Multivariate linear regression
returns
the estimated coefficients for a multivariate normal regression of
the ddimensional responses in beta
= mvregress(X
,Y
)Y
on
the design matrices in X
.
returns
the estimated coefficients using additional options specified by one
or more namevalue pair arguments. For example, you can specify the
estimation algorithm, initial estimate values, or maximum number of
iterations for the regression.beta
= mvregress(X
,Y
,Name,Value
)
Fit a multivariate regression model to panel data, assuming different intercepts and common slopes.
Load the sample data.
load('flu')
The dataset array flu
contains national CDC
flu estimates, and nine separate regional estimates based on Google^{®} query
data.
Extract the response and predictor data.
Y = double(flu(:,2:end1)); [n,d] = size(Y); x = flu.WtdILI;
The responses in Y
are the nine regional
flu estimates. Observations exist for every week over a oneyear period,
so n = 52. The dimension of the responses corresponds
to the regions, so d = 9. The predictors in x
are
the weekly national flu estimates.
Plot the flu data, grouped by region.
figure; regions = flu.Properties.VarNames(2:end1); plot(x,Y,'x') legend(regions,'Location','NorthWest')
Fit the multivariate regression model
$${y}_{ij}={\alpha}_{j}+\beta {x}_{ij}+{\epsilon}_{ij},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,\dots ,n;\text{\hspace{0.17em}}\text{\hspace{0.17em}}j=1,\dots ,d,$$
with betweenregion concurrent correlation
$$COV({\epsilon}_{ij},{\epsilon}_{i{j}^{\prime}})={\sigma}_{j{j}^{\prime}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}j=1,\dots ,d.$$
There are K = 10 regression coefficients
to estimate: nine intercept terms and a common slope. The input argument X
should
be an nelement cell array of dbyK design
matrices.
X = cell(n,1); for i=1:n X{i} = [eye(d) repmat(x(i),d,1)]; end [beta,Sigma] = mvregress(X,Y);
beta
contains estimates of the Kdimensional
coefficient vector
$${\left({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{9},\beta \right)}^{\prime}\text{\hspace{0.17em}}.$$
Sigma
contains estimates of the dbyd variancecovariance
matrix for the betweenregion concurrent correlations
$$\left(\begin{array}{ccc}{\sigma}_{11}& \dots & {\sigma}_{1,9}\\ \vdots & \ddots & \vdots \\ {\sigma}_{9,1}& \cdots & {\sigma}_{9,9}\end{array}\right)\text{\hspace{0.17em}}.$$
Plot the fitted regression model.
B = [beta(1:d)';repmat(beta(end),1,d)]; xx = linspace(.5,3.5)'; fits = [ones(size(xx)),xx]*B; figure; h = plot(x,Y,'x',xx,fits,''); for i = 1:d set(h(d+i),'color',get(h(i),'color')); end legend(regions,'Location','NorthWest');
The plot shows that each regression line has a different intercept but the same slope. Upon visual inspection, some regression lines appear to fit the data better than others.
Fit a multivariate regression model to panel data using least squares, assuming different intercepts and slopes.
Load the sample data.
load('flu');
The dataset array flu
contains national CDC
flu estimates, and nine separate regional estimates based on Google queries.
Extract the response and predictor data.
Y = double(flu(:,2:end1)); [n,d] = size(Y); x = flu.WtdILI;
The responses in Y
are the nine regional
flu estimates. Observations exist for every week over a oneyear period,
so n = 52. The dimension of the responses corresponds
to the regions, so d = 9. The predictors in x
are
the weekly national flu estimates.
Fit the multivariate regression model
$${y}_{ij}={\alpha}_{j}+{\beta}_{j}{x}_{ij}+{\epsilon}_{ij},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,\dots ,n;\text{\hspace{0.17em}}\text{\hspace{0.17em}}j=1,\dots ,d,$$
with betweenregion concurrent correlation
$$COV({\epsilon}_{ij},{\epsilon}_{i{j}^{\prime}})={\sigma}_{j{j}^{\prime}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}j=1,\dots ,d.$$
There are K = 18 regression coefficients
to estimate: nine intercept terms, and nine slope terms. X
is
an nelement cell array of dbyK design
matrices.
X = cell(n,1); for i=1:n X{i} = [eye(d) x(i)*eye(d)]; end [beta,Sigma] = mvregress(X,Y,'algorithm','cwls');
beta
contains estimates of the Kdimensional
coefficient vector,
$${\left({\alpha}_{1},{\alpha}_{2},\dots ,{\alpha}_{9},{\beta}_{1},{\beta}_{2},\dots ,{\beta}_{9}\right)}^{\prime}.$$
Plot the fitted regression model.
