fsrmrmr

Rank features for regression using minimum redundancy maximum relevance (MRMR) algorithm

Since R2022a

Description

fsrmrmr ranks features (predictors) using the MRMR algorithm to identify important predictors for regression problems.

To perform MRMR-based feature ranking for classification, see fscmrmr.

idx = fsrmrmr(Tbl,ResponseVarName) returns the predictor indices, idx, ordered by predictor importance (from most important to least important). The table Tbl contains the predictor variables and a response variable, ResponseVarName, which contains the response values. You can use idx to select important predictors for regression problems.

idx = fsrmrmr(Tbl,formula) specifies a response variable and predictor variables to consider among the variables in Tbl by using formula. For example, fsrmrmr(cartable,"MPG ~ Acceleration + Displacement + Horsepower") ranks the Acceleration, Displacement, and Horsepower predictors in cartable using the response variable MPG in cartable.

idx = fsrmrmr(Tbl,Y) ranks predictors in Tbl using the response variable Y.

example

idx = fsrmrmr(X,Y) ranks predictors in X using the response variable Y.

idx = fsrmrmr(___,Name=Value) specifies additional options using one or more name-value arguments in addition to any of the input argument combinations in the previous syntaxes. For example, you can specify observation weights.

example

[idx,scores] = fsrmrmr(___) also returns the predictor scores scores. A large score value indicates that the corresponding predictor is important.

Examples

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Simulate 1000 observations from the model $y={x}_{4}+2{x}_{7}+e$.

• $X=\left\{{x}_{1},...,{x}_{10}\right\}$ is a 1000-by-10 matrix of standard normal elements.

• e is a vector of random normal errors with mean 0 and standard deviation 0.3.

rng("default") % For reproducibility
X = randn(1000,10);
Y = X(:,4) + 2*X(:,7) + 0.3*randn(1000,1);

Rank the predictors based on importance.

idx = fsrmrmr(X,Y);

Select the top two most important predictors.

idx(1:2)
ans = 1×2

7     4

The function identifies the seventh and fourth columns of X as the most important predictors of Y.

Load the carbig data set, and create a table containing the different variables. Include the response variable MPG as the last variable in the table.

cartable = table(Acceleration,Cylinders,Displacement, ...
Horsepower,Model_Year,Weight,Origin,MPG);

Rank the predictors based on importance. Specify the response variable.

[idx,scores] = fsrmrmr(cartable,"MPG");

Note: If fsrmrmr uses a subset of variables in a table as predictors, then the function indexes the subset of predictors only. The returned indices do not count the variables that the function does not rank (including the response variable).

Create a bar plot of the predictor importance scores. Use the predictor names for the x-axis tick labels.

bar(scores(idx))
xlabel("Predictor rank")
ylabel("Predictor importance score")
predictorNames = cartable.Properties.VariableNames(1:end-1);
xticklabels(strrep(predictorNames(idx),"_","\_"))
xtickangle(45)

The drop in score between the second and third most important predictors is large, while the drops after the third predictor are relatively small. A drop in the importance score represents the confidence of feature selection. Therefore, the large drop implies that the software is confident of selecting the second most important predictor, given the selection of the most important predictor. The small drops indicate that the differences in predictor importance are not significant.

Select the top two most important predictors.

idx(1:2)
ans = 1×2

3     5

The third column of cartable is the most important predictor of MPG. The fifth column of cartable is the second most important predictor of MPG.

To improve the performance of a regression model, generate new features by using genrfeatures and then select the most important predictors by using fsrmrmr. Compare the test set performance of the model trained using only original features to the performance of the model trained using the most important generated features.

Read power outage data into the workspace as a table. Remove observations with missing values, and display the first few rows of the table.

