Primer on recursive portioning Using decision tress for regression and classification

1. Proprietary Information created by Parth Khare
Machine Learning
Classification & Decision Trees
04/01/2013

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Contents
 Recursive Partitioning
 Classification
 Regression/Decision
 Bagging
 Random Forest
 Boosting
 Gradient Boosting
 Questions
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Detail and flow
 What is the difference between supervised and unsupervised learning?
 What is ML? how is it different from classical statistics?
 Supervised learning: machine -> an application is Trees
 Most elementary analysis: CART
 Tree
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Basics
 Supervised Learning:
 Called “supervised” because of the presence of the outcome variable to guide learning process
 building a learner (/model) to predict the outcome for new unseen objects.
 Alternatively,
 Unsupervised Learning:
 observe only the features and have no measurements of the outcome
 task is rather to describe how the data are organized or clustered
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Machine Learning viz Statistics
‘learning’ viz ‘fitting’
 Machine learning: a branch of artificial intelligence, is about the construction and study
of systems that can learn from data.
 Statistics bases everything on probability models
 assuming your data are samples from a random variable with some
 distribution, then making
 inferences about the parameters of the distribution
 Machine learning may use probability models, and when it does, it overlaps with
statistics.
 isn't so committed to probability
 use other approaches to problem solving that are not based on probability
 The basic optimization concept is the same for trees is same as that of parametric
techniques, minimizing errors metrics. Instead of square error function or MLE,
Machine Learning supervises optimization of entropy, node impurity etc
 An application _-> Trees
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Decision Tree Approach: Parlance
 A decision tree represents a hierarchical segmentation of the data
 The original segment is called the root node and is the entire data set
 The root node is partitioned into two or more segments by applying a series of simple
rules over an input variables
 For example, risk = low, risk = not low
 Each rule assigns the observations to a segment based on its input value
 Each resulting segment can be further partitioned into sub-segments, and so on
 For example risk = low can be partitioned into income = low and income = not low
 The segments are also called nodes, and the final segments are called leaf nodes or
leaves
 Final node surviving the partitions called the terminal node

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Decision Tree Example: Risk
Assessment(Loan)
Income
< $30k >= $30k
Age Credit Score
< 25 >=25 < 600 >= 600
not on-time on-time not on-time on-time

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CART: Heuristic and Visual
 Generic supervised learning problem:
 given a bunch of data (x1, y1), (x2, y2)…(xn,yn), and a new point ‘xi ‘, supervised learning
objective: associates a ‘y’ with this new ‘x’
 Main Idea: form a binary tree and minimize error in each leaf
 Given dataset, a decision tree: choose a sequence of binary split of the data
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Growing the tree
 Growing the tree involves successively partitioning the data – recursively partitioning
 If an input variable is binary, then the two categories can be used to split the data
(relative concentration of ‘0’’s and ‘1’’s)
 If an input variable is interval, a splitting value is used to classify the data into two
segments
 For example, if household income is interval and there are 100 possible incomes in the
data set, then there are 100 possible splitting values
 For example, income < $30k, and income >= $30k

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Classification Tree: again (referrence)
 Represented by a series of binary splits.
 Each internal node represents a value
query on one of the variables — e.g. “Is
X3 > 0.4”. If the answer is “Yes”, go right,
else go left.
 The terminal nodes are the decision
nodes. Typically each terminal node is
dominated by one of the classes.
 The tree is grown using training data, by
recursive splitting.
 The tree is often pruned to an optimal
size, evaluated by cross-validation.
 New observations are classified by
passing their X down to a terminal node of
the tree, and then using majority vote.
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Evaluating the partitions
 When the target is categorical, for each partition of an input variable a chi-square
statistic is computed
 A contingency table is formed that maps responders and non-responders against the
partitioned input variable
 For example, the null hypothesis might be that there is no difference between people
with income <$30k and those with income >=$30k in making an on-time loan payment
 The lower the significance or p-value, the more likely that we reject this hypothesis,
meaning that this income split is a discriminating factor

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Splitting Criteria: Categorical
 Information Gain -> Entropy
 The rarity of an event is defined as: -log2(pi)
 Impurity Measure:
- Pr(Y=0) X log2 [Pr(Y=0)] - Pr(Y=1) X log2 [Pr(Y=1)]
e.g. check at Pr(Y=0) = 0.5??
 Entropy sums up the rarity of response and non-response over all observations
 Entropy ranges from the best case of 0 (all responders or all non-responders) to 1
(equal mix of responders and non-responders)
link
http://www.youtube.com/watch?v=p17C9q2M00Q
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Splitting Criteria :Continuous
 An F-statistic is used to measure the degree of separation of a split for an interval
target, such as revenue
 Similar to the sum of squares discussion under multiple regression,
F-statistic is based on the ratio of the sum of squares between the groups and the sum
of squares within groups, both adjusted for the number of degrees of freedom
 The null hypothesis is that there is no difference in the target mean between the two
groups