B = [beta(1:d)';beta(d+1:end)']; xx = linspace(.5,3.5)'; fits = [ones(size(xx)),xx]*B; figure; h = plot(x,Y,'x',xx,fits,''); for i = 1:d set(h(d+i),'color',get(h(i),'color')); end regions = flu.Properties.VarNames(2:end1); legend(regions,'Location','NorthWest');
The plot shows that each regression line has a different intercept and slope.
Fit a multivariate regression model using a single nbyP design matrix for all response dimensions.
Load the sample data.
load('flu');
The dataset array flu
contains national CDC
flu estimates, and nine separate regional estimates based on Google queries.
Extract the response and predictor data.
Y = double(flu(:,2:end1)); [n,d] = size(Y); x = flu.WtdILI;
The responses in Y
are the nine regional
flu estimates. Observations exist for every week over a oneyear period,
so n = 52. The dimension of the responses corresponds
to the regions, so d = 9. The predictors in x
are
the weekly national flu estimates.
Create an nbyP design
matrix X
. Add a column of ones to include a constant
term in the regression.
X = [ones(size(x)),x];
Fit the multivariate regression model
$${y}_{ij}={\alpha}_{j}+{\beta}_{j}{x}_{ij}+{\epsilon}_{ij},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,\dots ,n;\text{\hspace{0.17em}}\text{\hspace{0.17em}}j=1,\dots ,d,$$
with betweenregion concurrent correlation
$$COV({\epsilon}_{ij},{\epsilon}_{i{j}^{\prime}})={\sigma}_{j{j}^{\prime}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}j=1,\dots ,d.$$
There are 18 regression coefficients to estimate: nine intercept terms, and nine slope terms.
[beta,Sigma,E,CovB,logL] = mvregress(X,Y);
beta
contains estimates of the Pbyd coefficient
matrix. Sigma
contains estimates of the dbyd variancecovariance
matrix for the betweenregion concurrent correlations. E
is
a matrix of the residuals. CovB
is the estimated
variancecovariance matrix of the regression coefficients. logL
is
the value of the log likelihood objective function after the last
iteration.
Plot the fitted regression model.
B = beta; xx = linspace(.5,3.5)'; fits = [ones(size(xx)),xx]*B; figure; h = plot(x,Y,'x', xx,fits,''); for i = 1:d set(h(d+i),'color',get(h(i),'color')); end regions = flu.Properties.VarNames(2:end1); legend(regions,'Location','NorthWest');
The plot shows that each regression line has a different intercept and slope.
X
— Design matricesmatrix  cell array of matricesDesign matrices for the multivariate regression, specified as
a matrix or cell array of matrices. n is the number
of observations in the data, K is the number of
regression coefficients to estimate, p is the number
of predictor variables, and d is the number of
dimensions in the response variable matrix Y
.
If d = 1, then specify X
as
a single nbyK design matrix.
If d > 1 and all d dimensions
have the same design matrix, then you can specify X
as
a single nbyp design matrix
(not in a cell array).
If d > 1 and all n observations
have the same design matrix, then you can specify X
as
a cell array containing a single dbyK design
matrix.
If d > 1 and all n observations
do not have the same design matrix, then specify X
as
a cell array of length n containing dbyK design
matrices.
To include a constant term in the regression model, each design matrix should contain a column of ones.
mvregress
treats NaN
values
in X
as missing values, and ignores rows in X
with
missing values.
Data Types: single
 double
 cell
Y
— Response variablesmatrixResponse variables, specified as an nbyd matrix. n is
the number of observations in the data, and d is
the number of dimensions in the response. When d =
1, mvregress
treats the values in Y
like n independent
response values.
mvregress
treats NaN
values
in Y
as missing values, and handles them according
to the estimation algorithm specified using the namevalue pair argument algorithm
.
Data Types: single
 double
Specify optional commaseparated pairs of Name,Value
arguments.
Name
is the argument
name and Value
is the corresponding
value. Name
must appear
inside single quotes (' '
).
You can specify several name and value pair
arguments in any order as Name1,Value1,...,NameN,ValueN
.
'algorithm','cwls','covar0',C
specifies
covarianceweighted least squares estimation using the covariance
matrix C
.'algorithm'
— Estimation algorithm'mvn'
 'ecm'
 'cwls'
Estimation algorithm, specified as the commaseparated pair
consisting of 'algorithm'
and one of the following.
'mvn'  Ordinary multivariate normal maximum likelihood estimation. 
'ecm'  Maximum likelihood estimation via the ECM algorithm. 
'cwls'  Covarianceweighted least squares estimation. 
The default algorithm depends on the presence of missing data.
For complete data, the default is 'mvn'
.
If there are any missing responses (indicated by NaN
),
the default is 'ecm'
, provided the sample size
is sufficient to estimate all parameters. Otherwise, the default algorithm
is 'cwls'
.
