 10 5
14 13.7 108.5 67.6 x 10 5
Possible error in
measurements (%) ±0.2 ±0.2 ±1.5 ±1.0
A table of measurements and calculations should conform to several rules as illustrated in
Table 8.1:
The table should have a title that explains what data are being presented within the table.
Each column of figures in the table should refer to the measurements or calculations
associated with one quantity only.
Each column of figures should be headed by a title that identifies the data values
contained in the column.
Units in which quantities in each column are measured should be stated at the top ofthe
column.
All headings and columns should be separated by bold horizontal (and sometimes
vertical) lines.
Errors associated with each data value quoted in the table should be given. The form
shown in Table 8.1 is a suitable way to do this when the error level is the same for
all data values in a particular column. However, if error levels vary, then it is preferable
to write the error boundaries alongside each entry in the table.
8.4.2 Graphical Presentation ofData
Presentation of data in graphical form involves some compromise in the accuracy to which
data are recorded, as the exact values ofmeasurements are lost. However, graphical presentation
has important advantages over tabular presentation.
Graphs provide a pictorial representation ofresults that is comprehended more readily than
a set of tabular results.
Graphs are particularly useful for expressing the quantitative significance of results
and showing whether a linear relationship exists between two variables. Figure 8.6 shows a

Stress
(N/m2)
120
100
80
60
40
20
0 20 40 60 80 Strain ( x 10-5)
Figure 8.6
Sample graphical presentation of data: graph ofstress against strain.
graph drawn from the stress and strain values given in Table 8.1. Construction of the
graph involves first of all marking the points conesponding to the stress and strain values.
The next step is to draw some line through these data points that best represents the
relationship between the two variables. This line will normally be either a straight one or a
smooth curve. Data points will not usually lie exactly on this line but instead will lie
on either side of it. The magnitude of the excursions ofthe data points from the line drawn
will depend on the magnitude of the random measurement enors associated with data.
Graphs can sometimes show up on a data point that is clearly outside the straight line or
curve that seems to fit the rest of the data points. Such a data point is probably due either
to a human mistake in reading an instrument or to a momentary malfunction in the
measuring instrument itself. Ifthe graph shows such a data point where a human mistake or
instrument malfunction is suspected, the proper course of action is to repeat that particular
measurement and then discard the original data point if the mistake or malfunction is
confirmed.
Like tables, the proper representation of data in graphical form has to conform to certain rules:
The graph should have a title or caption that explains what data are being presented in
the graph.
Both axes ofthe graph should be labeled to express clearly what variable is associated with
each axis and to define the units in which the variables are expressed.

194 Chapter 8
The number ofpoints marked along each axis should be kept reasonably small-about five
divisions is often a suitable number.
No attempt should be made to draw the graph outside the boundaries corresponding to the
maximum and minimum data values measured, that is, in Figure 8.6, the graph stops at a
point corresponding to the highest measured stress value of 108.5.
Fitting curves to data points on a graph
The procedure ofdrawing a straight line or smooth curve as appropriate that passes close to all
data points on a graph, rather than joining data points by a jagged line that passes through
each data point, is justified on account of the random errors known to affect measurements.
Any line between data points is mathematically acceptable as a graphical representation of
data if the maximum deviation of any data point from the line is within the boundaries of the
identified level of possible measurement errors. However, within the range of possible lines
that could be drawn, only one will be the optimum one. This optimum line is where the sum of
negative errors in data points on one side of the line is balanced by the sum of positive errors
in data points on the other side of the line. The nature of data points is often such that a
perfectly acceptable approximation to the optimum can be obtained by drawing a line through
the data points by eye. In other cases, however, it is necessary to fit a line mathematically,
using regression techniques.
Regression techniques
Regression techniques consist offinding a mathematical relationship between measurements of
two variables, y and x, such that the value of variable y can be predicted from a measurement
of the other variable, x. However, regression techniques should not be regarded as a magic
formula that can fit a good relationship to measurement data in all circumstances, as the
characteristics of data must satisfy certain conditions. In determining the suitability of
measurement data for the application of regression techniques, it is recommended practice
to draw an approximate graph of the measured data points, as this is often the best means of
detecting aspects of data that make it unsuitable for regression analysis. Drawing a graph of data
will indicate, for example, whether any data points appearto be erroneous. This may indicate that
human mistakes or instrument malfunctions have affected the erroneous data points, and it is
assumed that any such data points will be checked for correctness.
Regression techniques cannot be applied successfully if the deviation of any particular
data point from the line to be fitted is greater than the maximum possible error calculated
for the measured variable (i.e., the predicted sum of all systematic and random errors).
The nature of some measurement data sets is such that this criterion cannot be satisfied,
and any attempt to apply regression techniques is doomed to failure. In that event, the only
valid course of action is to express the measurements in tabular form. This can then be
used as an x-y look-up table, from which values of the variable y corresponding to
particular values of x can be read off. In many cases, this problem of large errors in some

