Lecture 8: Feature Selection and Analysis

Introduction to Machine Learning Semester 1, 2022

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Features in Machine Learning

Machine Learning Workflow

Evaluation Task

??? ? Model

Depends on features

Evaluation Task

Data Preparation vs Feature Selection

GIGO: Garbage In, Garbage Out

Data Preparation and Cleaning (discussed before)

• Data Cleaning

• Data Aggregation

• Dealing with missing values

• Transformation (e.g., log transform) • Binarization

• Scaling or Normalization

Feature Selection (this lecture)

• Wrapper methods (aka recursive elimination) • Filtering (aka univariate filtering)

• Glance into some other common approaches

Data Preparation vs Feature Selection

Our job as Machine Learning experts:

• Inspect / clean the data

• Choose a model suitable for classifying the data according to the attributes

• Choose attributes suitable for classifying the data according to the model

• Inspection • Intuition

Data Preparation vs Feature Selection

Our job as Machine Learning experts:

• Inspect / clean the data

• Choose a model suitable for classifying the data according to the attributes

• Choose attributes suitable for classifying the data according to the model

• Inspection

• Intuition

• Neither possible in practice

Feature Selection

What makes features good?

Lead to better models

• Better performance according to some evaluation metric

Side-goal 1

• Seeing important features can suggest other important features • Tell us interesting things about the problem

Side-goal 2

• Fewer features → smaller models → faster answer • More accurate answer >> faster answer

Iterative feature selection: Wrappers

Choosing a good feature set

“Wrapper” methods

• Choose subset of attributes that give best performance on the development data

• For example: for the Weather data set:

• Train model on {Outlook}

• Train model on {Temperature}

• Train model on {Outlook, Temperature}

• Train model on {Outlook, Temperature, Humidity}

• Train model on {Outlook, Temperature, Humidity, Windy}

Choosing a good feature set

“Wrapper” methods

• Choose subset of attributes that give best performance on the development data

• For example: for the Weather data set:

• Evaluate model on {Outlook}

• Evaluate model on {Temperature}

• Evaluate model on {Outlook, Temperature}

• Evaluate model on {Outlook, Temperature, Humidity}

• Evaluate model on {Outlook, Temperature, Humidity, Windy}

• Best performance on data set → best feature set

Choosing a good feature set

“Wrapper” methods

• Choose subset of attributes that give best performance on the development data

• Advantages:

• Feature set with optimal performance on development data

• Disadvantages:

• Takes a long time

Aside: how long does the full wrapper method take?

Assume we have a fast method (e.g. Naive Bayes) over a data set of non-trivial size (∼10K instances):

• Assume: train–evaluate cycle takes 10 sec to complete How many cycles? For m features:

• 2m subsets = 2m minutes 6

• m=10→3hours

• m = 60 → heat death of universe

Only practical for very small data sets.

More practical wrapper methods: Greedy search

Greedy approach

• Train and evaluate model on each single attribute

• Choose best attribute

• Until convergence:

• Train and evaluate model on best attribute(s), plus each remaining

single attribute

• Choose best attribute out of the remaining set

• Iterate until performance (e.g. accuracy) stops increasing

More practical wrapper methods: Greedy search

Greedy approach

• Bad news:

• Takes 21 m2 cycles, for m attributes • In theory, 386 attributes → days

• Good news:

• In practice, converges much more quickly than this

• Bad news again:

• Convergences to a sub-optimal (and often very bad) solution

More practical wrapper methods: Ablation

“Ablation” approach

• Start with all attributes

• Remove one attribute, train and evaluate model

• Until divergence:

• From remaining attributes, remove each attribute, train and evaluate model

• Remove attribute that causes least performance degradation

• Termination condition usually: performance (e.g. accuracy) starts to degrade by more than ε

More practical wrapper methods: Ablation

“Ablation” approach

• Good news:

• Mostly removes irrelevant attributes (at the start)

• Bad news:

• Assumes independence of attributes

(Actually, both approaches do this)

• Takes O(m2) time; cycles are slower with more attributes • Not feasible on non-trivial data sets.

Feature Filtering

Feature filtering

Intuition: Evaluate the “goodness” of each feature, separate from other features

• Consider each feature separately: linear time in number of attributes • Possible (but difficult) to control for inter-dependence of features

• Typically most popular strategy

Feature “goodness”

What makes a feature set single feature good?

