- Training and test sets are used to measure classification success rates, with the test set being independent of the training set. The error rate on the training set is optimistic. Cross validation techniques like 10-fold stratified cross validation are used when data is limited. - True success rates are predicted using properties of statistics and normal distributions. Confidence levels determine the range within which the true rate is expected to lie. - Techniques like paired t-tests are used to statistically compare the performance of different algorithms or data mining methods. They determine if performance differences are statistically significant.

1. Credibility: Evaluating what’s Been Learned

2. Training and Testing We measure the success of a classification procedure by using error rates (or equivalent success rates) Measuring success rate using training set is highly optimistic The error rate on training set is called resubstitution error We have a separate test set for calculating success error Test set should be independent of the training set Also some time to improve our classification technique we use a validation set When we hold out some part of training set for testing (which is now not used for training), this process is called holdout procedure

3. Predicting performance Expected success rate = 100 – error rate (If error rate is also in percentage) We want the true success rate Calculation of true success rate Suppose we have expected success rate(f) = s/n, where s is the number of success out of a total n instances For large value of n, f follows a normal distribution Now we will predict the true success rate (p) based on the confidence percentage we want For example say our f = 75%, then p will lie in [73.2%,76.7%] with 80% confidence

4. Predicting performance Now using properties of statistics we know that the mean of f is p and the variance is p(1-p)/n To use normal distribution we will have to make the mean of f = 0 and standard deviation = 1 So suppose our confidence = c% and we want to calculate p We will use the two tailed property of normal distribution And also that the are covered by normal distribution is taken as 100% so the are we will leave is 100 - c

5. Predicting performance Finally after all the manipulations we have ,true success rate as: Here, p -> true success rate f - > expected success rate N -> Number of instances Z -> Factor derived from a normal distribution table using the 100-c measure

6. Cross validation We use cross validation when amount of data is small and we need to have independent training and test set from it It is important that each class is represented in its actual proportions in the training and test set: Stratification An important cross validation technique is stratified 10 fold cross validation, where the instance set is divided into 10 folds We have 10 iterations with taking a different single fold for testing and the rest 9 folds for training, averaging the error of the 10 iterations Problem: Computationally intensive

7. Other estimates Leave-one-out:Steps One instance is left for testing and the rest are used for training This is iterated for all the instances and the errors are averaged Leave-one-out:Advantage We use larger training sets Leave-one-out:Disadvantage Computationally intensive Cannot be stratified

8. Other estimates 0.632 Bootstrap Dataset of n samples is sampled n times, with replacements, to give another dataset with n instances There will be some repeated instances in the second set Here error is defined as: e = 0.632x(error in test instances) + 0.368x(error in training instances)

9. Comparing data mining methods Till now we were dealing with performance prediction Now we will look at methods to compare algorithms, to see which one did better We cant directly use Error rate to predict which algorithm is better as the error rate might have been calculated on different data sets So to compare algorithms we need some statistical tests We use Student’s t- test to do this. This test help us to figure out if the mean error of two algorithm are different or not for a given confidence level

10. Comparing data mining methods We will use paired t-test which is a slight modification of student’s t-test Paired t-test Suppose we have unlimited data, do the following: Find k data sets from the unlimited data we have Use cross validation with each technique to get the respective outcomes: x1, x2, x3,….,xk and y1,y2,y3,……,yk mx = mean of x values and similarly my di = xi – yi Using t-statistic:

11. Comparing data mining methods Based on the value of k we get a degree of freedom, which enables us to figure out a z for a particular confidence value If t <= (-z) or t >= (z) then, the two means differ significantly In case t = 0 then they don’t differ, we call this null hypothesis

12. Predicting Probabilities Till now we were considering a scheme which when applied, results in either a correct or an incorrect prediction. This is called 0 – loss function Now we will deal with the success incase of algorithms that outputs probability distribution for e.g. Naïve Bayes

13. Predicting Probabilities Quadratic loss function: For a single instance there are k out comes or classes Probability vector: p1,p2,….,pk The actual out come vector is: a1,a2,a3,…..ak (where the actual outcome will be 1, rest all 0) We have to minimize the quadratic loss function given by: The minimum will be achieved when the probability vector is the true probability vector

14. Predicting Probabilities Informational loss function: Given by: –log(pi) Minimum is again reached at true probabilities Differences between Quadratic loss and Informational loss While quadratic loss takes all probabilities under consideration, Informational loss is based only on the class probability While quadratic loss is bounded as its maximum output is 2, Informational loss is unbounded as it can output values up to infinity

15. Counting the cost Different outcomes might have different cost For example in loan decision, the cost of lending to a defaulter is far greater that the lost-business cost of refusing a loan to a non defaulter Suppose we have two class prediction. Outcomes can be:

