Performance Evaluation for Classifiers tutorial

Nathalie Japkowicz
School of Electrical Engineering
& Computer Science
University of Ottawa
nat@site.uottawa.ca

Motivation: My story
 A student and I designed a new algorithm for data that
had been provided to us by the National Institute of
Health (NIH).
 According to the standard evaluation practices in
machine learning, we found our results to be
significantly better than the state-of-the-art.
 The machine learning community agreed as we won a
best paper award at ISMIS’2008 for this work.
 NIH disagreed and would not consider our algorithm
because it was probably not truly better than the others.
2

Motivation: My story (cont’d)
 My reactions were:
 Surprise: Since my student and I properly applied the
evaluation methodology that we had been taught and
read about everywhere, how could our results be
challenged?
 Embarrassment: There is obviously much more to
evaluation than what I have been told. How can I call
myself a scientist and not know what the scientists of
other fields know so well?
 Determination: I needed to find out more about this
and share it with my colleagues and students.
3

Information about this tutorial
 This tutorial is based on the book I have co-written
after going through the experience I just described.
 It will give you an overview of the complexity and
uncertainty of evaluation.
 It will also give you a brief overview of the issues that
come up in different aspects of evaluation and what
the possible remedies may be.
 Finally, it will direct you to some resources available
that can help you perform more robust evaluations of
your systems.
4

Book Details
 Review:
"This treasure-trove of a book covers the
important topic of performance evaluation
of machine learning algorithms in a very
comprehensive and lucid fashion. As
Japkowicz and Shah point out, performance
evaluation is too often a formulaic affair in
machine learning, with scant appreciation
of the appropriateness of the evaluation
methods used or the interpretation of the
results obtained. This book makes
significant steps in rectifying this situation
by providing a reasoned catalogue of
evaluation measures and methods, written
specifically for a machine learning audience
and accompanied by concrete machine
learning examples and implementations in
R. This is truly a book to be savoured by
machine learning professionals, and
required reading for Ph.D students."
Peter A. Flach, University of Bristol
5
Evaluating Learning Algorithms:
A Classification Perspective
Nathalie Japkowicz & Mohak Shah
Cambridge University Press, 2011

The main steps of evaluation
6

What these steps depend on
 These steps depend on the purpose of the evaluation:
 Comparison of a new algorithm to other (may be generic or
application-specific) classifiers on a specific domain (e.g., when
proposing a novel learning algorithm)
 Comparison of a new generic algorithm to other generic ones on a
set of benchmark domains (e.g. to demonstrate general effectiveness
of the new approach against other approaches)
 Characterization of generic classifiers on benchmarks domains (e.g.
to study the algorithms' behavior on general domains for
subsequent use)
 Comparison of multiple classifiers on a specific domain (e.g. to find
the best algorithm for a given application task)

Outline of the tutorial:
 Part I (from now until the coffee break!)
 Choosing a performance measure.
 Choosing a statistical test.
 Part II (from after the coffee break to the lunch break!)
 What about sampling?
 What data sets should we use?
 Available resources
 Part III (if there is time before lunch and we’re not too
hungry!)
 Recent research
8

Topic 1: Choosing a Performance
Measure
9

Which Classifier is better?
Almost as many answers as there are
performance measures! (e.g., UCI Breast Cancer)
Algo Acc RMSE TPR FPR Prec Rec F AUC Info S
NB 71.7 .4534 .44 .16 .53 .44 .48 .7 48.11
C4.5 75.5 .4324 .27 .04 .74 .27 .4 .59 34.28
3NN 72.4 .5101 .32 .1 .56 .32 .41 .63 43.37
Ripp 71 .4494 .37 .14 .52 .37 .43 .6 22.34
SVM 69.6 .5515 .33 .15 .48 .33 .39 .59 54.89
Bagg 67.8 .4518 .17 .1 .4 .17 .23 .63 11.30
Boost 70.3 .4329 .42 .18 .5 .42 .46 .7 34.48
RanF 69.23 .47 .33 .15 .48 .33 .39 .63 20.78
10

Which Classifier is better?
Ranking the results
Algo Acc RMSE TPR FPR Prec Rec F AUC Info S
NB 3 5 1 7 3 1 1 1 2
C4.5 1 1 7 1 1 7 5 7 5
3NN 2 7 6 2 2 6 4 3 3
Ripp 4 3 3 4 4 3 3 6 6
SVM 6 8 4 5 5 4 6 7 1
Bagg 8 4 8 2 8 8 8 3 8
Boost 5 2 2 8 7 2 2 1 4
RanF 7 6 4 5 5 4 7 3 7
11

What should we make of that?
 Well, for certain pairs of measures, that makes sense,
since each measure focuses on a different aspect of
learning.
 For example, the TPR and the FPR are quite different, and
often, good results on one yields bad results on the other.
 Precision and Recall also seem to tradeoff each other.
 How about the global measures (Acc, RMSE, the F-
measure, AUC, the Information Score)?
 They too disagree as they each measure different (though
more difficult to pinpoint as they are composite
measures) aspects of learning.
12

Is this a problem?
 It is not a problem when working on a specific
application since the purposes of the application are
clear and the developer knows which performance
measure is best to optimize in that context.
 It does become a problem, however, when the general
usefulness of a new algorithm is assessed. In that case,
what performance measure should be chosen?
 If only one or a few measures are chosen, then it can be
argued that the analysis is incomplete, and misleading.
 If many measures are chosen, the results become too
mitigated for any clear statement to be issued.
13

What to do, then?
 One suggestion which tries to balance pragmatics
(getting the paper accepted!) with fairness
(acknowledging the weakness of the new algorithm!) is
to divide the evaluation procedure into two parts:
 In the first part, choose a few important metrics on which
the new method excels in order to demonstrate this new
method’s worth.
 In a second part, overview the results that the new method
obtains on a large selection of performance measures.
 Try to explain these observations; do not be too strict (i.e., if the new method
consistently ranks as one of the best three methods tested, then it is not that
bad. Similarly, if it ranks very badly on only one performance measure, then it
is not that bad either). 14

