DATA MINING OVERFITTING AND EVALUATION

Transcription

1 DATA MINING OVERFITTING AND EVALUATION 1

2 Overfitting Will cover mechanisms for preventing overfitting in decision trees But some of the mechanisms and concepts will apply to other algorithms 2

3 Occam s Razor William of Ockham ( ) Among competing hypotheses, the one with the fewest assumptions should be selected. For complex models, there is a greater chance that it was fitted accidentally by errors in data Therefore, one should include model complexity when evaluating a model 3

4 current (I) Overfitting Example The issue of overfitting had been known long before decision trees and data mining Experimentally measure 10 points Fit a curve to the Resulting data. voltage (V) In electrical circuits, Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference or voltage across the two points, and inversely proportional to the resistance between them. Perfect fit to training data with an 9 th degree polynomial (can fit n points exactly with an n-1 degree polynomial) Ohm was wrong, we have found a more accurate function! 4

5 current (I) Overfitting Example Testing Ohms Law: V = IR (I = (1/R)V) voltage (V) Better generalization with a linear function that fits training data less accurately. 5

6 Overfitting due to Noise Decision boundary is distorted by noise point 6

7 Overfitting due to Insufficient Examples Hollow red circles are test data Lack of data points in the lower half of the diagram makes it difficult to predict correctly the class labels of that region Insufficient number of training records in the region causes the decision tree to predict the test examples using other training records that are irrelevant to the classification task 7

8 x2: sepal width Decision Trees in Practice Growing to purity is bad (overfitting) x1: petal length 8

9 x2: sepal width Decision Trees in Practice Growing to purity is bad (overfitting) x1: petal length 9

10 x2: sepal width Decision Trees in Practice Growing to purity is bad (overfitting) Not statistically supportable leaf Remove split & merge leaves x1: petal length 10

11 Partitioning of Data We use a training set to build the model We use a test set to evaluate the model The test data is not used to build the model so the evaluation is fair and not biased The resubstitution error (error rate on training set) is a bad indicator of performance on new data Overfitting of training data will yield good resubstitution error but bad predictive accuracy We sometimes use a validation set to tune a model or choose between alternative models Often used for pruning and overfitting avoidance All three data sets may be generated from a single labeled data set 11

12 Underfitting and Overfitting Overfitting Underfitting: when model is too simple, both training and test errors are large Overfitting: when model is too complex, training error is low but test error rate is high How many decision tree nodes (x-axis) would you use? 12

13 The Right Fit Overfitting Best generalization performance seems to be achieved with around 130 nodes 13

14 Validation Set The prior chart shows the relationship between tree complexity and training and test set performance But you cannot look at it, find the best test set performance, and then say you can achieve that. Why? Because when you use the test set to tune the classifier by selecting the number of nodes, the test data is now used in the model building process Solution: use a validation set to find the tree that yields the best generalization performance. Then report performance of that tree on a independent test set. 14

15 How to Avoid Overfitting? Stop growing the tree before it reaches the point where it perfectly classifies the training data (prepruning) Such estimation is difficult Allow the tree to overfit the data, and then prune the tree back (postpruning) This is commonly used Although first approach is more direct, second approach found more successful in practice: because it is difficult to estimate when to stop Both need a criterion to determine final tree size 15

16 How to Address Overfitting Pre-Pruning (Early Stopping Rule) Stop the algorithm before it becomes a fully-grown tree Typical stopping conditions for a node: Stop if all instances belong to the same class Stop if all the attribute values are the same More restrictive conditions: Stop if number of instances is less than some user-specified threshold Stop if class distribution of instances are independent of the available features (e.g., using 2 test) Stop if expanding the current node does not improve impurity measures (e.g., Gini or information gain). Assign some penalty for model complexity and factor that in when deciding whether to refine the model (e.g., a penalty for each leaf node in a decision tree) 16

