predict
Description
specifies options using one or more name-value arguments. For example,
yFit
= predict(Mdl
,X
,Name,Value
)'IncludeInteractions',true
specifies to include interaction terms in
computations.
[
also returns the standard deviations and prediction intervals of the response variable,
evaluated at each observation in the predictor data yFit
,ySD
,yInt
] = predict(___)X
, using any of the
input argument combinations in the previous syntaxes. This syntax is valid only when you
specify 'FitStandardDeviation'
of fitrgam
as
true
for training Mdl
and the IsStandardDeviationFit
property of Mdl
is
true
.
Examples
Predict Test Sample Response
Train a generalized additive model using training samples, and then predict the test sample responses.
Load the patients
data set.
load patients
Create a table that contains the predictor variables (Age
, Diastolic
, Smoker
, Weight
, Gender
, SelfAssessedHealthStatus
) and the response variable (Systolic
).
tbl = table(Age,Diastolic,Smoker,Weight,Gender,SelfAssessedHealthStatus,Systolic);
Randomly partition observations into a training set and a test set. Specify a 10% holdout sample for testing.
rng('default') % For reproducibility cv = cvpartition(size(tbl,1),'HoldOut',0.10);
Extract the training and test indices.
trainInds = training(cv); testInds = test(cv);
Train a univariate GAM that contains the linear terms for the predictors in tbl
.
Mdl = fitrgam(tbl(trainInds,:),'Systolic')
Mdl = RegressionGAM PredictorNames: {'Age' 'Diastolic' 'Smoker' 'Weight' 'Gender' 'SelfAssessedHealthStatus'} ResponseName: 'Systolic' CategoricalPredictors: [3 5 6] ResponseTransform: 'none' Intercept: 122.7444 IsStandardDeviationFit: 0 NumObservations: 90
Mdl
is a RegressionGAM
model object.
Predict responses for the test set.
yFit = predict(Mdl,tbl(testInds,:));
Create a table containing the observed response values and the predicted response values.
table(tbl.Systolic(testInds),yFit, ... 'VariableNames',{'Observed Value','Predicted Value'})
ans=10×2 table
Observed Value Predicted Value
______________ _______________
124 126.58
121 123.95
130 116.72
115 117.35
121 117.45
116 118.5
123 126.16
132 124.14
125 127.36
124 115.99
Compare Predicted Responses
Predict responses for new observations using a generalized additive model that contains both linear and interaction terms for predictors. Use a memory-efficient model object, and specify whether to include interaction terms when predicting responses.
Load the carbig
data set, which contains measurements of cars made in the 1970s and early 1980s.
load carbig
Specify Acceleration
, Displacement
, Horsepower
, and Weight
as the predictor variables (X
) and MPG
as the response variable (Y
).
X = [Acceleration,Displacement,Horsepower,Weight]; Y = MPG;
Partition the data set into two sets: one containing training data, and the other containing new, unobserved test data. Reserve 10 observations for the new test data set.
rng('default')
n = size(X,1);
newInds = randsample(n,10);
inds = ~ismember(1:n,newInds);
XNew = X(newInds,:);
YNew = Y(newInds);
Train a GAM that contains all the available linear and interaction terms in X
.
Mdl = fitrgam(X(inds,:),Y(inds),'Interactions','all');
Mdl
is a RegressionGAM
model object.
Conserve memory by reducing the size of the trained model.
CMdl = compact(Mdl); whos('Mdl','CMdl')
Name Size Bytes Class Attributes CMdl 1x1 1255766 classreg.learning.regr.CompactRegressionGAM Mdl 1x1 1289882 RegressionGAM
CMdl
is a CompactRegressionGAM
model object.