Overview of Performance Measures
15

A Few Confusion Matrix-Based
Performance Measures
16
True class 
Hypothesized |
class V
Pos Neg
Yes TP FP
No FN TN
P=TP+FN N=FP+TN
 Accuracy =
(TP+TN)/(P+N)
 Precision = TP/(TP+FP)
 Recall/TP rate = TP/P
 FP Rate = FP/N
 ROC Analysis moves the
threshold between the
positive and negative class
from a small FP rate to a
large one. It plots the value
of the Recall against that of
the FP Rate at each FP Rate
considered.
A Confusion Matrix

17
Issues with Accuracy
True class 
Pos Neg
Yes 200 100
No 300 400
P=500 N=500
True class 
Pos Neg
Yes 400 300
No 100 200
P=500 N=500
Both classifiers obtain 60% accuracy
They exhibit very different behaviours:
On the left: weak positive recognition rate/strong
negative recognition rate
On the right: strong positive recognition rate/weak
negative recognition rate

18
Issues with Precision/Recall
True class 
Pos Neg
Yes 200 100
No 300 400
P=500 N=500
True class 
Pos Neg
Yes 200 100
No 300 0
P=500 N=100
Both classifiers obtain the same precision and recall values
of 66.7% and 40% (Note: the data sets are different)
They exhibit very different behaviours:
Same positive recognition rate
Extremely different negative recognition rate: strong on
the left / nil on the right
Note: Accuracy has no problem catching this!

Is the AUC the answer?
 Many researchers have now adopted the AUC (the area
under the ROC Curve).
 The principal advantage of the AUC is that it is more
robust than Accuracy in class imbalanced situations.
 Indeed, given a 95% imbalance (in favour of the
negative class, say), the accuracy of the default classifier
that issues “negative” all the time will be 95%, whereas a
more interesting classifier that actually deals with the
issue, is likely to obtain a worse score.
 The AUC takes the class distribution into consideration.
19

Is the AUC the Answer? (cont’)
 While the AUC has been generally adopted as a
replacement for accuracy, it met with a couple of
criticisms:
 The ROC curves on which the AUCs of different classifiers
are based may cross, thus not giving an accurate picture of
what is really happening.
 The misclassification cost distributions (and hence the
skew-ratio distributions) used by the AUC are different for
different classifiers. Therefore, we may be comparing
apples and oranges as the AUC may give more weight to
misclassifying a point by classifier A than it does by classifier B
(Hand, 2009) Answer: the H-Measure, but it has been
criticized too! 20

Some other measures that will be
discussed in this tutorial:
 Deterministic Classifiers:
 Chance Correction: Cohen’s Kappa
 Scoring Classifiers:
 Graphical Measures: Cost Curves (Drummond & Holte,
2006)
 Probabilistic Classifiers:
 Distance measure: RMSE
 Information-theoretic measure: Kononenko and
Bratko’s Information Score
 Multi-criteria Measures: The Efficiency Method
(Nakhaeizadeh & Schnabl, 1998) 21

Cohen’s Kappa Measure
 Agreement Statistics argue that accuracy does not take
into account the fact that correct classification could
be a result of coincidental concordance between the
classifier’s output and the label-generation process.
 Cohen’s Kappa statistics corrects for this problem. Its
formula is:
κ = (P0 – Pe
C )/ ( 1 – Pe
C) where
 P0 represents the probability of overall agreement over the label
assignments between the classifier and the true process, and
 Pe
C represents the chance agreement over the labels and is defined
as the sum of the proportion of examples assigned to a class times
the proportion of true labels of that class in the data set.
22

Cohen’s Kappa Measure: Example
Predicted ->
Actual
A B C Total
A 60 50 10 120
B 10 100 40 150
C 30 10 90 130
Total 100 160 140
23
Accuracy = P0 = (60 + 100 + 90) / 400 = 62.5%
Pe
C = 100/400 * 120/400 + 160/400 * 150/400 + 140/400 * 130/400 = 0.33875
κ = 43.29%
 Accuracy is overly optimistic in this example!

Cost Curves
24
Cost-curves are more practical than ROC
curves because they tell us for what class
probabilities one classifier is preferable
over the other.
ROC Curves only tell
us that sometimes
one classifier is
preferable over the
other

RMSE
 The Root-Mean Squared Error (RMSE) is usually used
for regression, but can also be used with probabilistic
classifiers. The formula for the RMSE is:
RMSE(f) = sqrt( 1/m Σi=1
m(f(xi) – yi)2))
where m is the number of test examples, f(xi), the classifier’s
probabilistic output on xi and yi the actual label.
RMSE(f) RMSE(f)
25
ID f(xi) yi (f(xi) – yi)2
1 .95 1 .0025
2 .6 0 .36
3 .8 1 .04
4 .75 0 .5625
5 .9 1 .01
RMSE(f) = sqrt(1/5 * (.0025+.36+.04+.5625+.01))
= sqrt(0.975/5) = 0.4416

Information Score
 Kononenko and Bratko’s Information Score assumes a prior P(y) on the
labels. This could be estimated by the class distribution of the training
data. The output (posterior probability) of a probabilistic classifier is
P(y|f), where f is the classifier. I(a) is the indicator fonction.
 IS(x) = I(P(y|f) ≥ P(y)) * (-log(P(y))+log(P(y|f)) +
+ I(P(y|f) < P(y)) * (- log(1-P(y))+log(1-P(y|f)))
 ISavg= 1/m Σi=1
m(IS(xi))
26
x P(yi|f) yi IS(x)
1 .95 1 0.66
2 .6 0 0
3 .8 1 .42
4 .75 0 .32
5 .9 1 .59
P(y=1) = 3/5 = 0.6
P(y=0) = 2/5 = 0.4
IS(x1) = 1 * (-log(.6) + log(.95)) +
0 * (-log(.4) + log).05))
= 0.66
ISavg= 1/5 (0.66+0+.42+.32+.59)
= 0.40

