predict.MuFicokm: Predictions and confidence intervals of Multi-Kriging...
In MuFiCokriging: Multi-Fidelity Cokriging models
Description
Usage
Arguments
Value
Author(s)
References
See Also
Examples
Description
Provide predictive mean, variance and covariance of a multi-fidelity cokriging model.
95 % confidence intervals are given based on Gaussian process assumption.
This might be abusive in particular in the case where the number of observations is small.
Usage
1
2
3
Arguments
object
an object of class S3 ("MuFicokm") provided by the function "MuFicokm" corresponding to the multi-fidelity cokriging model.
newdata
a vector, matrix or data frame containing the points where to perform predictions.
type
a list of character strings corresponding to the kriging family of the Gaussian processes δ_k(x) with k=1,...,nlevel (we use the convention δ_1(x) = Z_1(x)), to be chosen between simple kriging ("SK"), or universal kriging ("UK"). (see "MuFicokm")
se.compute
a list of optional booleans for each level. If FALSE, only the kriging mean is computed. If TRUE, the kriging variance and confidence intervals are computed too.
The length of the list must be either the number of levels or one. If the length is one, the same argument is repeated for all levels.
cov.compute
a list of optional booleans for each level. If TRUE, the conditional covariance matrix is computed.
The length of the list must be either the number of levels or one. If the length is one, the same argument is repeated for all levels.
checkNames
a list of optional booleans for each level. If TRUE, a consistency test is performed between the names of newdata and the names of the experimental design (contained in object@X). The length of the list must be either the number of levels or one. If the length is one, the same argument is repeated for all levels.
...
no other argument for this method.
Value
mean
multi-fidelity co-kriging predictive mean computed at newdata.
sig2
multi-fidelity co-kriging predictive variance computed at newdata.
C
multi-fidelity co-kriging predictive conditional covariance matrix. Not computed if cov.compute=FALSE (default).
mux
multi-fidelity co-kriging predictive means at each level.
varx
multi-fidelity co-kriging predictive variances at each level.
CovMat
multi-fidelity co-kriging predictive conditional covariance matrices at each level.
Author(s)
Loic Le Gratiet
References
KENNEDY, M.C. & O'HAGAN, A. (2000), Predicting the output from a complex computer code when fast approximations are available. Biometrika 87, 1-13
FORRESTER, A.I.J, SOBESTER A. & KEAN, A.J. (2007), Multi-Fidelity optimization via surrogate modelling. Proc. R. Soc. A 463, 3251-3269.
LE GRATIET, L. & GARNIER, J. (2012), Recursive co-kriging model for Design of Computer Experiments with multiple levels of fidelity, arXiv:1210.0686
LE GRATIET, L. (2012), Bayesian analysis of hierarchical multi-fidelity codes, arXiv:1112.5389
See Also
Examples
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#--- test functions (see [Le GRATIET, L. 2012])
Funcf <- function(x){return(0.5*(6*x-2)^2*sin(12*x-4)+sin(10*cos(5*x)))}
Funcc <- function(x){return((6*x-2)^2*sin(12*x-4)+10*(x-0.5)-5)}
#--- Data
Dc <- seq(0,1,0.1)
indDf <- c(1,3,7,11)
DNest <- NestedDesign(Dc, nlevel=2 , indices = list(indDf) )
zc <- Funcc(DNest$PX)
Df <- ExtractNestDesign(DNest,2)
zf <- Funcf(Df)
#--- Multi-fidelity cokriging creation without parameter estimations
#--- "SK" : Simple CoKriging, i.e. when parameters are known
#--- "UK" : Universal CoKriging, i.e. when parameters are estimated
#-- model creation
mymodelSK <- MuFicokm(
formula = list(~1,~1),
MuFidesign = DNest,
response = list(zc,zf),
nlevel = 2,
coef.trend=list(0,2),
coef.rho=list(0.5),
coef.var=list(2,2),
coef.cov=list(0.1,0.2))
#-- predictions with "SK"
predictionsSK <- predict(
object = mymodelSK,
newdata = seq(0,1,le=100),
type = "SK")
#--- Multi-fidelity co-kriging building with parameter estimations
#-- model creation
mymodelUK <- MuFicokm(
formula = list(~1,~1),
MuFidesign = DNest,
response = list(zc,zf),
nlevel = 2)
#-- predictions with "UK"
