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Giovanni Seni is currently a Senior Data Scientist with Intuit where he leads the Applied Data Sciences team. As an active data mining practitioner in Silicon Valley, he has over 15 years R&D experience in statistical pattern recognition and data mining applications. He has been a member of the technical staff at large technology companies, and a contributor at smaller organizations. He holds five US patents and has published over twenty conference and journal articles. His book with John Elder, “Ensemble Methods in Data Mining – Improving accuracy through combining predictions”, was published in February 2010 by Morgan & Claypool. Giovanni is also an adjunct faculty at the Computer Engineering Department of Santa Clara University, where he teaches an Introduction to Pattern Recognition and Data Mining class.

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- 1. How to Create Predictive Models in R using Ensembles Giovanni Seni, Ph.D. Intuit @IntuitInc Giovanni_Seni@intuit.com Santa Clara University GSeni@scu.edu Strata - Hadoop World, New York October 28, 2013
- 2. Reference © 2013 G.Seni 2013 Strata Conference + Hadoop World 2
- 3. Overview • Motivation, In a Nutshell & Timeline • Predictive Learning & Decision Trees • Ensemble Methods - Diversity & Importance Sampling – Bagging – Random Forest – Ada Boost – Gradient Boosting – Rule Ensembles • Summary © 2013 G.Seni 2013 Strata Conference + Hadoop World 3
- 4. Motivation Volume 9 Issue 2 © 2013 G.Seni 2013 Strata Conference + Hadoop World 4
- 5. Motivation (2) “1′st Place Algorithm Description: … 4. Classification: Ensemble classification methods are used to combine multiple classifiers. Two separate Random Forest ensembles are created based on the shadow index (one for the shadow-covered area and one for the shadow-free area). The random forest “Out of Bag” error is used to automatically evaluate features according to their impact, resulting in 45 features selected for the shadow-free and 55 for the shadow-covered part.” © 2013 G.Seni 2013 Strata Conference + Hadoop World 5
- 6. Motivation (3) • “What are the best of the best techniques at winning Kaggle competitions? – Ensembles of Decisions Trees – Deep Learning account for 90% of top 3 winners!” Jeremy Howard, Chief Scientist of Kaggle KDD 2013 ⇒ Key common characteristics: – Resistance to overfitting – Universal approximations © 2013 G.Seni 2013 Strata Conference + Hadoop World 6
- 7. Ensemble Methods in a Nutshell • “Algorithmic” statistical procedure • Based on combining the fitted values from a number of fitting attempts • Loosely related to: – Iterative procedures – Bootstrap procedures • Original idea: a “weak” procedure can be strengthened if it can operate “by committee” – e.g., combining low-bias/high-variance procedures • Accompanied by interpretation methodology © 2013 G.Seni 2013 Strata Conference + Hadoop World 7
- 8. Timeline • CART (Breiman, Friedman, Stone, Olshen, 1983) • Bagging (Breiman, 1996) – Random Forest (Ho, 1995; Breiman 2001) • AdaBoost (Freund, Schapire, 1997) • Boosting – a statistical view (Friedman et. al., 2000) – Gradient Boosting (Friedman, 2001) – Stochastic Gradient Boosting (Friedman, 1999) • Importance Sampling Learning Ensembles (ISLE) (Friedman, Popescu, 2003) © 2013 G.Seni 2013 Strata Conference + Hadoop World 8