data points only becomes apparent during the process of attempting to fit a relationship
by regression.
A further check that must be made before attempting to fit a line or curve to measurements
of two variables, x and y, is to examine data and look for any evidence that both variables are
subject to random errors. It is a clear condition for the validity ofregression techniques that only
one of the measured variables is subject to random errors, with no error in the other variable. If
random errors do exist in both measured variables, regression techniques cannot be applied
and recourse must be made instead to correlation analysis (covered later in this chapter).
Simple examples of a situation where both variables in a measurement data set are subject to
random errors are measurements of human height and weight, and no attempt should be made
to fit a relationship between them by regression.
Having determined that the technique is valid, the regression procedure is simplest if a
straight-line relationship exists between the variables, which allows a relationship ofthe form
y = a + bx to be estimated by linear least-squares regression. Unfortunately, in many cases,
a straight-line relationship between points does not exist, which is shown readily by plotting
raw data points on a graph. However, knowledge of physical laws governing data can
often suggest a suitable alternative form of relationship between the two sets of variable
measurements, such as a quadratic relationship or a higher order polynomial relationship.
Also, in some cases, the measured variables can be transformed into a form where a linear
relationship exists. For example, suppose that two variables, y and x, are related according
to y=axc. A linear relationship from this can be derived, using a logarithmic transformation,
as log(y) = log(a) + clog(x).
Thus, if a graph is constructed of log(y) plotted against log(x), the parameters of a straight-line
relationship can be estimated by linear least-squares regression.
All quadratic and higher order relationships relating one variable, y, to another variable, x, can
be represented by a power series of the form:
y = ao +a1x +a2x2
+ ···+apxP.
Estimation of the parameters a0 . .. aP is very difficult ifp has a large value. Fortunately, a
relationship where p only has a small value can be fitted to most data sets. Quadratic least-
squares regression is used to estimate parameters wherep has a value oftwo; for larger values of
p , polynomial least-squares regression is used for parameter estimation.
Where the appropriate form of relationship between variables in measurement data sets is
not obvious either from visual inspection or from consideration of physical laws, a method
that is effectively a trial and error one has to be applied. This consists of estimating the
parameters ofsuccessively higher order relationships betweeny and x until a curve is found that
fits data sufficiently closely. What level of closeness is acceptable is considered later in the
section on confidence tests.