16. Counting the cost True positive rate: TP/(TP+FN) False positive rate: FP/(FP+TN) Overall success rate: Number of correct classification / Total Number of classification Error rate = 1 – success rate In multiclass case we have a confusion matrix like (actual and a random one):

17. Counting the cost These are the actual and the random outcome of a three class problem The diagonal represents the successful cases Kappa statistic = (D-observed - D-actual) / (D-perfect - D-actual) Here kappa statistic = (140 – 82)/(200-82) = 49.2% Kappa is used to measure the agreement between predicted and observed categorizations of a dataset, while correcting for agreements that occurs by chance Does not take cost into account

18. Classification with costs Example Cost matrices (just gives us the number of errors): Success rate is measured by average cost per prediction We try to minimize the costs Expected costs: dot products of vectors of class probabilities and appropriate column in cost matrix

19. Classification with costs Steps to take cost into consideration while testing: First use a learning method to get the probability vector (like Naïve Bayes) Now multiple the probability vector to each column of a cost matrix one by one so as to get the cost for each class/column Select the class with the minimum(or maximum!!) cost

20. Cost sensitive learning Till now we included the cost factor during evaluation We will incorporate costs into the learning phase of a method We can change the ratio of instances in the training set so as to take care of costs For example we can do replication of a instances of particular class so that our learning method will give us a model with less errors of that class

21. Lift Charts In practice, costs are rarely known In marketing terminology the response rate is referred to as the lift factor We compare probable scenarios to make decisions A lift chart allows visual comparison Example: promotional mail out to 1,000,000 households Mail to all: 0.1%response (1000) Some data mining tool identifies subset of 100, 000 of which 0.4% respond (400) A lift of 4

22. Lift Charts Steps to calculate lift factor: We decide a sample size Now we arrange our data in decreasing order of the predicted probability of a class (the one which we will base our lift factor on: positive class) We calculate: Sample success proportion = Number of positive instances / Sample size Lift factor = Sample success proportion / Data success proportion We calculate lift factor for different sample size to get Lift Charts

23. Lift Charts A hypothetical lift chart

24. Lift Charts In the lift chart we will like to stay towards the upper left corner The diagonal line is the curve for random samples without using sorted data Any good selection will keep the lift curve above the diagonal

25. ROC Curves Stands for receiver operating characteristic Difference to lift charts: Y axis showspercentage of true positive X axis shows percentage of false positives in samples ROC is a jagged curve It can be smoothened out by cross validation

26. ROC Curves A ROC curve

27. ROC Curves Ways to generate cost curves (Consider the previous diagram for reference) First way: Get the probability distribution over different folds of data Sort the data in decreasing order of the probability of yes class Select a point on X-axis and for that number of no, get the number of yes for each probability distribution Average the number of yes from all the folds and plot it

28. ROC Curves Second way: Get the probability distribution over different folds of data Sort the data in decreasing order of the probability of yes class Select a point on X-axis and for that number of no, get the number of yes for each probability distribution Plot a ROC for each fold individually Average all the ROCs

29. ROC Curves ROC curves for two schemes

30. ROC Curves In the previous ROC curves: For a small, focused sample, use method A For a large one, use method B In between, choose between A and B with appropriate probabilities

31. Recall – precision curves In case of a search query: Recall = number of documents retrieved that are relevant / total number of documents that are relevant Precision = number of documents retrieved that are relevant / total number of documents that are retrieved

32. A summary Different measures used to evaluate the false positive versus the false negative tradeoff

33. Cost curves Cost curves plot expected costs directly Example for case with uniform costs (i.e. error):

34. Cost curves Example with costs:

35. Cost curves C[+|-] is the cost of predicting + when the instance is – C[-|+] is the cost of predicting - when the instance is +

36. Minimum Description Length Principle The description length is defined as: Space required to describe a theory + space required to describe the theory’s mistakes Theory = Classifier and mistakes = errors on the training data We try to minimize the description length MDL theory is the one that compresses the data the most. I.e to compress a data set we generate a model and then store the model and its mistakes We need to compute: Size of the model Space needed to encode the error

37. Minimum Description Length Principle The 2nd one is easy. Just use informational loss function For 1st we need a method to encode the model L[T] = “length” of the theory L[E|T] = training set encoded wrt the theory

38. Minimum Description Length Principle MDL and clustering Description length of theory: bits needed to encode the clusters. E.g. cluster centers Description length of data given theory: encode cluster membership and position relative to cluster. E.g. distance to cluster centers Works if coding scheme uses less code space for small numbers than for large ones

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