Efficiency Method
 The efficiency method is a framework for combining
various measures including qualitative metrics, such
as interestingness. It considers the positive metrics for
which higher values are desirable (e.g, accuracy) and
the negative metrics for which lower values are
desirable (e.g., computational time).
εS(f)= Σiwipmi
+(f) / Σjwjnmj
-(f)
pmi
+ are the positive metrics and nmj
- are the negative metrics. The wi’s
and wj’s have to be determined and a solution is proposed that uses linear
programming.
27

Topic 2: Choosing a Statistical Test
28

The purpose of Statistical
Significance Testing
 The performance metrics just discussed allow us to
make observations about different classifiers.
 The question we ask here is: can the observed results
be attributed to real characteristics of the classifiers
under scrutiny or are they observed by chance?
 The purpose of statistical significance testing is to help
us gather evidence of the extent to which the results
returned by an evaluation metric are representative of
the general behaviour of our classifiers.
29

Hypothesis Testing
 Hypothesis testing consists of stating a null hypothesis which usually is
the opposite of what we wish to test (for example, classifiers A and B
perform equivalently)
 We then choose a suitable statistical test and statistic that will be used to
reject the null hypothesis.
 We also choose a critical region for the statistic to lie in that is extreme
enough for the null hypothesis to be rejected.
 We calculate the observed test statistic from the data and check whether
it lies in the critical region. If so, reject the null hypothesis. If not, we fail
to reject the null hypothesis, but do not accept it either.
 Rejecting the null hypothesis gives us some confidence in the belief that
our observations did not occur merely by chance.
30

Issues with Hypothesis Testing
 Hypothesis testing never constitutes a proof that our observation is valid.
It provides added support for our observations. We can never be 100%
sure about them.
 Statistical tests come in two forms: parametric and non parametric.
Parametric tests make strong assumptions about the distribution of the
underlying data. Non-parametric ones make weaker assumptions about
the data, but are also typically less powerful (less apt at rejecting the null
hypothesis when it is false) than their parametric counterparts. It is often
difficult, if not impossible, to verify that all the assumptions hold.
 The results of statistical tests are often misinterpreted:
 (1-p) does not represent P(H|D)
 (1-p) does not represent the probability of replication of successful replication
of the observations
 It is always possible to show that a difference between two alternatives,
no matter how small, is significant, provided that enough data is used.
31

To test or not to test?
 Given the serious issues associated with statistical testing, some
researchers (Drummond, 2006, Demsar, 2008) argue that
Machine Learning and Data Mining researchers should drop the
habit of performing statistical tests. They argue that, this process:
 Overvalues the results, and
 Limits the search for novel ideas (because of the excess (not
always warranted) confidence in the results).
 The alternative, however, is to train researchers properly in the
understanding and application of statistical methods, so that they
can decide on their own when a statistical test is warranted, what
its limitations are and when the search for new ideas is necessary.
32

How to choose a statistical test?
 There are several aspects to consider when choosing a
statistical test.
 What kind of problem is being handled?
 Whether we have enough information about the
underlying distributions of the classifiers’ results to apply
a parametric test?
 Regarding the type of problem, we distinguish between
 The comparison of 2 algorithms on a single domain
 The comparison of 2 algorithms on several domains
 The comparison of multiple algorithms on multiple
domains
33

Statistical Tests we will describe and
illustrate in this tutorial
 Two classifiers, one domain:
 The t-test (parametric)
 McNemar’s test (non-parametric)
 The Sign Test
 Two classifiers, multiple domains
 The Sign Test (non-parametric)
 Wilcoxon’s signed-Rank Test (non-parametric)
 Multiple classifiers, multiple domains:
 Friedman’s Test (non-parametric)
 The Nemenyi Test
 We will also discuss and illustrate the concept of the effect size.
35

Two classifiers, one domain
 The t-test (parametric)
 McNemar’s test (non-parametric)
 The Sign Test (usually used for multiple domains but
can also be used on a single domain).
36

The (2-matched samples) t-test
 Given two matched samples (e.g., the results of two
classifiers applied to the same data set with matching
randomizations and partitions), we want to test whether the
difference in means between these two sets is significant.
 In other words, we want to test whether the two samples
come from the same population.
 We look at the difference in observed means and standard
deviation.
 We assume that the difference between these means is zero
(the null hypothesis) and see if we can reject this hypothesis.
37

The t-statistic

38
The degree of freedom is n-1 = 9; Using a 2-sided test, the null hypothesis can
therefore be rejected at the 0.001 significance level (since the obtained t has to be
greater than 4.781for that to be possible).

