More Methodology; Nearest-Neighbor Classifiers Sec 4.7

Review: Properties of DTs Axis orthagonal, hyperrectangular, piecewise- constant models Categorical labels Non-metric

Separation of train & test Fundamental principle (1st amendment of ML): Don’t evaluate accuracy (performance) of your classifier (learning system) on the same data used to train it!

Holdout data Usual to “hold out” a separate set of data for testing; not used to train classifier A.k.a., test set, holdout set, evaluation set, etc. E.g., is training set accuracy is test set (or generalization) accuracy

Gotchas... What if you’re unlucky when you split data into train/test? E.g., all train data are class A and all test are class B? No “red” things show up in training data Best answer: stratification Try to make sure class (+feature) ratios are same in train/test sets (and same as original data) Why does this work? Almost as good: randomization Shuffle data randomly before split Why does this work?

More gotchas... What if your data set is small? Might not be able to get perfect stratification Can’t get really representative accuracy from any single train/test split A: cross-validation for (i=0;i<k;++i) { [Xtrain,Ytrain,Xtest,Ytest]= splitData(X,Y,N/k,i); model[i]=train(Xtrain,Ytrain); cvAccs[i]=measureAcc(model[i],Xtest,Ytest); } avgAcc=mean(cvAccs); stdAcc=stddev(cvAccs);

CV in pix [X;Y][X;Y] Original data [X’;Y’] Random shuffle k -way partition [X1’ Y1’] [X2’ Y2’] [Xk’ Yk’]... k train/ test sets k accuracies 53.7%85.1%73.2%

But is it really learning? Now we know how well our models are performing But are they really learning? Maybe any classifier would do as well E.g., a default classifier (pick the most likely class) or a random classifier How can we tell if the model is learning anything? Go back to first definitions What does it mean to learn something?

The learning curve Train on successively larger fractions of data Watch how accuracy (performance) changes

Measuring variance Cross validation helps you get better estimate of accuracy for small data Randomization (shuffling the data) helps guard against poor splits/ordering of the data Learning curves help assess learning rate/asymptotic accuracy Still one big missing component: variance Definition: Variance of a classifier is the fraction of error due to the specific data set it’s trained on

Measuring variance Variance tells you how much you expect your classifier/performance to change when you train it on a new (but similar) data set E.g., take 5 samplings of a data source; train/test 5 classifiers Accuracies: 74.2, 90.3, 58.1, 80.6, 90.3 Mean accuracy: 78.7% Std dev of acc: 13.4% Variance is usually a function of both classifier and data source High variance classifiers are very susceptible to small changes in data

Putting it all together Suppose you want to measure the expected accuracy of your classifier, assess learning rate, and measure variance all at the same time? for (i=0;i<10;++i) { // variance reps shuffle data do 10-way CV partition of data for each train/test partition { // xval for (pct=0.1;pct+=0.1;pct<=0.9) { // LC Subsample pct fraction of training set train on subsample, test on test set } avg across all folds of CV partition generate learning curve for this partition } get mean and std across all curves

Putting it all together “hepatitis” data

5 minutes of math... Decision trees are non-metric Don’t know anything about relations between instances, except sets induced by feature splits Often, we have well-defined distances between points Idea of distance encapsulated by a metric

5 minutes of math... Definition: a metric function is a function that obeys the following properties: Identity: Symmetry: Triangle inequality:

5 minutes of math... Examples: Euclidean distance * Note: omitting the square root still yields a metric and usually won’t change our results

5 minutes of math... Examples: Manhattan (taxicab) distance Distance travelled along a grid between two points No diagonals allowed

5 minutes of math... Examples: What if some attribute is categorical?

5 minutes of math... Examples: What if some attribute is categorical? Typical answer is 0/1 distance: For each attribute, add 1 if the instances differ in that attribute, else 0

Distances in classification Nearest neighbor: find the nearest instance to the query point in feature space, return the class of that instance Simplest possible distance-based classifier With more notation:

Properties of NN Training time of NN? Classification time? Geometry of model?

Properties of NN Training time of NN? Classification time? Geometry of model?

Properties of NN Training time of NN? Classification time? Geometry of model?

NN miscellaney Slight generalization: k -Nearest neighbors ( k - NN) Find k training instances closest to query point Vote among them for label Q: How does this affect system? Gotcha: unscaled dimensions What happens if one axis is measured in microns and one in lightyears? Usual trick is to scale each axis to [-1,1] range