Download CSI5388 Functional Elements of Statistics for Machine Learning

CSI5388:
Functional Elements of
Statistics for Machine
Learning
Part II
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Contents of the Lecture
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Part I (This set of lecture notes):
• Definition and Preliminaries
• Hypothesis Testing: Parametric Approaches
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Part II (The next set of lecture notes)
• Hypothesis Testing: Non-Parametric
Approaches
• Power of a Test
• Statistical Tests for Comparing Multiple
Classifiers
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Non-parametric approaches to
Hypothesis testing
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The hypothesis testing procedures discussed in the
previous lecture are called parametric.
This means that they are based on assumptions
regarding the distribution of the population for
which the test was ran, and rely on the estimation of
parameters from these distributions.
In our cases, we assumed that the distributions
were either normal or followed a Student t
distribution. The parameters we estimated were the
mean and the variance.
The problem we now turn to is the issue of
hypothesis testing that is not based on assumptions
regarding the distribution and do not rely on the
estimation of parameters.
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The Different Types of non-parametric
hypothesis testing approaches I
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There are two important families of tests that do
not involve distributional assumptions and
parameter estimations:
• Nonparametric tests, which rely on ranking
the data and performing a statistical test on
the ranks.
• Resampling statistics which consist of
drawing samples repeatedly from a population
and evaluating the distribution of the result.
Resampling Statistics will be discussed in the
next lecture.
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The Different Types of non-parametric
hypothesis testing approaches II
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The nonparametric tests are quite useful
in populations for which outliers skew the
distribution too much.
Ranking eliminates the problem.
However, they typically are less powerful
(see further) than parametric tests.
Resampling statistics are useful when the
statistics of interest cannot be derived
analytically (e.g., statistics about the
median of a population), unless we
assume a normal distribution.
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Non-Parametric Tests
Wilcoxon’s Rank-Sum Test
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The case of independent samples
The case of matched pairs
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Wilcoxon’s Rank-Sum Tests
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Wilcoxon’s Rank-Sum Tests are equivalent
to the t-test, but apply when the normality
assumption of the distribution is not met.
As a result of their non-parametric nature,
however, power is lost (see further for a
formal discussion of power). In particular,
the tests are not as specific as their
parametric equivalent.
This means that, although we interpret the
result of these non-parametric tests to
mean one thing of a central nature to the
distributions under study, they could mean
something else.
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Wilcoxon’s Rank-Sum Test
(for two independent samples)
Informal Description I
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Given two populations with n1 observations
in group 1 and n2 observations in group 2.
The null hypothesis we are trying to reject
is: “H0: The two samples come from
identical populations (not just populations
with the same mean)”.
We consider two cases:
• Case 1: The null hypothesis is false (to a
substantial degree) and the scores from
population 1 are generally lower than those of
population 2.
• Case 2: The null hypothesis is true. This means
that the two samples came from the same
population.
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Wilcoxon’s Rank-Sum Test
(for two independent samples)
Informal Description II
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In both cases, the procedure consists of
ranking the scores of the two populations
taken together.
• Case 1: In the first case, we assume that the
ranks from population 1 should be generally
lower than those of population 2. Actually, we
could also expect that the sum of the ranks in
group 1 is smaller than the sum of the ranks in
group 2.
• Case 2: In the second case, we assume that
the sum of ranks of the first group is about
equal to the sum of ranks of the second group.9
Wilcoxon’s Rank-Sum Test
(for two independent samples)
n1 and n2 ≤ 25:
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Consider the two groups of data sets of size n1 and n2,
respectively, where n1 is the smallest sample size.
Rank their scores together from lowest to highest.
In case of an x-way tie just after rank y, then assign (y+1
+ y +2 + … + y+ x)/x to all the tied elements.
Add the scores of the group containing the smallest number
of samples (n1) (if both groups contain as many samples,
choose the smallest value). Call this sum Ws.
Find the value V in the Wilcoxon table, for n1 and n2 and
the significance level s required, where n1 in the table
corresponds to the smallest value, as well.
