CpSc 810: Machine Learning

Evaluation of Classifier

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Copy Right Notice

Most slides in this presentation are adopted from slides of text book and various sources. The Copyright belong to the original authors. Thanks!

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Classifier Accuracy Measures

C1 C2

C1 True positive (TP) False negative (FN)

C2 False positive (FP) True negative (TN)

classes buy_computer = yes

buy_computer = no

total

buy_computer = yes 6954 46 7000

buy_computer = no 412 2588 3000

total 7366 2634 10000

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Classifier Accuracy Measures

The sensitivity: the percentage of correctly predicted positive data over the total number of positive data

The specificity: the percentage of correctly identified negative data over the total number of negative data.

The accuracy: the percentage of correctly predicted positive and negative data over the sum of positive and negative data

Positive

TP

FNTP

TPySensitivit

Negative

TN

FPTN

TNySpecificit

FNFPTNTP

TNTPAccuracy

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Classifier Accuracy Measures

The precision: the percentage of correctly predicted positive data over the total number of predicted positive data.

The F-measure is also called F-score. As a weighted average of the precision and recall, it considers both the precision and recall of the test to computer the score.

FPTP

TPPrecision

Recall)(Precision

Recall)(Precision2F

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ROC curves

ROC = Receiver Operating Characteristic

Started in electronic signal detection theory (1940s - 1950s)

Has become very popular method used in machine learning applications to assess classifiers

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ROC curves: simplest case

Consider diagnostic test for a disease

Test has 2 possible outcomes:

‘positive’ = suggesting presence of disease

‘negative’ = suggesting un-presence of disease

An individual can test either positive or negative for the disease

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Hypothesis testing refresher

2 ‘competing theories’ regarding a population parameter:

NULL hypothesis HALTERNATIVE hypothesis A

H: NO DIFFERENCE any observed deviation from what we expect to see is due to chance variability

A: THE DIFFERENCE IS REAL

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Test Statistic

Measure how far the observed data are from what is expected assuming the NULL H by computing the value of a test statistic (TS) from the data

The particular TS computed depends on the parameter

For example, to test the population mean , the TS is the sample mean (or standardized sample mean)

The NULL is rejected if the TS falls in a user-specified ‘rejection region’

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True disease state vs. Test result

not rejected rejected

No disease (D = 0)

specificity

XType I error (False +)

Disease (D = 1) X

Type II error (False -)

Power 1 - ; sensitivity

DiseaseTest

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Specific Example

Test Result

Pts Pts with with diseasdiseasee

Pts Pts without without the the diseasedisease

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Test Result

Call these patients “negative”

Call these patients “positive”

Threshold

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Test Result

Call these patients “negative”

Call these patients “positive”

without the disease

with the disease

True Positives

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Test Result

Call these patients “negative”

Call these patients “positive”

False Positives

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Test Result

Call these patients “negative”

Call these patients “positive”

True negatives

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Test Result

Call these patients “negative”

Call these patients “positive”

False negatives

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Test Result

‘‘‘‘-’-’’’

‘‘‘‘+’+’’’

Moving the Threshold: right

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Test Result

‘‘‘‘-’-’’’

‘‘‘‘+’+’’’

Moving the Threshold: left

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ROC curve

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False Positive Rate (1-specificity)

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A good test: A poor test:

ROC curve comparison

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Best Test: Worst test:T

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The distributions don’t overlap at all

The distributions overlap completely

ROC curve extremes

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Area under ROC curve (AUC)

Overall measure of test performance

Comparisons between two tests based on differences between (estimated) AUC

For continuous data, AUC equivalent to Mann-Whitney U-statistic (nonparametric test of difference in location between two populations)

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False Positive Rate

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AUC = 50%

AUC = 90% AUC =

65%

AUC = 100%

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False Positive Rate

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AUC for ROC curves

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Interpretation of AUC

AUC can be interpreted as the probability that the test result from a randomly chosen diseased individual is more indicative of disease than that from a randomly chosen nondiseased individual: P(Xi Xj | Di = 1, Dj = 0)

