Multivariate normal regression (ignore missing data)
[Parameters,Covariance,Resid,Info] = mvnrmle(Data,Design,MaxIterations,TolParam,TolObj,Covar0,CovarFormat)


 Matrix or a cell array that handles two model structures:

 (Optional) Maximum number of iterations for the estimation
algorithm. Default value is 
 (Optional) Convergence tolerance for estimation algorithm
based on changes in model parameter estimates. Default value is 
$$\Vert Para{m}_{k}Para{m}_{k1}\Vert <TolParam\times \left(1+\Vert Para{m}_{k}\Vert \right)$$  
where  
 (Optional) Convergence tolerance for estimation algorithm based on changes in the objective function. Default value is eps ∧ 3/4 which is about 1.0e12 for double precision. The convergence test for changes in the objective function is $$\leftOb{j}_{k}Ob{j}_{k1}\right<\text{\hspace{0.17em}}TolObj\times \left(1+\leftOb{j}_{k}\right\right)$$ for iteration k =
2, 3, ... . Convergence is assumed when both the 
 (Optional) 
 (Optional) Character vector that specifies the format for the covariance matrix. The choices are:

[Parameters,Covariance,Resid,Info] = mvnrmle(Data,Design,MaxIterations,TolParam,TolObj,Covar0,CovarFormat)
estimates
a multivariate normal regression model without missing data. The model
has the form
$$Dat{a}_{k}\sim N\left(Desig{n}_{k}\times Parameters,\text{\hspace{0.17em}}Covariance\right)$$
for samples k = 1, ... , NUMSAMPLES
.
mvnrmle
estimates a NUMPARAMS
by1
column vector of model parameters called Parameters
, and a
NUMSERIES
byNUMSERIES
matrix of covariance
parameters called Covariance
.
mvnrmle(Data, Design)
with no output arguments
plots the loglikelihood function for each iteration of the algorithm.
To summarize the outputs of mvnrmle
:
Parameters
is a NUMPARAMS
by1
column
vector of estimates for the parameters of the regression model.
Covariance
is a NUMSERIES
byNUMSERIES
matrix
of estimates for the covariance of the regression model's residuals.
Resid
is a NUMSAMPLES
byNUMSERIES
matrix
of residuals from the regression. For any row with missing values
in Data
, the corresponding row of residuals is
represented as all NaN
missing values, since this
routine ignores rows with NaN
values.
Another output, Info
, is a structure that
contains additional information from the regression. The structure
has these fields:
Info.Obj
– A variableextent
column vector, with no more than MaxIterations
elements,
that contain each value of the objective function at each iteration
of the estimation algorithm. The last value in this vector, Obj
(end)
,
is the terminal estimate of the objective function. If you do maximum
likelihood estimation, the objective function is the loglikelihood
function.
Info.PrevParameters
– NUMPARAMS
by1
column
vector of estimates for the model parameters from the iteration just
before the terminal iteration.
Info.PrevCovariance
– NUMSERIES
byNUMSERIES
matrix
of estimates for the covariance parameters from the iteration just
before the terminal iteration.
mvnrmle
does not accept an initial parameter vector, because the parameters
are estimated directly from the first iteration onward.
You can configure Design
as a matrix if NUMSERIES
= 1
or as a cell array if NUMSERIES
≥ 1
.
If Design
is a cell array and NUMSERIES
= 1
,
each cell contains a NUMPARAMS
row
vector.
If Design
is a cell array and NUMSERIES
> 1
, each cell
contains a NUMSERIES
byNUMPARAMS
matrix.
These points concern how Design
handles missing
data:
Although Design
should not have NaN
values,
ignored samples due to NaN
values in Data
are
also ignored in the corresponding Design
array.
If Design
is a 1
by1
cell
array, which has a single Design
matrix for each
sample, no NaN
values are permitted in the array.
A model with this structure must have NUMSERIES
≥ NUMPARAMS
with rank(Design{1})
= NUMPARAMS
.
Two functions for handling missing data, ecmmvnrmle
and ecmlsrmle
,
are stricter about the presence of NaN
values in Design
.
Use the estimates in the optional output structure Info
for
diagnostic purposes.
See Multivariate Normal Regression, LeastSquares Regression, CovarianceWeighted Least Squares, Feasible Generalized Least Squares, and Seemingly Unrelated Regression.
Roderick J. A. Little and Donald B. Rubin. Statistical Analysis with Missing Data., 2nd Edition. John Wiley & Sons, Inc., 2002.
XiaoLi Meng and Donald B. Rubin. “Maximum Likelihood Estimation via the ECM Algorithm.” Biometrika. Vol. 80, No. 2, 1993, pp. 267–278.
ecmmvnrmle
 mvnrstd
 mvnrobj
 mvregress