Assumptions of the t-test
 The Normality or Pseudo-Normality Assumption: The t-test requires that
the samples come from normally distributed population. Alternatively,
the sample size of the testing set should be greater than 30.
 The Randomness of the Samples: The sample should be representative of
the underlying population. Therefore, the instances of the testing set
should be randomly chosen from their underlying distribution.
 Equal Variance of the populations: The two sample come from
populations with equal variance.
 Labour Example:
 Normality: The labour data set contains only 57 instances altogether. At each
fold of the CV process, only 6 or 7 data points are tested. However, since each
trial is a run of 10-fold CV, all 57 examples are tested, so we may be able to
assume pseudo-normality.
 Randomness: The labour data was collected in the 1987-1988 period. All the
collective agreements for this period were collected. There is no reason to
assume that 1987-1988 was a non-representative year, so we assume that the
data was randomly collected. 39

Assumptions of the t-test (cont’d)
 Equal Variance: The
variance of C4.5 (1) and NB
(1) cannot be considered
equal. (See figure)
 We were not warranted
to use the t-test to compare
C4.5 to NB on the Labour
data.
A better test to use in
this case is the non-
parametric alternative to
the t-test: McNemar’s Test
(See Japkowicz & Shah,
2011 for a description)
40

The Sign Test
 Since the Sign test is usually used on multiple domains
(though can be used on a single one with several trial
(e.g., 10 folds of cross-validation)), we will discuss it in
the next section which looks at multiple domains.
42

Two classifiers multiple domains
 There are no clear parametric way to deal with the problem
of comparing the performance of two classifiers on multiple
domains:
 The t-test is not a very good alternative because it is not clear that
we have commensurability of the performance measures in such a
setting.
 The normality assumption is difficult to establish as the number
of domains on which the test is run must exceed 30.
 The t-test is susceptible to outliers, which is more likely when
many different domains are considered.
 Therefore we will describe two non-parametric alternatives
 The Sign Test
 Wilcoxon’s signed-Rank Test
43

The Sign test
 The sign test can be used either to compare two classifiers
on a single domain (using the results at each fold as a trial)
or more commonly, to compare two classifiers on multiple
domains.
 We count the number of times that f1 outperforms f2, nf1
and the number of times that f2 outperforms f1, nf2.
 The null hypothesis (stating that the two classifiers perform
equally well) holds if the number of wins follows a binomial
distribution.
 Practically speaking, a classifier should perform better on at
least wα datasets to be considered statistically significantly
better at the α significance level, where wα is the critical
value for the sign test at the α significance level .
44

Illustration of the sign test
Dataset NB SVM Adaboost Rand Forest
Anneal 96.43 99.44 83.63 99.55
Audiology 73.42 81.34 46.46 79.15
Balance Scale 72.30 91.51 72.31 80.97
Breast Cancer 71.70 66.16 70.28 69.99
Contact Lenses 71.67 71.67 71.67 71.67
Pima Diabetes 74.36 77.08 74.35 74.88
Glass 70.63 62.21 44.91 79.87
Hepatitis 83.21 80.63 82.54 84.58
Hypothyroid 98.22 93.58 93.21 99.39
Tic-Tac-Toe 69.62 99.90 72.54 93.94
45
• NB vs SVM: nf1= 4.5, nf2= 5.5 and w0.05 = 8  we cannot reject the null hypothesis
stating that NB and SVM perform similarly on these data sets for level α=0.05 (1-tailed)
• Ada vs RF: nf1=1 and nf2=8.5  we can reject the null hypothesis at level α=0.05 (1-tailed)
and conclude that RF is significantly better than Ada on these data sets at that significance level.

Wilcoxon’s signed-Rank Test
 Wilcoxon’s signed-Rank Test, like the sign test, deals with two
classifiers on multiple domains. It is also non-parametric,
however, it is more powerful than the sign test. Here is its
description:
 For each domain, we calculate the difference in performance of the two
classifiers.
 We rank the absolute values of these differences and graft the signs in
front of the ranks.
 We calculate the sum of positive and negative ranks, respectively (WS1
and WS2)
 TWilcox = min(WS1 ,WS2)
 Compare to critical value Vα. If Vα ≥ TWilcox we reject the null hypothesis
that the performance of the two classifiers is the same, at the α
confidence level.
46

Wilcoxon’s signed-Rank Test: Illustration
Data NB SVM NB-SVM |NB-SVM| Ranks
1 .9643 .9944 -0.0301 0.0301 3 -3
2 .7342 .8134 -0.0792 0.0792 6 -6
3 .7230 .9151 -0.1921 0.1921 8 -8
4 .7170 .6616 +0.0554 0.0554 5 +5
5 .7167 .7167 0 0 Remove Remove
6 .7436 .7708 -0.0272 0.0272 2 -2
7 .7063 .6221 +0.0842 0.0842 7 +7
8 .8321 .8063 +0.0258 0.0258 1 +1
9 .9822 .9358 +0.0464 0.0464 4 +4
10 .6962 .9990 -0.3028 0.3028 9 -9
47
WS1 = 17 and WS2 = 28  TWilcox = min(17, 28) = 17
For n= 10-1 degrees of freedom and α = 0.005, V = 8 for the 1-sided test. V must be larger than
TWilcox in order to reject the hypothesis. Since 17 > 8, we cannot reject the hypothesis that NB’s
performance is equal to that of SVM at the 0.005 level.

Multiple classifiers, Multiple Domains
 For the case of multiple classifiers and multiple domains, two
alternatives are possible. The parametric alternative is (one-way
repeated measure) ANOVA and the non-parametric alternative is
Friedman’s Test.
 These two tests are multiple-hypothesis tests, also called Omnibus
tests. Their null hypotheses is that all the classifiers perform equally,
and rejection of that null hypothesis means that: there exists at least
one pair of classifiers with significantly different performances.
 In case of rejection of this null hypothesis, the omnibus test is
followed by a Post-hoc test whose job is to identify the significantly
different pairs of classifiers.
 In this tutorial, we will discuss Friedman’s Test (omnibus) and the
Nemenyi test (post-hoc test).
48

Illustration of the Friedman test
Domain Classifier
fA
Classifier
fB
Classifier
fC
1 85.83 75.86 84.19
2 85.91 73.18 85.90
3 86.12 69.08 83.83
4 85.82 74.05 85.11
5 86.28 74.71 86.38
6 86.42 65.90 81.20
7 85.91 76.25 86.38
8 86.10 75.10 86.75
9 85.95 70.50 88.03
19 86.12 73.95 87.18
Domain Classifier
fA
Classifier
fB
Classifier
fC
1 1 3 2
2 1.5 3 1.5
3 1 3 2
4 1 3 2
5 2 3 1
6 1 3 2
7 2 3 1
8 2 3 1
9 2 3 1
10 2 3 1
R .j 15.5 30 14.5
50

