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What is it?• Large datasets• Fast methods• Not significance testing• Topics
– Trees (recursive splitting)– Nearest Neighbor– Neural Networks– Clustering– Association Analysis
Trees
• A “divisive” method (splits)
• Start with “root node” – all in one group
• Get splitting rules
• Response often binary
• Result is a “tree”
• Example: Loan Defaults
• Example: Framingham Heart Study
Recursive Splitting
X1=DebtToIncomeRatio
X2 = Age
Pr{default} =0.007 Pr{default} =0.012
Pr{default} =0.0001
Pr{default} =0.003
Pr{default} =0.006
No defaultDefault
Some Actual Data
• Framingham Heart Study
• First Stage Coronary Heart Disease – P{CHD} = Function of:
• Age - no drug yet! • Cholesterol• Systolic BP
Import
How to make splits?
• Which variable to use?
• Where to split?– Cholesterol > ____– Systolic BP > _____
• Goal: Pure “leaves” or “terminal nodes”
• Ideal split: Everyone with BP>x has problems, nobody with BP<x has problems
Where to Split?
• First review Chi-square tests• Contingency tables
95 5
55 45
Heart DiseaseNo Yes
LowBP
HighBP
100
100
DEPENDENT
75 25
75 25
INDEPENDENT
Heart DiseaseNo Yes
2 Test Statistic • Expect 100(150/200)=75 in upper left if
independent (etc. e.g. 100(50/200)=25)
95
(75)
5
(25)
55
(75)
45
(25)
Heart DiseaseNo Yes
LowBP
HighBP
100
100
150 50 200
allcells ected
ectedobserved
exp
)exp( 22
2(400/75)+2(400/25) = 42.67
Compare to Tables – Significant!WHERE IS HIGH BP CUTOFF???
Measuring “Worth” of a Split
• P-value is probability of Chi-square as great as that observed if independence is true. (Pr {2>42.67} is 6.4E-11)
• P-values all too small.
• Logworth = -log10(p-value) = 10.19
• Best Chi-square max logworth.
How to make splits?
• Which variable to use?
• Where to split?– Cholesterol > ____– Systolic BP > _____
• Idea – Pick BP cutoff to minimize p-value for 2
• What does “signifiance” mean now?
Multiple testing
• 50 different BPs in data, 49 ways to split • Sunday football highlights always look
good!• If he shoots enough baskets, even 95%
free throw shooter will miss.• Jury trial analogy• Tried 49 splits, each has 5% chance of
declaring significance even if there’s no relationship.
Multiple testing
= Pr{ falsely reject hypothesis 1}
= Pr{ falsely reject hypothesis 2}
Pr{ falsely reject one or the other} < 2Desired: 0.05 probabilty or lessSolution: use = 0.05/2Or – compare 2(p-value) to 0.05
Multiple testing
• 50 different BPs in data, m=49 ways to split
• Multiply p-value by 49
• Bonferroni – original idea
• Kass – apply to data mining (trees)
• Stop splitting if minimum p-value is large.
• For m splits, logworth becomes
-log10(m*p-value)
Other Split Evaluations
• Gini Diversity Index – { A A A A B A B B C B}– Pick 2, Pr{different} = 1-Pr{AA}-Pr{BB}-Pr{CC}
• 1-[10+6+0]/45=29/45=0.64
– { A A B C B A A B C C }• 1-[6+3+3]/45 = 33/45 = 0.73 MORE DIVERSE, LESS
PURE
• Shannon Entropy– Larger more diverse (less pure)
– -i pi log2(pi) {0.5, 0.4, 0.1} 1.36{0.4, 0.2, 0.3} 1.51 (more diverse)
Goals
• Split if diversity in parent “node” > summed diversities in child nodes
• Observations should be – Homogeneous (not diverse) within leaves– Different between leaves– Leaves should be diverse
• Framingham tree used Gini for splits
Cross validation
• Traditional stats – small dataset, need all observations to estimate parameters of interest.
• Data mining – loads of data, can afford “holdout sample”
• Variation: n-fold cross validation– Randomly divide data into n sets– Estimate on n-1, validate on 1– Repeat n times, using each set as holdout.
