ecmnhess

Hessian of negative log-likelihood function

Syntax

Hessian = ecmnhess(Data,Covariance)

Hessian = ecmnhess(___,InvCovar,MatrixType)

Description

Hessian = ecmnhess(Data,Covariance) computes an NUMPARAMS-by-NUMPARAMS Hessian matrix of the observed negative log-likelihood function based on current parameter estimates.

Use ecmnhess after estimating the mean and covariance of Data with ecmnmle.

Hessian = ecmnhess(___,InvCovar,MatrixType) adds optional arguments for InvCovar and MatrixType.

Examples

Compute Hessian for Negative Log-Likelihood Function for Data

This example shows how to compute the Hessian for the negative log-likelihood function for five years of daily total return data for 12 computer technology stocks, with six hardware and six software companies

load ecmtechdemo.mat

The time period for this data extends from April 19, 2000 to April 18, 2005. The sixth stock in Assets is Google (GOOG), which started trading on August 19, 2004. So, all returns before August 20, 2004 are missing and represented as NaNs. Also, Amazon (AMZN) had a few days with missing values scattered throughout the past five years.

[ECMMean, ECMCovar] = ecmnmle(Data)

ECMMean = 12×1

    0.0008
    0.0008
   -0.0005
    0.0002
    0.0011
    0.0038
   -0.0003
   -0.0000
   -0.0003
   -0.0000
   -0.0003
    0.0004
      ⋮

ECMCovar = 12×12

    0.0012    0.0005    0.0006    0.0005    0.0005    0.0003    0.0005    0.0003    0.0006    0.0003    0.0005    0.0006
    0.0005    0.0024    0.0007    0.0006    0.0010    0.0004    0.0005    0.0003    0.0006    0.0004    0.0006    0.0012
    0.0006    0.0007    0.0013    0.0007    0.0007    0.0003    0.0006    0.0004    0.0008    0.0005    0.0008    0.0008
    0.0005    0.0006    0.0007    0.0009    0.0006    0.0002    0.0005    0.0003    0.0007    0.0004    0.0005    0.0007
    0.0005    0.0010    0.0007    0.0006    0.0016    0.0006    0.0005    0.0003    0.0006    0.0004    0.0007    0.0011
    0.0003    0.0004    0.0003    0.0002    0.0006    0.0022    0.0001    0.0002    0.0002    0.0001    0.0003    0.0016
    0.0005    0.0005    0.0006    0.0005    0.0005    0.0001    0.0009    0.0003    0.0005    0.0004    0.0005    0.0006
    0.0003    0.0003    0.0004    0.0003    0.0003    0.0002    0.0003    0.0005    0.0004    0.0003    0.0004    0.0004
    0.0006    0.0006    0.0008    0.0007    0.0006    0.0002    0.0005    0.0004    0.0011    0.0005    0.0007    0.0007
    0.0003    0.0004    0.0005    0.0004    0.0004    0.0001    0.0004    0.0003    0.0005    0.0006    0.0004    0.0005
    0.0005    0.0006    0.0008    0.0005    0.0007    0.0003    0.0005    0.0004    0.0007    0.0004    0.0013    0.0007
    0.0006    0.0012    0.0008    0.0007    0.0011    0.0016    0.0006    0.0004    0.0007    0.0005    0.0007    0.0020
      ⋮

To evaluate the negative log-likelihood function for ecmnmle, use ecmnhess based on the current maximum likelihood parameter estimates for ECMCovar.

Hessian = ecmnhess(Data,ECMCovar)