Tbl = rmmissing(outages);
Region           OutageTime        Loss     Customers     RestorationTime            Cause
_____________    ________________    ______    __________    ________________    ___________________

{'SouthWest'}    2002-02-01 12:18    458.98    1.8202e+06    2002-02-07 16:50    {'winter storm'   }
{'SouthEast'}    2003-02-07 21:15     289.4    1.4294e+05    2003-02-17 08:14    {'winter storm'   }
{'West'     }    2004-04-06 05:44    434.81    3.4037e+05    2004-04-06 06:10    {'equipment fault'}
{'MidWest'  }    2002-03-16 06:18    186.44    2.1275e+05    2002-03-18 23:23    {'severe storm'   }
{'West'     }    2003-06-18 02:49         0             0    2003-06-18 10:54    {'attack'         }
{'NorthEast'}    2003-07-16 16:23    239.93         49434    2003-07-17 01:12    {'fire'           }
{'MidWest'  }    2004-09-27 11:09    286.72         66104    2004-09-27 16:37    {'equipment fault'}
{'SouthEast'}    2004-09-05 17:48    73.387         36073    2004-09-05 20:46    {'equipment fault'}

Some of the variables, such as OutageTime and RestorationTime, have data types that are not supported by regression model training functions like fitrensemble.

Partition the data set into a training set and a test set by using cvpartition. Use approximately 70% of the observations as training data and the other 30% as test data.

rng("default") % For reproducibility of the data partition
c = cvpartition(length(Tbl.Loss),"Holdout",0.30);
trainTbl = Tbl(training(c),:);
testTbl = Tbl(test(c),:);

Identify and remove outliers of Customers from the training data by using the isoutlier function.

[customersIdx,customersL,customersU] = isoutlier(trainTbl.Customers);
trainTbl(customersIdx,:) = [];

Remove the outliers of Customers from the test data by using the same lower and upper thresholds computed on the training data.

testTbl(testTbl.Customers < customersL | testTbl.Customers > customersU,:) = [];

Generate 35 features from the predictors in trainTbl that can be used to train a bagged ensemble. Specify the Loss variable as the response and MRMR as the feature selection method.

[Transformer,newTrainTbl] = genrfeatures(trainTbl,"Loss",35, ...
TargetLearner="bag",FeatureSelectionMethod="mrmr");

The returned table newTrainTbl contains various engineered features. The first three columns of newTrainTbl are the original features in trainTbl that can be used to train a regression model using the fitrensemble function, and the last column of newTrainTbl is the response variable Loss.

originalIdx = 1:3;
c(Region)    Customers        c(Cause)         Loss
_________    __________    _______________    ______

SouthEast    1.4294e+05    winter storm        289.4
West         3.4037e+05    equipment fault    434.81
MidWest      2.1275e+05    severe storm       186.44
West                  0    attack                  0
MidWest           66104    equipment fault    286.72
SouthEast         36073    equipment fault    73.387
SouthEast    1.0698e+05    winter storm       46.918
NorthEast    1.0444e+05    winter storm       255.45

Rank the predictors in newTrainTbl. Specify the response variable.

[idx,scores] = fsrmrmr(newTrainTbl,"Loss");

Note: If fsrmrmr uses a subset of variables in a table as predictors, then the function indexes the subset only. The returned indices do not count the variables that the function does not rank (including the response variable).

Create a bar plot of the predictor importance scores.

bar(scores(idx))
xlabel("Predictor rank")
ylabel("Predictor importance score")

Because there is a large gap between the scores of the seventh and eighth most important predictors, select the seven most important features to train a bagged ensemble model.

importantIdx = idx(1:7);
fsMdl = fitrensemble(newTrainTbl(:,importantIdx),newTrainTbl.Loss, ...
Method="Bag");

For comparison, train another bagged ensemble model using the three original predictors that can be used for model training.

originalMdl = fitrensemble(newTrainTbl(:,originalIdx),newTrainTbl.Loss, ...
Method="Bag");

Transform the test data set.

newTestTbl = transform(Transformer,testTbl);

Compute the test mean squared error (MSE) of the two regression models.

fsMSE = loss(fsMdl,newTestTbl(:,importantIdx), ...
newTestTbl.Loss)
fsMSE = 1.0867e+06

originalMSE = loss(originalMdl,newTestTbl(:,originalIdx), ...
newTestTbl.Loss)
originalMSE = 1.0961e+06

fsMSE is less than originalMSE, which suggests that the bagged ensemble trained on the most important generated features performs slightly better than the bagged ensemble trained on the original features.