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Contents
 Recursive Partitioning
 Classification
 Regression/Decision
 Bagging
 Random Forest
 Boosting
 Gradient Boosting
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Bagging
 Ensemble Models : Combines the results from different models
 An ensemble classifier using many decision tree models
 Bagging: Bootstrapped Samples of data
Working: Random Forest
 A different subset of the training data are selected (~2/3), with replacement, to train
each tree
 Remaining training data (OOB) are used to estimate error and variable importance
 Class assignment is made by the number of votes from all of the trees and for
regression the average of the results is used
 A randomly selected subset of variables is used to split each node
 The number of variables used is decided by the user (mtry parameter in R)
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Bagging: Stanford
 Suppose
 C(S, x) is a classifier, such as a tree, based
on our training data S, producing a
predicted class label at input point x.
 ‘To bag C, we draw bootstrap samples
S∗1,...S∗B each of size N with replacement
from the training data.
 Then
 Cˆbag(x) = Majority Vote{C(S∗b, x)}B
b =1.
 Bagging can dramatically reduce the
variance of unstable procedures (like
trees), leading to improved prediction.
 However any simple structure in C (e.g a
tree) is lost.
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Bootstrapped samples
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Contents
 Recursive Partitioning
 Classification
 Regression/Decision
 Bagging
 Random Forest
 Boosting
 Gradient Boosting
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Boosting
Make Copies of Data
 Boosting idea: Based on "strength of weak learn ability" principles
 Example:
IF Gender=MALE AND Age<=25 THEN claim_freq.=‘high’
Combination of weak learners increased accuracy
 Simple or “weak" learners are not perfect!
 Every “boosting” algorithm can be interpreted as optimizing the loss function in a “greedy stage-
wise” manner
Working: Gradient Descent
 First tree is created, residuals observed
 Now, a tree is fitted on the residuals of the first tree and so on
 In this way, boosting grows trees in series, with later trees dependent on the results of previous
trees
 Shrinkage, CV folds, Interaction Depth
 Adaboost, DirectBoost, Laplace Loss(Gaussian Boost)
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GBM
 Gradient Tree Boosting is a generalization of boosting to arbitrary differentiable loss functions.
GBRT is an accurate and effective off-the-shelf procedure that can be used for both regression and
classification problems.
 What it does essentially
 By sequentially learning form the errors of the previous trees Gradient Boosting, in a way tries to
‘learn’ the unconditional distribution of the target variable. So, analogus to how we use different
types of distributions in GLM modeling, GBM creates/replicates the distribution in the given data
as close as possible.
 This comes with an additional risk of over-fitting, resolved by methods like cross validation
within, min observation per node etc.
 Parameters working: OOB data/error
 We know that the first tree of GBM is build on training data and the subsequent trees are
developed on the error form the first tree. This process carries on.
 For OOB, the training data is also split in two parts, on one part the trees and developed, and on
the other part the tree developed on the first part is tested. This second part is called the OOB
data and the error obtained is known as OOB error.
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Summary: Rf and GBM
Main similarities:
 Both derive many benefits from ensembling, with few disadvantages
 Both can be applied to ensembling decision trees
Main differences:
 Boosting performs an exhaustive search for best predictor to split on; RF searches
only a small subset
 Boosting grows trees in series, with later trees dependent on the results of
previous trees
 RF grows trees in parallel independently of one another.
 RF cannot work with missing values GBM can
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More diff b/w RF and GBM
 Algorithmic difference is;
 Random Forests are trained with random sample of data (even more randomized
cases available like feature randomization) and it trusts randomization to have better
generalization performance on out of train set.
 On the other spectrum, Gradient Boosted Trees algorithm additionally tries to find
optimal linear combination of trees (assume final model is the weighted sum of
predictions of individual trees) in relation to given train data. This extra tuning might
be deemed as the difference. Note that, there are many variations of those
algorithms as well.
 At the practical side; owing to this tuning stage,
 Gradient Boosted Trees are more susceptible to jiggling data. This final stage makes
GBT more likely to overfit therefore if the test cases are inclined to be so verbose
compared to train cases this algorithm starts lacking.
 On the contrary, Random Forests are better to strain on overfitting although it is
lacking on the other way around.
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Questions
 Concept/ Interpretation
 Application
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24. For further details contact:
Parth Khare
https://www.linkedin.com/profile/view?id=43877647&trk=nav_responsive_tab_profile