17 How to Address Overfitting Post-pruning Grow decision tree to its entirety Trim the nodes of the decision tree in bottom-up fashion If generalization error improves after trimming (validation set), replace sub-tree by a leaf node. Class label of leaf node is determined from majority class of instances in the sub-tree Can use Minimum Description Length for postpruning 17

18 Minimum Description Length (MDL) X y X 1 1 X 2 0 X 3 0 X 4 1 X n 1 A Yes 0 A? No B? B 1 B 2 C? 1 C 1 C B X y X 1? X 2? X 3? X 4? X n? Cost(Model,Data) = Cost(Data Model) + Cost(Model) Cost(Data Model) encodes the misclassification errors. If you have the model, you only need to remember the examples that do not agree with the model. Cost(Model) is the cost of encoding the model (in bits) General idea is to trade off model complexity and number of errors while assigning objective costs to both Costs are based on bit encoding 18

19 Methods for Determining Tree Size Training and Validation Set Approach: Use a separate set of examples, distinct from the training examples, to evaluate the utility of post-pruning nodes from the tree. Use all available data for training, but apply a statistical test (Chi-square test) to estimate whether expanding (or pruning) a particular node is likely to produce an improvement. Use an explicit measure of the complexity for encoding the training examples and the decision tree, halting growth when this encoding size is minimized. 19

20 Validation Set Provides a safety check against overfitting spurious characteristics of data Needs to be large enough to provide a statistically significant sample of instances Typically validation set is one half size of training set Reduced Error Pruning: Nodes are removed only if the resulting pruned tree performs no worse than the original over the validation set. 20

21 Reduced Error Pruning Properties When pruning begins tree is at maximum size and lowest accuracy over test set As pruning proceeds number of nodes is reduced and accuracy over test set increases Disadvantage: when data is limited, number of samples available for training is further reduced 21

22 test accuracy Issues with Reduced Error Pruning The problem with this approach is that it potentially wastes training data on the validation set. Severity of this problem depends where we are on the learning curve: number of training examples 22

23 EVALUATION 23

24 Model Evaluation Metrics for Performance Evaluation How to evaluate the performance of a model? Methods for Performance Evaluation How to obtain reliable estimates? 24

25 Metrics for Performance Evaluation Focus on the predictive capability of a model Rather than how fast it takes to classify or build models, scalability, etc. Confusion Matrix: PREDICTED CLASS Class=Yes Class=No a: TP (true positive) ACTUAL CLASS Class=Yes a b Class=No c d b: FN (false negative) c: FP (false positive) d: TN (true negative) 25

26 Metrics for Performance Evaluation PREDICTED CLASS Class=P Class=N ACTUAL CLASS Class=P Class=N a (TP) c (FP) b (FN) d (TN) Accuracy a a b d c d TP TP TN TN FP FN Error Rate = 1 - accuracy 26

27 Limitation of Accuracy Consider a 2-class problem Number of Class 0 examples = 9990 Number of Class 1 examples = 10 If model predicts everything to be class 0, accuracy is 9990/10000 = 99.9 % Accuracy is misleading because model does not detect any class 1 example 27

28 Cost-Sensitive Measures a Precision (p) a c a Recall (r) a b 2rp F - measure (F) r p 2a 2a b c PREDICTED CLASS Class=Yes Class=No ACTUAL CLASS Class=Yes Class=No a (TP) c (FP) b (FN) d (TN)

29 More on the F measure We just saw the F1-measure or F1-score 2RP/(R+P) F-measure is more general and allows you to vary the relative importance of precision and recall: F= ( 2 +1)RP / 2 P + R 29

30 Measuring predictive ability Can count number (percent) of correct predictions or errors in Weka percent correctly classified instances In business applications, different errors (different decisions) have different costs and benefits associated with them Usually need either to rank cases or to compute probability of the target (class probability estimation rather than just classification) 30

31 Costs Matter The error rate is an inadequate measure of the performance of an algorithm, it doesn t take into account the cost of making wrong decisions. Example: Based on chemical analysis of the water try to detect an oil slick in the sea. False positive: wrongly identifying an oil slick if there is none. False negative: fail to identify an oil slick if there is one. Here, false negatives (environmental disasters) are much more costly than false negatives (false alarms). We have to take that into account when we evaluate our model. 31