Toy example

a1 a2 c YYY YNY NYN NNN

Which of a1, a2 is good?

Toy example

a1 a2 c YYY YNY NYN NNN

Toy example

a1 a2 c YYY YNY NYN NNN

Pointwise Mutual Information

Discrepancy between the observed joint probability of two random variables A and C and the expected joint probability if A and C were independent.

Recall independence: P(C|A) = P(C)

Pointwise Mutual Information

Discrepancy between the observed joint probability of two random variables A and C and the expected joint probability if A and C were independent.

Recall independence: P(C|A) = P(C)

PMI is defined as

We want to find attributes that are not independent of the class.

• If PMI >> 0, attribute and class occur together much more often than randomly.

• If RHS ∼ 0, attribute and class occur together as often as we would expect from random chance

• If RHS << 0, attribute and class are negatively correlated. (More on that later!)
Attributes with greatest PMI: best attributes
PMI(A, C) = log P(A, C) 2 P(A)P(C)
Toy example, revisited
a1 a2 c YYY YNY NYN NNN
Calculate PMI of a1, a2 with respect to c
Toy example, revisited
a1 a2 c YYY YNY NYN NNN
P(a1) = P(c) = P(a1,c) = PMI(a1,c) =
Toy example, revisited
a1 a2 c YYY YNY NYN NNN
P(a1) = P(c) = P(a1,c) = PMI(a1,c) =
Toy example, revisited
a1 a2 c YYY YNY NYN NNN
P ( a 2 ) = 42 P ( c ) = 42 P ( a 2 , c ) = 14
Toy example, revisited
a1 a2 c YYY YNY NYN NNN
P ( a 2 ) = 42
P ( c ) = 42
P ( a 2 , c ) = 14
PMI(a2,c) = =
log2(1) = 0
Feature “goodness”, revisited
What makes a single feature good?
• Well correlated with class
• Knowing a lets us predict c with more confidence
• Reverse correlated with class
• Knowing a ̄ lets us predict c with more confidence
• Well correlated (or reverse correlated) with not class
• Knowing a lets us predict c ̄ with more confidence • Usually not quite as good, but still useful
Mutual Information
• Expected value of PMI over all possible events
• For our example: Combine PMI of all possible combinations: a, a ̄, c, c ̄
Aside: Contingency tables
Contingency tables: compact representation of these frequency counts
a ̄ Total σ(c) σ(c ̄)
σ(a, c) σ(a ̄, c) σ(a, c ̄) σ(a ̄, c ̄)
P(a, c) = σ(a,c) , etc. N
Aside: Contingency tables
Contingency tables for toy example:
a1 a=Y a=N Total c=Y 2 c=N 2 Total 4
Total c=Y 1 1 2 c=N 1 1 2 Total 2 2 4
22 a=Y a=N
Mutual Information
Combine PMI of all possible combinations: a, a ̄, c, c ̄
MI(A, C) =P(a, c)PMI(a, c) + P(a ̄, c)PMI(a ̄, c)+ P(a, c ̄)PMI(a, c ̄) + P(a ̄, c ̄)PMI(a ̄, c ̄)
MI(A, C) =P(a, c) log P(a, c) 2 P(a)P(c)
P(a, c ̄) log P(a, c ̄) 2 P(a)P(c ̄)
+ P(a ̄, c) log + P(a ̄, c ̄) log
P(a ̄, c) + 2 P(a ̄)P(c)
P(a ̄, c ̄) 2 P(a ̄)P(c ̄)
Mutual Information
Combine PMI of all possible combinations: a, a ̄, c, c ̄
MI(A, C) =P(a, c)PMI(a, c) + P(a ̄, c)PMI(a ̄, c)+ P(a, c ̄)PMI(a, c ̄) + P(a ̄, c ̄)PMI(a ̄, c ̄)
MI(A, C) =P(a, c) log P(a, c) 2 P(a)P(c)
P(a, c ̄) log P(a, c ̄) 2 P(a)P(c ̄)
Often written more compactly as:
MI(A,C) = i∈{a,a ̄} j∈{c,c ̄}
We define that 0log0 ≡ 0.
+ P(a ̄, c) log + P(a ̄, c ̄) log
P(a ̄, c) + 2 P(a ̄)P(c)
P(a ̄, c ̄) 2 P(a ̄)P(c ̄)
P(i,j) 2 P(i)P(j)
Mutual Information Example
Contingency Table for attribute a1
a1 a=Y a=N Total
2 Total 2 2 4
P(a,c)= 24; P(a)= 42; P(c)= 42; P(a,c ̄)=0 P(a ̄,c ̄) = 42; P(a ̄) = 42; P(c ̄) = 42; P(a ̄,c) = 0
Mutual Information Example
Contingency Table for attribute a1
