BSSML16 L2. Ensembles and Logistic Regressions

D E C E M B E R 8 - 9 , 2 0 1 6

BigML, Inc 2

Poul Petersen CIO, BigML, Inc.

EnsemblesMaking Trees Unstoppable

BigML, Inc 3Ensembles

Ensemble Idea

• Rather than build a single model… • Combine the output of several “weaker” models

into a powerful ensemble… • Q1: Why would this work? • Q2: How do we build “weaker” models? • Q3: How do we “combine” models?


Why Ensembles1. Every “model” is an approximation of the “real” function

and there may be several good approximations.

2. ML Algorithms use random processes to solve NP-hard problems and may arrive at different “models” depending on the starting conditions, local optima, etc.

3. A given ML algorithm may not be able to exactly “model” the real characteristics of a particular dataset.

4. Anomalies in the data may cause over-fitting, that is trying to model behavior that should be ignored. By using several models, the outliers may be averaged out.

In any case, if we find several accurate “models”, the combination may be closer to the real “model”

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Ensemble Demo #1


Weaker Models?1. Bootstrap Aggregating - aka “Bagging” If there are “n”

instances, each tree is trained with “n” instances, but they are sampled with replacement.

2. Random Decision Forest - In addition to sampling with replacement, the tree randomly selects a subset of features to consider when making each split. This introduces a new parameter, the random candidates which is the number of features to randomly select before making the split.


Over-Fitting ExampleDiameter Color Shape Fruit

4 red round plum

5 red round apple

5 red round apple

6 red round plum

7 red round appleBagging!

Random Decision Forest!

All Data: “plum”

Sample 2: “apple”

Sample 3: “apple”

Sample 1: “plum”}“apple”

What is a round, red 6cm fruit?


Voting Methods1. Plurality - majority wins.

2. Confidence Weighted - majority wins but each vote is weighted by the confidence.

3. Probability Weighted - each tree votes the distribution at it’s leaf node.

4. K Threshold - only votes if the specified class and required number of trees is met. For example, allowing a “True” vote if and only if at least 9 out of 10 trees vote “True”.

5. Confidence Threshold - only votes the specified class if the minimum confidence is met.

Linear and non-linear combinations of votes using stacking

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Ensemble Demo #2


Model vs Bagging vs RF

Model Bagging Random Forest

Increasing Performance

Decreasing Interpretability

Increasing Stochasticity

Increasing Complexity

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Ensemble Demo #3


SMACdown

• How many trees? • How many nodes? • Missing splits? • Random candidates? • Too many parameters?

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Poul Petersen CIO, BigML, Inc.

Logistic RegressionModeling Probabilities

BigML, Inc 14Logistic Regression

Logistic Regression

• Classification implies a discrete objective. How can this be a regression?

• Why do we need another classification algorithm?

• more questions….

Logistic Regression is a classification algorithm


Linear Regression


Polynomial Regression


Regression

• What function can we fit to discrete data?

Key Take-Away: Fitting a function to the data


Discrete Data Function?


Discrete Data Function?

????


Logistic Function

• x→-∞ : f(x)→0

• x→∞ : f(x)→1

• Looks promising, but still not

"discrete"


Probabilities

P≈0 P≈10<P<1


Logistic Regression

• Assumes that output is linearly related to "predictors" … but we can "fix" this with feature engineering

• How do we "fit" the logistic function to real data?

LR is a classification algorithm … that models the probability of the output class.


Logistic Regressionβ₀ is the "intercept"

β₁ is the "coefficient"

The inverse of the logistic function is called the "logit":

In which case solving is now a linear regression


Logistic RegressionIf we have multiple dimensions, add more coefficients:

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Logistic Regression Demo #1


LR Parameters1. Bias: Allows an intercept term.

Important if P(x=0) != 0 2. Regularization:

• L1: prefers zeroing individual coefficients • L2: prefers pushing all coefficients towards zero

3. EPS: The minimum error between steps to stop. 4. Auto-scaling: Ensures that all features contribute

equally. • Unless there is a specific need to not auto-scale,

it is recommended.


Logistic Regression

• How do we handle multiple classes?

• What about non-numeric inputs?

Questions:


LR - Multi Class• Instead of a binary class ex: [ true, false ], we have multi-

class ex: [ red, green, blue, … ]

• k classes

• solve one-vs-rest LR • coefficients βᵢ for

each class


LR - Field Codings• LR is expecting numeric values to perform regression. • How do we handle categorical values, or text?

Class color=red color=blue color=green color=NULL

red 1 0 0 0

blue 0 1 0 0

green 0 0 1 0

NULL 0 0 0 1

One-hot encoding

Only one feature is "hot" for each class


LR - Field Codings

Dummy Encoding

Chooses a *reference class* requires one less degree of freedom

Class color_1 color_2 color_3

*red* 0 0 0

blue 1 0 0

green 0 1 0

NULL 0 0 1


LR - Field Codings

Contrast Encoding

Field values must sum to zero Allows comparison between classes …. so which one?

Class field "influence"

red 0,5 positive

blue -0,25 negative

green -0,25 negative

NULL 0 excluded


LR - Field Codings

• The "text" type gives us new features that have counts of the number of times each token occurs in the text field. "Items" can be treated the same way.

token "hippo" "safari" "zebra"

instance_1 3 0 1

instance_2 0 11 4

instance_3 0 0 0

instance_4 1 0 3

Text / Items ?

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Curvilinear LRInstead of

We could add a feature

Where

????

Possible to add any higher order terms or other functions to match shape of data

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LR versus DT

• Expects a "smooth" linear relationship with predictors.

• LR is concerned with probability of a discrete outcome.

• Lots of parameters to get wrong:

regularization, scaling, codings

• Slightly less prone to over-fitting

• Because fits a shape, might work

better when less data available.

• Adapts well to ragged non-linear relationships

• No concern: classification, regression, multi-class all fine.

• Virtually parameter free

• Slightly more prone to over-fitting

• Prefers surfaces parallel to

parameter axes, but given enough

data will discover any shape.

Logistic Regression Decision Tree

BigML, Inc 38


BSSML16 L2. Ensembles and Logistic Regressions

Data & Analytics

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