predictionsUK <- predict(
object = mymodelUK,
newdata = seq(0,1,le=100),
type = "UK")
#--- Multi-fidelity co-kriging building with known and unknown parameters
#-- model creation
mymodelSK_UK <- MuFicokm(
formula = list(~1,~1),
MuFidesign = DNest,
response = list(zc,zf),
nlevel = 2,
coef.trend=list(-5,NULL),
coef.rho=list(NULL),
coef.var=list(2,NULL),
coef.cov=list(0.1,NULL))
#-- predictions with "UK"
predictionsSK_UK <- predict(
object = mymodelSK_UK,
newdata = seq(0,1,le=100),
type = list("SK","UK"))
#--- plot
op <- par(mfrow=c(3,1))
x <- seq(0,1,le=100)
curve(Funcf,ylim=c(-5,10),main="SK")
polygon(c(x,rev(x)), c(predictionsSK$mean+2*sqrt(predictionsSK$sig2),
rev(predictionsSK$mean-2*sqrt(predictionsSK$sig2))),
col="gray", border = "red",density=20 )
lines(seq(0,1,le=100), predictionsSK$mean,col=2,lty=2,lwd=2)
curve(Funcf,ylim=c(-5,10),main="UK")
polygon(c(x,rev(x)), c(predictionsUK$mean+2*sqrt(predictionsUK$sig2),
rev(predictionsUK$mean-2*sqrt(predictionsUK$sig2))),
col="gray", border = "red",density=20 )
lines(seq(0,1,le=100), predictionsUK$mean,col=2,lty=2,lwd=2)
curve(Funcf,ylim=c(-5,10),main="mix SK & UK")
polygon(c(x,rev(x)), c(predictionsSK_UK$mean+2*sqrt(predictionsSK_UK$sig2),
rev(predictionsSK_UK$mean-2*sqrt(predictionsSK_UK$sig2))),
col="gray", border = "red",density=20 )
lines(seq(0,1,le=100), predictionsSK_UK$mean,col=2,lty=2,lwd=2)
par(op)
Example output
Loading required package: DiceKriging
optimisation start
------------------
* estimation method : MLE
* optimisation method : BFGS
* analytical gradient : used
* trend model : ~1
* covariance model :
- type : matern5_2
- nugget : NO
- parameters lower bounds : 1e-10
- parameters upper bounds : 2
- best initial criterion value(s) : -30.18028
N = 1, M = 5 machine precision = 2.22045e-16
At X0, 0 variables are exactly at the bounds
At iterate 0 f= 30.18 |proj g|= 0.3502
At iterate 1 f = 30.111 |proj g|= 0.33614
At iterate 2 f = 30.042 |proj g|= 1.6348
At iterate 3 f = 30.034 |proj g|= 0.26938
At iterate 4 f = 30.033 |proj g|= 0.013115
At iterate 5 f = 30.033 |proj g|= 0.00011532
At iterate 6 f = 30.033 |proj g|= 4.8591e-08
iterations 6
function evaluations 8
segments explored during Cauchy searches 6
BFGS updates skipped 0
active bounds at final generalized Cauchy point 0
norm of the final projected gradient 4.85908e-08
final function value 30.0335
F = 30.0335
final value 30.033468
converged
optimisation start
------------------
* estimation method : MLE
* optimisation method : BFGS
* analytical gradient : used
* trend model : ~1
* covariance model :
- type : matern5_2
- nugget : NO
- parameters lower bounds : 1e-10
- parameters upper bounds : 2
- best initial criterion value(s) : -3.813998
N = 1, M = 5 machine precision = 2.22045e-16
At X0, 0 variables are exactly at the bounds
At iterate 0 f= 3.814 |proj g|= 0.00016814
At iterate 1 f = 3.814 |proj g|= 0.0001474
At iterate 2 f = 3.814 |proj g|= 5.3894e-05
At iterate 3 f = 3.814 |proj g|= 2.8356e-05
At iterate 4 f = 3.814 |proj g|= 1.3074e-05
At iterate 5 f = 3.814 |proj g|= 6.3667e-06
iterations 5
function evaluations 6
segments explored during Cauchy searches 5
BFGS updates skipped 0
active bounds at final generalized Cauchy point 0
norm of the final projected gradient 6.36673e-06
final function value 3.814
F = 3.814
final value 3.813998
converged
optimisation start
------------------
* estimation method : MLE
* optimisation method : BFGS
* analytical gradient : used
* trend model : ~1
* covariance model :
- type : matern5_2
- nugget : NO
- parameters lower bounds : 1e-10
- parameters upper bounds : 2
- best initial criterion value(s) : -4.732139
N = 1, M = 5 machine precision = 2.22045e-16
At X0, 0 variables are exactly at the bounds
At iterate 0 f= 4.7321 |proj g|= 0.226
At iterate 1 f = 3.814 |proj g|= 0
iterations 1
function evaluations 2
segments explored during Cauchy searches 1
BFGS updates skipped 0
active bounds at final generalized Cauchy point 1
norm of the final projected gradient 0
final function value 3.814
F = 3.814
final value 3.813998
converged
MuFiCokriging documentation built on May 2, 2019, 3:33 p.m.