Hessian = 90×90
107 ×

    0.0001    0.0000   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000    0.0000   -0.0000   -0.0000         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
    0.0000    0.0001   -0.0000   -0.0000   -0.0000    0.0000   -0.0000    0.0000   -0.0000   -0.0000    0.0000   -0.0000         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
   -0.0000   -0.0000    0.0002   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
   -0.0000   -0.0000   -0.0000    0.0003   -0.0000    0.0000   -0.0000   -0.0000   -0.0001   -0.0001   -0.0000   -0.0000         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
   -0.0000   -0.0000   -0.0000   -0.0000    0.0001   -0.0000   -0.0000   -0.0000    0.0000   -0.0000   -0.0000   -0.0000         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
   -0.0000    0.0000   -0.0000    0.0000   -0.0000    0.0000    0.0000   -0.0000    0.0000    0.0000    0.0000   -0.0000         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000    0.0000    0.0002   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
   -0.0000    0.0000   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000    0.0004   -0.0000   -0.0000   -0.0000   -0.0000         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
   -0.0000   -0.0000   -0.0000   -0.0001    0.0000    0.0000   -0.0000   -0.0000    0.0002   -0.0001   -0.0000   -0.0000         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
    0.0000   -0.0000   -0.0000   -0.0001   -0.0000    0.0000   -0.0000   -0.0000   -0.0001    0.0004   -0.0000   -0.0000         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
   -0.0000    0.0000   -0.0000   -0.0000   -0.0000    0.0000   -0.0000   -0.0000   -0.0000   -0.0000    0.0001   -0.0000         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000   -0.0000    0.0001         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0
         0         0         0         0         0         0         0         0         0         0         0         0    0.0756    0.0014    0.0000   -0.0036   -0.0000    0.0000   -0.0392   -0.0004    0.0009    0.0051   -0.0086   -0.0001    0.0002    0.0022    0.0002   -0.0004   -0.0000    0.0000    0.0001    0.0000    0.0000   -0.0189   -0.0002    0.0005    0.0049    0.0011    0.0001    0.0012   -0.0142   -0.0001    0.0003    0.0037    0.0008    0.0000    0.0018    0.0007   -0.0315   -0.0003
         0         0         0         0         0         0         0         0         0         0         0         0    0.0014    0.0828    0.0008   -0.0115   -0.0021    0.0003   -0.0034   -0.0215    0.0030    0.0008   -0.0231   -0.0049    0.0012    0.0061    0.0013    0.0088   -0.0002   -0.0002   -0.0023   -0.0004   -0.0002   -0.0047   -0.0104    0.0015    0.0015    0.0031   -0.0011    0.0006    0.0118   -0.0076    0.0008   -0.0028    0.0015   -0.0008   -0.0011   -0.0011   -0.0036   -0.0173
         0         0         0         0         0         0         0         0         0         0         0         0    0.0000    0.0008    0.0229   -0.0001   -0.0063    0.0004   -0.0000   -0.0016    0.0002    0.0001   -0.0002   -0.0127    0.0018    0.0004    0.0018    0.0000    0.0058   -0.0007    0.0003   -0.0016    0.0019   -0.0001   -0.0023    0.0003    0.0002    0.0006    0.0004    0.0001    0.0001    0.0064   -0.0009   -0.0003   -0.0018    0.0000   -0.0005    0.0005   -0.0000   -0.0017
      ⋮

Input Arguments

Data — Data

matrix

Data, specified as an NUMSAMPLES-by-NUMSERIES matrix with NUMSAMPLES samples of a NUMSERIES-dimensional random vector. Missing values are indicated by NaNs.

Data Types: double

Covariance — Maximum likelihood parameter estimates for covariance of Data

matrix

Maximum likelihood parameter estimates for the covariance of the Data using the ECM algorithm, specified as a NUMSERIES-by-NUMSERIES matrix.

InvCovar — Cholesky decomposition of covariance matrix

[ ] (default) | matrix

(Optional) Inverse of covariance matrix, specified as a matrix using inv as:

inv(Covariance)

Data Types: double

MatrixType — Matrix format

'full' (default) | character vector

(Optional) Matrix format, specified as a character vector with a value of:

• 'full' — Computes the full Hessian matrix.

• 'meanonly' — Computes only the components of the Hessian matrix associated with the mean.

Data Types: char

Output Arguments

Hessian — Hessian matrix

matrix

Hessian matrix, returned as an NUMPARAMSNUMPARAMS matrix of the observed log-likelihood function based on current parameter estimates, where NUMPARAMS = NUMSERIES * (NUMSERIES + 3)/2 if the MatrixFormat = 'full'. If the MatrixFormat = 'meanonly', then the NUMPARAMS = NUMSERIES.

More About

Hessian Matrix

The Hessian matrix is a square matrix of second-order partial derivatives of a scalar-valued function.

The Hessian matrix is a key concept in optimization, particularly in the context of multivariable calculus, and is used to study the local curvature of functions. The Hessian matrix provides important information about the behavior of functions near critical points, which are points where the gradient (first derivative) is zero.

Version History

Introduced before R2006a