a1 a=Y a=N Total
2 Total 2 2 4
P(a,c)= 24; P(a)= 42; P(c)= 42; P(a,c ̄)=0 P(a ̄,c ̄) = 42; P(a ̄) = 42; P(c ̄) = 42; P(a ̄,c) = 0
MI(A1, C) = P(a1, c) log P(a1, c) 2 P(a1)P(c)
P(a1, c ̄) log P(a1, c ̄) 2 P(a1)P(c ̄)
+ P(a ̄1, c) log + P(a ̄1, c ̄) log
P(a ̄1, c) + 2 P(a ̄1)P(c)
P(a ̄1, c ̄) 2 P(a ̄1)P(c ̄)
Mutual Information Example
Contingency Table for attribute a1
a1 a=Y a=N Total
2 Total 2 2 4
P(a,c)= 24; P(a)= 42; P(c)= 42; P(a,c ̄)=0 P(a ̄,c ̄) = 42; P(a ̄) = 42; P(c ̄) = 42; P(a ̄,c) = 0
MI(a1, C) = P(a1, c) log P(a1, c) 2 P(a1)P(c)
P(a1, c ̄) log P(a1, c ̄) 2 P(a1)P(c ̄)
+ P(a ̄1, c) log + P(a ̄1, c ̄) log
P(a ̄1, c) + 2 P(a ̄1)P(c)
P(a ̄1, c ̄) 2 P(a ̄1)P(c ̄)
112 00112 = 2log2 11 +0log2 11 +0log2 11 +2log2 11
22222222 = 12(1)+0+0+12(1)=1
Mutual Information Example continued
Contingency Table for attribute a2
a2 a=Y a=N Total
2 Total 2 2 4
Mutual Information Example continued
Contingency Table for attribute a2
a2 a=Y a=N Total
2 Total 2 2 4
P(a) = 42; P(c) = 42; P(a ̄,c) = 41 P(a ̄) = 42; P(c ̄) = 42; P(a,c ̄) = 41
P(a,c) = 14; P(a ̄,c ̄) = 41;
Mutual Information Example continued
Contingency Table for attribute a2
a2 a=Y a=N Total
2 Total 2 2 4
P(a) = 42; P(c) = 42; P(a ̄,c) = 41 P(a ̄) = 42; P(c ̄) = 42; P(a,c ̄) = 41
P(a,c) = 14; P(a ̄,c ̄) = 41;
= P(a2, c) log P(a2, c) 2 P(a2)P(c)
P(a2, c ̄) log P(a2, c ̄) 2 P(a2)P(c ̄)
+ P(a ̄2, c) log + P(a ̄2, c ̄) log
P(a ̄2, c) + 2 P(a ̄2)P(c)
P(a ̄2, c ̄) 2 P(a ̄2)P(c ̄)
1 14 1 14 1 41 1 14 = 4log2 11 +4log2 11 +4log2 11 +4log2 11 22222222
= 41(0)+ 14(0)+ 14(0)+ 41(0) = 0
Mutual Information Example continued
Contingency Table for attribute a2
a2 a=Y a=N Total
2 Total 2 2 4
P(a) = 42; P(c) = 42; P(a ̄,c) = 41 P(a ̄) = 42; P(c ̄) = 42; P(a,c ̄) = 41
P(a,c) = 14; P(a ̄,c ̄) = 41;
χ2 (“Chi-square”)
Similar idea, different solution:
Contingency table (shorthand):
σ(c) σ(c ̄) N
σ(a, c) σ(a ̄, c) σ(a, c ̄) σ(a ̄, c ̄)
c W+X c ̄ Y+Z
Total W+Y X+Z N=W+X+Y+Z
If a, c were independent (uncorrelated), what value would you expect in W ?
Denote the expected value as E(W).
χ2 (“Chi-square”)
If a, c were independent, then P(a, c) = P(a)P(c) P(a, c) = P(a)P(c)
σ(a, c) = σ(a) σ(c) NNN
σ(a,c) = E(W) =
σ(a)σ(c) N
(W +Y)(W +X) W+X+Y+Z
χ2 (“Chi-square”)
Compare the value we actually observed O(W) with the expected value E(W):
• If the observed value is much greater than the expected value, a occurs more often with c than we would expect at random — predictive
• If the observed value is much smaller than the expected value, a occurs less often with c than we would expect at random — predictive
• If the observed value is close to the expected value, a occurs as often with c as we would expect randomly — not predictive
Similarly with X, Y, Z
χ2 (“Chi-square”)
Actual calculation (to fit to a chi-square distribution)
2 (O(W) − E(W))2 (O(X) − E(X))2
χ = E(W) + E(X) +
(O(Y) − E(Y))2 (O(Z) − E(Z))2 E(Y) + E(Z)
r c (Oi,j −Ei,j)2
i=1 j=1 Ei,j
• i sums over rows and j sums over columns.
• Because the values are squared, χ2 becomes much greater when |O−E |islarge,evenifE isalsolarge.
Chi-square Example
Contingency table for toy example (observed values):
a1 a=Y a=N Total c =Y 2
Total 2 2 4
Contingency table for toy example (expected values): a1 a=Y a=N Total
2 Total 2 2 4
Chi-square Example
(Oa,c −Ea,c)2 Ea,c
(Oa,c ̄ − Ea,c ̄)2 Ea,c ̄
(Oa ̄,c −Ea ̄,c)2 Ea ̄,c (Oa ̄,c ̄ − Ea ̄,c ̄)2 Ea ̄,c ̄
