Introduction to Machine Learning and Cross-Validation

Transcription

1 Introduction to Machine Learning and Cross-Validation Jonathan Hersh 1 February 27, 2019 J.Hersh (Chapman ) Intro & CV February 27, / 29

2 Plan 1 Introduction 2 Preliminary Terminology 3 Bias-Variance Trade-off 4 Cross-Validation 5 Conclusion J.Hersh (Chapman ) Intro & CV February 27, / 29

3 Machine learning versus econometrics Machine Learning Developed to solve problems in computer science Econometrics Developed to solve problems in economics J.Hersh (Chapman ) Intro & CV February 27, / 29

4 Machine learning versus econometrics Machine Learning Developed to solve problems in computer science Prediction/classification Econometrics Developed to solve problems in economics Explicitly testing a theory J.Hersh (Chapman ) Intro & CV February 27, / 29

5 Machine learning versus econometrics Machine Learning Developed to solve problems in computer science Prediction/classification Want: goodness of fit Econometrics Developed to solve problems in economics Explicitly testing a theory Statistical significance more important than model fit J.Hersh (Chapman ) Intro & CV February 27, / 29

6 Machine learning versus econometrics Machine Learning Developed to solve problems in computer science Prediction/classification Want: goodness of fit Huge datasets (many terabytes), large # variables (1000s) Econometrics Developed to solve problems in economics Explicitly testing a theory Statistical significance more important than model fit Small datasets, few variables J.Hersh (Chapman ) Intro & CV February 27, / 29

7 Machine learning versus econometrics Machine Learning Developed to solve problems in computer science Prediction/classification Want: goodness of fit Huge datasets (many terabytes), large # variables (1000s) Whatever works Econometrics Developed to solve problems in economics Explicitly testing a theory Statistical significance more important than model fit Small datasets, few variables It works in practice, but what about in theory? J.Hersh (Chapman ) Intro & CV February 27, / 29

8 Machine learning versus econometrics Machine Learning Developed to solve problems in computer science Prediction/classification Want: goodness of fit Huge datasets (many terabytes), large # variables (1000s) Whatever works Econometrics Developed to solve problems in economics Explicitly testing a theory Statistical significance more important than model fit Small datasets, few variables It works in practice, but what about in theory? Can we utilize some of this machinery to solve problems in development economics? J.Hersh (Chapman ) Intro & CV February 27, / 29

9 Applications of Machine Learning in economics (due to Sendhil Mullainathan) 1. New data 2. Predictions for policy 3. Better econometrics See Machine learning: an applied econometric approach, JEP See Athey Beyond Prediction: Using Big Data for Policy Problems (2017) Science See Hersh, Jonathan, and Matthew Harding. "Big Data in economics." IZA World of Labor (2018) J.Hersh (Chapman ) Intro & CV February 27, / 29

10 This course Primer in statistical learning theory, which grew out of statistics How does this differ from ML? Machine learning places more emphasis on large scale applications and prediction accuracy. Statistical learning covers J.Hersh (Chapman ) Intro & CV February 27, / 29

11 This course Primer in statistical learning theory, which grew out of statistics How does this differ from ML? Machine learning places more emphasis on large scale applications and prediction accuracy. Statistical learning covers There is much overlap and cross-fertilization J.Hersh (Chapman ) Intro & CV February 27, / 29

12 This course Primer in statistical learning theory, which grew out of statistics How does this differ from ML? Machine learning places more emphasis on large scale applications and prediction accuracy. Statistical learning covers There is much overlap and cross-fertilization Very little coding, but example code provided: jonathan-hersh.com/machinelearningdev J.Hersh (Chapman ) Intro & CV February 27, / 29

13 Topics covered 1. Cross-validation [Chapter 2] J.Hersh (Chapman ) Intro & CV February 27, / 29

14 Topics covered 1. Cross-validation [Chapter 2] 2. Shrinkage methods (Ridge and LASSO) [Chapter 6] J.Hersh (Chapman ) Intro & CV February 27, / 29

15 Topics covered 1. Cross-validation [Chapter 2] 2. Shrinkage methods (Ridge and LASSO) [Chapter 6] 3. Classification [Chapter 4, APM Chapter 11-12] J.Hersh (Chapman ) Intro & CV February 27, / 29

16 Topics covered 1. Cross-validation [Chapter 2] 2. Shrinkage methods (Ridge and LASSO) [Chapter 6] 3. Classification [Chapter 4, APM Chapter 11-12] 4. Tree-based methods (Decision trees, bagging, random forest boosting) [Chapter 8] J.Hersh (Chapman ) Intro & CV February 27, / 29