Compare Ws to V and conclude that the difference between
the two groups at the chosen level, L1 for a one-tailed test
or 2*L1 for the two-tailed test is significant only if Ws < V.
If Ws ≥ V, the null hypothesis cannot be rejected.
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Wilcoxon’s Rank-Sum Test
(for two independent samples)
n1 and n2 > 25:
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Compute Ws as before
Use the fact that Ws approaches a normal
distribution as size increases with:
• A mean of
m= n1(n1+n2+1)/2, and
• A standard error of
std= sqrt(n1n2(n1+n2+1)/12)
Compute the z statistic
z = (Ws – m)/std
Use the tables of the normal distribution.
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Wilcoxon’s Matched Pairs Signed
Ranks Test (for paired scores)
Informal Description
Logic of the Test:
 Given the same population tested under
different circumstances C1 and C2.
 If there is improvement in C2, then most
of the results recorded in C2 will be
greater than those recorded in C1 and
those that are not greater will be smaller
by only a small amount.
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Wilcoxon’s Matched Pairs Signed
Ranks Test (for paired scores)
n ≤ 50
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We calculate the difference score for each pair of
measurement
We rank all the difference scores without paying
attention to their signs (i.e., we rank their
absolute values)
We assign the algebraic sign of the differences to
the ranks
We sum the positive and negative ranks
separately
We choose as test statistic T, the smaller of the
absolute values of the two sums.
We compare T to a Wilcoxon T table
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Wilcoxon’s Matched Pairs Signed
Ranks Test (for paired scores)
n > 50
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Compute T as before
Use the fact that T approaches a normal
distribution as size increases with:
• A mean of
m= n(n+1)/4 and
• A standard error
std= sqrt(n(n+1)(2n+1)/24)
And compute the z statistic
z = (T – m)/std
Use the tables of the normal distribution.
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Power Analysis
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Type I and Type II Errors
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Definition: A Type I error (α)
corresponds to the error of rejecting
H0, the null hypothesis, when it is, in
fact, true. A Type II error (β)
corresponds to the error of failing to
reject H0 when it is false.
Definition: The power of a test is
the probability of rejecting H0 given
that it is false. Power = 1- β
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Why does Power Matter? I
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All the hypothesis tests described in the
previous three sections are only concerned
about reducing the Type I error.
i.e., they try to ascertain the conditions
under which we are rejecting a hypothesis
rightly.
They are not at all concerned about the
case where the null hypothesis is really
false, but we do not reject it.
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Why does Power Matter? II
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In the case of Machine Learning, reducing the
type I error means reducing the probability of us
saying that there is a difference in the
performance of the 2 classifiers, when in fact,
there isn’t.
Reducing the type II error means reducing the
probability of us saying that there is no difference
in the performance of the two classifiers, when,
in fact, there is.
Power matters because we do not want to discard
a classifier that shouldn’t have been discarded. If
a test does not have enough power, then this
kind of situation can arise
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What is the Effect Size?
The effect size measures how strong the
relationship between two entities is.
 In particular, if we consider a particular procedure,
in addition to knowing how statistically significant
the effect of that procedure is, we may want to
know what the size of this effect is.
 There are different measures of effect sizes,
including:
• Pearsons’ correlation coefficient
• Odd’s ratio
• Cohen’s d statistics
 Cohen's d statistic is appropriate in the context of a
t-test on means. It is thus the effect size measure
we concentrate on here.
[Wikipedia: http://en.wikipedia.org/wiki/Effect_size]
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Cohen’s d-statistics
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Cohen’s d-statistic is expressed as:
d = (X1 – X2)/ sp
Where sp2, the pooled variance estimate is:
sp2= ((n1-1)*s12 + (n2-1)*s22)
(n1+n2-2)
and sp, its square root.
[Note this is not exactly Cohen’s d measure
which was expressed in terms of
parameters. What we show above is an
estimate of d].
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Usefulness of the d statistic
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d is useful in that it standardizes the difference
between the two means. We can talk about
deviations in terms of proportions of standard
deviation points that are more useful than actual
differences that are domain dependent.