- 9. Timeline (2) • Regularization – variance control techniques: – Lasso (Tibshirani, 1996) – LARS (Efron, 2004) – Elastic Net (Zou, Hastie, 2005) – GLMs via Coordinate Descent (Friedman, Hastie, Tibshirani, 2008) • Rule Ensembles (Friedman, Popescu, 2008) © 2013 G.Seni 2013 Strata Conference + Hadoop World 9
- 10. Overview • Motivation, In a Nutshell & Timeline Ø Predictive Learning & Decision Trees • Ensemble Methods • Summary © 2013 G.Seni 2013 Strata Conference + Hadoop World 10
- 11. Predictive Learning Procedure Summary N N • Given "training" data D = { yi , xi1 , xi 2 ,, xin }1 = { yi , x i }1 – D is a random sample from some unknown (joint) distribution • Build a functional model y = F ( x1 , x2 ,, xn ) = F ( x ) – Offers adequate and interpretable description of how the inputs affect the outputs – Parsimony is an important criterion: simpler models are preferred for the sake of scientific insight into the x - y relationship • Need to specify: < model, score criterion, search strategy > © 2013 G.Seni 2013 Strata Conference + Hadoop World 11
- 12. Predictive Learning Procedure Summary (2) • Model: underlying functional form sought from data F (x) = F (x; a) ∈ ℱ family of functions indexed by a • Score criterion: judges (lack of) quality of fitted model – Loss function L( y, F ): penalizes individual errors in prediction – Risk R(a) = E y ,x L( y, F (x; a)) : the expected loss over all predictions • Search Strategy: minimization procedure of score criterion a* = arg min R(a) a © 2013 G.Seni 2013 Strata Conference + Hadoop World 12
- 13. Predictive Learning Procedure Summary (3) • “Surrogate” Score criterion: N – Training data: { yi , x i }1 ~ p( x, y ) * – p ( x, y ) unknown ⇒ a unknown ⇒ Use approximation: Empirical Risk 1 • R (a) = N ∑ L( y, F (xi ; a)) N i =1 • If not N >> n , © 2013 G.Seni ⇒ a = arg min R(a) a R(a) >> R(a* ) 2013 Strata Conference + Hadoop World 13
- 14. Predictive Learning Example • A simple data set Attribute-1 Attribute-2 Class ( x1 ) ( x2 ) 1.0 2.0 blue 2.0 1.0 green … … … 4.5 3.5 x2 ? (y) • What is the class of new point x1 ? • Many approaches… no method is universally better; try several / use committee © 2013 G.Seni 2013 Strata Conference + Hadoop World 14
- 15. Predictive Learning Example (2) • Ordinary Linear Regression (OLR) x2 x1 n – Model: F(x) = a0 + ∑ a j x j j=1 ; ⎧ F (x) ≥ 0 ⎨ ⎩else ⇒ Not flexible enough © 2013 G.Seni 2013 Strata Conference + Hadoop World 15
- 16. Decision Trees Overview x2 R2 x1 ≥ 5 R1 x2 ≥ 3 3 R4 x1 ≥ 2 R3 2 x1 5 M ˆ ˆ ˆ • Model: y = T (x ) = ∑ cm I R (x ) m =1 m M {Rm }m=1 = Sub-regions of input variable space where I R (x) = 1 if x ∈ R , 0 otherwise © 2013 G.Seni 2013 Strata Conference + Hadoop World 16
- 17. Decision Trees Overview (2) • Score criterion: – Classification – "0-1 loss" ⇒ misclassification error (or surrogate) N M ˆ { cˆm, Rm } = argmin 1 M cm ,Rm 1 TM ={ } ∑I (y ≠ T i M (x i )) i=1 2 ˆ ˆ – Regression – least squares – i.e., L( y , y ) = ( y − y ) M ˆ ˆ { cm, Rm } = argmin 1 M N ∑( y − T i TM ={cm ,Rm }1 i=1 M (x i )) R(TM ) 2 • Search: Find T = arg min T R(T ) – i.e., find best regions Rm and constants cm © 2013 G.Seni 2013 Strata Conference + Hadoop World 17