Illustration of the Nemenyi Test

52

What is the Purpose of Resampling?
 Ideally, we would have access to the entire population
or a lot of representative data from it.
 This is usually not the case, and the limited data
available has to be re-used in clever ways in order to be
able to estimate the error of our classifiers as reliably as
possible, i.e., to be re-used in clever ways in order to
obtain sufficiently large numbers of samples.
 Resampling is divided into two categories: Simple re-
sampling (where each data point is used for testing
only once) and Multiple re-sampling (which allows the
use of the same data point more than once for testing)
55

What are the dangers of Resampling?
 Re-sampling is usually followed by Statistical testing.
Yet statistical testing relies on the fundamental
assumption that the data used to obtain a sample
statistics must be independent.
 However, if data is re-used, then this important
independence assumption is broken and the result of
the statistical test risks being invalid.
 In addition to discussing a few re-sampling
approaches, we will underline the issues that may arise
when applying them.
56

Overview of Re-Sampling Methods
57

Resampling Approaches Discussed in
this Tutorial
 Simple Resampling:
 Cross-Validation and its variants
 Multiple Resampling:
 The 0.632 Bootstrap
 The Permutation Test
 Repeated k-fold Cross-Validation
58

k-fold Cross-Validation
59
…..
In Cross-Validation, the data set is divided into k folds and at each iteration, a
different fold is reserved for testing and all the others, used for training the
classifiers.
: training data subset
: testing data subset
Fold 1:
Fold 2:
Fold k-1:
Fold k:

Some variations of Cross-Validation
 Stratified k-fold Cross-Validation:
 This variation is useful when the class-distribution of
the data is skewed. It ensures that the distribution is
respected in the training and testing sets created at
every fold. This would not necessarily be the case if a
pure random process were use.
 Leave-One-Out
 In k-fold Cross-Validation, each fold contains m/k data
points where m is the overall size of the data set. In
Leave-one-out, k =m and therefore, each fold contains a
single data point.
60

Considerations about k-fold Cross-
Validation and its variants
 k-fold Cross-Validation is the best known and most
commonly used resampling technique.
 K-fold Cross-Validation is less computer intensive than
Leave-One-Out.
 The testing sets are independent of one another, as
required, by many statistical testing methods, but the
training sets are highly overlapping. This can affect the bias
of the error estimates.
 Leave-One-Out produces error estimates with high
variance given that the testing set at each fold contains only
one example. The classifier is practically unbiased since
each fold trains on almost all the data.
61

Boootsrapping
 Bootstrapping assumes that the available sample is
representative and creates a large number of new samples
by drawing from replacement from the available sample.
 Bootstrapping is useful in practice when the sample is too
small for Cross-Validation or Leave-One-Out approaches to
yield a good estimate
 There are two Bootstrap estimates that are useful in the
context of classification: the Є0 and the e632 Bootstrap.
 The Є0 Bootstrap tends to be pessimistic because it is only
trained on 63.2% of the data in each run. The e632 attempts
to correct for this.
62

The Є0 and e632 Bootstraps

63

Considerations about Bootstrapping

64

 Given the error estimate found on a data set, the question is:
 Is this error estimate significant, or
 Could it also have been obtained on ‘bogus’ data?
 The ‘bogus’ data is created by taking the genuine samples and
randomly choosing to either leave their label intact or switch
them.
 Once this ‘bogus’ data set is created, the classifier is ran on it
and its error estimated.
 This process is repeated a very large number of times in an
attempt to establish whether the error estimate obtained on
the true data is truly different from those obtained on large
numbers of ‘bogus’ data sets.
The Permutation Test

Repeated k-Fold Cross-Validation(1)
 In order to obtain more stable estimates of an
algorithm’s performance, it is useful to perform
multiple runs of simple re-sampling schemes. This can
also enhance replicability of the results.
 Two specific schemes have been suggested in the
context of Cross-Validation: 5x2 CV and 10x10 CV
 K-fold CV does not estimate the mean of the difference
between 2 learning algorithms properly. The mean at a
single fold behaves better. This lead Dietterich (1998)
to propose the 5x2 CV fold, in which 2-fold CV is
repeated 5 times.
66

 Dietterich found that the paired t-test based on the the
5x2 CV scheme had lower probability of issuing a type-I
error but had less power than the k-fold CV paired t-test.
 Alpaydyn (1999) proposed to substitute the t-test at the
end of the 5x2CV scheme by an F-test. That test had an
even lower chance of issuing a type-I error and had
increased power (though that new test was not compared
to k-fold CV in terms of type I error or power)
 Bouckaert (2003) proposed several variations of a 10x10
CV scheme. Generally speaking these schemes show a
higher probability of Type-I error than 10-fold CV, but
higher power.
67
Repeated k-Fold Cross-Validation(2)

Topic 2: Choosing Appropriate Data
Sets for Testing Classifiers
68

Considerations to keep in mind while
choosing an appropriate test bed
 Wolpert’s “No Free Lunch” Theorems: if one algorithm
tends to perform better than another on a given class of
problems, then the reverse will be true on a different
class of problems.
 LaLoudouana and Tarate (2003) showed that even
mediocre learning approaches can be shown to be
competitive by selecting the test domains carefully.
The purpose of data set selection should not be to
demonstrate an algorithm’s superiority to another in all
cases, but rather to identify the areas of strengths of
various algorithms with respect to domain characteristics
or on specific domains of interest. 69

