MATLAB Examples

## Contents

% DEMO_FEATURE_SELECT - demonstrate feature selection algorithms on some
% toy examples.

## Introduction

The problem of Feature Selection is: Given a (usually large) number of noisy and partly redundant variables and a target that we would like to predict, choose a small but indicative subset as input to a classification or regression technique. While wrappers employ one specific such technique, filters try to come up with a "most informative subset" (in some sense to be defined). Several such criteria are based on Shannon information (mutual information between two variables, or interaction information between larger subsets). The Matlab function select_features captures several criteria previously proposed in the literature, and some generalizations thereof. For some more background and comparisons, see e.g.: Gavin Brown, A New Perspective for Information Theoretic Feature Selection, Artificial Intelligence and Statistics, 2009.

In the first example, we generate a very simple dependence: X1,X2,X3 are
normally distributed variables. Our target, X3, is a noisy observation
of X1. X2 is uncorrelated with either of them.
X1
\
X2  X3
nsamples      = 1000;
data          = zeros(nsamples,3);
data(:,[1 2]) = randn(nsamples,2);
data(:,3)    =  0.5*data(:,1) + 0.5*randn(nsamples,1);

For calculating mutual information, continuous variables have to be discretized. The function quantize offers several options to do this. We choose the simplest case of binary variables.

data_quant=quantize(data,'levels',2);

Then, we run the default feature selection algorithm, called 'first order utility'. It is based on approximating the mutual information between the set of selected variables and the target by expanding interaction information terms of up to degree 2. In each step, the variable with the highest estimated incremental gain is selected greedily. The output distinguishes between relevance, i.e., mutual information between a feature and the target; redundancy, i.e., mutual information between different variables; and conditional redundancy, which measures the increase of mutual information between the previously selected variables and the target, conditional on a selected variable.

[steps,sel_flag,rel,red,cond_red] = select_features(data_quant(:,1:2),data_quant(:,3),2);
using settings: degree 2, pessimistic 0, dir_fwd 1, dir_bwd 0, prior 0.000000, red_wt 1.000000, cond_red_wt 1.000000
1. select 1, score 0.201650 (relevance 0.201650, redundancy 0.000000, conditional redundancy 0.000000, red weight 1.000000, cond red weight 1.000000)
2. select 2, score 0.000419 (relevance 0.000416, redundancy 0.002263, conditional redundancy 0.002266, red weight 1.000000, cond red weight 1.000000)
Elapsed time is 1.622600 seconds.

As expected, X1 gets a significantly higher score than X2. You can inspect the results more closely in the output arguments (steps for the sequence of selections, scores, and (conditional) redundancies; sel_flag for the finally selected variables; and the final values of relevance and conditional redundancy, for all variables. The final scores are computed as sum([rel; - red; cond_red]).

## Smoothing

A general issue with (especially higher-order) interaction information is sparsity of data. By subdividing our observed data into many categories, we are led to believe spurious associations. For example, look what happens to the above example if we increase the quantization level and decrease the number of samples:

nsamples      = 100;
data          = zeros(nsamples,3);
data(:,[1 2]) = randn(nsamples,2);
data(:,3)    =  0.5*data(:,1) + 0.5*randn(nsamples,1);
data_quant=quantize(data,'levels',5);
[steps,sel_flag,rel,red,cond_red] = select_features(data_quant(:,1:2),data_quant(:,3),2);
using settings: degree 2, pessimistic 0, dir_fwd 1, dir_bwd 0, prior 0.000000, red_wt 1.000000, cond_red_wt 1.000000
1. select 1, score 0.355160 (relevance 0.355160, redundancy 0.000000, conditional redundancy 0.000000, red weight 1.000000, cond red weight 1.000000)
2. select 2, score 0.611538 (relevance 0.209419, redundancy 0.142521, conditional redundancy 0.544640, red weight 1.000000, cond red weight 1.000000)
Elapsed time is 0.008645 seconds.

Notice that the conditional redundancy of the uncorrelated variable X2 (after addition of X1) now seems to be higher than the mutual information between X1 and X2. Several remedies have been suggested, e.g., downweighting the (conditional) redundancy terms (You can explore these options in the select_features function). In contrast, we propose to use a common method in Bayesian statistics, namely adding a prior in the form of "pseudo-samples' drawn from the marginal distributions.

[steps,sel_flag,rel,red,cond_red] = select_features(data_quant(:,1:2),data_quant(:,3),2,'prior',1);
using settings: degree 2, pessimistic 0, dir_fwd 1, dir_bwd 0, prior 0.010000, red_wt 1.000000, cond_red_wt 1.000000
1. select 1, score 0.206970 (relevance 0.206970, redundancy 0.000000, conditional redundancy 0.000000, red weight 1.000000, cond red weight 1.000000)
2. select 2, score 0.123660 (relevance 0.123831, redundancy 0.090154, conditional redundancy 0.089983, red weight 1.000000, cond red weight 1.000000)
Elapsed time is 0.048254 seconds.

In this case, we are adding one pseudo-sample to each possible combination of joint variable values. As a result, while all scores (including the one for X1) decrease, the irrelevant X2 is reduced much more rapidly.

## Diamond

This example contains 4 variables in the well-known diamond shape. X4 is our target.