- 18. Decision Trees Overview (3) • Join optimization with respect to Rm and cm simultaneously is very difficult ⇒ use a greedy iterative procedure R0 R4 R1 R5 R6 R2 R3 • • j 1 , s1 R0 j 2 , s2 • • j 1 , s1 R0 j 2 , s2 R1 • R0 • R1 • R3 • j 1 , s1 • R4 j 3 , s3 j 2 , s2 • j 1 , s1 R0 • R3 •• • R4 R5 R7 2013 Strata Conference + Hadoop World j 4 , s4 R6 • © 2013 G.Seni j 3 , s3 R2 R1 R2 • • R8 18
- 19. Decision Trees What is the “right” size of a model? y y y ο ο ο ο ο ο ο ο ο ο ο ο ο ο ο ⇒ ο ο ο c1 ο ο ο x ο ο ο ο ο ο ο ο ο c2 ο vs ο ο c1 ο c2 ο ο ο ο ο ο ο ο ο ο c3 ο ο ο ο x x • Dilemma – If model (# of splits) is too small, then approximation is too crude (bias) ⇒ increased errors – If model is too large, then it fits the training data too closely (overfitting, increased variance) ⇒ increased errors © 2013 G.Seni 2013 Strata Conference + Hadoop World 19
- 20. Decision Trees What is the “right” size of a model? (2) High Bias Low Bias Low Variance Prediction Error High Variance Test Sample Training Sample Low M* High Model Complexity – Right sized tree, M * when test error is at a minimum , – Error on the training is not a useful estimator! • If test set is not available, need alternative method © 2013 G.Seni 2013 Strata Conference + Hadoop World 20
- 21. Decision Trees Pruning to obtain “right” size • Two strategies – Prepruning - stop growing a branch when information becomes unreliable • #(Rm) – i.e., number of data points, too small ⇒ same bound everywhere in the tree • Next split not worthwhile ⇒ Not sufficient condition – Postpruning - take a fully-grown tree and discard unreliable parts (i.e., not supported by test data) • C4.5: pessimistic pruning • CART: cost-complexity pruning © 2013 G.Seni (more statistically grounded) 2013 Strata Conference + Hadoop World 21
- 22. Decision Trees 1.0 Hands-on Exercise Start Rstudio • 0.8 • Navigate to directory: example.1.LinearBoundary Load and run “fitModel_CART.R” • If curious, also see “gen2DdataLinear.R” • After boosting discussion, load and run “fitModel_GBM.R 0.0 0.2 0.4 x2 Set working directory: use setwd() or with GUI • 0.6 • 0.0 0.2 0.4 0.6 0.8 1.0 x1 © 2013 G.Seni 2013 Strata Conference + Hadoop World 22
- 23. Decision Trees Key Features • Ability to deal with irrelevant inputs – i.e., automatic variable subset selection – Measure anything you can measure – Score provided for selected variables ("importance") • No data preprocessing needed - Naturally handle all types of variables • numeric, binary, categorical - Invariant under monotone transformations: x j = g j (x j ) • • © 2013 G.Seni Variable scales are irrelevant Immune to bad x j −distributions (e.g., outliers) 2013 Strata Conference + Hadoop World 23
- 24. Decision Trees Key Features (2) • Computational scalability – Relatively fast: O(nN log N ) • Missing value tolerant - Moderate loss of accuracy due to missing values - Handling via "surrogate" splits • "Off-the-shelf" procedure - Few tunable parameters • Interpretable model representation - Binary tree graphic © 2013 G.Seni 2013 Strata Conference + Hadoop World 24
- 25. Decision Trees Limitations • Discontinuous piecewise constant model F (x) x – In order to have many splits you need to have a lot of data • In high-dimensions, you often run out of data after a few splits – Also note error is bigger near region boundaries © 2013 G.Seni 2013 Strata Conference + Hadoop World 25