Where can we get our data from?
 Repository Data: Data Repositories such as the UCI
repository and the UCI KDD Archive have been extremely
popular in Machine Learning Research.
 Artificial Data: The creation of artificial data sets
representing the particular characteristic an algorithm was
designed to deal with is also a common practice in Machine
Learning
 Web-Based Exchanges: Could we imagine a multi-
disciplinary effort conducted on the Web where researchers
in need of data analysis would “lend” their data to macine
learning researchers in exchange for an analysis?
70

Pros and Cons of Repository Data
 Pros:
 Very easy to use: the data is available, already processed and the
user does not need any knowledge of the underlying field.
 The data is not artificial since it was provided by labs and so on. So
in some sense, the research is conducted in real-world setting
(albeit a limited one)
 Replication and comparisons are facilitated, since many researchers
use the same data set to validate their ideas.
 Cons:
 The use of data repositories does not guarantee that the results will
generalize to other domains.
 The data sets in the repository are not representative of the data
mining process which involves many steps other than classification.
 Community experiment/Multiplicity effect: since so many
experiments are run on the same data set, by chance, some will
yield interesting (though meaningless) results 71

Pros and Cons of Artificial Data
 Pros:
 Data sets can be created to mimic the traits that are expected to be
present in real data (which are unfortunately unavailable)
 The researcher has the freedom to explore various related situations
that may not have been readily testable from accessible data.
 Since new data can be generated at will, the multiplicity effect will
not be a problem in this setting.
 Cons:
 Real data is unpredictable and does not systematically behave
according to a well defined generation model.
 Classifiers that happen to model the same distribution as the one
used in the data generation process have an unfair advantage.
72

Pros and Cons of Web-Based Exchanges
 Pros:
 Would be useful for three communities:
 Domain experts, who would get interesting analyses of their
problem
 Machine Learning researchers who would be getting their
hand on interesting data, and thus encounter new problems
 Statisticians, who could be studying their techniques in an
applied context.
 Cons:
 It has not been organized. Is it truly feasible?
 How would the quality of the studies be controlled?
73

Topic 3: Available Resources
74

What help is available for
conducting proper evaluation?
 There is no need for researchers to program all the code necessary to
conduct proper evaluation nor to do the computations by hand.
 Resources are readily available for every steps of the process, including:
 Evaluation metrics
 Statistical tests
 Re-sampling
 And of course, Data repositories
 Pointers to these resources and explanations on how to use them are
discussed in our book: <“Evaluating Learning Algorithms: A
Classification Perspective” by Japkowicz and Shah, Cambridge
University Press, 2011>.
75

Where to look for evaluation
metrics?
Actually, as most
people know, Weka
is a great source for,
not only, classifiers,
but also
computation of the
results according to
a variety of
evaluation metrics
76

WEKA even performs ROC Analysis
and draws Cost-Curves
77
Although better graphical analysis packages are available in R, namely the
ROCR package, which permits the visualization of many versions of ROC and
Cost Curves, and also Lift curves, P-R Curves, and so on

Where to look for Statistical Tests?
78
The R Software
Package contains
implementations
of all the statistical
tests discussed
here and many
more. They are
very simple to run.

Where to look for Re-sampling methods?
 In our book, we have implemented all the re-sampling schemes
described in this tutorial. The actual code is also available upon request.
(e-mail me at <nat@site.uottawa.ca>)
e0Boot = function(iter, dataSet, setSize, dimension, classifier1, classifier2){
classifier1e0Boot <- numeric(iter)
classifier2e0Boot <- numeric(iter)
for(i in 1:iter) {
Subsamp <- sample(setSize, setSize, replace=TRUE)
Basesamp <- 1:setSize
oneTrain <- dataSet[Subsamp ,1:dimension ]
oneTest <- dataSet[setdiff(Basesamp,Subsamp), 1:dimension]
classifier1model <- classifier1(class~., data=oneTrain)
classifier2model <- classifier2(class~., data=oneTrain)
classifier1eval <- evaluate_Weka_classifier(classifier1model, newdata=oneTest)
classifier1acc <- as.numeric(substr(classifier1eval$string, 70,80))
classifier2eval <- evaluate_Weka_classifier(classifier2model,
newdata=oneTest)
classifier2acc <- as.numeric(substr(classifier2eval$string, 70,80))
classifier1e0Boot[i]= classifier1acc
classifier2e0Boot[i]= classifier2acc
}
return(rbind(classifier1e0Boot, classifier2e0Boot))}
79

Recent Research
Topic 1: A Visualization-based framework for
classifier evaluation
(Nathalie Japkowicz, Pritika Sanghi and Peter Tischer)
[ECML’2008, ISAIM’2008]
80

81
A New Framework for Classifier
Evaluation
 Classifier evaluation can be viewed as a problem of
analyzing high-dimensional data.
 The performance measures currently used are but one
class of projections that can be applied to these data.
 Why not apply other (standard or not) projections to
the data with various kinds (standard or not) distance
measures?

82
Some Advantages of this new Framework
 Projection approaches are typically intended for
visualization. This yields two advantages:
 A quick and easy way for human-beings to assess classifier
performance results.
 The possibility to offer simultaneous multiple views of
classifier performance evaluation.
 The framework offers a solution to the problem of
aggregating the results obtained by a classifier on
several domains.
 The framework offers a way to deal with multi-class
domains.

83
The Framework and its Implementation
 The framework is implemented as a function of
the following steps:
1. All the classifiers in the study are run on all the
domains of the study.
2. The performance matrices (e.g., confusion matrix) of
a single classifiers on every domain are aggregated
into a single vector. This is repeated for each classifier
3. A projection and a distance measure for that
projection are chosen and applied to the vectors of
Step 2.