(2−1)2 (0−1)2 (0−1)2 (2−1)2 =1+1+1+1
= 1+1+1+1=4
χ2(A2, C) is obviously 0, because all observed values are equal to expected values.
Common Issues
Types of Attribute
So far, we’ve only looked at binary (Y/N) attributes:
• Nominal attributes
• Continuous attributes • Ordinal attributes
Types of Attributes: Nominal
Two common strategies
1. Treat as multiple binary attributes:
• e.g. sunny=Y, overcast=N, rainy=N, etc.
• Can just use the formulae as given
• Results sometimes difficult to interpret
• Forexample,Outlook=sunnyisuseful,butOutlook=overcast and Outlook=rainy are not useful... Should we use Outlook?
2. Modify contingency tables (and formulae)
Osor c=Y U V W c=N X Y Z
Types of Attributes: Nominal
Modified MI:
MI(O,C) = P(i,j)log P(i,j)
i∈{s,o,r} j∈{c,c ̄}
= P(s,c)log P(s,c)
2 P(i)P(j) + P(s,c ̄)log + P(o, c ̄) log
2 P(s)P(c) P(o, c) log P(o, c)
P(r, c) log
P(s,c ̄ + 2 P(s)P(c ̄)
P(o, c ̄) + 2 P(o)P(c ̄)
P(r, c ̄) 2 P(r)P(c ̄)
2 P(o)P(c) P(r, c)
+ P(r, c ̄) log
2 P(r)P(c)
• Biased towards attributes with many values.
Types of Attributes: Nominal
Chi-square can be used as normal, with 6 observed/expected values.
• To control for score inflation, we need to consider “number of degrees of freedom”, and then use the significance test explicitly (beyond the scope of this subject)
Types of Attributes: Continuous
Continuous attributes
• Usually dealt with by estimating probability based on a Gaussian (normal) distribution
• With a large number of values, most random variables are normally distributed due to the Central Limit Theorem
• For small data sets or pathological features, we may need to use binomial/multinomial distributions
All of this is beyond the scope of this subject
Types of Attributes: Ordinal
Three possibilities, roughly in order of popularity:
1. Treat as binary
• Particularly appropriate for frequency counts where events are
low-frequency (e.g. words in tweets) 2. Treat as continuous
• The fact that we haven’t seen any intermediate values is usually not important
• Does have all of the technical downsides of continuous attributes, however
3. Treat as nominal (i.e. throw away ordering)
Multi-class problems
So far, we’ve only looked at binary (Y/N) classification tasks.
Multiclass (e.g. LA, NY, C, At, SF) classification tasks are usually much more difficult.
What makes a single feature good?
• Highly correlated with class
• Highly reverse correlated with class
• Highly correlated (or reverse correlated) with not class
... What if there are many classes?
What makes a feature bad?
• Irrelevant
• Correlated with other features
• Good at only predicting one class (but is this truly bad?)
Multi-class problems
So far, we’ve only looked at binary (Y/N) classification tasks.
Multiclass (e.g. LA, NY, C, At, SF) classification tasks are usually much more difficult.
• PMI, MI, χ2 are all calculated per-class
• (Some other feature selection metrics, e.g. Information Gain, work for all
classes at once)
• Need to make a point of selecting (hopefully uncorrelated) features for each class to give our classifier the best chance of predicting everything correctly.
Multi-class problems
So far, we’ve only looked at binary (Y/N) classification tasks.
Multiclass (e.g. LA, NY, C, At, SF) classification tasks are usually much more difficult.
Actual example (MI):
LA NY C At SF
chicago atlanta sf
hollywood atlanta atlanta yankees