32 Cost Matrix PREDICTED CLASS C(i j) Class=Yes Class=No ACTUAL CLASS Class=Yes C(Yes Yes) C(No Yes) Class=No C(Yes No) C(No No) C(i j): Cost of misclassifying class j example as class i 32

33 Computing Cost of Classification Cost Matrix ACTUAL CLASS PREDICTED CLASS C(i j) Model M 1 PREDICTED CLASS Model M 2 PREDICTED CLASS ACTUAL CLASS ACTUAL CLASS Accuracy = 80% Cost = 3910 Accuracy = 90% Cost =

34 Cost-Sensitive Learning Cost sensitive learning algorithms can utilize the cost matrix to try to find an optimal classifier given those costs This can be implemented via in several ways Simulate the costs by modifying the training distribution Modify the probability threshold for making a decision if the costs are 2:1 you can modify the threshold from 0.5 to 0.33 Weka uses these two methods to allow you to do costsensitive learning Foundations of Cost-Sensitive Learning by Charles Elkan shows the equivalence between these methods 34

35 Model Evaluation Metrics for Performance Evaluation How to evaluate the performance of a model? Methods for Performance Evaluation How to obtain reliable estimates? Methods for Model Comparison How to compare the relative performance among competing models? 35

36 Assumptions Standard Cost Model correct classifications: zero cost cost of misclassification depends only on the class, not on the individual example over a set of examples costs are additive Costs or Class Distributions: are not known precisely at evaluation time may vary with time may depend on where the classifier is deployed True FP and TP do not vary with time or location, and are accurately estimated. 36

37 How to Evaluate Performance? Scalar Measures: make comparisons easy since only a single number involved Accuracy Expected cost Area under the ROC curve Visualization Techniques ROC Curves Lift Chart 37

38 What s Wrong with Scalars? A scalar does not tell the whole story. There are fundamentally two numbers of interest (FP and TP), a single number invariably loses some information. How are errors distributed across the classes? How will each classifier perform in different testing conditions (costs or class ratios other than those measured in the experiment)? A scalar imposes a linear ordering on classifiers. what we want is to identify the conditions under which each is better. Why Performance evaluation is useful Shape of curves more informative than a single number 38

39 ROC Curves Receiver operator characteristic Summarize & present performance of any binary classification model Models ability to distinguish between false & true positives 39

40 ROC Curve Analysis Signal Detection Technique Traditionally used to evaluate diagnostic tests Now employed to identify subgroups of a population at differential risk for a specific outcome (clinical decline, treatment response)

41 ROC Analysis: Historical Development Derived from early radar in WW2 Battle of Britain to address problem of accurately identifying planes Using the signals on the radar screen, predict the outcome of interest Enemy planes when there are many extraneous signals (e.g. Geese)?

42 ROC Analysis: Historical Development True Positives: Radar operator interpreted signal as Enemy Planes and there were Enemy planes Good Result: No wasted Resources True Negatives : Radar operator said no planes and correct Good Result: No wasted resources False Positives = Radar operator said planes, but none Geese: wasted resources False Negatives = Radar operator said no planes, but there were planes Bombs dropped: very bad outcome

43 Definition of TPR and FPR True class Predicted class positive negative positive (#P) #TP #P - #TP negative (#N) #FP #N - #FP Reduce the 4 numbers to two rates true positive rate = TPR = (#TP)/(#P) false positive rate = FPR = (#FP)/(#N) Rates are independent of class ratio* 43

44 Which Classifier Is Best? Predicted True pos neg pos neg Predicted True pos neg pos neg Predicted True pos neg pos neg Classifier 1 TP = 0.4 FP = 0.3 Classifier 2 TP = 0.7 FP = 0.5 Classifier 3 TP = 0.6 FP =