- 26. Decision Trees Limitations (2) • Not good for low interaction F * (x ) n * – e.g., F (x ) = ao + ∑ a j x j is worst function for trees j =1 n = ∑ f j* (x j ) (no interaction, additive) j =1 – In order for xl to enter model, must split on it • Path from root to node is a product of indicators • Not good for F * (x ) that has dependence on many variables - Each split reduces training data for subsequent splits (data fragmentation) © 2013 G.Seni 2013 Strata Conference + Hadoop World 26
- 27. Decision Trees Limitations (3) • High variance caused by greedy search strategy (local optima) – Errors in upper splits are propagated down to affect all splits below it ⇒ Small changes in data (sampling fluctuations) can cause big changes in tree - Very deep trees might be questionable - Pruning is important • What to do next? – Live with problems – Use other methods (when possible) – Fix-up trees: use ensembles © 2013 G.Seni 2013 Strata Conference + Hadoop World 27
- 28. Overview • In a Nutshell & Timeline • Predictive Learning & Decision Trees Ø Ensemble Methods – In a Nutshell, Diversity & Importance Sampling – Generic Ensemble Generation – Bagging, RF, AdaBoost, Boosting, Rule Ensembles • Summary © 2013 G.Seni 2013 Strata Conference + Hadoop World 28
- 29. Ensemble Methods In a Nutshell M • Model: F (x) = c0 + ∑m=1 cmTm (x) M – { m (x)}1 : “basis” functions (or “base learners”) T – i.e., linear model in a (very) high dimensional space of derived variables • Learner characterization: Tm (x) = T (x; p m ) – p m : a specific set of joint parameter values – e.g., split definitions at internal nodes and predictions at terminal nodes – {T (x; p)}p∈P : function class – i.e., set of all base learners of specified family © 2013 G.Seni 2013 Strata Conference + Hadoop World 29
- 30. Ensemble Methods In a Nutshell (2) • Learning: two-step process; approximate solution to N M M {cm , p m }o = arg min ∑ L yi , c0 + ∑ cmT (x;p m ) M {cm , p m }o i=1 ( m=1 ) M – Step 1: Choose points {p m }1 M • i.e., select {Tm (x)}1 ⊂ {T (x; p)}p∈P M – Step 2: Determine weights {cm }0 • e.g., via regularized LR © 2013 G.Seni 2013 Strata Conference + Hadoop World 30
- 31. Ensemble Methods Importance Sampling (Friedman, 2003) • How to judiciously choose the “basis” functions (i.e., {pm }1M )? M • Goal: find “good” {pm }1 so that M M F (x;{p m }1 , {cm }1 ) ≅ F * (x ) • Connection with numerical integration: – ∫ Ρ M I (p) ∂p ≈ ∑m =1 w m I (p m ) vs. © 2013 G.Seni 2013 Strata Conference + Hadoop World Accuracy improves when we choose more points from this region… 31
- 32. Importance Sampling Numerical Integration via Monte Carlo Methods M • r (p) = sampling pdf of p ∈ P -- i.e, {p m ~ r (p)}1 – Simple approach: r (p m ) iid -- i.e., uniform – In our problem: inversely related to p m’s “risk” • i.e., T (x; p m ) has high error ⇒ lack of relevance of p m ⇒ low r (pm ) • “Quasi” Monte Carlo: – with/out knowledge of the other points that will be used • i.e., single point vs. group importance – Sequential approximation: p’s relevance judged in the context of the (fixed) previously selected points © 2013 G.Seni 2013 Strata Conference + Hadoop World 32