Cohen came up with a set of guidelines
concerning d:
• d=.2 has a small effect, but is probably
meaningful;
• d= .5 is a medium effect that is
noticeable.
• d= .8 shows a large effect size.
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Statistical Tests for
Comparing Multiple
Classifiers
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What is the Analysis of Variance
(ANOVA)?
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The analysis of variance is similar to the t-test in
that it deals with differences between sample
means.
However, unlike the t-test that is restricted to the
difference between two means, ANOVA allows us
to assess whether the differences observed
between any number of means are statistically
significant.
In addition, ANOVA allows us to deal with more
than one independent variable. For example, we
could choose, as two independent variables, 1) the
learning algorithm and 2) the domain to which
the learning algorithm is applied.
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Why is ANOVA useful?
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One may wonder why ANOVA is useful in
the context of classifier evaluation.
Very simply, if we want to answer the
following common question : "How do
various classifiers fare on different data
sets?", then we have 2 independent
variables: the learning algorithm and the
domain, and a lot of results.
ANOVA makes it easy to tell whether the
difference observed are indeed significant.
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Variations on the ANOVA Theme
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There are different implementations of
ANOVA:
• One-way ANOVA is a linear model trying to
assess if the difference in the performance
measures of classifiers over different datasets is
statistically significant, but does not distinguish
between the performance measures’ variability
within-datasets and the performance measure
variability between-datasets.
• Two-way/Multi-way ANOVA can deal with
more than one independent variable. For
instance, two performance measures over
different classifiers over various datasets.
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Then there are other related tests as well:
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• Friedman’s test, Post-hoc tests, Tukey Test, etc…
How does One-Way ANOVA work? I
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It considers various groups of
observations and sets as a hypothesis that
all the means are equal.
The opposite hypothesis is that they are
not all equal.
The ANOVA model is as follows:
xij = μi + eij
• where xij is the jth observation from group i, μi
is the mean of group i and eij is the noise that
is normally distributed with mean 0 and
common standard deviation σ
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How does One-Way ANOVA work? II
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ANOVA monitors three different kinds of variation
in the data:
• Within-group variation
• Between-group variation
• Total variation = within-group variation + between-group
variation
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Each of the above variations are represented by
sums of squares (SS) of the variations.
The statistics of interest in ANOVA is F, where
F = Between-group variation
Within-group variation
Larger F’s demonstrate greater statistical
significance than smaller ones. Like for z and t,
there are tables of significance levels associated
with the F-ratio.
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How does One-Way ANOVA work? III
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The goal of ANOVA is to find out whether or not
the differences in means between different
groups are statistically significant.
To do so, ANOVA partitions the total variance into
variance caused by random error (the within
group SS) and variance caused by actual
differences between means (the between-group
SS).
If the null hypothesis holds, then the withingroup SS should be about the same as the
between-groups SS.
We can compare these two SS using the F test,
which checks whether the ratio of the two SSs is
significantly greater than 1.
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What is Multi-Way ANOVA?
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In One-Way ANOVA, we simply considered several
groups.
For example this could correspond the comparing
the performance of 10 different classifiers on one
domain.
How about the case where we compare the
performance of these same 10 different classifiers
on 5 domains?
Two-Way ANOVA can help with that
If we were to use an additional dimension such as
the consideration of 6 different (but matched)
threshold levels (as in AUC) for each classifier on
the same 5 domains, then Three-way ANOVA could
be used, and so on…
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How Does Multi-Way ANOVA work?
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In our example, the difference between One-Way
ANOVA and Two-Way ANOVA can be illustrated as
follows:
• in One-Way ANOVA, we would calculate the within-group
SS by collapsing the results obtained on all the data sets
together within each classifier results.
• In Two-Way ANOVA, with would calculate all the withinclassifier, within-domain variances separately and group
the results together.
• As a result, the spooled within-group SS of two-way
ANOVA would be smaller than the spooled within-group
SS of one-way ANOVA.
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Multi-way ANOVA is thus a more statistically
powerful test than One-way ANOVA since we need
fewer observations to find significant effects.
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