Multivariate normal regression with missing data
[Parameters,Covariance,Resid,Info] = ecmmvnrmle(Data,Design,MaxIterations,TolParam,TolObj,Param0,Covar0,CovarFormat)


 A matrix or a cell array that handles two model structures:

 (Optional) Maximum number of iterations for the estimation algorithm. Default value is 100. 
 (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)

 (Optional) Character vector that specifies the format for the covariance matrix. The choices are:

[Parameters,Covariance,Resid,Info] =
ecmmvnrmle(Data,Design,MaxIterations,TolParam,TolObj,Param0,Covar0,CovarFormat)
estimates a multivariate normal regression model with 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
.
ecmmvnrmle
estimates a
NUMPARAMS
by1
column vector of model
parameters called Parameters
, and a
NUMSERIES
byNUMSERIES
matrix of covariance
parameters called Covariance
.
ecmmvnrmle(Data, Design)
with no output arguments plots the
loglikelihood function for each iteration of the algorithm.
To summarize the outputs of ecmmvnrmle
:
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 missing values in
Data
, the corresponding residual is the difference between
the conditionally imputed value for Data
and the model, that
is, the imputed residual.
Note
The covariance estimate Covariance
cannot be
derived from the residuals.
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 prior to the terminal
iteration.nfo.PrevCovariance
–
NUMSERIES
byNUMSERIES
matrix of
estimates for the covariance parameters from the iteration just prior to the
terminal iteration.
ecmmvnrmle
does not accept an initial parameter vector, since 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
.
ecmmvnrmle
is more strict than mvnrmle
about the presence of
NaN
values in the Design
array.
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.
Joe Sexton and Anders Rygh Swensen. “ECM Algorithms that Converge at the Rate of EM.” Biometrika. Vol. 87, No. 3, 2000, pp. 651–662.
A. P. Dempster, 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.