196 Chapter 8
Linear least-squares regression
If a linear relationship between y and x exists for a set of n measurements, y1 .. ·Yn, x1 .. . Xm
then this relationship can be expressed as y =a+ bx, where coefficients a and b are constants.
The purpose of least-squares regression is to select optimum values for a and b such that the line
gives the best fit to the measurement data.
The deviation of each point (x; ,y;) from the line can be expressed as d;, where d; = Y; - (a+ bx;).
The best-fit line is obtained when the sum of squared deviations, S, is a minimum, that is,
when
n n
S= L (d?) = L (Y; - a - bx1)
2
i= I i= I
is a mmrmum.
The minimum can be found by setting partial derivatives oS/Oa and oS/ob to zero and solving
the resulting two simultaneous (normal) equations:
oS/oa = L2(y; - a - bx;)(-1) = 0
oS/ob = L2(y; - a - bx1)(-x1) = 0
(8.1)
(8.2)
Values of the coefficients a and b at the minimum point can be represented by aand b,
which are known as the least-squares estimates of a and b. These can be calculated as
follows.
From Equation (8.1),
and thus,
(8.3)
From Equation (8.2),
(8 .4)
Now substitute for ain Equation (8.4) using Equation (8.3):

Collecting terms in b,
Rearranging gives
which can be expressed as
where Xmand Ym are the mean values of x and y. Thus,
b = L (X1Y1) - nxmYm
2=x? - nxm2 ·
(8.5)
And, from Equation (8.3):
a= Ym - bxm. (8.6)
Example 8.1
In an experiment to determine the characteristics ofa displacement sensor with a voltage
output, the following output voltage values were recorded when a set of standard
displacements was measured:
Displacement (cm) 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
Voltage 2.1 4.3 6.2 8.5 10.7 12.6 14.5 16.3 18.3 21.2
Fit a straight line to this set of data using least-sciuares regression and estimate the
output voltage when a displacement of 4.5 cm is measured.
•
Solution
Lety represent the output voltage and x represent the displacement. Then a suitable
straight line is given byy=a+bx. We can now proceed to calculate estimates for the
coefficients a and busing Eciuations (8.5) and (8.6). The first step is to calculate the mean
values ofx andy. These are found to bexm = 5.5 andym = 11.47. Next, we need to
tabulate Xi)'i and x/ for each pair of data values:

198 Chapter 8
X; y; x;y;
1.0 2.1 2.1
2.0 4.3 8.6
3.0 6.2 18.6
10.0 21.2 212.0
Now calculate the values needed from this table-n = 1O; 2=(xJ';)= 801.0;
2=(x;2)= 385-and enter these values into Equations (8.5) and (8.6).
2
X;
4
9
100
~ 801.0 - (10 x 5.5 x 11.47) ~
b = ( 2 ) = 2.067; a= 11.47 - (2.067 x 5.5) = 0.1033;
385 - 10 x 5.5
that is,y = 0.1033 + 2.067x.
Hence, for x = 4.5, y= 0.1033 + (2.067 x 4.5) = 9.40 volts. Note that in this solution
we have only specified the answer to an accuracy of three figures, which is the same
accuracy as the measurements. Any greater number of figures in the answer would be
meaningless.
•
Least-squares regression is often appropriate for situations where a straight-line relationship
is not immediately obvious, for example, where y ocx2
or y oc exp(x).
Example 8.2
From theoretical considerations, it is known that the voltage (V) across a charged
capacitor decays with time (t) according to the relationship V= K exp(-t/1:). Estimate
values for Kand 1: if the following values of V and tare measured.
v 8.67 6.55 4.53 3.29 2.56 1.95 1.43 1.04 0.76
0 2 3 4 5 6 7 8
•
Solution
lfV= K exp(- T/c), then loge(V) = loge(K)- t/1:. Now lety= loge(V),a = log(K), b= -1/'L,
and x = t. Hence,y=a+ bx, which is the equation ofa straight line whose coefficients can be
estimated by applying Equations (8.5) and (8.6). Therefore, proceed in the same way as
Example 8.1 and tabulate the values required:

