Create partial dependence plot (PDP) and individual conditional expectation (ICE) plots
plotPartialDependence(___,
uses additional options specified by one or more namevalue pair arguments in
addition to any of the arguments in the previous syntaxes. For example, if you
specify Name,Value
)'Conditional','absolute'
, the
plotPartialDependence
function creates a figure including a
PDP, a scatter plot of the selected feature and predicted responses, and an ICE plot
for each observation.
Train a regression tree using the carsmall
data set, and create a PDP that shows the relationship between a feature and the predicted responses in the trained regression tree.
Load the carsmall
data set.
load carsmall
Specify Weight
, Cylinders
, and Horsepower
as the predictor variables (X
), and MPG
as the response variable (Y
).
X = [Weight,Cylinders,Horsepower]; Y = MPG;
Construct a regression tree using X
and Y
.
Mdl = fitrtree(X,Y);
View a graphical display of the trained regression tree.
view(Mdl,'Mode','graph')
Create a PDP of the first predictor variable, Weight
.
plotPartialDependence(Mdl,1)
The blue line represents averaged partial relationships between Weight
(labeled as x1
) and MPG
(labeled as Y
) in the trained regression tree Mdl
.
The regression tree viewer shows that the first decision is whether x1
is smaller than 3085.5. The PDP also shows a large change near x1
= 3085.5. The tree viewer visualizes each decision at each node based on predictor variables. You can find several nodes split based on the values of x1
, but it is not easy to figure out the dependence of Y
on x1
. However, the PDP plots averaged predicted responses against x1
, so you can clearly see the partial dependence of Y
on x1
.
The labels x1
and Y
are the default values of the predictor names and the response name. You can modify these names by specifying the namevalue pair arguments 'PredictorNames'
and 'ResponseName'
when you train Mdl
using fitrtree
. You can also modify axis labels by using the xlabel
and ylabel
functions.
Train a Gaussian process regression model using generated sample data where a response variable includes interactions between predictor variables. Then, create ICE plots that show the relationship between a feature and the predicted responses for each observation.
Generate sample predictor data x1
and x2
.
rng('default') % For reproducibility n = 200; x1 = rand(n,1)*21; x2 = rand(n,1)*21;
Generate response values that include interactions between x1
and x2
.
Y = x12*x1.*(x2>0)+0.1*rand(n,1);
Construct a Gaussian process regression model using [x1 x2]
and Y
.
Mdl = fitrgp([x1 x2],Y);
Create a figure including a PDP (red line) for the first predictor x1
, a scatter plot (black circle markers) of x1
and predicted responses, and a set of ICE plots (gray lines) by specifying 'Conditional'
as 'centered'
.
plotPartialDependence(Mdl,1,'Conditional','centered')
When 'Conditional'
is 'centered'
, plotPartialDependence
offsets plots so that all plots start from zero, which is helpful in examining the cumulative effect of the selected feature.
A PDP finds averaged relationships, so it does not reveal hidden dependencies especially when responses include interactions between features. However, the ICE plots clearly show two different dependencies of responses on x1
.
Train a regression ensemble using the carsmall
data set, and create a PDP plot and ICE plots for each predictor variable using a new data set, carbig
. Then, compare the figures to analyze the importance of predictor variables. Also, compare the results with the estimates of predictor importance returned by the predictorImportance
function.
Load the carsmall
data set.
load carsmall
Specify Weight
, Cylinders
, Horsepower
, and Model_Year
as the predictor variables (X
), and MPG
as the response variable (Y
).
X = [Weight,Cylinders,Horsepower,Model_Year]; Y = MPG;
Train a regression ensemble using X
and Y
.
Mdl = fitrensemble(X,Y, ... 'PredictorNames',{'Weight','Cylinders','Horsepower','Model Year'}, ... 'ResponseName','MPG');
Compare the importance of predictor variables by using the plotPartialDependence
and predictorImportance
functions. The plotPartialDependence
function visualizes the relationships between a selected predictor and predicted responses. predictorImportance
summarizes the importance of a predictor with a single value.
