# fsrmrmr

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

## Syntax

``idx = fsrmrmr(Tbl,ResponseVarName)``
``idx = fsrmrmr(Tbl,formula)``
``idx = fsrmrmr(Tbl,Y)``
``idx = fsrmrmr(X,Y)``
``idx = fsrmrmr(___,Name=Value)``
``[idx,scores] = fsrmrmr(___)``

## Description

````idx = fsrmrmr(Tbl,ResponseVarName)` ranks features (predictors) using the MRMR algorithm. The table `Tbl` contains predictor variables and a response variable, and `ResponseVarName` is the name of the response variable in `Tbl`. The function returns `idx`, which contains the indices of predictors ordered by predictor importance. 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.

```load carbig 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.

```outages = readtable("outages.csv"); Tbl = rmmissing(outages); head(Tbl)```
``` 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; head(newTrainTbl(:,[originalIdx end]))```
``` 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 in 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 .

## Algorithms

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### Minimum Redundancy Maximum Relevance (MRMR) Algorithm

The MRMR algorithm  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.

 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.

 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.