ForK Library

# krigingeipredict.f90 File Reference

Subroutine to Predict Expected improvement from Kriging Surface based on a given minimum value (requires buildkriging to be called first) More...

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## Functions/Subroutines

subroutine krigingeipredict (ndim, ntot, X, stot, H, beta, V, hyper, mtot, Xm, Hm, EI, covarflagi, Ymin)
Due to the stochastic nature of Kriging models, the variance and mean predictions can be used to predict the expected outcomes for various scenarios. A typical scenario is to determine the expected improvement of a current minimum value if a new sample point is generated at a particular location in the domain. This expected improvement is useful within the context of optimization, where the expected improvement is used as a criteria for determining new sample locations with a high probability of finding a new minimum. Because the Kriging surface is based on a gaussian process, the expected improvement can be computed analytically using the mean function predictions and the variance predictions from the Kriging model:

## Detailed Description

Subroutine to Predict Expected improvement from Kriging Surface based on a given minimum value (requires buildkriging to be called first)

Definition in file krigingeipredict.f90.

## Function Documentation

 subroutine krigingeipredict ( integer,intent(in) ndim, integer,intent(in) ntot, real(8),dimension(ndim,ntot),intent(in) X, integer,intent(in) stot, real(8),dimension(stot,ntot),intent(in) H, real(8),dimension(stot),intent(in) Beta, real(8),dimension(ntot),intent(in) V, real(8),dimension(ndim+2),intent(in) hyper, integer,intent(in) mtot, real(8),dimension(ndim,mtot),intent(in) Xm, real(8),dimension(stot,mtot),intent(in) Hm, real(8),dimension(mtot),intent(out) EI, integer,intent(in) covarflagi, real(8),intent(in) Ymin )

Due to the stochastic nature of Kriging models, the variance and mean predictions can be used to predict the expected outcomes for various scenarios. A typical scenario is to determine the expected improvement of a current minimum value if a new sample point is generated at a particular location in the domain. This expected improvement is useful within the context of optimization, where the expected improvement is used as a criteria for determining new sample locations with a high probability of finding a new minimum. Because the Kriging surface is based on a gaussian process, the expected improvement can be computed analytically using the mean function predictions and the variance predictions from the Kriging model:

$EI(x) = \begin{cases} (y_{min} - y^{*}(x)) \Phi \left( \frac{y_{min}-y^{*}(x)}{s(x)} \right) + s(x) \phi \left( \frac{y_{min}-y^{*}(x)}{s(x)} \right) & \text{if s(x)>0,} \\ 0 & \text{if s(x)=0} \end{cases}$

where is the probability density function and is the cummulative distribution function for the Normal distribution.

Date:
May 2, 2012
Parameters:
 in) ndim : The dimension of the problem in) ntot : The number of Training points in) X : The location of the training points (size=[ndimxntot]) in) stot : Number of Terms in the regression in) H: The collocation matrix for the regression (size=[stotxntot]) in) beta: Regression coefficients based on the optimal estimate for the Kriging model (size=[stot]) Supplied by buildkriging subroutine in) V: Processed Training Data (size=[ntot]) Supplied by buildkriging subroutine in) hyper: Hyperparameters for the Kriging Model (size=[ndim+2]) Supplied by buildkriging subroutine in) mtot : The number of test points, the places where function prediction are desired in) Xm : The location of the test points (size=[ndimxmtot]) in) Hm : The collocation matrix evaluated at the test points (size=[stotxmtot]) out) EI: The predicted expected improvement value (size=[mtot]) in) covarflagi: Flag to govern which covariance function is used covarflag==0 Uses Matern function with $$\nu=1/2$$ covarflag==1 Uses Matern function with $$\nu=3/2$$ covarflag==2 Uses Matern function with $$\nu=5/2$$ The parameter $$\nu$$ governs the smoothness and differentiability of the covariance function. For function only Kriging, all three options are available. Must supply the same covariance flag as used in buildkriging. in) Ymin : Minimum value to be improved upon

Definition at line 47 of file krigingeipredict.f90.