Feature Selection, Feature Extraction

Need for reduction

• Classification of leukemia tumors from microarray gene expression data1

– 72 patients (data points)– 7130 features (expression levels of different genes)

• Text mining, document classification– features are words

• Quantitative Structure-Activity Relationship (QSAR)– features are molecular descriptors, there exist plenty

of them

1 Xing, Jordan, Karp, Feature Selection for High-Dimensional Genomic Microarray Data, 2001

QSAR

• biological activity– an expression describing the beneficial or adverse

effects of a drug on living matter• Structure-Activity Relationship (SAR)– hypotheses that similar molecules have similar

activities• molecular descriptor– mathematical procedure transforms chemical

information encoded within a symbolic representation of a molecule into a useful number

Molecular descriptor

adjacency (connectivity) matrix

total adj. index AV – sum all aij

measure of the graph connectedness

2.183

Randic connectivity indices measure of the molecular branching

QSAR• Form a mathematical/statistical relationship (model) between structural

(physiochemical) properties and activity.• The mathematical expression can then be used to predict the biological

response of other chemical structures.

descriptor

biol

ogic

al a

ctivi

ty

Selection vs. Extraction

• In feature selection we try to find the best subset of the input feature set.

• In feature extraction we create new features based on transformation or combination of the original feature set.

• Both selection and extraction lead to the dimensionality reduction.

• No clear cut evidence that one of them is superior to the other on all types of task.

Why to do it?1. We’re interested in features – we want to

know which are relevant. If we fit a model, it should be interpretable.• facilitate data visualization and data understanding• reduce experimental costs (measurements)

2. We’re interested in prediction – features are not interesting in themselves, we just want to build a good predictor.• faster training• defy the curse of dimensionality

Feature selection(FS)

Classification of FS methods

• Filter– Assess the relevance of features only by looking at the

intrinsic properties of the data.– Usually, calculate the feature relevance score and remove

low-scoring features.• Wrapper– Bundle the search for best model with the FS.– Generate and evaluate various subsets of features. The

evaluation is obtained by training and testing a specific ML model.

• Embedded– The search for an optimal subset is built into the classifier

construction (e.g. decision trees).

Filter methods

• Two steps (score-and-filter approach)1. assess each feature individually for ist potential in

discriminating among classes in the data2. features falling beyong threshold are eliminated

• Advantages:– easily scale to high-dimensional data– simple and fast– independent of the classification algorithm

• Disadvantages:– ignore the interaction with the classifier– most techniques are univariate (each feature is

considered separately)

Scores in filter methods• Distance measures– Euclidean distance

• Dependence measures– Pearson correlation coefficient– χ2-test – t-test– AUC

• Information measures– information gain– mutual information

• complexity: O(d)

Wrappers

• Search for the best feature subset in combination with a fixed classification method.

• The goodness of a feature subset is determined using cross-validation (k-fold, LOOCV)

• Advantages:– interaction between feature subset and model selection– take into account feature dependencies– generally more accurate

• Disadvantages:– higher risk of overfitting than filter methods– very computationally intensive

Exhaustive search

• Evaluate all possible subsets using exhaustive search – this leads to the optimum subset.

• For a total of d variables, and subset of size p, the total number of possible subsets is

• complexity: O(2d) (exponential)• Various strategies how to reduce the search space.– They are still O(2d), but much faster (at least 1000-times)– e.g. “branch and bound”

!

! !

d

d p pe.g. d = 100, p = 10 → ≈2×1013

Stochastic

• Genetic algorithms• Simulated Annealing

Difficulty in Searching Global Optima

startingpoint

descenddirection

local minima

global minima

barrier to local search

Introduction to Simulated Annealing, Dr. Gildardo Sánchez ITESM Campus Guadalajara

Consequences of the Occasional Ascents

Help escaping the local optima.

desired effect

Might pass global optima after reaching it

adverse effect

Introduction to Simulated Annealing, Dr. Gildardo Sánchez ITESM Campus Guadalajara

Simulated annealing

• Slowly cool down a heated solid, so that all particles arrange in the ground energy state.

• At each temperature wait until the solid reaches its thermal equilibrium.