Create a figure including a PDP plot (red line) and ICE plots (gray lines) for each predictor by using plotPartialDependence
and specifying 'Conditional','absolute'
. Each figure also includes a scatter plot (black circle markers) of the selected predictor and predicted responses. Also, load the carbig
data set and use it as new predictor data, Xnew
. When you provide Xnew
, the plotPartialDependence
function uses Xnew
instead of the predictor data in Mdl
.
load carbig Xnew = [Weight,Cylinders,Horsepower,Model_Year]; f = figure; f.Position = [100 100 1.75*f.Position(3:4)]; % Enlarge figure for visibility. for i = 1 : 4 subplot(2,2,i) plotPartialDependence(Mdl,i,Xnew,'Conditional','absolute') end
Compute estimates of predictor importance by using predictorImportance
. This function sums changes in the meansquared error (MSE) due to splits on every predictor, and then divides the sum by the number of branch nodes.
imp = predictorImportance(Mdl); figure; bar(imp); title('Predictor Importance Estimates'); ylabel('Estimates'); xlabel('Predictors'); ax = gca; ax.XTickLabel = Mdl.PredictorNames;
The variable Weight
has the most impact on MPG
according to predictor importance. The PDP of Weight
also shows that MPG
has high partial dependence on Weight
. The variable Cylinders
has the least impact on MPG
according to predictor importance. The PDP of Cylinders
also shows that MPG
does not change much depending on Cylinders
.
Train a support vector machine (SVM) regression model using the carsmall
data set, and create a PDP for two predictor variables. Then, extract partial dependence estimates from the output of plotPartialDependence
.
Load the carsmall
data set.
load carsmall
Specify Weight
, Cylinders
, and Horsepower
as the predictor variables (Tbl
).
Tbl = table(Weight,Cylinders,Horsepower);
Construct a support vector machine (SVM) regression model using Tbl
and the response variable MPG
. Use a Gaussian kernel function with an automatic kernel scale.
Mdl = fitrsvm(Tbl,MPG,'ResponseName','MPG', ... 'KernelFunction','gaussian','KernelScale','auto');
Create a PDP that visualizes partial dependence of predicted responses (MPG
) on the predictor variables Weight
and Horsepower
. Use the QueryPoints
namevalue pair argument to specify the points to compute partial dependence.
pt1 = linspace(min(Weight),max(Weight),50)'; pt2 = linspace(min(Horsepower),max(Horsepower),50)'; ax = plotPartialDependence(Mdl,{'Weight','Horsepower'},'QueryPoints',[pt1 pt2]); view(140,30) % Modify the viewing angle
The PDP shows an interaction effect between Weight
and Horsepower
.
Extract the estimated partial dependence of MPG
on Weight
and Horsepower
. The XData
, YData
, and ZData
values of ax.Children
are xaxis values (the first selected predictor values), yaxis values (the second selected predictor values), and zaxis values (the corresponding partial dependence values), respectively.
xval = ax.Children.XData; yval = ax.Children.YData; zval = ax.Children.ZData;
If you specify 'Conditional'
as 'absolute'
, plotPartialDependence
creates a figure including a PDP, a scatter plot, and a set of ICE plots. ax.Children(1)
and ax.Children(2)
correspond to the PDP and scatter plot, respectively. The remaining elements of ax.Children
correspond to the ICE plots. The XData
and YData
values of ax.Children(i)
are xaxis values (the selected predictor values) and yaxis values (the corresponding partial dependence values), respectively.
Mdl
— Trained regression modelTrained regression model, specified as a full or compact regression model object described in the following table.
Trained Model Type  Regression Model Object  Returned By 

Bootstrap aggregation for ensemble of decision trees  TreeBagger ,
CompactTreeBagger  TreeBagger ,
compact 
Ensemble of regression models  RegressionEnsemble , RegressionBaggedEnsemble , CompactRegressionEnsemble  fitrensemble ,
compact 
Gaussian process regression  RegressionGP ,
CompactRegressionGP  fitrgp ,
compact 
Generalized linear mixedeffect model  GeneralizedLinearMixedModel  fitglme 
Generalized linear model  GeneralizedLinearModel , CompactGeneralizedLinearModel  fitglm ,
stepwiseglm ,
compact 
Linear mixedeffect model  LinearMixedModel  fitlme , fitlmematrix 
Linear regression  LinearModel ,
CompactLinearModel  fitlm ,
stepwiselm ,
compact 
Linear regression for highdimensional data  RegressionLinear  fitrlinear 
Nonlinear regression  NonLinearModel  fitnlm 
Regression tree  RegressionTree ,
CompactRegressionTree  fitrtree ,
compact 
Support vector machine regression  RegressionSVM ,
CompactRegressionSVM  fitrsvm ,
compact 
If Mdl
is a compact regression model object, you must
provide X
.
plotPartialDependence
does not support a trained
model object with a sparse matrix. If you train Mdl
by
using fitrlinear
, use a full numeric
matrix for predictor data where rows correspond to individual
observations.