httpbitlyczmk lol san cubs u u
la georgia lol chi chicago save
bears atl il ga
httpdealnaycom
Multi-class problems
So far, we’ve only looked at binary (Y/N) classification tasks.
Multiclass (e.g. LA, NY, C, At, SF) classification tasks are usually much more difficult.
Intuitive features:
LA NY C At SF
la angeles los chicago hollywood atlanta lakers
nyc chicago atlanta sf
york bears atl ny il ga
httpdealnaycom
francisco chicago httpbitlyczmk lol san
atlanta cubs u u
yankees la sf chi
georgia lol chicago save
Multi-class problems
So far, we’ve only looked at binary (Y/N) classification tasks.
Multiclass (e.g. LA, NY, C, At, SF) classification tasks are usually much more difficult.
Features for predicting not class:
LA NY C At SF
la angeles los chicago hollywood atlanta lakers
nyc chicago atlanta sf
york bears atl ny il ga
httpdealnaycom
chicago httpbitlyczmk lol san atlanta cubs u u
yankees la sf chi
georgia lol chicago save
Multi-class problems
So far, we’ve only looked at binary (Y/N) classification tasks.
Multiclass (e.g. LA, NY, C, At, SF) classification tasks are usually much more difficult.
Unintuitive features:
LA NY C At SF
chicago atlanta sf
hollywood atlanta atlanta yankees
cubs u u la georgia lol
chi chicago save
bears atl il ga
httpdealnaycom
francisco httpbitlyczmk lol san
What’s going on with MI?
Mutual Information is biased toward rare, uninformative features
• All probabilities: no notion of the raw frequency of events
• If a feature is seen rarely, but always with a given class, it will be seen as “good”
• Best features in the Twitter dataset only had MI of about 0.01 bits; 100th best for a given class had MI of about 0.002 bits
Glance into a few other common approaches to feature selection
A common (unsupervised) alternative
Term Frequency Inverse Document Frequency (TFIDF)
• Detect important words / Natural Language Processing
• Find words that are relevant to a document in a given document
collection
• To be relevant, a word should be
• Frequent enough in the corpus (TF). A word that occurs only 5 times in a corpus of 5,000,000 words is probably not too interesting
• Special enough (IDF). A very common word (“the”, “you”, ...) that occurs in (almost) every document is probably not too interesting
A common (unsupervised) alternative
Term Frequency Inverse Document Frequency (TFIDF)
• Detect important words / Natural Language Processing
• Find words that are relevant to a document in a given document
collection
• To be relevant, a word should be
• Frequent enough in the corpus (TF). A word that occurs only 5 times in a corpus of 5,000,000 words is probably not too interesting
• Special enough (IDF). A very common word (“the”, “you”, ...) that occurs in (almost) every document is probably not too interesting
tfidf(d,t,D)=tf +idf
tf =log(1+freq(t,d))
idf = log |D| count(d ∈ D : t ∈ d)
d=document, t=term, D=document collection; |D|=number of documents in D
Embedded Methods
Some ML models include feature selection inherently 1. Decision trees: Generalization of 1-R
2. Regression models with regularization
house price = β0 +β1 ×size+β2 ×location+β3 ×age
Regularization (or ‘penalty’) nudges the weight β of unimportant features towards zero
https://towardsdatascience.com/a- beginners- guide- to- decision- tree- classification- 6d3209353ea?gi=e0
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