• Probability of being in a state with energy E:

1

expB

EP E

Z T k T

E … energyT … temperaturekB … Boltzmann constantZ(T) … normalization factor

Cooling simulation

• At a fixed temperature T :– Perturb (randomly) the current state to a new state– E is the difference in energy between current and new state– If E < 0 (new state is lower), accept new state as current state– If E 0 , accept new state with probability

• Eventually the systems evolves into thermal equilibrium at temperature T

• When equilibrium is reached, temperature T can be lowered and the process can be repeated

expB

EP

k T

Metropolis, 1953Metropolis algorithm

Simulated annealing

• Same algorithm can be used for combinatorial optimization problems:– Energy E corresponds to the objective function C– Temperature T is parameter controlled within the

algorithm

Algorithm

initialize;

REPEAT

REPEAT

perturb ( config.i config.j, Cij);

IF Cij < 0 THEN accept

ELSE IF exp(-Cij/T) > random[0,1) THEN accept;

IF accept THEN update(config.j);

UNTIL equilibrium is approached sufficient closely;

T := next_lower(T);

UNTIL system is frozen or stop criterion is reached

Parameters

• Choose the start value of T so that in the beginning nearly all perturbations are accepted (exploration), but not too big to avoid long run times

• At each temperature, search is allowed to proceed for a certain number of steps, L(k).

• The function next_lower (T(k)) is generally a simple function to decrease T, e.g. a fixed part (80%) of current T.

• At the end T is so small that only a very small number of the perturbations is accepted (exploitation).

• The choice of parameters {T(k), L(k)} is called the cooling schedule.

• If possible, always try to remember explicitly the best solution found so far; the algorithm itself can leave its best solution and not find it again.

Deterministic

• Sequential Forward Selection (SFS)• Sequential Backward Selection (SBS)• “ Plus q take away r ” Selection• Sequential Forward Floating Search (SFFS)• Sequential Backward Floating Search (SBFS)

Sequential Forward Selection

• SFS• At the beginning select the best feature using

a scalar criterion function.• Add one feature at a time which along with

already selected features maximizes the criterion function.

• A greedy algorithm, cannot retract (also called nesting effect).

• Complexity is O(d)

Sequential Backward Selection

• SBS• At the beginning select all d features.• Delete one feature at a time and select the

subset which maximize the criterion function.• Also a greedy algorithm, cannot retract.• Complexity is O(d).

“Plus q take away r” Selection

• At first add q features by forward selection, then discard r features by backward selection

• Need to decide optimal q and r• No subset nesting problems Like SFS and SBS

Sequential Forward Floating Search

• SFFS• It is a generalized “plus q take away r”

algorithm• The value of q and r are determined

automatically• Close to optimal solution• Affordable computational cost • Also in backward disguise

Embedded FS

• The feature selection process is done inside the ML algorithm.

• Decision trees– In final tree, only a subset of features are used

• Regularization– It effectively “shuts down” unnecessary features.– Pruning in NN.

Feature extraction(FE)

• FS – indetify and select the “best” features with respect to the target task.– Selected features retain their original physical

interpretation.• FE – create new features as a transformation

(combination) of original features. Usually followed by FS.– May provide better discriminatory ability than the

best subset.– Do not retain the original physical interpretation,

may not have clear meaning.

Principal Component Analysis(PCA)

x1

x2

x1

x2

Make datato have zero mean (i.e. move data into [0, 0] point).

centering

x1

x2

This is a line given by equation w0 + w1x1 + w2x2

This is another linew’0 + w’1x1 + w’2x2

x1

x2

The variability in data is highest along this line. It is called 1st principal component.

And this is 2nd principal component.

x1

x2

Principal components (PC’s) are linear combinations of original coordinates.

The coefficients of linear combination (w0, w1, …) are called loadings.

In the transformed coordinate system, individual data points have different coordinates, these are called scores.

w0 + w1x1 + w2x2

w’0 + w’1x1 + w’2x2

• PCA - orthogonal linear transformation that changes the data into a new coordinate system such that the variance is put in order from the greatest to the least.

• Solve the problem = find new orthogonal coordinate system = find loadings

• PC’s (vectors) and their corresponding variances (scalars) are found by eigenvalue decompositions of the covariance matrix C = XXT of the xi variables.– Eigenvector corresponding to the largest eigenvalue is 1st PC.– The 2nd eigenvector (the 2nd largest eigenvalue) is orthogonal

to the 1st one. …• Eigenvalue decomposition is computed using standard

algorithms: eigen decomposition of covariance matrix (e.g. QR algorithm), SVD of mean centered data matrix.

Interpretation of PCA

• New variables (PCs) have a variance equal to their corresponding eigenvalue

Var(Yi)= i for all I = 1…p• Small i small variance data changes little

in the direction of component Yi

• The relative variance explained by each PC is given by li / li

How many components?

• Enough PCs to have a cumulative variance explained by the PCs that is >50-70%

• Kaiser criterion: keep PCs with eigenvalues >1• Scree plot: represents the ability of PCs to

explain de variation in data