17 Topics covered 1. Cross-validation [Chapter 2] 2. Shrinkage methods (Ridge and LASSO) [Chapter 6] 3. Classification [Chapter 4, APM Chapter 11-12] 4. Tree-based methods (Decision trees, bagging, random forest boosting) [Chapter 8] 5. Unsupervised learning (PCA, k-means clustering, hierarchical clustering) [Chapter 10] J.Hersh (Chapman ) Intro & CV February 27, / 29

18 Topics covered 1. Cross-validation [Chapter 2] 2. Shrinkage methods (Ridge and LASSO) [Chapter 6] 3. Classification [Chapter 4, APM Chapter 11-12] 4. Tree-based methods (Decision trees, bagging, random forest boosting) [Chapter 8] 5. Unsupervised learning (PCA, k-means clustering, hierarchical clustering) [Chapter 10] 6. Caret Automated Machine Learning [APM] J.Hersh (Chapman ) Intro & CV February 27, / 29

19 1 Introduction 2 Preliminary Terminology 3 Bias-Variance Trade-off 4 Cross-Validation 5 Conclusion J.Hersh (Chapman ) Intro & CV February 27, / 29

20 Plan 1 Introduction 2 Preliminary Terminology 3 Bias-Variance Trade-off 4 Cross-Validation 5 Conclusion J.Hersh (Chapman ) Intro & CV February 27, / 29

21 Supervised vs. unsupervised learning Def: Supervised learning for every x i we also observe a response y i J.Hersh (Chapman ) Intro & CV February 27, / 29

22 Supervised vs. unsupervised learning Def: Supervised learning for every x i we also observe a response y i Ex: Estimating housing values by OLS or random forest; J.Hersh (Chapman ) Intro & CV February 27, / 29

23 Supervised vs. unsupervised learning Def: Supervised learning for every x i we also observe a response y i Ex: Estimating housing values by OLS or random forest; Def: Unsupervised learning for each observation we only observe x i, but do not observe y i J.Hersh (Chapman ) Intro & CV February 27, / 29

24 Supervised vs. unsupervised learning Def: Supervised learning for every x i we also observe a response y i Ex: Estimating housing values by OLS or random forest; Def: Unsupervised learning for each observation we only observe x i, but do not observe y i Ex: Clustering customers into segments; using principle component analysis for dimension reduction J.Hersh (Chapman ) Intro & CV February 27, / 29

25 Test, Training, and Validation Set Training set: (observation-wise) subset of data used to develop models J.Hersh (Chapman ) Intro & CV February 27, / 29

26 Test, Training, and Validation Set Training set: (observation-wise) subset of data used to develop models Validation set: subset of data used during intermediate stages to tune model parameters J.Hersh (Chapman ) Intro & CV February 27, / 29

27 Test, Training, and Validation Set Training set: (observation-wise) subset of data used to develop models Validation set: subset of data used during intermediate stages to tune model parameters Test set: subset of data (used sparingly) to approximate out of sample fit J.Hersh (Chapman ) Intro & CV February 27, / 29

28 Assessing model accuracy Mean squared error measures how well model predictions match observed data MSE (ˆf (x)) = 1 N N y i i=1 }{{} data 2 ˆf (x i ) }{{} model J.Hersh (Chapman ) Intro & CV February 27, / 29

29 Assessing model accuracy Mean squared error measures how well model predictions match observed data MSE (ˆf (x)) = 1 N N y i i=1 }{{} data 2 ˆf (x i ) }{{} model Training MSE vs Test MSE: good in-sample fit (low training MSE) can often obscure poor out of sample fit (high test MSE) J.Hersh (Chapman ) Intro & CV February 27, / 29

30 Plan 1 Introduction 2 Preliminary Terminology 3 Bias-Variance Trade-off 4 Cross-Validation 5 Conclusion J.Hersh (Chapman ) Intro & CV February 27, / 29

31 Quick example on out-of-sample fit Let s compare three estimators to see how estimator complexity affects out of sample fit 1. f 1 = linear regression (in orange) 2. f 2 = third order polynomial (in black) 3. f 3 = very flexible smoothing spline (in green) J.Hersh (Chapman ) Intro & CV February 27, / 29