Predict the responses using both linear and interaction terms, and then using only linear terms. To exclude interaction terms, specify 'IncludeInteractions',false
.
yFit = predict(CMdl,XNew);
yFit_nointeraction = predict(CMdl,XNew,'IncludeInteractions',false);
Create a table containing the observed response values and the predicted response values.
t = table(YNew,yFit,yFit_nointeraction, ... 'VariableNames',{'Observed Response', ... 'Predicted Response','Predicted Response Without Interactions'})
t=10×3 table
Observed Response Predicted Response Predicted Response Without Interactions
_________________ __________________ _______________________________________
27.9 23.04 23.649
NaN 37.163 35.779
NaN 25.876 21.978
13 12.786 14.141
36 28.889 27.281
19.9 22.199 18.451
24.2 23.995 24.885
12 14.247 13.982
38 33.797 33.528
13 12.225 11.127
Plot Prediction Intervals
Train a generalized additive model (GAM), and then compute and plot the prediction intervals of response values.
Load the patients
data set.
load patients
Create a table that contains the predictor variables (Age
, Diastolic
, Smoker
, Weight
, Gender
, SelfAssessedHealthStatus
) and the response variable (Systolic
).
tbl = table(Age,Diastolic,Smoker,Weight,Gender,SelfAssessedHealthStatus,Systolic);
Train a univariate GAM that contains the linear terms for the predictors in tbl
. Specify the FitStandardDeviation
name-value argument as true
so that you can use the trained model to compute prediction intervals. A recommended practice is to use optimal hyperparameters when you fit the standard deviation model for the accuracy of the standard deviation estimates. Specify 'OptimizeHyperparameters'
as 'all-univariate'
. For reproducibility, use the 'expected-improvement-plus'
acquisition function. Specify 'ShowPlots'
as false
and 'Verbose'
as 0 to disable plot and message displays, respectively.
rng('default') % For reproducibility Mdl = fitrgam(tbl,'Systolic','FitStandardDeviation',true, ... 'OptimizeHyperparameters','all-univariate', ... 'HyperparameterOptimizationOptions',struct('AcquisitionFunctionName','expected-improvement-plus', ... 'ShowPlots',false,'Verbose',0))
Mdl = RegressionGAM PredictorNames: {'Age' 'Diastolic' 'Smoker' 'Weight' 'Gender' 'SelfAssessedHealthStatus'} ResponseName: 'Systolic' CategoricalPredictors: [3 5 6] ResponseTransform: 'none' Intercept: 122.7800 IsStandardDeviationFit: 1 NumObservations: 100 HyperparameterOptimizationResults: [1x1 BayesianOptimization]
Mdl
is a RegressionGAM
model object that uses the best estimated feasible point. The best estimated feasible point indicates the set of hyperparameters that minimizes the upper confidence bound of the objective function value based on the underlying objective function model of the Bayesian optimization process. For more details on the optimization process, see Optimize GAM Using OptimizeHyperparameters.
Predict responses for the training data in tbl
, and compute the 99% prediction intervals of the response variable. Specify the significance level ('Alpha'
) as 0.01 to set the confidence level of the prediction intervals to 99%.
[yFit,~,yInt] = predict(Mdl,tbl,'Alpha',0.01);
Plot the sorted true responses together with the predicted responses and prediction intervals.
figure yTrue = tbl.Systolic; [sortedYTrue,I] = sort(yTrue); plot(sortedYTrue,'o') hold on plot(yFit(I)) plot(yInt(I,1),'k:') plot(yInt(I,2),'k:') legend('True responses','Predicted responses', ... 'Prediction interval limits','Location','best') hold off
Input Arguments
Mdl
— Generalized additive model
RegressionGAM
model object | CompactRegressionGAM
model object
Generalized additive model, specified as a RegressionGAM
or a CompactRegressionGAM
model object.
X
— Predictor data
numeric matrix | table
Predictor data, specified as a numeric matrix or table.
Each row of X
corresponds to one observation, and each column corresponds to one variable.
For a numeric matrix:
The variables that make up the columns of
X
must have the same order as the predictor variables that trainedMdl
.If you trained
Mdl
using a table, thenX
can be a numeric matrix if the table contains all numeric predictor variables.