v loge(V) t
(y;) (x;) (x;y;) (xl)
8.67 2.16 0 0 0
6.55 1.88 1.88
4.53 1.51 2 3.02 4
0.76 -0.27 8 -2.16 64
Now calculate the values needed from this table-n = 9; 2._)xJ';)= 15.86; l:(x/)=
204; Xm = 4.0; Ym = 0.9422-and enter these values into Equations (8.5) and (8.6).
A 15,86 - (9 x 4,0 x 0.9422) A
b = ( 2 ) = -0.301; a = 0.9422 + (0.301 x 4.0) = 2.15
204 - 9 x 4.0
K =exp(a) = exp(2.15) = 8.58; 1: = -1 /b = -1 /(-0.301) = 3.32
Quadratic least-squares regression
Quadratic least-squares regression is used to estimate the parameters of a relationship,
y=a+bx+ cx2
, between two sets of measurements, YI ...Yn , XI ... Xn.
The deviation of each point (x; ,y;) from the line can be expressed as d;, where d; =
y, -(a+bx,+ex?).
•
The best-fit line is obtained when the sum of the squared deviations, S, is a minimum, that is,
when
n n
S = L (d?) = L (Yt - a - bx1 +cx?)
2
i=l i= l
is a minimum.
The minimum can be found by setting the partial derivatives 8S/8a, 8S/8b, and 8S/8c to
zero and solving the resulting simultaneous equations, as for the linear least-squares regression
case given earlier. Standard computer programs to estimate the parameters a, b, and c by
numerical methods are widely available and therefore a detailed solution is not presented here.
Polynomial least-squares regression
Polynomial least-squares regression is used to estimate the parameters ofthepthorder relationship
y=a0 + aix+a2x2+ ···+a/ between two sets of measurements, YI .. ·Yn, XI .. . Xn·

200 Chapter 8
The deviation of each point (x; ,y;) from the line can be expressed as d;, where
d; = y; - (ao +a1x; +a2x? +···+apx/).
The best-fit line is obtained when the sum of squared deviations given by
n
s = 2= (d?)
i=l
1s a mimmum.
The minimum can be found as before by setting p partial derivatives 8S/Oa0 ... oS/oaP
to zero and solving the resulting simultaneous equations. Again, as for the quadratic least-
squares regression case, standard computer programs to estimate the parameters a0. .. aP by
numerical methods are widely available and therefore a detailed solution is not presented here.
Confidence tests in curve fitting by least-squares regression
Once data have been collected and a mathematical relationship that fits the data points has been
determined by regression, the level of confidence that the mathematical relationship fitted is
correct must be expressed in some way. The first check that must be made is whether the
fundamental requirement for the validity of regression techniques is satisfied, that is, whether
the deviations of data points from the fitted line are all less than the maximum error level
predicted for the measured variable. Ifthis condition is violated by any data point that a line or
curve has been fitted to, then use ofthe fitted relationship is unsafe and recourse must be made
to tabular data presentation, as described earlier.
The second check concerns whether random errors affect both measured variables. If attempts
are made to fit relationships by regression to data where both measured variables contain
random errors, any relationship fitted will only be approximate and it is likely that one or more
data points will have a deviation from the fitted line or curve greater than the maximum
error level predicted for the measured variable. This will show up when the appropriate checks
are made.
Having carried out the aforementioned checks to show that there are no aspects of data that
suggest that regression analysis is not appropriate, the next step is to apply least-squares
regression to estimate the parameters of the chosen relationship (linear, quadratic, etc.). After
this, some form of follow-up procedure is clearly required to assess how well the estimated
relationship fits the data points. A simple curve-fitting confidence test is to calculate the sum of
squared deviations S for the chosen y/x relationship and compare it with the value of S
calculated for the next higher order regression curve that could be fitted to data. Thus if a
straight-line relationship is chosen, the value ofS calculated should be ofa similar magnitude to

