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
(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
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) String 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
for samples k = 1, ... ,
ecmlsrmle estimates a
vector of model parameters called
of covariance parameters called
ecmlsrmle(Data, Design) with no output arguments
plots the log-likelihood function for each iteration of the algorithm.
To summarize the outputs of
Parameters is a
vector of estimates for the parameters of the regression model.
Covariance is a
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
of residuals from the regression.
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
that contains each value of the objective function at each iteration
of the estimation algorithm. The last value in this vector,
is the terminal estimate of the objective function. If you do least-squares,
the objective function is the least-squares objective function.
column vector of estimates for the model parameters from the iteration
just prior to the terminal iteration.
of estimates for the covariance parameters from the iteration just
prior to the terminal iteration.
If doing covariance-weighted least-squares,
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 need not be a diagonal matrix
You can configure
Design as a matrix if
= 1 or as a cell array if
Design is a cell array and
1, each cell contains a
Design is a cell array and
NUMSERIES > 1, each cell contains a
These points concern how
Design handles missing
Design should not have
ignored samples due to
NaN values in
also ignored in the corresponding
Design is a
array, which has a single
Design matrix for each
NaN values are permitted in the array.
A model with this structure must have
Use the estimates in the optional output structure
Roderick J. A. Little and Donald B. Rubin, Statistical Analysis with Missing Data, 2nd ed., 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.