X1
/  \
X2   X4
\   /
X3
nsamples=10000;
data=zeros(nsamples,4);
data(:,1)=randn(nsamples,1);
data(:,2)= 0.5*data(:,1) + 0.5*randn(nsamples,1);
data(:,4)= 0.5*data(:,1) + 0.5*randn(nsamples,1);
data(:,3)= 0.5*data(:,2) + 0.5*data(:,4) + 0.5*randn(nsamples,1);

data_quant=quantize(data,'levels',2);
[steps,sel_flag,rel,red,cond_red] = select_features(data_quant(:,1:3),data_quant(:,4),3, 'degree', 3);
using settings: degree 3, pessimistic 0, dir_fwd 1, dir_bwd 0, prior 0.000000, red_wt 1.000000, cond_red_wt 1.000000
1. select 1, score 0.177202 (relevance 0.177202, redundancy 0.000000, conditional redundancy 0.000000, red weight 1.000000, cond red weight 1.000000)
2. select 3, score 0.065645 (relevance 0.156249, redundancy 0.142378, conditional redundancy 0.051774, red weight 1.000000, cond red weight 1.000000)
3. select 2, score 0.000596 (relevance 0.081238, redundancy 0.260985, conditional redundancy 0.180343, red weight 1.000000, cond red weight 1.000000)
Elapsed time is 0.482361 seconds.

Since X3 depends on X2 as well, X1 receives a higher score. Clearly, due to the common dependency, X2 bears some mutual information on X4. Note, however, how this is outweighed by the interaction term (redundancy - conditional redundancy): In fact, once we know X1, X2 cannot provide any additional information about X4.

## Higher-Order interaction

Assume binary variables X1 .. X5, where our target, X5, is the logical exclusive or of X1..X3. We include the independent X4 for illustration.

X1   X2  X3  X4
\   |  /
X5
data=zeros(nsamples,5);
data(:,1:4) = floor(rand(nsamples,4)/0.5);
data(:,5)   = xor(xor(data(:,1),data(:,2)),data(:,3));
[steps,sel_flag,rel,red,cond_red] = select_features(data(:,1:4),data(:,5),4);
using settings: degree 2, pessimistic 0, dir_fwd 1, dir_bwd 0, prior 0.000000, red_wt 1.000000, cond_red_wt 1.000000
1. select 1, score 0.000072 (relevance 0.000072, redundancy 0.000000, conditional redundancy 0.000000, red weight 1.000000, cond red weight 1.000000)
2. select 2, score 0.000146 (relevance 0.000002, redundancy 0.000067, conditional redundancy 0.000211, red weight 1.000000, cond red weight 1.000000)
3. select 3, score 0.000242 (relevance 0.000062, redundancy 0.000080, conditional redundancy 0.000260, red weight 1.000000, cond red weight 1.000000)
4. select 4, score 0.000046 (relevance 0.000012, redundancy 0.000158, conditional redundancy 0.000191, red weight 1.000000, cond red weight 1.000000)
Elapsed time is 0.179574 seconds.

The default algorithm only considers interactions of degree 2, and therefore cannot find the relationship. We have to switch to degree 3:

[steps,sel_flag,rel,red,cond_red] = select_features(data(:,1:4),data(:,5),4,'degree', 3);
using settings: degree 3, pessimistic 0, dir_fwd 1, dir_bwd 0, prior 0.000000, red_wt 1.000000, cond_red_wt 1.000000
1. select 1, score 0.000072 (relevance 0.000072, redundancy 0.000000, conditional redundancy 0.000000, red weight 1.000000, cond red weight 1.000000)
2. select 2, score 0.000146 (relevance 0.000002, redundancy 0.000067, conditional redundancy 0.000211, red weight 1.000000, cond red weight 1.000000)
3. select 3, score 0.999352 (relevance 0.000062, redundancy 0.000499, conditional redundancy 0.999789, red weight 1.000000, cond red weight 1.000000)
4. select 4, score 0.000206 (relevance 0.000012, redundancy 0.000196, conditional redundancy 0.000390, red weight 1.000000, cond red weight 1.000000)
Elapsed time is 0.327354 seconds.

The algorithm selects variables more or less randomly, until two of the three determinants have been included. At this point, it discovers the strong significance of the missing one. In cases like this, the function will have more guidance when going backwards, i.e., starting with all variables, and iteratively dropping the least significant one. It will delay discarding any of the relevant variables as long as possible:

[steps,sel_flag,rel,red,cond_red] = select_features(data(:,1:4),data(:,5),4,'degree', 3, 'init', [ 1 2 3 4], 'direction', 'b');
using settings: degree 3, pessimistic 0, dir_fwd 0, dir_bwd 1, prior 0.000000, red_wt 1.000000, cond_red_wt 1.000000
done initial selection
1. drop 4, score -0.000206 (relevance 0.000012, redundancy 0.000196, conditional redundancy 0.000390, red weight 1.000000, cond red weight 1.000000)
2. drop 1, score -0.999328 (relevance 0.000072, redundancy 0.000487, conditional redundancy 0.999743, red weight 1.000000, cond red weight 1.000000)
3. drop 2, score -0.000181 (relevance 0.000002, redundancy 0.000078, conditional redundancy 0.000257, red weight 1.000000, cond red weight 1.000000)
4. drop 3, score -0.000062 (relevance 0.000062, redundancy -0.000000, conditional redundancy -0.000000, red weight 1.000000, cond red weight 1.000000)
Elapsed time is 0.648296 seconds.

This concludes our simple selection of feature selection examples. There is a lot to explore - hope you have fun with your own experiments!