that obtained by fitting a quadratic relationship. If the value of S were substantially lower
for a quadratic relationship, this would indicate that a quadratic relationship was a better fit to
data than a straight-line one and further tests would be needed to examine whether a cubic or
higher order relationship was a better fit still.
Other more sophisticated confidence tests exist, such as the F-ratio test. However, these are
outside the scope of this book.
Correlation tests
Where both variables in a measurement data set are subject to random fluctuations, correlation
analysis is applied to determine the degree of association between the variables. For example,
in the case already quoted of a data set containing measurements of human height and
weight, we certainly expect some relationship between the variables of height and weight
because a tall person is heavier on average than a short person. Correlation tests determine
the strength of the relationship (or interdependence) between the measured variables, which
is expressed in the form of a correlation coefficient.
For two sets of measurements Y1 ...Yn, X1 ...Xn with means Xm and Ym, the correlation
coefficient is given by
= 2=(x1 - Xm)(Yt - Ym)
J[L(Xt - Xm)
2
] [ L(Yt - Ym)
2
]
The value of lI always lies between 0 and 1, with 0 representing the case where the
variables are completely independent of one another and 1 is the case where they are totally
related to one another. For 0 < lI < 1, linear least-squares regression can be applied to
find relationships between the variables, which allows x to be predicted from a measurement of
y, and y to be predicted from a measurement ofx. This involves finding two separate regression
lines of the form:
y = a + bx and x = c +dy.
These two lines are not normally coincident, as shown in Figure 8.7. Both lines pass through the
centroid of the data points but their slopes are different.
As lI ____, 1, the lines tend to coincidence, representing the case where the two variables are
totally dependent on one another.
As l<l>I ____, 0, the lines tend to orthogonal ones parallel to thex and y axes. In this case, the two sets
of variables are totally independent. The best estimate of x given any measurement of y is Xm,
and the best estimate of y given any measurement of x is Ym·

202 Chapter 8
y
variable
Figure 8.7
x
variable
Relationship between two variables with random fluctuations.
For the general case, the best fit to data is the line that bisects the angle between the lines on
Figure 8.7.
8.5 Summary
This chapter began by looking at the various ways that measurement data can be displayed,
either using electronic display devices or using a computer monitor. We then went on to
consider how measurement data could be recorded in a way that allows future analysis. We
noted that this facility was particularly useful when faults develop in systems, as analysis ofthe
changes in measured parameters in the time before the fault is discovered can often quickly
indicate the reason for the fault. Options available for recording data are numerous and include
chart recorders, digital oscilloscopes, digital data recorders, and hard-copy devices such as
ink-jet and laser printers. We gave consideration to each of these and indicated some of the
circumstances in which each alternative recording device might be used.
The next subject of study in the chapter was recommendations for good practice in the
presentation ofdata. We looked at both graphical and tabular forms of presentation using either
paper or a computer monitor screen as the display medium. We then went on to consider the
best way offitting lines through data points on a graph. This led to a discussion ofmathematical
regression techniques and the associated confidence tests necessary to assess the correctness of
the line fitted using regression. Finally, we looked at correlation tests. These are used to
determine the degree of association between two sets of data when they are both subject to
random fluctuations.

8.6 Problems
8.1. What are the main ways available for displaying parameter values to human operators
responsible for controlling industrial manufacturing systems? (Discussion on electronic
displays and computer monitors is expected.)
8.2. Discuss the range of instruments and techniques available for recording measurement
signals, mentioning particularly the frequency response characteristics ofeach instrument
or technique and the upper frequency limit for signals in each case.
8.3. Discuss the features of the main types of chart recorders available for recording
measurement signals.
8.4. What is a digital data recorder and how does it work?
8.5. (a) Explain the derivation of the expression
e K?B Ks9 _ KiVt
+JR +J- JR
describing the dynamic response ofa galvanometric chart recorder following a step
change in the electrical voltage output of a transducer connected to its input.
Explain also what all the terms in the expression stand for. (Assume that impedances
of both the transducer and the recorder have a resistive component only and that
there is negligible friction in the system.)
(b) Derive expressions for the measuring system natural frequency, Wm the damping
factor, ~' and the steady-state sensitivity.
(c) Explain simple ways of increasing and decreasing the damping factor and describe
the corresponding effect on measurement sensitivity.
(d) What damping factor gives the best system bandwidth?
(e) What aspects ofthe design ofa chart recorder would you modify in order to improve
the system bandwidth? What is the maximum bandwidth typically attainable in
chart recorders, and if such a maximum-bandwidth instrument is available, what is
the highest frequency signal that such an instrument would be generally regarded as
being suitable for measuring if the accuracy ofthe signal amplitude measurement is
important?
8.6. Discuss the relative merits of tabular and graphical methods of recording measurement
data.
8.7. What would you regard as good practice in recording measurement data in graphical
form?
8.8. What wouldyou regard as goodpractice in recording measurement data in tabular form?
8.9. Explain the technique of linear least-squares regression for finding a relationship
between two sets of measurement data.
8.10. Explain the techniques of (a) quadratic least-squares regression and (b) polynomial
least-squares regression. How would you determine whether either quadratic or