- 33. Ensemble Methods Importance Sampling – Characterization of • Let p∗ = arg minp Risk(p) Narrow r (p) Broad r (p) M T • Ensemble { (x; p m )}1 of “strong” base learners - i.e., all with Risk (p m ) ≈ Risk (p∗ ) • Diverse ensemble - i.e., predictions are not highly correlated with each other • T (x; p m ) yield similar highly correlated ’s predictions ⇒ unexceptional performance • However, many “weak” base learners - i.e., Risk (p m ) >> Risk (p ∗ ) ⇒ poor performance © 2013 G.Seni 2013 Strata Conference + Hadoop World 33
- 34. Ensemble Methods Approximate Process of Drawing from • Heuristic sampling strategy: sampling around p by iteratively applying small perturbations to existing problem structure ∗ – Generating ensemble members Tm (x) = T (x; p m ) For m = 1 to M { pm = PERTURBm { minp Ε xy L( y, T (x; p) )} arg } ⋅ – PERTURB {} is a (random) modification of any of • Data distribution - e.g., by re-weighting the observations • Loss function - e.g., by modifying its argument • Search algorithm (used to find minp) – Width of r (p ) is controlled by degree of perturbation © 2013 G.Seni 2013 Strata Conference + Hadoop World 34
- 35. Generic Ensemble Generation Step 1: Choose Base Learners p! ! ! ! • Forward Stagewise Fitting Procedure: 𝐹0 (x) = 0 For 𝑚 = 1 to 𝑀 { // Fit a single base learner p Modification of data distribution 𝑚 = argmin . p 𝐿0𝑦 𝑖 , 𝐹 𝑚 −1 + 𝑇(x 𝑖 ; p)8 𝑖∈𝑆 𝑚 ( 𝜂 ) // Update additive expansion 𝑇 𝑚 ( 𝑥 ) = 𝑇0x; p 𝒎 8 𝐹 𝑚 (x) = 𝐹 𝑚 −1 (x) + 𝜐 ∙ 𝑇 𝑚 (x) } write { 𝑇 𝑚 (x)}1𝑀 – Algorithm control: L, η , υ Modification of loss function (“sequential” approximation) • Sm (η ) : random sub-sample of size η ≤ N ⇒ impacts ensemble "diversity" m −1 • Fm−1 (x) = υ ⋅ ∑k =1Tk (x) : “memory” function (0 ≤ υ ≤ 1 ) © 2013 G.Seni 2013 Strata Conference + Hadoop World 35
- 36. Generic Ensemble Generation Step 2: Choose Coefficients c! ! !! M M • Given {Tm (x)}m=1 = {T (x; pm )}m=1 , coefficients can be obtained by a regularized linear regression N M ⎛ ⎞ {cm } = arg min ∑ L⎜ yi , c0 + ∑ cmTm (xi ) ⎟ + λ ⋅ P(c ) {cm } i =1 ⎝ m =1 ⎠ – Regularization here helps reduce bias (in addition to variance) of the model – New iterative fast algorithms for various loss/penalty combinations • “GLMs via Coordinate Descent” (2008) © 2013 G.Seni 2013 Strata Conference + Hadoop World 36
- 37. Bagging (Breiman, 1996) • Bagging = Bootstrap Aggregation ˆ • L(y, y) : as available for single tree F0 (x) = 0 For m = 1 to M { • υ = 0 ⇒ no memory p m = arg min p • η = N / 2 i m −1 i∈S m ( ) ( x i ) + T ( x i ; p )) Tm (x) = T (x; p m ) • Tm (x) ⇒ are large un-pruned trees ∑ηL(y , F Fm (x) = Fm −1 (x) + υ ⋅ Tm (x) υ } • co = 0, {cm = 1 / M }1M M i.e., not fit to the data (avg) write {Tm (x)}1 – i.e., perturbation of the data distribution only – Potential improvements? – R package: ipred © 2013 G.Seni 2013 Strata Conference + Hadoop World 37
- 38. Bagging Hands-on Exercise 1.0 • Navigate to directory: example.2.EllipticalBoundary 0.0 Load and run – fitModel_Bagging_by_hand.R -0.5 – fitModel_CART.R (optional) • If curious, also see gen2DdataNonLinear.R -1.0 x2 Set working directory: use setwd() or with GUI • 0.5 • • After class, load and run fitModel_Bagging.R -2 -1 0 1 2 x1 © 2013 G.Seni 2013 Strata Conference + Hadoop World 38