32 Case 1: True f (x) slightly complicated J.Hersh (Chapman ) Intro & CV February 27, / 29

33 Case 2: True f (x) not complicated at all J.Hersh (Chapman ) Intro & CV February 27, / 29

34 Case 2: True f (x) not complicated at all J.Hersh (Chapman ) Intro & CV February 27, / 29

35 Case 3: True f (x) very complicated J.Hersh (Chapman ) Intro & CV February 27, / 29

36 How to select right model complexity? J.Hersh (Chapman ) Intro & CV February 27, / 29

37 Generalizing this problem: Bias-Variance tradeoff J.Hersh (Chapman ) Intro & CV February 27, / 29

38 Bias Variance Tradeoff in Math Prediction Error: E y i }{{} data 2 ˆf (x i ) }{{} model

39 Bias Variance Tradeoff in Math [ ( ) ] 2 E y ˆf Prediction Error: E y i }{{} data [ ] = E y 2 + ˆf 2 2yˆf 2 ˆf (x i ) }{{} model (expanding terms)

40 Bias Variance Tradeoff in Math [ ( ) ] 2 E y ˆf Prediction Error: E y i }{{} data [ ] = E y 2 + ˆf 2 2yˆf = E [y 2] }{{} Var(y)+E[y] 2 2 ˆf (x i ) }{{} model (expanding terms) + E [ˆf 2] [ ] E 2yˆf }{{} Var(ˆf )+E[ˆf ] 2

41 Bias Variance Tradeoff in Math [ ( ) ] 2 E y ˆf Prediction Error: E y i }{{} data [ ] = E y 2 + ˆf 2 2yˆf = E [y 2] }{{} Var(y)+E[y] 2 2 ˆf (x i ) }{{} model (expanding terms) + E [ˆf 2] [ ] E 2yˆf }{{} Var(ˆf )+E[ˆf ] 2 = Var (y) + ) ] E [y] 2 2 [ ] + Var (ˆf + E [ˆf E 2yˆf }{{} y f (x)+ε, E[ε]=0 (def n var)

42 Bias Variance Tradeoff in Math [ ( ) ] 2 E y ˆf Prediction Error: E y i }{{} data [ ] = E y 2 + ˆf 2 2yˆf = E [y 2] }{{} Var(y)+E[y] 2 2 ˆf (x i ) }{{} model (expanding terms) + E [ˆf 2] [ ] E 2yˆf }{{} Var(ˆf )+E[ˆf ] 2 = Var (y) + ) ] E [y] 2 2 [ ] + Var (ˆf + E [ˆf E 2yˆf }{{} y f (x)+ε, E[ε]=0 (def n var) = Var (y) + Var ) ( ] 2 [ ] ) (ˆf + E [ˆf E 2yˆf + f 2 (def n y)

43 Bias Variance Tradeoff in Math ) ( ]) 2 = Var (y) + Var (ˆf + f E [ˆf }{{} =E[((y E[y]) 2 ]=E[((y f ) 2 ]=E[ε 2 ]=Var(ε)=σε 2 J.Hersh (Chapman ) Intro & CV February 27, / 29

44 Bias Variance Tradeoff in Math ) ( ]) 2 = Var (y) + Var (ˆf + f E [ˆf }{{} =E[((y E[y]) 2 ]=E[((y f ) 2 ]=E[ε 2 ]=Var(ε)=σε 2 Prediction Error [( )] E y i ˆf = σε 2 }{{} irreducible error ) + Var (ˆf }{{} variance ([ + ]]) 2 f E [ˆf } {{ } bias J.Hersh (Chapman ) Intro & CV February 27, / 29

45 Bias Variance Tradeoff in Math ) ( ]) 2 = Var (y) + Var (ˆf + f E [ˆf }{{} =E[((y E[y]) 2 ]=E[((y f ) 2 ]=E[ε 2 ]=Var(ε)=σε 2 Prediction Error [( )] E y i ˆf = σε 2 }{{} irreducible error ] Bias is minimized when f = E [ˆf ) + Var (ˆf }{{} variance ([ + ]]) 2 f E [ˆf } {{ } bias But total error (variance + bias) may be minimized by some other ˆf J.Hersh (Chapman ) Intro & CV February 27, / 29

46 Bias Variance Tradeoff in Math ) ( ]) 2 = Var (y) + Var (ˆf + f E [ˆf }{{} =E[((y E[y]) 2 ]=E[((y f ) 2 ]=E[ε 2 ]=Var(ε)=σε 2 Prediction Error [( )] E y i ˆf = σε 2 }{{} irreducible error ] Bias is minimized when f = E [ˆf ) + Var (ˆf }{{} variance ([ + ]]) 2 f E [ˆf } {{ } bias But total error (variance + bias) may be minimized by some other ˆf ˆf (x) with smaller variance fewer variables, smaller magnitude coefficients J.Hersh (Chapman ) Intro & CV February 27, / 29

47 Plan 1 Introduction 2 Preliminary Terminology 3 Bias-Variance Trade-off 4 Cross-Validation 5 Conclusion J.Hersh (Chapman ) Intro & CV February 27, / 29