Vars
— Features to visualizeFeatures to visualize, specified as a vector of positive integers, character vector or string scalar, string array, or cell array of character vectors. You can choose one or two features, as shown in the following tables.
One Feature
Value  Description 

positive integer  Index value corresponding to the column of the predictor data to visualize. 
character vector or string scalar 
Name of a predictor variable to visualize. The
name must match the entry in

Two Features
Value  Description 

vector of two positive integers  Index values corresponding to the columns of the predictor data to visualize. 
string array or cell array of character vectors 
Names of the predictor variables to visualize.
Each element in the array is the name of a predictor
variable. The names must match the entries in

Example: {'x1','x3'}
Data Types: single
 double
 char
 string
 cell
X
— Predictor dataPredictor data, specified as a numeric matrix or table. Each row of
X
corresponds to one observation, and each column
corresponds to one variable.
If Mdl
is a full regression model object,
plotPartialDependence
uses the
predictor data in Mdl
. If you provide
X
, then
plotPartialDependence
does not use the
predictor data in Mdl
and uses
X
only.
If Mdl
is a compact regression model
object, you must provide X
.
X
must be consistent with the predictor data that
trained Mdl
, stored in either Mdl.X
or Mdl.Variables
.
If you trained Mdl
using a numeric
matrix, then X
must be a numeric matrix.
The variables making up the columns of X
must have the same number and order as the predictor variables
that trained Mdl
.
If you trained Mdl
using a table (for
example, Tbl
), then X
must be a table. All predictor variables in
X
must have the same variable names and
data types as the names and types in Tbl
.
However, the column order of X
does not
need to correspond to the column order of
Tbl
.
plotPartialDependence
does not support a
sparse matrix.
Data Types: single
 double
 table
Specify optional
commaseparated pairs of Name,Value
arguments. Name
is
the argument name and Value
is the corresponding value.
Name
must appear inside quotes. You can specify several name and value
pair arguments in any order as
Name1,Value1,...,NameN,ValueN
.
plotPartialDependence(Mdl,Vars,X,'NumObservationsToSample',100,'UseParallel',true)
creates a PDP by using 100 sampled observations in
X
and executing
for
loop iterations in parallel.'Conditional'
— Plot type'none'
(default)  'absolute'
 'centered'
Plot type, specified as the commaseparated pair consisting of
'Conditional'
and 'none'
,
'absolute'
, or 'centered'
.
Value  Description 

'none' 

'absolute' 
This value is valid when you select only one feature. 
'centered' 
This value is valid when you select only one feature. 
For details, see Partial Dependence Plot and Individual Conditional Expectation Plots.
Example: 'Conditional','absolute'
'NumObservationsToSample'
— Number of observations to sampleNumber of observations to sample, specified as the commaseparated
pair consisting of 'NumObservationsToSample'
and a
positive integer. The default value is the number of total observations
in either Mdl
or X
. If you
specify a value larger than the number of total observations, then
plotPartialDependence
uses all
observations.
plotPartialDependence
samples observations
without replacement by using the function datasample
and uses the
sampled observations to compute partial dependence. If you specify
'Conditional'
as either
'absolute'
or 'centered'
,
plotPartialDependence
creates a figure
including an ICE plot for each sampled observation.
Example: 'NumObservationsToSample',100
Data Types: single
 double
'ParentAxisHandle'
— Axes in which to plotgca
(default)  axes objectAxes in which to plot, specified as the commaseparated pair
consisting of 'ParentAxisHandle'
and an axes object.
If you do not specify the axes and if the current axes are Cartesian,
then plotPartialDependence
uses the current axes
(gca
). If axes do not exist,
plotPartialDependence
plots in a new
figure.
Example: 'ParentAxisHandle',ax
'QueryPoints'
— Points to compute partial dependencePoints to compute partial dependence, specified as the commaseparated
pair consisting of 'QueryPoints'
and a numeric column
vector, a numeric twocolumn matrix, or a cell array of two numeric
column vectors.
The default value is a numeric column vector or a numeric twocolumn matrix, depending on the number of selected features, where a column contains 100 evenly spaced points between the minimum and maximum values of the selected feature.