Pruning
• Grow bushy tree on the “fit data”
• Classify holdout data
• Likely farthest out branches do not improve, possibly hurt fit on holdout data
• Prune non-helpful branches.
• What is “helpful”? What is good discriminator criterion?
Goals
• Want diversity in parent “node” > summed diversities in child nodes
• Goal is to reduce diversity within leaves
• Goal is to maximize differences between leaves
• Use same evaluation criteria as for splits
• Costs (profits) may enter the picture for splitting or evaluation.
Accounting for Costs
• Pardon me (sir, ma’am) can you spare some change?
• Say “sir” to male +$2.00
• Say “ma’am” to female +$5.00
• Say “sir” to female -$1.00 (balm for slapped face)
• Say “ma’am” to male -$10.00 (nose splint)
Including Probabilities
True Gender
M
F
Leaf has Pr(M)=.7, Pr(F)=.3. You say:
M F
0.7 (2)
0.3 (-1)
0.7 (-10)
0.3 (5)
Expected profit is 2(0.7)-1(0.3) = $1.10 if I say “sir” Expected profit is -7+1.5 = -$5.50 (a loss) if I say “Ma’am”Weight leaf profits by leaf size (# obsns.) and sumPrune (and split) to maximize profits.
Additional Ideas
• Forests – Draw samples with replacement (bootstrap) and grow multiple trees.
• Random Forests – Randomly sample the “features” (predictors) and build multiple trees.
• Classify new point in each tree then average the probabilities, or take a plurality vote from the trees
• “Bagging” – Bootstrap aggregation• “Boosting” – Similar, iteratively reweights points
that were misclassified to produce sequence of more accurate trees.
* Lift Chart - Go from leaf of most to least response. - Lift is cumulative proportion responding.
Regression Trees
• Continuous response (not just class)
• Predicted response constant in regions
Predict 50
Predict 80
Predict 100
Predict 130 Predict
20
X1
X2
• Predict Pi in cell i.
• Yij jth response in cell i.
• Split to minimize i j (Yij-Pi)2
Predict 50
Predict 80
Predict 100
Predict 130 Predict
20
Logistic Regression
• “Trees” seem to be main tool.
• Logistic – another classifier
• Older – “tried & true” method
• Predict probability of response from input variables (“Features”)
• Linear regression gives infinite range of predictions
• 0 < probability < 1 so not linear regression.
• Logistic idea: Map p in (0,1) to L in whole real line
• Use L = ln(p/(1-p))• Model L as linear in temperature• Predicted L = a + b(temperature)• Given temperature X, compute a+bX then p
= eL/(1+eL)• p(i) = ea+bXi/(1+ea+bXi) • Write p(i) if response, 1-p(i) if not• Multiply all n of these together, find a,b to
maximize
Example: Ignition
• Flame exposure time = X
• Ignited Y=1, did not ignite Y=0– Y=0, X= 3, 5, 9 10 , 13, 16 – Y=1, X = 11, 12 14, 15, 17, 25, 30
• Q=(1-p)(1-p)(1-p)(1-p)pp(1-p)pp(1-p)ppp
• P’s all different p=f(exposure)
• Find a,b to maximize Q(a,b)
DATA LIKELIHOOD; ARRAY Y(14) Y1-Y14; ARRAY X(14) X1-X14; DO I=1 TO 14; INPUT X(I) y(I) @@; END; DO A = -3 TO -2 BY .025; DO B = 0.2 TO 0.3 BY .0025; Q=1; DO i=1 TO 14; L=A+B*X(i); P=EXP(L)/(1+EXP(L)); IF Y(i)=1 THEN Q=Q*P; ELSE Q=Q*(1-P); END; IF Q<0.0006 THEN Q=0.0006; OUTPUT; END;END; CARDS; 3 0 5 0 7 1 9 0 10 0 11 1 12 1 13 0 14 1 15 1 16 0 17 1 25 1 30 1 ;
Generate Q for array of (a,b) values
IGNITION DATA The LOGISTIC Procedure Analysis of Maximum Likelihood Estimates Standard WaldParameter DF Estimate Error Chi-Square Pr > ChiSqIntercept 1 -2.5879 1.8469 1.9633 0.1612TIME 1 0.2346 0.1502 2.4388 0.1184
Association of Predicted Probabilities and Observed Responses
