fsrmrmr
Rank features for regression using minimum redundancy maximum relevance (MRMR) algorithm
Since R2022a
Syntax
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
.
returns the predictor indices, idx
= fsrmrmr(Tbl
,ResponseVarName
)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.
specifies a response variable and predictor variables to consider among the variables in
idx
= fsrmrmr(Tbl
,formula
)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
.
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.idx
= fsrmrmr(___,Name=Value
)
Examples
Rank Predictors by Importance
Simulate 1000 observations from the model .
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
.
Rank Predictors and Visualize Scores
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
.
Improve Regression Model Performance by Generating and Selecting Features
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
Tbl
— Sample data
table
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 inTbl
as predictors, then specify the response variable by usingResponseVarName
. IfTbl
also contains the observation weights, then you can specify the weights by usingWeights
.If
Tbl
contains the response variable, and you want to use only a subset of the remaining variables inTbl
as predictors, then specify the subset of variables by usingformula
.If
Tbl
does not contain the response variable, then specify a response variable by usingY
. The response variable andTbl
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
ResponseVarName
— Response variable name
character vector or string scalar containing name of variable in
Tbl
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
formula
— Explanatory model of response variable and subset of predictor variables
character vector | string scalar
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
Y
— Response variable
numeric vector
X
— Predictor data
numeric matrix
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.
CategoricalPredictors
— List of categorical predictors
vector of positive integers | logical vector | character matrix | string array | cell array of character vectors | "all"
List of categorical predictors, specified as one of the values in this table.
Value | Description |
---|---|
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 If |
Logical vector |
A |
Character matrix | Each 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 vectors | Each 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
UseMissing
— Indicator for whether to use missing values in predictors
false
(default) | true
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
Verbose
— Verbosity level
0
(default) | nonnegative integer
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 theVerbose
value.
Example: Verbose=1
Data Types: single
| double
Weights
— Observation weights
ones(size(X,1),1)
(default) | vector of scalar values | name of variable in Tbl
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
idx
— Indices of predictors ordered by predictor importance
numeric vector
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.
scores
— Predictor scores
numeric vector
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.
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.
More About
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
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
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:
|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.
where Vx and Wx are the relevance and redundancy of a feature, respectively:
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:
Select the feature with the largest relevance, . Add the selected feature to an empty set S.
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, . Add the selected feature to the set S.
Repeat Step 2 until the redundancy is not zero for all features in Sc.
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.
Repeat Step 4 until the relevance is zero for all features in Sc.
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
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