84
Illustration of the Framework
True
class 
Pos Neg
Yes 82 17
No 12 114
True
class 
Pos Neg
Yes 15 5
No 25 231
Ideal
True
class 
Pos Neg
Yes 99 6
No 1 94
82 17 12 114 15 5 25 231 99 6 1 94
Confusion matrices
for a single classifier
on three domains

85
A Few Remarks about our Framework
 It decomposes the evaluation problem neatly,
separating the issue of projection from that of
distance measure.
 By going from a projection (or two) into a one-
dimensional space to one into a two-dimensional
space, we allow for two rather than one
relationships to be established:
 The ranking of classifiers with respect to the ideal
classifier.
 The comparison of each classifier to the others.

86
Specific Implementation Details
 Our framework can be used with any projection technique
and any distance function associated to the projection.
 In this work, we experimented with:
 A Minimal Cost Spanning Tree (MCST) distance-preserving
projection [Yang, 2004].
 Two distance functions: the Euclidean Distance (L2-norm) and the
Manhattan Distance (L1-Norm). In our experiments, the L2 norm is
the one normally used, unless otherwise specified.
 We focus on the results obtained
 When combining the confusion matrices of various classifiers on
several domains,
 When dealing with a multi-class domain.

87
Experimental Set-Up
 We tested our approach on four UCI domains:
 3 Binary ones: Breast Cancer, Labour and Liver.
 1 Multi-Class One: Anneal.
 We compared the performance of eight WEKA classifiers on
these domains: NB, J48, Ibk, JRip, SMO, Bagging, AdaBoost,
RandFor.
 The focus of our study is not to discover which classifier wins
or loses on our data sets. Rather, we are using our
experiments to illustrate the advantages and disadvantages of
our framework over other evaluation methods. We, thus,
used Weka’s default parameters in all cases.
 Also, though we test our approach with the MCST projection,
others could have been used. This is also true of our distance
functions.

88
Illustration on Multiple domains: Breast Cancer, Labour and Liver
Acc. F-
Meas.
AUC
NB BC:
La:
Li:
Avg:
71.7
89.5
55.4
72.2
.48
.92
.6
.67
.7
.97
.64
.77
SMO BC:
La:
Li:
Avg:
69.6
89.5
58.3
72.46
.39
.92
.014
.44
.59
.87
.5
.65
Boost. BC:
La:
Li:
Avg:
70.3
87.7
66.1
74.7
.46
.91
.534
.64
.7
.87
.68
.75
8: SVM (SMO)
9: NB 1: Ideal
Abnormality detection with our
new approach is a lot easier and
accurate than it is, when relying on Accuracy,
F-Measure, or AUC listings on each domain
or their average on all domains.
Also, our new approach
Allows us to mix binary
and multi-class domains.
Averaging does not!

89
Illustration on a Multiclass domain:
Anneal (L2-Norm)
Adaboost:
a b c d e f  classified as
0 0 8 0 0 0 | a
0 0 99 0 0 0 | b
0 0 684 0 0 0 | c
0 0 0 0 0 0 | d
0 0 0 0 67 0 | e
0 0 40 0 0 0 | f
NB:
a b c d e f classified as
7 0 1 0 0 0 | a
0 99 0 0 0 0 | b
3 38 564 0 0 79 | c
0 0 0 0 0 0 | d
0 0 0 0 67 0 | e
0 0 2 0 0 38 | f
8: Adaboost
9: NB 1: Ideal
NB Other
Classifiers
Ada
boost
86.3 97.4 to
99.3
83.6
Accuracy
Accuracy
does not
tell us
whether
NB and
Adaboost
make the
same kind
of errors!

90
Illustration on Anneal
using the L1-Norm
 When using the L2-norm,
NB and Adaboost were at
approximately the same
distance to Ideal.
 When using the L1-norm,
NB is significantly closer.
NB makes fewer errors than
Adaboost, but the majority
of its errors are
concentrated on one or
several large classes.
1: Ideal, 8: NB
9: Adaboost
Because our new evaluation
framework is visual in nature,
we can process quickly the results
obtained using various distance
Measures (evaluation measure), and, thus, interpret our results
In a more informed manner. It is easier done this way than by
staring at large tables of numbers!

91
Summary
 We present a new framework for classifier evaluation
that recognizes that classifier evaluation consists of
projecting high-dimensional data into low-
dimensional ones.
 By using a projection into 2-dimensional space rather
than one, we propose a visualization approach to the
problem. This allows for quick assessments of the
classifiers’ behaviour based on the results obtained
using multiple performance measures.
 Each entry of the evaluation vectors we project is
compared in pair-wise fashion to its equivalent in
other vectors. Thus, our aggregation technique is
more precise than that used with traditional
performance measures. This is an advantage when
considering results over various domains, or in the
case of multi-class domains.

92
Future Work
 As presented, our approach seems limited to the
comparison of single classifier’s performance.
 How about threshold-insensitive classifiers?
 How about the computation of statistical guarantees on
our results?
This can be solved by plotting either the results
obtained at various thresholds, or the results
obtained at various folds of a cross-validation
regimen, thus plotting clouds of classifiers that
could then be analyzed.
We also plan to experiment with other distance
measures and projection methods.

Recent Research
Topic 2: Assessing the Impact of Changing
environments on Classifier Performance
(Rocio Alaiz-Rodriguez and Nathalie Japkowicz)
[Canadian AI ’ 2008]
93

94
Purpose of the Work
 Direct purpose: To test the hypothesis by David
Hand (2006) that simple classifiers are more robust to
changing environments than complex ones.
 Indirectly: To demonstrate the feasibility and value of
generating artificial, but realistic domains.
 More generally: To propose an alternative to the use
of the UCI domains.

95
Specific hypotheses under review
 Preliminaries: Different kinds of changing environments:
 Population Drift — p(d|x) remains unchanged, but p(x)
differs from training to testing set. Also known as: covariate
shift or sample selection bias.
 Class Definition Change — p(x) does not change, but p(d|x)
varies from training to testing set. Also known as: concept drift
or functional relation change.
 Hypotheses under review:
 Hypothesis 1: When either or both a population drift and a
class definition change occurs, can we generally observe a drop
in performance by all kinds of classifiers?
 Hypothesis 2: Do simpler classifiers maintain their
performance more reliably than more complex ones in such
cases?