Input Arguments

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Sample data, specified as a table. Multicolumn variables and cell arrays other than cell arrays of character vectors are not allowed.

Each row of Tbl corresponds to one observation, and each column corresponds to one predictor variable. Optionally, Tbl can contain additional columns for a response variable and observation weights. The response variable must be a numeric vector.

• If Tbl contains the response variable, and you want to use all remaining variables in Tbl as predictors, then specify the response variable by using ResponseVarName. If Tbl also contains the observation weights, then you can specify the weights by using Weights.

• If Tbl contains the response variable, and you want to use only a subset of the remaining variables in Tbl as predictors, then specify the subset of variables by using formula.

• If Tbl does not contain the response variable, then specify a response variable by using Y. The response variable and Tbl must have the same number of rows.

If fsrmrmr uses a subset of variables in Tbl as predictors, then the function indexes the predictors using only the subset. The values in the CategoricalPredictors name-value argument and the output argument idx do not count the predictors that the function does not rank.

If Tbl contains a response variable, then fsrmrmr considers NaN values in the response variable to be missing values. fsrmrmr does not use observations with missing values in the response variable.

Data Types: table

Response variable name, specified as a character vector or string scalar containing the name of a variable in Tbl.

For example, if a response variable is the column Y of Tbl (Tbl.Y), then specify ResponseVarName as "Y".

Data Types: char | string

Explanatory model of the response variable and a subset of the predictor variables, specified as a character vector or string scalar in the form "Y ~ x1 + x2 + x3". In this form, Y represents the response variable, and x1, x2, and x3 represent the predictor variables.

To specify a subset of variables in Tbl as predictors, use a formula. If you specify a formula, then fsrmrmr does not rank any variables in Tbl that do not appear in formula.

The variable names in the formula must be both variable names in Tbl (Tbl.Properties.VariableNames) and valid MATLAB® identifiers. You can verify the variable names in Tbl by using the isvarname function. If the variable names are not valid, then you can convert them by using the matlab.lang.makeValidName function.

Data Types: char | string

Response variable, specified as a numeric vector. Each row of Y represents the response of the corresponding row of X or Tbl.

fsrmrmr considers NaN values in Y to be missing values. fsrmrmr does not use observations with missing values for Y.

Data Types: single | double

Predictor data, specified as a numeric matrix. Each row of X corresponds to one observation, and each column corresponds to one predictor variable.

Data Types: single | double

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Example: fsrmrmr(Tbl,"y",CategoricalPredictors=[1 2 4],Weights="w") specifies that the y column of Tbl is the response variable, the w column of Tbl contains the observation weights, and the first, second, and fourth columns of Tbl (with the y and w columns removed) are categorical predictors.

List of categorical predictors, specified as one of the values in this table.

ValueDescription
Vector of positive integers

Each entry in the vector is an index value indicating that the corresponding predictor is categorical. The index values are between 1 and p, where p is the number of predictors used to train the model.

If fsrmrmr uses a subset of input variables as predictors, then the function indexes the predictors using only the subset. The CategoricalPredictors values do not count the response variable, observation weights variable, or any other variables that the function does not use.

Logical vector

A true entry means that the corresponding predictor is categorical. The length of the vector is p.

Character matrixEach row of the matrix is the name of a predictor variable. The names must match the names in Tbl. Pad the names with extra blanks so each row of the character matrix has the same length.
String array or cell array of character vectorsEach element in the array is the name of a predictor variable. The names must match the names in Tbl.
"all"All predictors are categorical.

By default, if the predictor data is a table (Tbl), fsrmrmr assumes that a variable is categorical if it is a logical vector, unordered categorical vector, character array, string array, or cell array of character vectors. If the predictor data is a matrix (X), fsrmrmr assumes that all predictors are continuous. To identify any other predictors as categorical predictors, specify them by using the CategoricalPredictors name-value argument.