So can think of this as a nonparametric distance between disease/nondisease test results

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Predictor Error Measures

Measure predictor accuracy: measure how far off the predicted value is from the actual known value

Loss function: measures the error betw. yi and the

predicted value yi’

Absolute error: | yi – yi’|

Squared error: (yi – yi’)2

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Predictor Error Measures

Test error (generalization error): the average loss over the test set

Mean absolute error: Mean squared error:

Relative absolute error: Relative squared error:

The mean squared-error exaggerates the presence of outliers

Popularly use (square) root mean-square error, similarly, root relative squared error

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Evaluating the Accuracy of a Classifier or Predictor (I)

Holdout methodGiven data is randomly partitioned into two independent sets

Training set (e.g., 2/3) for model construction

Test set (e.g., 1/3) for accuracy estimationRandom sampling: a variation of holdout

Repeat holdout k times, accuracy = avg. of the accuracies obtained

Cross-validation (k-fold, where k = 10 is most popular)Randomly partition the data into k mutually exclusive subsets, each approximately equal sizeAt i-th iteration, use Di as test set and others as training setLeave-one-out: k folds where k = # of examples, for small sized dataStratified cross-validation: folds are stratified so that class distribution in each fold is approximately the same as that in the initial data

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Evaluating the Accuracy of a Classifier or Predictor (II)

BootstrapWorks well with small data setsSamples the given training examples uniformly with replacement

i.e., each time a example is selected, it is equally likely to be selected again and re-added to the training set

Several bootstrap methods, and a common one is .632 bootstrap

Suppose we are given a data set of d examples. The data set is sampled d times, with replacement, resulting in a training set of d samples. The data examples that did not make it into the training set end up forming the test set. About 63.2% of the original data will end up in the bootstrap, and the remaining 36.8% will form the test set (since (1 – 1/d)d ≈ e-1 = 0.368)

Repeat the sampling procedure k times, overall accuracy of the model:

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More About Bootstrap

The bootstrap method attempts to determine The bootstrap method attempts to determine the probability distribution from the data itself, the probability distribution from the data itself, without recourse to CLT (without recourse to CLT (Central Limit Theorem)..

The bootstrap method is not a way of The bootstrap method is not a way of reducing the error ! It only tries to estimate it.reducing the error ! It only tries to estimate it.

Basic idea of BootstrapOriginally, from some list of data, one computes an object.Create an artificial list by randomly drawing elements from that list. Some elements will be picked more than once. Compute a new object.Repeat 100-1000 times and look at the distribution of these objects.

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More About Bootstrap

How many bootstraps ? No clear answer to this. Lots of theorems on asymptotic convergence, but no real estimates !Rule of thumb: try it 100 times, then 1000 times, and see if your answers have changed by much.Anyway have NN possible subsamples

Is it reliable ?A very good question !Jury still out on how far it can be applied, but for now nobody is going to shoot you down for using it.Good agreement for Normal (Gaussian) distributions, skewed distributions tend to more problematic, particularly for the tails, (boot strap underestimates the errors).

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Sampling

Sampling is the main technique employed for data selection.

It is often used for both the preliminary investigation of the data and the final data analysis.

Statisticians sample because obtaining the entire set of data of interest is too expensive or time consuming.

Sampling is used in data mining because processing the entire set of data of interest is too expensive or time consuming.

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Sampling …

The key principle for effective sampling is the following:

Using a sample will work almost as well as using the entire data sets, if the sample is representative

A sample is representative if it has approximately the same property (of interest) as the original set of data

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Sample Size

8000 points 2000 Points 500 Points

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Types of Sampling

Simple Random SamplingThere is an equal probability of selecting any particular itemSampling without replacement

As each item is selected, it is removed from the population

Sampling with replacement

Objects are not removed from the population as they are selected for the sample.

In sampling with replacement, the same object can be picked up more than once

Stratified samplingSplit the data into several partitions; then draw random samples from each partition