Percent Concordant 79.2 Somers' D 0.583Percent Discordant 20.8 Gamma 0.583Percent Tied 0.0 Tau-a 0.308Pairs 48 c 0.792
Framingham
The LOGISTIC Procedure
Analysis of Maximum Likelihood Estimates
Standard WaldParameter DF Estimate Error Chi-Square Pr>ChiSq
Intercept 1 -5.4639 0.5563 96.4711 <.0001age 1 0.0630 0.0110 32.6152 <.0001
Example: Shuttle Missions
• O-rings failed in Challenger disaster• Low temperature• Prior flights “erosion” and “blowby” in O-rings• Feature: Temperature at liftoff• Target: problem (1) - erosion or blowby vs. no
problem (0)
Neural Networks
• Very flexible functions• “Hidden Layers” • “Multilayer Perceptron”
Logistic function of
Logistic functions
Of data
outputinputs
Arrows represent linear combinations of “basis functions,” e.g. logistics
b1
Example:
Y = a + b1 p1 + b2 p2 + b3 p3
Y = 4 + p1+ 2 p2 - 4 p3
• Should always use holdout sample
• Perturb coefficients to optimize fit (fit data)– Nonlinear search algorithms
• Eliminate unnecessary arrows using holdout data.
• Other basis sets– Radial Basis Functions– Just normal densities (bell shaped) with
adjustable means and variances.
Terms• Train: estimate coefficients• Bias: intercept a in Neural Nets• Weights: coefficients b • Radial Basis Function: Normal density• Score: Predict (usually Y from new Xs)• Activation Function: transformation to target• Supervised Learning: Training data has
response.
Hidden LayerL1 = -1.87 - .27*Age – 0.20*SBP22H11=exp(L1)/(1+exp(L1))
L2 = -20.76 -21.38*H11Pr{first_chd} = exp(L2)/(1+exp(L2))“Activation Function”
Demo (optional)
• Compare several methods using SAS Enterprise Miner– Decision Tree – Nearest Neighbor– Neural Network
Unsupervised Learning
• We have the “features” (predictors)
• We do NOT have the response even on a training data set (UNsupervised)
• Clustering– Agglomerative
• Start with each point separated
– Divisive • Start with all points in one cluster then spilt
EM PROC FASTCLUS
• Step 1 – find “seeds” as separated as possible
• Step 2 – cluster points to nearest seed– Drift: As points are added, change seed
(centroid) to average of each coordinate– Alternatively: Make full pass then recompute
seed and iterate.
Cubic Clustering Criterion (to decide # of Clusters)
• Divide random scatter of (X,Y) points into 4 quadrants
• Pooled within cluster variation much less than overall variation
• Large variance reduction• Big R-square despite no real clusters• CCC compares random scatter R-square
to what you got to decide #clusters• 3 clusters for “macaroni” data.
Association Analysis
• Market basket analysis – What they’re doing when they scan your “VIP”
card at the grocery– People who buy diapers tend to also buy
_________ (beer?)– Just a matter of accounting but with new
terminology (of course ) – Examples from SAS Appl. DM Techniques, by
Sue Walsh:
Termnilogy
• Baskets: ABC ACD BCD ADE BCE
• Rule Support Confidence
• X=>Y Pr{X and Y} Pr{Y|X}
• A=>D 2/5 2/3
• C=>A 2/5 2/4
• B&C=>D 1/5 1/3
Don’t be Fooled!• Lift = Confidence /Expected Confidence if Independent
Checking->
Saving V
No
(1500)
Yes
(8500) (10000)
No 500 3500 4000
Yes 1000 5000 6000
SVG=>CHKG Expect 8500/10000 = 85% if independentObserved Confidence is 5000/6000 = 83%Lift = 83/85 < 1. Savings account holders actually LESS likely than others to have checking account !!!
Summary
• Data mining – a set of fast stat methods for large data sets
• Some new ideas, many old or extensions of old• Some methods:
– Decision Trees– Nearest Neighbor– Neural Nets– Clustering– Association