Jacobian matrix: array of 2x2 first-order partial derivatives, ordered as follows and the Jacobian determinant is |J| = x2 − x1 Then the Hessian matrix is. You did not do anything wrong in your calculation. If you directly compute the Jacobian of the gradient of f with the conventions you used, you. The Hessian is symmetric if the second partials are continuous. The Jacobian of a function f: n → m is the matrix of its first partial derivatives. Note that the Hessian of a function f: n → is the Jacobian of its gradient.
The Hessian is defined as the second partial derivatives of the discrepancy function with respect to the model parameters: Suppose that the mean and covariance structures fit perfectly with in the population.
The expected information matrix is defined as where the expectation is taken over the sampling space of and. Hence, the expected information matrix does not contain any sample values. The expected information matrix plays a significant role in statistical theory.
Under certain regularity conditions, the inverse of the information matrix is the asymptotic covariance matrix forwhere N denotes the sample size and is an estimator. In practice, is never known and can only be estimated.
For a sample of size N, PROC CALIS computes the estimated covariance matrix of by It then computes approximate standard errors for as the square roots of the diagonal elements of this estimated covariance matrix.Hessian matrix
Kenward and Molenberghs show that the use of the expected information leads to biased standard errors when the missing data mechanism satisfies only the missing at random MAR; see Rubin condition but not the missing completely at random MCAR condition. Under the MAR condition, the observed information matrix is the correct choice.
Because the FIML estimation is mostly applied when the data contain missing values, using the observed information by default is quite reasonable. It computes by the finite-difference method based on the analytic formulas of the first-order partial derivatives of F. If a particular information matrix is singular, PROC CALIS offers two ways to compute a generalized inverse of the matrix and, therefore, two ways to compute approximate standard errors of implicitly constrained parameter estimates, t valuesand modification indices.
The computationally expensive Moore-Penrose inverse calculates an estimate of the null space by using an eigenvalue decomposition. The computationally cheaper G2 inverse is produced by sweeping the linearly independent rows and columns and zeroing out the dependent ones. Satorra-Bentler Sandwich Formula for Standard Errors In addition to the scaled chi-square statistics, Satorra and Bentler propose the so-called sandwich formula for computing standard errors.
The CALIS Procedure
For ML estimation, let be the estimated covariance matrix of the parameter estimates, obtained through either the expected or observed information matrix formula. The Satorra-Bentler sandwich formula for the estimated covariance matrix is of the form where depends on the model Jacobian, the weight matrix under the normal distribution theory, and the weight matrix under general distribution theory—all evaluated at the sample estimates or the sample data values.
See Satorra and Bentler for detailed formulas. For all other estimation methods that can produce standard error estimates, it uses the unadjusted formula by default. Theoretically, if the population is truly multivariate normal, the weight matrix under normal distribution theory is correctly specified.
Asymptotically, the first two terms in the formula for cancel out, so that That is, you can use the unadjusted covariance formula to compute standard error estimates if the multivariate normality assumption is satisfied. If the multivariate normal assumption is not true, then the full sandwich formula has to be involved.
Hessian matrix - Wikipedia
Note that for positive semidefinite and negative semidefinite Hessians the test is inconclusive a critical point where the Hessian is semidefinite but not definite may be a local extremum or a saddle point. However, more can be said from the point of view of Morse theory. The second derivative test for functions of one and two variables is simple. In one variable, the Hessian contains just one second derivative; if it is positive then x is a local minimum, and if it is negative then x is a local maximum; if it is zero then the test is inconclusive.
In two variables, the determinant can be used, because the determinant is the product of the eigenvalues. If it is positive then the eigenvalues are both positive, or both negative.
If it is negative then the two eigenvalues have different signs. If it is zero, then the second derivative test is inconclusive.
Equivalently, the second-order conditions that are sufficient for a local minimum or maximum can be expressed in terms of the sequence of principal upper-leftmost minors determinants of sub-matrices of the Hessian; these conditions are a special case of those given in the next section for bordered Hessians for constrained optimization—the case in which the number of constraints is zero. Critical points[ edit ] If the gradient the vector of the partial derivatives of a function f is zero at some point x, then f has a critical point or stationary point at x.