If you select one feature in Vars
, use a
numeric column vector.
If you select two features in Vars
:
Use a numeric twocolumn matrix to specify the same number of points for each feature.
Use a cell array of two numeric column vectors to specify a different number of points for each feature.
The default values for a categorical feature are all categorical
values in the selected feature. You cannot modify
'QueryPoints'
for a categorical feature. If you
select one numeric feature and one categorical feature, you can specify
'QueryPoints'
for a numeric feature by using a
cell array consisting of a numeric column vector and an empty array.
Example: 'QueryPoints',{pt,[]}
Data Types: single
 double
 cell
'UseParallel'
— Flag to run in parallelfalse
(default)  true
Flag to run in parallel, specified as the commaseparated pair
consisting of 'UseParallel'
and
true
or false
. If you specify
'UseParallel'
as true
,
plotPartialDependence
executes
for
loop iterations in parallel by using
parfor
when predicting
responses for each observation and averaging them.
Example: 'UseParallel',true
Data Types: logical
ax
— Axes of the plotAxes of the plot, returned as an axes object. For details on how to modify the appearance of the axes and extract data from plots, see Axes Appearance (MATLAB) and Extract Partial Dependence Estimates from Plots.
A partial dependence plot[1] (PDP) visualizes
relationships between features and predicted responses in a trained regression
model. plotPartialDependence
creates either a line plot or a
surface plot of predicted responses against a single feature or a pair of features,
respectively, by marginalizing over the other variables.
Consider a PDP for a subset X^{S} of the whole feature set X = {x_{1}, x_{2}, …, x_{m}}. A subset X^{S} includes either one feature or two features: X^{S} = {x_{S1}} or X^{S} = {x_{S1}, x_{S2}}. Let X^{C} be the complementary set of X^{S} in X. A predicted response f(X) depends on all features in X:
f(X) = f(X^{S}, X^{C}).
The partial dependence of predicted responses on X^{S} is defined by the expectation of predicted responses with respect to X^{C}:
$${f}^{S}\left({X}^{S}\right)={E}_{C}\left[f\left({X}^{S},{X}^{C}\right)\right]={\displaystyle \int f\left({X}^{S},{X}^{C}\right)}{p}_{C}\left({X}^{C}\right)d{X}^{C},$$
where
p_{C}(X^{C})
is the marginal probability of X^{C},
that is, $${p}_{C}\left({X}^{C}\right)\approx {\displaystyle \int p}\left({X}^{S},{X}^{C}\right)d{X}^{S}$$. Assuming that each observation is equally likely, and the
dependence between X^{S} and
X^{C} and the interactions of
X^{S} and
X^{C} in responses are not
strong, plotPartialDependence
estimates the partial dependence
by using observed predictor data as follows:
$${f}^{S}\left({X}^{S}\right)\approx \frac{1}{N}{\displaystyle \sum _{i=1}^{N}f\left({X}^{S},{X}_{i}{}^{C}\right)},$$  (1) 
where N is the number of observations and X_{i} = (X_{i}^{S}, X_{i}^{C}) is the ith observation.
plotPartialDependence
creates a PDP by using Equation 1. Input a trained
model (f(·)) and select features
(X^{S}) to visualize by using the
input arguments Mdl
and Vars
,
respectively. plotPartialDependence
computes partial dependence
at 100 evenly spaced points of X^{S} or
the points that you specify by using the 'QueryPoints'
namevalue pair argument. You can specify the number (N) of
observations to sample from given predictor data by using the
'NumObservationsToSample'
namevalue pair argument.
An individual conditional expectation (ICE) plot[2], as an extension of a PDP, visualizes the relationship between a feature and the predicted responses for each observation. While a PDP visualizes the averaged relationship between features and predicted responses, a set of ICE plots disaggregates the averaged information and visualizes an individual dependence for each observation.
plotPartialDependence
creates an ICE plot
for each observation. A set of ICE plots is useful to investigate heterogeneities of
partial dependence originating from different observations.
plotPartialDependence
can also create ICE plots with any
predictor data provided through the input argument X
. You can
use this feature to explore predicted response space.