- 39. Bagging Why it helps? ˆ • Under L( y, y) = ( y − y) 2, averaging reduces variance and leaves bias unchanged • Consider “idealized” bagging (aggregate) estimator: f (x) = Ε f Z (x) – f Z fit to bootstrap data set Z = {yi , xi }1N – Z is sampled from actual population distribution (not training data) – We can write: Ε[Y − f Z (x)] = Ε[Y − f (x) + f (x) − f Z (x)] 2 2 2 = Ε Y − f ( x) + Ε f Z ( x) − f ( x) [ ] ≥ Ε[Y − f (x)] [ ] 2 2 ⇒ true population aggregation never increases mean squared error! ⇒ Bagging will often decrease MSE… © 2013 G.Seni 2013 Strata Conference + Hadoop World 39
- 40. Random Forest (Ho, 1995; Breiman, 2001) • Random Forest = Bagging + algorithm randomizing – Subset splitting As each tree is constructed… • Draw a random sample of predictors before each node is split ns = ⎣log 2 (n) + 1⎦ • Find best split as usual but selecting only from subset of predictors M ⇒ Increased diversity among {Tm (x)}1 - i.e., wider r (p) • Width (inversely) controlled by ns – Speed improvement over Bagging – R package: randomForest © 2013 G.Seni 2013 Strata Conference + Hadoop World 40
- 41. Bagging vs. Random Forest vs. ISLE 100 Target Functions Comparison (Popescu, 2005) • ISLE improvements: – Different data sampling strategy (not fixed) – Fit coefficients to data Comparative RMS Error • xxx_6_5%_P : 6 terminal nodes trees 5% samples without replacement Post-processing – i.e., using estimated “optimal” quadrature coefficients ⇒ Significantly faster to build! Bag © 2013 G.Seni RF Bag_6_5%_P RF_6_5%_P 2013 Strata Conference + Hadoop World 41
- 42. AdaBoost (Freund & Schapire, 1997) observation weights : wi( 0 ) = 1 N For m = 1 to M { a. Fit a classifier Tm (x) to training data with wi( m ) b. Compute errm = ∑ N i =1 (cm , p m ) = arg min w I ( yi ≠ Tm (x i )) ∑ N ∑ηL( y , F i m −1 (x i ) + c ⋅ T (x i ; p) ) i∈S m ( ) Tm (x) = T (x; p m ) Fm (x) = Fm −1 (x) + υ ⋅ cm ⋅ Tm (x) d. Set wi( m +1) = wi( m ) ⋅ exp[α m ⋅ I ( yi ≠ Tm (x i )] } Output sign ∑m =1α mTm (x) c, p wi( m ) c. Compue α m = log((1 − errm ) errm ) M For m = 1 to M { (m) i i =1 ( F0 (x) = 0 } M write {cm , Tm (x)}1 ) – We need to show p m = arg min (⋅) is equivalent to line a. above p Book • Equivalence to Forward Stagewise Fitting Procedure – cm = arg min (⋅) is equivalent to line c. c • R package adabag © 2013 G.Seni 2013 Strata Conference + Hadoop World 42
- 43. AdaBoost Hands-on Exercise 1.0 • Navigate to directory: example.2.EllipticalBoundary Set working directory: use setwd() or with GUI • Load and run 0.0 – fitModel_Adaboost_by_hand.R -0.5 • After class, load and run fitModel_Adaboost.R and fitModel_RandomForest.R -1.0 x2 0.5 • -2 -1 0 1 2 x1 © 2013 G.Seni 2013 Strata Conference + Hadoop World 43
- 44. Stochastic Gradient Boosting (Friedman, 2001) • Boosting with any differentiable loss criterion ˆ • General L( y, y ) F0 (x) = c00 • υ = 0.1 ⇒ Sequential sampling For m = 1 to M { (cm , p m ) = arg min m c, p • η = N 2 ∑ηL( y , F i m −1 i∈S m ( ) (x i ) + c ⋅ T (x i ; p)) Tm (x) = T (x; p m ) • Tm (x) ⇒ Any “weak” learner N • co = arg minc ∑i =1 L( yi , c) Fm (x) = Fm −1 (x) +υ ⋅ cm ⋅ Tm (x) υ } M write {(υ ⋅ cm ), Tm (x)}1 M • {cm }1 ⇒ “shrunk” sequential partial regression coefficients – Potential improvements? – R package: gbm © 2013 G.Seni 2013 Strata Conference + Hadoop World 44