Example: "CategoricalPredictors","all"

Example: CategoricalPredictors=[1 5 6 8]

Data Types: single | double | logical | char | string | cell

Indicator for whether to use missing values in predictors, specified as either true to use the values for ranking, or false to discard the values.

fsrmrmr considers NaN, '' (empty character vector), "" (empty string), <missing>, and <undefined> values to be missing values.

If you specify UseMissing as true, then fsrmrmr uses missing values for ranking. For a categorical variable, fsrmrmr treats missing values as an extra category. For a continuous variable, fsrmrmr places NaN values in a separate bin for binning.

If you specify UseMissing as false, then fsrmrmr does not use missing values for ranking. Because fsrmrmr computes mutual information for each pair of variables, the function does not discard an entire row when values in the row are partially missing. fsrmrmr uses all pair values that do not include missing values.

Example: "UseMissing",true

Example: UseMissing=true

Data Types: logical

Verbosity level, specified as a nonnegative integer. The value of Verbose controls the amount of diagnostic information that the software displays in the Command Window.

• 0 — fsrmrmr does not display any diagnostic information.

• 1 — fsrmrmr displays the elapsed times for computing mutual information and ranking predictors.

• ≥ 2 — fsrmrmr displays the elapsed times and more messages related to computing mutual information. The amount of information increases as you increase the Verbose value.

Example: Verbose=1

Data Types: single | double

Observation weights, specified as a vector of scalar values or the name of a variable in Tbl. The function weights the observations in each row of X or Tbl with the corresponding value in Weights. The size of Weights must equal the number of rows in X or Tbl.

If you specify the input data as a table Tbl, then Weights can be the name of a variable in Tbl that contains a numeric vector. In this case, you must specify Weights as a character vector or string scalar. For example, if the weight vector is the column W of Tbl (Tbl.W), then specify Weights="W".

fsrmrmr normalizes the weights to add up to one.

Data Types: single | double | char | string

Output Arguments

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Indices of predictors in X or Tbl ordered by predictor importance, returned as a 1-by-r numeric vector, where r is the number of ranked predictors.

If fsrmrmr uses a subset of variables in Tbl as predictors, then the function indexes the predictors using only the subset. For example, suppose Tbl includes 10 columns and you specify the last five columns of Tbl as the predictor variables by using formula. If idx(3) is 5, then the third most important predictor is the 10th column in Tbl, which is the fifth predictor in the subset.

Predictor scores, returned as a 1-by-r numeric vector, where r is the number of ranked predictors.

A large score value indicates that the corresponding predictor is important. Also, a drop in the feature importance score represents the confidence of feature selection. For example, if the software is confident of selecting a feature x, then the score value of the next most important feature is much smaller than the score value of x.

• If you use X to specify the predictors or use all the variables in Tbl as predictors, then the values in scores have the same order as the predictors in X or Tbl.

• If you specify a subset of variables in Tbl as predictors, then the values in scores have the same order as the subset.

For example, suppose Tbl includes 10 columns and you specify the last five columns of Tbl as the predictor variables by using formula. Then, score(3) contains the score value of the 8th column in Tbl, which is the third predictor in the subset.

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Mutual Information

The mutual information between two variables measures how much uncertainty of one variable can be reduced by knowing the other variable.

The mutual information I of the discrete random variables X and Z is defined as

$I\left(X,Z\right)={\sum }_{i,j}P\left(X={x}_{i},Z={z}_{j}\right)\mathrm{log}\frac{P\left(X={x}_{i},Z={z}_{j}\right)}{P\left(X={x}_{i}\right)P\left(Z={z}_{j}\right)}.$

If X and Z are independent, then I equals 0. If X and Z are the same random variable, then I equals the entropy of X.