For a table:
If you trained
Mdl
using a table (for example,Tbl
), then all predictor variables inX
must have the same variable names and data types as those inTbl
. However, the column order ofX
does not need to correspond to the column order ofTbl
.If you trained
Mdl
using a numeric matrix, then the predictor names inMdl.PredictorNames
and the corresponding predictor variable names inX
must be the same. To specify predictor names during training, use the'PredictorNames'
name-value argument. All predictor variables inX
must be numeric vectors.X
can contain additional variables (response variables, observation weights, and so on), butpredict
ignores them.predict
does not support multicolumn variables or cell arrays other than cell arrays of character vectors.
Data Types: table
| double
| single
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.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.
Example: 'Alpha',0.01,'IncludeInteractions',false
specifies the
confidence level as 99% and excludes interaction terms from computations.
Alpha
— Significance level
0.05 (default) | numeric scalar in [0,1]
Significance level for the confidence level of the prediction intervals
yInt
, specified as a numeric scalar in the range
[0,1]
. The confidence level of yInt
is equal
to 100(1 – Alpha)%
.
This argument is valid only when the IsStandardDeviationFit
property of Mdl
is
true
. Specify the 'FitStandardDeviation'
name-value argument of
fitrgam
as true
to fit the model for the
standard deviation.
Example: 'Alpha',0.01
specifies to return 99% prediction
intervals.
Data Types: single
| double
IncludeInteractions
— Flag to include interaction terms
true
| false
Flag to include interaction terms of the model, specified as true
or
false
.
The default 'IncludeInteractions'
value is true
if Mdl
contains interaction terms. The value must be false
if the model does not contain interaction terms.
Example: 'IncludeInteractions',false
Data Types: logical
Output Arguments
yFit
— Predicted responses
column vector
Predicted responses, returned as a column vector of length n,
where n is the number of observations in the predictor data
X
.
ySD
— Standard deviations of response variable
column vector
Standard deviations of the response variable, evaluated at each observation in the
predictor data X
, returned as a column vector of length
n, where n is the number of observations in
X
. The i
th element ySD(i)
contains the standard deviation of the i
th response for the
i
th observation
,
estimated using the trained standard deviation model in X
(i,:)Mdl
.
This argument is valid only when the IsStandardDeviationFit
property of Mdl
is
true
. Specify the 'FitStandardDeviation'
name-value argument of fitrgam
as true
to fit the model for the standard deviation.
yInt
— Prediction intervals of response variable
two-column matrix
Prediction intervals of the response variable, evaluated at each observation in the
predictor data X
, returned as an n-by-2 matrix,
where n is the number of observations in X
. The
i
th row yInt(i,:)
contains the
100(1–
prediction interval of the
Alpha
)%i
th response for the i
th observation
. The X
(i,:)Alpha
value
is the probability that the prediction interval does not contain the true response value
for
. The first column of
X
(i,:)yInt
contains the lower limits of the prediction intervals, and
the second column contains the upper limits.
This argument is valid only when the IsStandardDeviationFit
property of Mdl
is
true
. Specify the 'FitStandardDeviation'
name-value argument of fitrgam
as true
to fit the model for the standard deviation.
Algorithms
Standard Deviation and Prediction Interval
predict
returns the predicted responses (yFit
)
and, optionally, the standard deviations (ySD
) and prediction intervals
(yInt
) of the response variable, estimated at each observation in
X
.
A Generalized Additive Model (GAM) for Regression
assumes that the response variable y follows the normal distribution with
mean μ and standard deviation σ. If you specify
'FitStandardDeviation'
of fitrgam
as
false
(default), then fitrgam
trains a model for
μ. If you specify 'FitStandardDeviation'
as
true
, then fitrgam
trains an additional model for
σ and sets the IsStandardDeviationFit
property of
the GAM object to true
. The outputs yFit
and
ySD
correspond to the estimated mean μ and
standard deviation σ, respectively.
Version History
Introduced in R2021a
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