- 45. Stochastic Gradient Boosting LAD Regression – L !, ! = ! − ! • More robust than ( y − F )2 • Resistant to outliers in y …trees already providing resistance to outliers in x ! N F0 (x) = median{yi }1 For m = 1 to M { // Step1 : find Tm (x) ~ = sign ( y − F (x ) ) yi i m −1 i • Note: {R } J jm 1 – Trees are fitted to pseudoresponse ( // Step2 : find coefficients ⇒ Can’t interpret interpret γˆ jm = median{yi − Fm −1 (x i )}1N x i ∈R jm individual trees – “shrunk” version of tree gets added to ensemble j = 1… J // Update expansion – Original tree constants are overwritten © 2013 G.Seni N = J − terminal node LS - regression tree {~i , x i }1 y Fm (x) = Fm −1 (x) + υ ⋅ ∑ γˆ jm I x i ∈ R jm J j =1 ( ) } 2013 Strata Conference + Hadoop World 45 )
- 46. Parallel vs. Sequential Ensembles 100 Target Functions Comparison (Popescu, 2005) Comparative RMS Error • xxx_6_5%_P : 6 terminal nodes trees 5% samples without replacement Post-processing – i.e., using estimated “optimal” quadrature coefficients “Sequential” “Parallel” Bag RF Boost Seq_0.01_20%_P • Seq_υ_η%_P : “Sequential” ensemble 6 terminal nodes trees υ : “memory” factor η % samples without replacement Post-processing Bag_6_5%_P RF_6_5%_P Seq_0.1_50%_P • Sequential ISLE tend to perform better than parallel ones – Consistent with results observed in classical Monte Carlo integration © 2013 G.Seni 2013 Strata Conference + Hadoop World 46
- 47. Rule Ensembles (Friedman & Popescu, 2005) J ˆ ˆ • Trees as collection of conjunctive rules: Tm (x) = ∑ c jm I (x ∈ R jm ) j =1 R1 27 R4 15 R2 ⇒ R5 15 22 x1 r2 (x) = I ( x1 > 22) ⋅ I (0 ≤ x2 ≤ 27) r3 (x) = I (15 < x1 ≤ 22) ⋅ I (0 ≤ x2 ) R4 ⇒ r4 (x) = I (0 ≤ x1 ≤ 15) ⋅ I ( x2 > 15) R5 ⇒ x2 r1 (x) = I ( x1 > 22) ⋅ I ( x2 > 27) R3 ⇒ R3 ˆ y R2 R1 ⇒ r5 (x) = I (0 ≤ x1 ≤ 15) ⋅ I (0 ≤ x2 ≤ 15) – These simple rules, rm (x) ∈ {0,1} can be used as base learners , – Main motivation is interpretability © 2013 G.Seni 2013 Strata Conference + Hadoop World 47
- 48. Rule Ensembles ISLE Procedure • Rule-based model: F (x) = a0 + ∑ am rm (x) m – Still a piecewise constant model ⇒ complement the non-linear rules with purely linear terms: • Fitting – Step 1: derive rules from tree ensemble (shortcut) • Tree size controls rule “complexity” (interaction order) – Step 2: fit coefficients using linear regularized procedure: ( N P K ˆ ˆ ({ak },{b j }) = arg min ∑ L yi , F x; {ak }0 , {b j }1 {ak },{b j } © 2013 G.Seni i=1 ( )) +!!! ⋅ 2013 Strata Conference + Hadoop World !(a) + !(b) ! 48
- 49. Boosting & Rule Ensembles Hands-on Exercise 2500 • example.3.Diamonds Load and run 1500 Set working directory: use setwd() or with GUI • 2000 • – viewDiamondData.R – fitModel_GBM.R 1000 – fitModel_RE.R • 500 Absolute loss Navigate to directory: After class, go to: example.1.LinearBoundary 0 200 400 600 800 1000 Run fitModel_GBM.R Iteration © 2013 G.Seni 2013 Strata Conference + Hadoop World 49
- 50. Overview • Motivation, In a Nutshell & Timeline • Predictive Learning & Decision Trees • Ensemble Methods Ø Summary © 2013 G.Seni 2013 Strata Conference + Hadoop World 50
- 51. Summary • Ensemble methods have been found to perform extremely well in a variety of problem domains • Shown to have desirable statistical properties • Latest ensemble research brings together important foundational strands of statistics • Emphasis on accuracy but significant progress has been made on interpretability Go build Ensembles and keep in touch! © 2013 G.Seni 2013 Strata Conference + Hadoop World 51

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