Note:
If 
Example: 'algorithm','ecm'
'beta0'
— Initial estimates for regression coefficientsvectorInitial estimates for the regression coefficients, specified
as the commaseparated pair consisting of 'beta0'
and
a vector with K elements. The default value is
a vector of 0s.
The beta0
argument is not used if the estimation algorithm
is 'mvn'
.
'covar0'
— Initial estimate for variancecovariance matrixmatrixInitial estimate for the variancecovariance matrix, Sigma
,
specified as the commaseparated pair consisting of 'covar0'
and
a symmetric, positive definite, dbyd matrix.
The default value is the identity matrix.
If the estimation algorithm
is 'cwls'
,
then mvregress
uses covar0
as
the weighting matrix at each iteration, without changing it.
'covtype'
— Type of variancecovariance matrix'full'
(default)  'diagonal'
Type of variancecovariance matrix to estimate for Y
,
specified as the commaseparated pair consisting of 'covtype'
and
one of the following.
'full'  Estimate all d(d + 1)/2 variancecovariance elements. 
'diagonal'  Estimate only the d diagonal elements of the variancecovariance matrix. 
Example: 'covtype','diagonal'
'maxiter'
— Maximum number of iterations100
(default)  positive integerMaximum number of iterations for the estimation algorithm, specified
as the commaseparated pair consisting of 'maxiter'
and
a positive integer.
Iterations continue until estimates are within the convergence
tolerances tolbeta
and tolobj
,
or the maximum number of iterations specified by maxiter
is
reached. If both tolbeta
and tolobj
are
0, then mvregress
performs maxiter
iterations
with no convergence tests.
Example: 'maxiter',50
'outputfcn'
— Function to evaluate each iterationfunction handleFunction to evaluate at each iteration, specified as the commaseparated
pair consisting of 'outputfcn'
and a function handle.
The function must return a logical true
or false
.
At each iteration, mvregress
evaluates the function.
If the result is true
, iterations stop. Otherwise,
iterations continue. For example, you could specify a function that
plots or displays current iteration results, and returns true
if
you close the figure.
The function must accept three input arguments, in this order:
Vector of current coefficient estimates
Structure containing these three fields:
Covar  Current value of the variancecovariance matrix 
iteration  Current iteration number 
fval  Current value of the loglikelihood objective function 
Text string that takes these three values:
'init'  When the function is called during initialization 
'iter'  When the function is called after an iteration 
'done'  When the function is called after completion 
'tolbeta'
— Convergence tolerance for regression coefficientssqrt(eps)
(default)  positive scalar valueConvergence tolerance for regression coefficients, specified
as the commaseparated pair consisting of 'tolbeta'
and
a positive scalar value.
Let $${b}^{t}$$ denote the estimate of the coefficient
vector at iteration t, and $${\tau}_{\beta}$$ be the tolerance specified by tolbeta
.
The convergence criterion for regression coefficient estimation is
$$\Vert {b}^{t}{b}^{t1}\Vert <{\tau}_{\beta}\sqrt{K}\left(1+\Vert {b}^{t}\Vert \right),$$
where K is the length of $${b}^{t}$$ and $$\Vert v\Vert $$ is the norm of a vector $$v.$$
Iterations continue until estimates are within the convergence
tolerances tolbeta
and tolobj
,
or the maximum number of iterations specified by maxiter
is
reached. If both tolbeta
and tolobj
are
0, then mvregress
performs maxiter
iterations
with no convergence tests.
Example: 'tolbeta',1e5
'tolobj'
— Convergence tolerance for loglikelihood objective functioneps^(3/4)
(default)  positive scalar valueConvergence tolerance for the loglikelihood objective function,
specified as the commaseparated pair consisting of 'tolobj'
and
a positive scalar value.
Let $${L}^{t}$$ denote the value of the loglikelihood
objective function at iteration t, and $${\tau}_{\ell}$$ be the tolerance specified by tolobj
.
The convergence criterion for the objective function is
$$\left{L}^{t}{L}^{t1}\right<{\tau}_{\ell}\left(1+\left{L}^{t}\right\right).$$
Iterations continue until estimates are within the convergence
tolerances tolbeta
and tolobj
,
or the maximum number of iterations specified by maxiter
is
reached. If both tolbeta
and tolobj
are
0, then mvregress
performs maxiter
iterations
with no convergence tests.
Example: 'tolobj',1e5
'varformat'
— Format for parameter estimate variancecovariance matrix'beta'
(default)  'full'
Format for the parameter estimate variancecovariance matrix, CovB
,
specified as the commaseparated pair consisting of 'varformat'
and
one of the following.
'beta'  Return the variancecovariance matrix for only the regression
coefficient estimates, beta . 
'full'  Return the variancecovariance matrix for both the regression
coefficient estimates, beta , and the variancecovariance
matrix estimate, Sigma . 