204 Chapter 8
polynomial least-squares regression provides a better fit to a set of measurement data
than linear least-squares regression?
8.11. During calibration of a platinum resistance thermometer, the following temperature and
resistance values were measured:
Resistance (Q) 21 2.8 218.6 225.3
Temperature (°C) 300 320 340
233.6
360
240.8 246.6
380 400
The temperature measurements were made using a primary reference standard
instrument for which the measurement errors can be assumed to be zero. The
resistance measurements were subject to random errors but it can be assumed that
there are no systematic errors in them.
(a) Determine the sensitivity of measurement in Q/°C in as accurate a manner as
possible.
(b) Write down the temperature range that this sensitivity value is valid for.
(c) Explain the steps that you would take to test the validity ofthe type of mathematical
relationship that you have used for data.
8.12. Theoretical considerations show that quantities x and y are related in a linear fashion
such that y =ax+b. Show that the best estimate of the constants a and b are given by
b = Ym - Ctxm
Explain carefully the meaning of all the terms in the aforementioned two equations.
8.13. The characteristics of a chromel-constantan thermocouple is known to be approximately
linear over the temperature range of 300-800°C. The output e.m.f. was measured
practically at arangeoftemperatures, andthe following table ofresultswas obtained.Using
least-squares regression, calculate coefficients aand b for the relationship T=a+bE that
best describes the temperature-e.m.f. characteristic.
Temp (°C) 300 325 350 375 400 425 450 475 500 525 550
e.m.f (mV) 21.0 23.2 25.0 26.9 28.6 31.3 32.8 35.0 37.2 38.5 40.7
Temp (°C) 575 600 625 650 675 700 725 750 775 800
e.m.f (mV) 43.0 45.2 47.6 49.5 51.1 53.0 55.5 57.2 59.0 61.0
8.14. Measurements of the current (I) flowing through a resistor and the corresponding
voltage drop (V) are shown:
I (amps) 2 3 4 5
V (volts) 10.8 20.4 30.7 40.5 50.0
fustruments used to measure voltage and current were accurate in all respects except that
they each had a zero error that the observer failed to take account of or to correct at the
time of measurement. Determine the value of the resistor from data measured.

8.15. A measured quantity y is known from theoretical considerations to depend on variable
x according to the relationship y=a+bx2
. For the following set of measurements of
x and y, use linear least-squares regression to determine the estimates of parameters
a and b that fit data best.
x 0 2 3 4 5
y 0.9 9.2 33.4 72.5 130.1 200.8
8.16. The mean time to failure (MITF) of an integrated circuit is known to obey a law of
the following form: MITF = C exp T0/T, where Tis the operating temperature and
C and T0 are constants. The following values ofMITF at various temperatures were
obtained from accelerated life tests.
MTTF (hours) 54 105 206 411 941 2145
Temperature (0
1<) 600 580 560 540 520 500
(a) Estimate the values of C and T0 • [Hint: loge(MITF) = loge(C) +To/T. This equation
is now a straight-line relationship between log(MITF) and 1IT, where log(C) and T0
are constants.]
(b) For an MITF of 10 years, calculate the maximum allowable temperature.

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