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Description Usage Arguments Value Author(s) References See Also Examples
Description
Provide predictive mean, variance and covariance of a multi-fidelity cokriging model. 95 % confidence intervals are given based on Gaussian process assumption. This might be abusive in particular in the case where the number of observations is small.
Usage
1 2 3 |
Arguments
object |
an object of class S3 ( |
newdata |
a vector, matrix or data frame containing the points where to perform predictions. |
type |
a list of character strings corresponding to the kriging family of the Gaussian processes δ_k(x) with k=1,...,nlevel (we use the convention δ_1(x) = Z_1(x)), to be chosen between simple kriging ( |
se.compute |
a list of optional booleans for each level. If |
cov.compute |
a list of optional booleans for each level. If |
checkNames |
a list of optional booleans for each level. If |
... |
no other argument for this method. |
Value
mean |
multi-fidelity co-kriging predictive mean computed at |
sig2 |
multi-fidelity co-kriging predictive variance computed at |
C |
multi-fidelity co-kriging predictive conditional covariance matrix. Not computed if |
mux |
multi-fidelity co-kriging predictive means at each level. |
varx |
multi-fidelity co-kriging predictive variances at each level. |
CovMat |
multi-fidelity co-kriging predictive conditional covariance matrices at each level. |
Author(s)
Loic Le Gratiet
References
KENNEDY, M.C. & O'HAGAN, A. (2000), Predicting the output from a complex computer code when fast approximations are available. Biometrika 87, 1-13
FORRESTER, A.I.J, SOBESTER A. & KEAN, A.J. (2007), Multi-Fidelity optimization via surrogate modelling. Proc. R. Soc. A 463, 3251-3269.
LE GRATIET, L. & GARNIER, J. (2012), Recursive co-kriging model for Design of Computer Experiments with multiple levels of fidelity, arXiv:1210.0686
LE GRATIET, L. (2012), Bayesian analysis of hierarchical multi-fidelity codes, arXiv:1112.5389
See Also
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 | #--- test functions (see [Le GRATIET, L. 2012])
Funcf <- function(x){return(0.5*(6*x-2)^2*sin(12*x-4)+sin(10*cos(5*x)))}
Funcc <- function(x){return((6*x-2)^2*sin(12*x-4)+10*(x-0.5)-5)}
#--- Data
Dc <- seq(0,1,0.1)
indDf <- c(1,3,7,11)
DNest <- NestedDesign(Dc, nlevel=2 , indices = list(indDf) )
zc <- Funcc(DNest$PX)
Df <- ExtractNestDesign(DNest,2)
zf <- Funcf(Df)
#--- Multi-fidelity cokriging creation without parameter estimations
#--- "SK" : Simple CoKriging, i.e. when parameters are known
#--- "UK" : Universal CoKriging, i.e. when parameters are estimated
#-- model creation
mymodelSK <- MuFicokm(
formula = list(~1,~1),
MuFidesign = DNest,
response = list(zc,zf),
nlevel = 2,
coef.trend=list(0,2),
coef.rho=list(0.5),
coef.var=list(2,2),
coef.cov=list(0.1,0.2))
#-- predictions with "SK"
predictionsSK <- predict(
object = mymodelSK,
newdata = seq(0,1,le=100),
type = "SK")
#--- Multi-fidelity co-kriging building with parameter estimations
#-- model creation
mymodelUK <- MuFicokm(
formula = list(~1,~1),
MuFidesign = DNest,
response = list(zc,zf),
nlevel = 2)
#-- predictions with "UK"
predictionsUK <- predict(
object = mymodelUK,
newdata = seq(0,1,le=100),
type = "UK")
#--- Multi-fidelity co-kriging building with known and unknown parameters
#-- model creation
mymodelSK_UK <- MuFicokm(
formula = list(~1,~1),
MuFidesign = DNest,
response = list(zc,zf),
nlevel = 2,
coef.trend=list(-5,NULL),
coef.rho=list(NULL),
coef.var=list(2,NULL),
coef.cov=list(0.1,NULL))
#-- predictions with "UK"
predictionsSK_UK <- predict(
object = mymodelSK_UK,
newdata = seq(0,1,le=100),