Consider an ICE plot for a selected feature x_{S} with a given observation X_{i}^{C}, where X^{S} = {x_{S}}, X^{C} is the complementary set of X^{S} in the whole feature set X, and X_{i} = (X_{i}^{S}, X_{i}^{C}) is the ith observation. The ICE plot corresponds to the summand of the summation in Equation 1:
$${f}^{S}{}_{i}\left({X}^{S}\right)=f\left({X}^{S},{X}_{i}{}^{C}\right).$$
plotPartialDependence
plots $${f}^{S}{}_{i}\left({X}^{S}\right)$$ for each observation i when you specify
'Conditional'
as 'absolute'
. If you
specify 'Conditional'
as 'centered'
,
plotPartialDependence
draws all plots after removing level
effects due to different observations:
$${f}^{S}{}_{i,\text{centered}}\left({X}^{S}\right)=f\left({X}^{S},{X}_{i}{}^{C}\right)f\left(\mathrm{min}\left({X}^{S}\right),{X}_{i}{}^{C}\right).$$
This subtraction ensures that each plot starts from zero, so that you can examine the cumulative effect of X^{S} and the interactions between X^{S} and X^{C}.
The weighted traversal algorithm[1] is a method to estimate partial dependence for a treebased regression model. The estimated partial dependence is the weighted average of response values corresponding to the leaf nodes visited during the tree traversal.
Let X^{S} be a subset of the whole feature set X and X^{C} be the complementary set of X^{S} in X. For each X^{S} value to compute partial dependence, the algorithm traverses a tree from the root (beginning) node down to leaf (terminal) nodes and finds the weights of leaf nodes. The traversal starts by assigning a weight value of one at the root node. If a node splits by X^{S}, the algorithm traverses to the appropriate child node depending on the X^{S} value. The weight of the child node becomes the same value as its parent node. If a node splits by X^{C}, the algorithm traverses to both child nodes. The weight of each child node becomes a value of its parent node multiplied by the fraction of observations corresponding to each child node. After completing the tree traversal, the algorithm computes the weighted average by using the assigned weights.
For an ensemble of regression trees, the estimated partial dependence is an average of the weighted averages over the individual regression trees.
plotPartialDependence
uses a predict
function
to predict responses. plotPartialDependence
chooses the proper
predict
function according to Mdl
and runs
predict
with its default settings. For details about each
predict
function, see the predict
function in
the following table. If Mdl
is a treebased model (not including a
boosted ensemble of trees) and 'Conditional'
is
'none'
, then plotPartialDependence
uses the
weighted traversal algorithm instead of the predict
function. For
details, see Weighted Traversal Algorithm.
Trained Model Type  Regression Model Object  Function to Predict Responses 

Bootstrap aggregation for ensemble of decision trees  CompactTreeBagger  predict 
Bootstrap aggregation for ensemble of decision trees  TreeBagger  predict 
Ensemble of regression models  RegressionEnsemble , RegressionBaggedEnsemble ,
CompactRegressionEnsemble  predict 
Gaussian process regression  RegressionGP , CompactRegressionGP  predict 
Generalized linear mixedeffect model  GeneralizedLinearMixedModel  predict 
Generalized linear model  GeneralizedLinearModel , CompactGeneralizedLinearModel  predict 
Linear mixedeffect model  LinearMixedModel  predict 
Linear regression  LinearModel , CompactLinearModel  predict 
Linear regression for highdimensional data  RegressionLinear  predict 
Nonlinear regression  NonLinearModel  predict 
Regression tree  RegressionTree , CompactRegressionTree  predict 
Support vector machine regression  RegressionSVM , CompactRegressionSVM  predict 
[1] Friedman, J. H. “Greedy function approximation: a gradient boosting machine.” The Annals of Statistics. Vol. 29, No. 5, 2001, pp. 11891232.
[2] Goldstein, A., A. Kapelner, J. Bleich, and E. Pitkin. “Peeking inside the black box: Visualizing statistical learning with plots of individual conditional expectation.” Journal of Computational and Graphical Statistics. Vol. 24, No. 1, 2015, pp. 4465.
[3] Hastie, T., R. Tibshirani, and J. H. Friedman. The Elements of Statistical Learning. New York: Springer, 2001.
To run in parallel, set the 'UseParallel'
option to true
.
Set the 'UseParallel',true
namevalue pair argument in the call to this function.
For more general information about parallel computing, see Run MATLAB Functions with Automatic Parallel Support (Parallel Computing Toolbox).
oobPermutedPredictorImportance
 predictorImportance (RegressionEnsemble)
 predictorImportance (RegressionTree)
 relieff
 sequentialfs
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