45 ROC plot for the 3 Classifiers Ideal classifier always positive chance always negative 45

46 ROC Curves more generally, ranking models produce a range of possible (FP,TP) tradeoffs Separates classifier performance from costs, benefits and target class distributions Generated by starting with best rule and progressively adding more rules Last case is when always predict positive class and TP =1 and FP = 1 46

47 Using ROC for Model Comparison No model consistently outperform the other M 1 better for small FPR M 2 better for large FPR Area Under ROC curve Ideal AUC: AUC = 1.0 Random guess: AUC =

48 Cumulative Response Curve Cumulative response curve more intuitive than ROC curve Plots TP rate (% of positives targeted) on the y- axis vs. percentage of population targeted (x-axis) Formed by ranking the classification rules from most to least accurate. Start with most accurate and plot point, add next most accurate, etc. Eventually include all rules and cover all examples Common in marketing applications 48

49 Cumulative Response Curve The chart below calls the one curve the lift curve but the name is a bit ambiguous (as we shall see on next slide) 49

50 Lift Chart Generated by dividing the cumulative response curve by the baseline curve for each x-value. Lift of 3 means that your prediction is 3X better than baseline (guessing) Data Mining tools also generate non-cumulative curves that may be more insightful 50

51 Learning Curve Learning curve shows how accuracy changes with varying sample size Requires a sampling schedule for creating learning curve: Arithmetic sampling (Langley, et al) Geometric sampling (Provost et al) 51

52 Methods for Partitioning Data Need to partition labelled data to form a training and test set (sometimes validation set) Holdout Reserve fixed amount for training (2/3) and testing (1/3) Random subsampling Repeated holdout Cross validation Partition data into k disjoint subsets k-fold: train on k-1 partitions, test on the remaining one Leave-one-out: k=n 52

53 Cross-validation Partition data into k folds (randomly) Run training/test evaluation k times K=10 for 10-fold cross validation common 53

54 Cross Validation Example: data set with 20 instances, 5-fold cross validation training test d 1 d 2 d 3 d 4 d 1 d 2 d 3 d 4 d 1 d 2 d 3 d 4 d 5 d 6 d 7 d 8 d 5 d 6 d 7 d 8 d 5 d 6 d 7 d 8 d 9 d 10 d 11 d 12 d 13 d 14 d 15 d 16 d 17 d 18 d 19 d 20 d 9 d 10 d 11 d 12 d 13 d 14 d 15 d 16 d 17 d 18 d 19 d 20 d 9 d 10 d 11 d 12 d 13 d 14 d 15 d 16 d 17 d 18 d 19 d 20 compute error rate for each fold then compute average error rate d 1 d 2 d 3 d 4 d 5 d 6 d 7 d 8 d 9 d 10 d 11 d 12 d 13 d 14 d 15 d 16 d 1 d 2 d 3 d 4 d 5 d 6 d 7 d 8 d 9 d 10 d 11 d 12 d 13 d 14 d 15 d 16 d 17 d 18 d 19 d 20 d 17 d 18 d 19 d 20 54

55 Leave-one-out Cross Validation Leave-one-out cross validation is simply k-fold cross validation with k set to n, the number of instances in the data set. The test set only consists of a single instance, which will be classified either correctly or incorrectly. Advantages: maximal use of training data, i.e., training on n 1 instances. The procedure is deterministic, no sampling involved. Disadvantages: unfeasible for large data sets: large number of training runs required, high computational cost. 55

56 Multiple Comparisons Beware the multiple comparisons problem The example in Data Science for Business is telling: Create 1000 stock funds by randomly choosing stocks See how they do and liquidate all but the top 3 Now you can report that these top 3 funds perform very well (and hence you might infer they will in the future). But the stocks were randomly picked! If you generate large numbers of models then the ones that do really well may just be due to luck or statistical variations. If you picked the top fund after this weeding out process and then evaluated it over the next year and reported that performance, that would be fair. Note: stock funds actually use this trick. If a stock fund does poorly at the start it is likely to be terminated while good ones will not be. 56