48 Cross-Validation Cross-validation is a tool to approximate out of sample fit J.Hersh (Chapman ) Intro & CV February 27, / 29

49 Cross-Validation Cross-validation is a tool to approximate out of sample fit In machine learning, many models have parameters that must be tuned We adjust these parameters using using cross-validation J.Hersh (Chapman ) Intro & CV February 27, / 29

50 Cross-Validation Cross-validation is a tool to approximate out of sample fit In machine learning, many models have parameters that must be tuned We adjust these parameters using using cross-validation Also useful to select between large classes of models e.g random forest vs lasso J.Hersh (Chapman ) Intro & CV February 27, / 29

51 K-fold Cross-Validation (CV) K-fold CV Algorithm 1. Randomly divide the data into K equal sized parts or folds. J.Hersh (Chapman ) Intro & CV February 27, / 29

52 K-fold Cross-Validation (CV) K-fold CV Algorithm 1. Randomly divide the data into K equal sized parts or folds. 2. Leave out part k, fit the model to the other K 1 parts. J.Hersh (Chapman ) Intro & CV February 27, / 29

53 K-fold Cross-Validation (CV) K-fold CV Algorithm 1. Randomly divide the data into K equal sized parts or folds. 2. Leave out part k, fit the model to the other K 1 parts. 3. Use fitted model to obtain predictions for left-out k-th part J.Hersh (Chapman ) Intro & CV February 27, / 29

54 K-fold Cross-Validation (CV) K-fold CV Algorithm 1. Randomly divide the data into K equal sized parts or folds. 2. Leave out part k, fit the model to the other K 1 parts. 3. Use fitted model to obtain predictions for left-out k-th part 4. Repeat until k = 1,..., K and combine results J.Hersh (Chapman ) Intro & CV February 27, / 29

55 K-Fold CV Example 2

56 Details of K-Fold CV Let n k be the number of test observations in fold k, where n k = N/K Cross-Validation Error for fold k: CV {k} = K k=1 n k N MSE k where MSE k = i C k (y i ŷ) 2 /n k is the mean squared error of fold k J.Hersh (Chapman ) Intro & CV February 27, / 29

57 Details of K-Fold CV Let n k be the number of test observations in fold k, where n k = N/K Cross-Validation Error for fold k: CV {k} = K k=1 n k N MSE k where MSE k = i C k (y i ŷ) 2 /n k is the mean squared error of fold k Setting k = N is referred to as leave-one out cross-validation (LOOCV) J.Hersh (Chapman ) Intro & CV February 27, / 29

58 Classical frequentist model selection Akaike information criterion (AIC) AIC = 2 N loglik + 2 d N where d is the number of parameters in our model Bayesian information criterion (BIC) BIC = 2 loglik + (log N) d J.Hersh (Chapman ) Intro & CV February 27, / 29

59 Classical frequentist model selection Akaike information criterion (AIC) AIC = 2 N loglik + 2 d N where d is the number of parameters in our model Bayesian information criterion (BIC) BIC = 2 loglik + (log N) d Both penalize models with more parameters in a somewhat arbitrary fashion This usually helps with model selection, but still does not answer the important question of model assessment J.Hersh (Chapman ) Intro & CV February 27, / 29

60 What is the Optimal Number of Folds, K? Do you want big or a small training folds? J.Hersh (Chapman ) Intro & CV February 27, / 29

61 What is the Optimal Number of Folds, K? Do you want big or a small training folds? Because training set is only (K 1)/K as big as the full dataset, the estimates of the prediction error will be biased upward. Bias is minimized when K = N (LOOCV) But LOOCV has higher variance! J.Hersh (Chapman ) Intro & CV February 27, / 29

62 What is the Optimal Number of Folds, K? Do you want big or a small training folds? Because training set is only (K 1)/K as big as the full dataset, the estimates of the prediction error will be biased upward. Bias is minimized when K = N (LOOCV) But LOOCV has higher variance! No clear statistical rules for how to set k Convention is to set K = 5 or 10 in practice is a good trade-off between bias and variance for most problems J.Hersh (Chapman ) Intro & CV February 27, / 29

63 Plan 1 Introduction 2 Preliminary Terminology 3 Bias-Variance Trade-off 4 Cross-Validation 5 Conclusion J.Hersh (Chapman ) Intro & CV February 27, / 29

64 Conclusion Econometric models are at risk for overfitting But what we want are theories that extend beyond the dataset we have on our computer Cross-validation is a key tool that allows us to adjust models so that they more closely match reality J.Hersh (Chapman ) Intro & CV February 27, / 29