Example: 'varformat','full'
'vartype'
— Type of variancecovariance matrix for parameter estimates'hessian'
(default)  'fisher'
Type of variancecovariance matrix for parameter estimates,
specified as the commaseparated pair consisting of 'vartype'
and
either 'hessian'
or 'fisher'
.
If the value is 'hessian'
, then mvregress
uses
the Hessian, or observed information, matrix to compute CovB
.
If the value is 'fisher'
, then mvregress
uses
the completedata Fisher, or expected information, matrix to compute CovB
.
The 'hessian'
method takes into account the
increase uncertainties due to missing data, while the 'fisher'
method
does not.
Example: 'vartype','fisher'
beta
— Estimated regression coefficientscolumn vector  matrixEstimated regression coefficients, returned as a column vector or matrix.
If you specify X
as a single nbyK design
matrix, then mvregress
returns beta
as
a column vector of length K. For example, if X
is
a 20by5 design matrix, then beta
is a 5by1
column vector.
If you specify X
as a cell array
containing one or more dbyK design
matrices, then mvregress
returns beta
as
a column vector of length K. For example, if X
is
a cell array containing 2by10 design matrices, then beta
is
a 10by1 column vector.
If you specify X
as a single nbyp design
matrix (not in a cell array), and Y
has dimension d >
1, then mvregress
returns beta
as
a pbyd matrix. For example,
if X
is a 20by5 design matrix, and Y
has
two dimensions such that d = 2, then beta
is
a 5by2 matrix, and the fitted Y
values are X
× beta
.
Sigma
— Estimated variancecovariance matrixsquare matrixE
— ResidualsmatrixResiduals for the fitted regression model, returned as an nbyd matrix.
If algorithm
has the value 'ecm'
or 'cwls'
,
then mvregress
computes the residual values corresponding
to missing values in Y
as the difference between
the conditionally
imputed values and the fitted values.
Note:
If 
CovB
— Parameter estimate variancecovariance matrixsquare matrixParameter estimate variancecovariance matrix, returned as a square matrix.
logL
— Loglikelihood objective function valuescalar valueLoglikelihood objective function value after the last iteration, returned as a scalar value.
Multivariate normal regression is the regression of a ddimensional response on a design matrix of predictor variables, with normally distributed errors. The errors can be heteroscedastic and correlated.
The model is
$${y}_{i}={X}_{i}\beta +{e}_{i},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,\dots ,n,$$
where
$${y}_{i}$$ is a ddimensional vector of responses.
$${X}_{i}$$ is a design matrix of predictor variables.
$$\beta $$ is vector or matrix of regression coefficients.
$${e}_{i}$$ is a ddimensional vector of error terms, with multivariate normal distribution
$${e}_{i}~MV{N}_{d}(0,\Sigma ).$$
The expectation/conditional maximization ('ecm'
)
and covarianceweighted least squares ('cwls'
)
estimation algorithms include imputation of missing response values.
Let $$\tilde{y}$$ denote missing observations. The conditionally imputed values are the expected value of the missing observation given the observed data, $${\rm E}\left(\tilde{y}y\right).$$
The joint distribution of the missing and observed responses is a multivariate normal distribution,
$$\left(\begin{array}{l}\tilde{y}\\ y\end{array}\right)~MVN\left\{\left(\begin{array}{l}\tilde{X}\beta \\ X\beta \end{array}\right),\left(\begin{array}{cc}{\Sigma}_{\tilde{y}}& {\Sigma}_{\tilde{y}y}\\ {\Sigma}_{y\tilde{y}}& {\Sigma}_{y}\end{array}\right)\right\}.$$
Using properties of the multivariate normal distribution, the imputed conditional expectation is given by
$${\rm E}\left(\tilde{y}y\right)=\tilde{X}\beta +{\Sigma}_{\tilde{y}y}{\Sigma}_{y}^{1}(yX\beta ).$$
Note:

[1] Little, Roderick J. A., and Donald B. Rubin. Statistical Analysis with Missing Data. 2nd ed., Hoboken, NJ: John Wiley & Sons, Inc., 2002.
[2] Meng, XiaoLi, and Donald B. Rubin. "Maximum Likelihood Estimation via the ECM Algorithm." Biometrika. Vol. 80, No. 2, 1993, pp. 267–278.
[3] Sexton, Joe, and A. R. Swensen. "ECM Algorithms that Converge at the Rate of EM." Biometrika. Vol. 87, No. 3, 2000, pp. 651–662.
[4] Dempster, A. P., N. M. Laird, and D. B. Rubin. "Maximum Likelihood from Incomplete Data via the EM Algorithm." Journal of the Royal Statistical Society. Series B, Vol. 39, No. 1, 1977, pp. 1–37.
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