The fsrmrmr function uses this definition to compute the mutual information values for both categorical (discrete) and continuous variables. For each continuous variable, including the response, fsrmrmr discretizes the variable into 256 bins or the number of unique values in the variable if it is less than 256. The function finds optimal bivariate bins for each pair of variables using the adaptive algorithm [2].

Algorithms

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Minimum Redundancy Maximum Relevance (MRMR) Algorithm

The MRMR algorithm [1] finds an optimal set of features that is mutually and maximally dissimilar and can represent the response variable effectively. The algorithm minimizes the redundancy of a feature set and maximizes the relevance of a feature set to the response variable. The algorithm quantifies the redundancy and relevance using the mutual information of variables—pairwise mutual information of features and mutual information of a feature and the response. You can use this algorithm for regression problems.

The goal of the MRMR algorithm is to find an optimal set S of features that maximizes VS, the relevance of S with respect to a response variable y, and minimizes WS, the redundancy of S, where VS and WS are defined with mutual information I:

${V}_{S}=\frac{1}{|S|}{\sum }_{x\in S}I\left(x,y\right),$

${W}_{S}=\frac{1}{{|S|}^{2}}{\sum }_{x,z\in S}I\left(x,z\right).$

|S| is the number of features in S.

Finding an optimal set S requires considering all 2|Ω| combinations, where Ω is the entire feature set. Instead, the MRMR algorithm ranks features through the forward addition scheme, which requires O(|Ω|·|S|) computations, by using the mutual information quotient (MIQ) value.

${\text{MIQ}}_{x}=\frac{{V}_{x}}{{W}_{x}},$

where Vx and Wx are the relevance and redundancy of a feature, respectively:

${V}_{x}=I\left(x,y\right),$

${W}_{x}=\frac{1}{|S|}{\sum }_{z\in S}I\left(x,z\right).$

The fsrmrmr function ranks all features in Ω and returns idx (the indices of features ordered by feature importance) using the MRMR algorithm. Therefore, the computation cost becomes O(|Ω|2). The function quantifies the importance of a feature using a heuristic algorithm and returns a score (scores). A large score value indicates that the corresponding predictor is important. Also, a drop in the feature importance score represents the confidence of feature selection. For example, if the software is confident of selecting a feature x, then the score value of the next most important feature is much smaller than the score value of x. You can use the outputs to find an optimal set S for a given number of features.

fsrmrmr ranks features as follows:

1. Select the feature with the largest relevance, $\underset{x\in \Omega }{\mathrm{max}}{V}_{x}$. Add the selected feature to an empty set S.

2. Find the features with nonzero relevance and zero redundancy in the complement of S, Sc.

• If Sc does not include a feature with nonzero relevance and zero redundancy, go to step 4.

• Otherwise, select the feature with the largest relevance, $\underset{x\in {S}^{c},\text{\hspace{0.17em}}{W}_{x}=0}{\mathrm{max}}{V}_{x}$. Add the selected feature to the set S.

3. Repeat Step 2 until the redundancy is not zero for all features in Sc.

4. Select the feature that has the largest MIQ value with nonzero relevance and nonzero redundancy in Sc, and add the selected feature to the set S.

$\underset{x\in {S}^{c}}{\mathrm{max}}{\text{MIQ}}_{x}=\underset{x\in {S}^{c}}{\mathrm{max}}\frac{I\left(x,y\right)}{\frac{1}{|S|}{\sum }_{z\in S}I\left(x,z\right)}.$

5. Repeat Step 4 until the relevance is zero for all features in Sc.

6. Add the features with zero relevance to S in random order.

The software can skip any step if it cannot find a feature that satisfies the conditions described in the step.

References

[1] Ding, C., and H. Peng. "Minimum redundancy feature selection from microarray gene expression data." Journal of Bioinformatics and Computational Biology. Vol. 3, Number 2, 2005, pp. 185–205.

[2] Darbellay, G. A., and I. Vajda. "Estimation of the information by an adaptive partitioning of the observation space." IEEE Transactions on Information Theory. Vol. 45, Number 4, 1999, pp. 1315–1321.

Version History

Introduced in R2022a