96
Our Experimental Framework I
 Our domain is a simulated medical domain that states
the prognostic of patients infected with the flu and
described as follows:
 Patient’s age [Infant, Teenager, YoungAdult, Adult,
OldAdult, Elderly]
 Severity of flu symptoms [Light, Medium, Strong]
 Patient’s general health [Good, Medium, Poor]
 Patient’s social position [Rich, MiddleClass, Poor]
 Class: NormalRemission, Complications

97
Our Experimental Framework II
 In order to make the
problem interesting for
classifiers, we assumed
that the features are not
independent of one
another.
 We also assumed that
certain feature values
were irrelevant.
Attribute Dependency Graph

98
Our Experimental Framework III
We used different distributions to model the various features
and the class:
 Age: we assumed a region with negative growth which,
according to the Population Reference Bureau contains
many people in the Adult and Young Adult categories, and
few people in the other categories. We used uniform
distributions to model the first five categories, and an
exponential distribution for the elderly category (which is
not bounded upwards).
 Social Status: Normal Distribution distributed around the
Middle class (= 2), with variance .75. [Poor=3 and Rich= 1]
 Severity of Flu Symptoms: (= Severity of the virus strain)
Same distribution as Social Status.

99
Our Experimental Framework IV
 General Health: We used an unobserved binary
variable called “delicate person” whose probability
increases with age, and generated rules based on the
assumptions that (1) delicate people have worse
general health than other members of the
population; and (2) poorer people have worse
geberal health than the richer members of society.
 Class: The class labels were assigned automatically
and abided by the following general principles:
 In case of stronger flu symptoms, the chances of
complications are greater.
 Infants and Elderly have greater chances of complications
 People with poorer general health are more susceptible to
complications

100
Our Experimental Framework V
 Class Distribution, or
probability of normal
remission as a function
of age, severity of the
flu symptoms and
general health. When
two lines appear, the
discontinuous one
applies to instances
with poor social status

Experimental Set Up
 We compared two pairs of simple/complex classifiers:
1R/Decision Tree and Simple Perceptron/Multi-Layer
Perceptron
 We looked at three situations:
 Population drifts with full representation
 Population drifts with some cases not represented
 Concept drifts
 We studied the performance deterioration that occurs
when going from no drift to one of the three types of
drifts above.
101

102
Changes to the Testing Set I
 Population Drifts with full representation:
 Developing Population (DP): high birth and death rates.
 Zero Growth Population (ZGP): Similar birth and death
rates.
 Season changes (NGP/W): Winter: Flu symptoms get
stronger
 Season changes (NGP/SW): Soft Winter: Flu Symptoms
get milder
 Season changes (NGP/DW): Drastic Winter: Flu
Symptoms get stronger and general health declines
 Population is much poorer (NGP/P)
 Population is much poorer and the winter is trastic
(NGP/P+DW)

103
Changes to the Testing Set II
 Population Drifts with Non-Represented cases
 We considered several situations where one or two
population groups are not represented.
 Class Definition Changes:
 More Complications (MC): the probability of normal
remission decreases for certain ages, social statuses and
flu symptoms.
 Fewer Complications (FC): the age group for which the
probability of normal remission is high is widened.

104
Evaluation Measure
 In order to measure the changes in performance
caused by environmental changes, we introduce a
new metric, called performance Deterioration
(pD), defined as follows:
Etest – Eideal , if Etest <= E0
pD = E0 – Eideal
1, Otherwise
With E0 representing the error rate of the trivial
classifier, Etest is the classifier’s error rate on the
test set, and Eideal is the classifier’s error rate when
trained and tested on data abiding by the same
distribution.

105
Results I: Population Drifts with Full
Representation
Verification of our hypotheses:
(a) A drop in performance is observed by all classifiers
(b) Simpler classifiers suffer much more.

106
Results II: Population Drifts with Non-
Represented Cases
(b) Simpler classifiers suffer slightly more.

107
Results III: Class Definition Changes
(b) Simpler classifiers don’t necessarily suffer more than complex
ones. (SimpleNN does not. 1R does)

108
Summary
 Our results show that the trend hypothesized by David Hand
does happen in some cases, but does not happen in others.
 In all cases, however, complex classifiers that generally obtain
lower error rates in the original scenario (with no changing
conditions) remain the best choice since their performance
remains higher than that of the simple classifiers even though
their performance deterioration are sometimes equivalent.
 Given the dearth of data sets representing changing
environments, none of the results we present here could have
been obtained had we not generated artificial though realistic
data sets simulating various conditions.

109
Future Work
 Develop a systematic way to generate realistic artificial
data sets that could replace, or, at least, supplement
the UCI domains.
 Find a way to verify the realistic nature of these data
sets.
 Rather than generate data sets from intuition as we’ve
done it here, start from actual real-world data sets and
expand them artificially.

110
General Conclusions
 A first step to improve the evaluation process in
machine learning is to be more aware of the
tradeoffs involved in choosing one evaluation
method over another. This tutorial as well as the
book I co-wrote will help researchers understand
these issues more deeply.
 In particular, the tutorial overviewed:
 Evaluation Metrics
 Statistical Tests
 Re-sampling methods
 Data set selection criteria
 Available resources
 Some recent research from our group

If you need help, advice, etc…
 Please, contact me at:
nat@site.uottawa.ca
 Thank you for attending the workshop!!!
111

References
 Too many to put down here, but pp. 393-402 of the
book.
 If you want specific ones, come and see me at the end
of the tutorial or send me an e-mail.
112

Performance Evaluation for Classifiers tutorial

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