Least-squares regression with missing data
[Parameters,Covariance,Resid,Info] = ecmlsrmle(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 |
| |
where | |
| (Optional) Convergence tolerance for estimation algorithm based on changes in the objective function. Default value is eps ∧ 3/4 which is about 1.0e-12 for double precision. The convergence test for changes in the objective function is for iteration k =
2, 3, ... . Convergence is assumed when both the |
| (Optional) |
| (Optional) For covariance-weighted least-squares calculations, this matrix corresponds with weights for each series in the regression. The matrix also serves as an initial guess for the residual covariance in the expectation conditional maximization (ECM) algorithm. |
| (Optional) Character vector that specifies the format for the covariance matrix. The choices are:
|
[Parameters, Covariance, Resid, Info] = ecmlsrmle(Data,
Design, MaxIterations, TolParam, TolObj, Param0, Covar0, CovarFormat)
estimates
a least-squares regression model with missing data. The model has
the form
for samples k = 1, ... , NUMSAMPLES
.
ecmlsrmle
estimates a NUMPARAMS
-by-1
column vector of model parameters called Parameters
, and a
NUMSERIES
-by-NUMSERIES
matrix of covariance
parameters called Covariance
.
ecmlsrmle(Data, Design)
with no output arguments
plots the log-likelihood function for each iteration of the algorithm.
To summarize the outputs of ecmlsrmle
:
Parameters
is a NUMPARAMS
-by-1
column
vector of estimates for the parameters of the regression model.
Covariance
is a NUMSERIES
-by-NUMSERIES
matrix
of estimates for the covariance of the regression model's residuals.
For least-squares models, this estimate may not be a maximum likelihood
estimate except under special circumstances.
Resid
is a NUMSAMPLES
-by-NUMSERIES
matrix
of residuals from the regression.
Another output, Info
, is a structure that
contains additional information from the regression. The structure
has these fields:
Info.Obj
— A variable-extent
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 least-squares,
the objective function is the least-squares objective function.
Info.PrevParameters
— NUMPARAMS
-by-1
column
vector of estimates for the model parameters from the iteration just
prior to the terminal iteration.
Info.PrevCovariance
— NUMSERIES
-by-NUMSERIES
matrix
of estimates for the covariance parameters from the iteration just
prior to the terminal iteration.
If doing covariance-weighted least-squares, Covar0
should
usually be a diagonal matrix. Series with greater influence should
have smaller diagonal elements in Covar0
and series
with lesser influence should have larger diagonal elements. Note that
if doing CWLS, Covar0
do not need to be a diagonal
matrix even if CovarFormat
= 'diagonal'
.
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
-by-NUMPARAMS
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
-by-1
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
.
ecmlsrmle
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, Least-Squares Regression, Covariance-Weighted 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.
Xiao-Li 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.