type = list("SK","UK"))
#--- plot
op <- par(mfrow=c(3,1))
x <- seq(0,1,le=100)
curve(Funcf,ylim=c(-5,10),main="SK")
polygon(c(x,rev(x)), c(predictionsSK$mean+2*sqrt(predictionsSK$sig2),
rev(predictionsSK$mean-2*sqrt(predictionsSK$sig2))),
col="gray", border = "red",density=20 )
lines(seq(0,1,le=100), predictionsSK$mean,col=2,lty=2,lwd=2)
curve(Funcf,ylim=c(-5,10),main="UK")
polygon(c(x,rev(x)), c(predictionsUK$mean+2*sqrt(predictionsUK$sig2),
rev(predictionsUK$mean-2*sqrt(predictionsUK$sig2))),
col="gray", border = "red",density=20 )
lines(seq(0,1,le=100), predictionsUK$mean,col=2,lty=2,lwd=2)
curve(Funcf,ylim=c(-5,10),main="mix SK & UK")
polygon(c(x,rev(x)), c(predictionsSK_UK$mean+2*sqrt(predictionsSK_UK$sig2),
rev(predictionsSK_UK$mean-2*sqrt(predictionsSK_UK$sig2))),
col="gray", border = "red",density=20 )
lines(seq(0,1,le=100), predictionsSK_UK$mean,col=2,lty=2,lwd=2)
par(op)
|
Example output
Loading required package: DiceKriging
optimisation start
------------------
* estimation method : MLE
* optimisation method : BFGS
* analytical gradient : used
* trend model : ~1
* covariance model :
- type : matern5_2
- nugget : NO
- parameters lower bounds : 1e-10
- parameters upper bounds : 2
- best initial criterion value(s) : -30.18028
N = 1, M = 5 machine precision = 2.22045e-16
At X0, 0 variables are exactly at the bounds
At iterate 0 f= 30.18 |proj g|= 0.3502
At iterate 1 f = 30.111 |proj g|= 0.33614
At iterate 2 f = 30.042 |proj g|= 1.6348
At iterate 3 f = 30.034 |proj g|= 0.26938
At iterate 4 f = 30.033 |proj g|= 0.013115
At iterate 5 f = 30.033 |proj g|= 0.00011532
At iterate 6 f = 30.033 |proj g|= 4.8591e-08
iterations 6
function evaluations 8
segments explored during Cauchy searches 6
BFGS updates skipped 0
active bounds at final generalized Cauchy point 0
norm of the final projected gradient 4.85908e-08
final function value 30.0335
F = 30.0335
final value 30.033468
converged
optimisation start
------------------
* estimation method : MLE
* optimisation method : BFGS
* analytical gradient : used
* trend model : ~1
* covariance model :
- type : matern5_2
- nugget : NO
- parameters lower bounds : 1e-10
- parameters upper bounds : 2
- best initial criterion value(s) : -3.813998
N = 1, M = 5 machine precision = 2.22045e-16
At X0, 0 variables are exactly at the bounds
At iterate 0 f= 3.814 |proj g|= 0.00016814
At iterate 1 f = 3.814 |proj g|= 0.0001474
At iterate 2 f = 3.814 |proj g|= 5.3894e-05
At iterate 3 f = 3.814 |proj g|= 2.8356e-05
At iterate 4 f = 3.814 |proj g|= 1.3074e-05
At iterate 5 f = 3.814 |proj g|= 6.3667e-06
iterations 5
function evaluations 6
segments explored during Cauchy searches 5
BFGS updates skipped 0
active bounds at final generalized Cauchy point 0
norm of the final projected gradient 6.36673e-06
final function value 3.814
F = 3.814
final value 3.813998
converged
optimisation start
------------------
* estimation method : MLE
* optimisation method : BFGS
* analytical gradient : used
* trend model : ~1
* covariance model :
- type : matern5_2
- nugget : NO
- parameters lower bounds : 1e-10
- parameters upper bounds : 2
- best initial criterion value(s) : -4.732139
N = 1, M = 5 machine precision = 2.22045e-16
At X0, 0 variables are exactly at the bounds
At iterate 0 f= 4.7321 |proj g|= 0.226
At iterate 1 f = 3.814 |proj g|= 0
iterations 1
function evaluations 2
segments explored during Cauchy searches 1
BFGS updates skipped 0
active bounds at final generalized Cauchy point 1
norm of the final projected gradient 0
final function value 3.814
F = 3.814
final value 3.813998
converged
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