crossValidation: K-fold Cross Validation

Description Usage Arguments Value Note Author(s) See Also Examples

View source: R/crossValidation.R

Description

This function calculates the predicted values at each point of the design and gives an estimation of criterion using K-fold cross-validation.

Usage

1
crossValidation(model, K)

Arguments

model

an output of the modelFit function. This argument is the initial model fitted with all the data.

K

the number of groups into which the data should be split to apply cross-validation

Value

A list with the following components:

Ypred

a vector of predicted values obtained using K-fold cross-validation at the points of the design

Q2

a real which is the estimation of the criterion R2 obtained by cross-validation

folds

a list which indicates the partitioning of the data into the folds

RMSE_CV

RMSE by K-fold cross-validation (see more details below)

MAE_CV

MAE by K-fold cross-validation (see more details below)

In the case of a Kriging model, other components to test the robustess of the procedure are proposed:

theta

the range parameter theta estimated for each fold,

trend

the trend parameter estimated for each fold,

shape

the estimated shape parameter if the covariance structure is of type powerexp.

The principle of cross-validation is to split the data into K folds of approximately equal size A_{1}{A1}, ..., A_{K}{AK}. For k=1 to K, a model Y^(-k) is fitted from the data A1 U ... U AK and this model is validated on the fold Ak. Given a criterion of quality L (here, L could be the RMSE or the MAE criterion), the "evaluation" of the model consists in computing :

Lk = 1/(n/K) Sum (i in Ak) L (yi,Y^(-k)(xi).

The cross-validation criterion is the mean of the K criterion: L_CV1/K (L1+...+LK).

The Q2 criterion is defined as: Q2=\code{R2}(\code{Y},\code{Ypred}) with Y the response value and Ypred the value fit by cross-validation.

Note

When K is equal to the number of observations, leave-one-out cross-validation is performed.

Author(s)

D. Dupuy

See Also

R2, modelFit, MAE, RMSE, foldsComposition, testCrossValidation

Examples

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## Not run: 
rm(list=ls())
# A 2D example
Branin <- function(x1,x2) {
  x1 <- x1*15-5   
  x2 <- x2*15
  (x2 - 5/(4*pi^2)*(x1^2) + 5/pi*x1 - 6)^2 + 10*(1 - 1/(8*pi))*cos(x1) + 10
}

# Linear model on 50 points
n <- 50
X <- matrix(runif(n*2),ncol=2,nrow=n)
Y <- Branin(X[,1],X[,2])
modLm <- modelFit(X,Y,type = "Linear",formula=Y~X1+X2+X1:X2+I(X1^2)+I(X2^2))
R2(Y,modLm$model$fitted.values)
crossValidation(modLm,K=10)$Q2


# kriging model : gaussian covariance structure, no trend, no nugget effect
# on 16 points 
n <- 16
X <- data.frame(x1=runif(n),x2=runif(n))
Y <- Branin(X[,1],X[,2])
mKm <- modelFit(X,Y,type="Kriging",formula=~1, covtype="powexp")
K <- 10
out   <- crossValidation(mKm, K)
par(mfrow=c(2,2))
plot(c(0,1:K),c(mKm$model@covariance@range.val[1],out$theta[,1]),
 	xlab='',ylab='Theta1')
 plot(c(0,1:K),c(mKm$model@covariance@range.val[2],out$theta[,2]),
 	xlab='',ylab='Theta2')
 plot(c(0,1:K),c(mKm$model@covariance@shape.val[1],out$shape[,1]),
 	xlab='',ylab='p1',ylim=c(0,2))
 plot(c(0,1:K),c(mKm$model@covariance@shape.val[2],out$shape[,2]),
 	xlab='',ylab='p2',ylim=c(0,2))
par(mfrow=c(1,1))

## End(Not run)

Example output

Loading required package: DiceKriging
[1] 0.7593807
[1] 0.6678448

optimisation start
------------------
* estimation method   : MLE 
* optimisation method : BFGS 
* analytical gradient : used
* trend model : ~1
* covariance model : 
  - type :  powexp 
  - nugget : NO
  - parameters lower bounds :  1e-10 1e-10 1e-10 1e-10 
  - parameters upper bounds :  1.481862 1.776442 2 2 
  - best initial criterion value(s) :  -78.89193 

N = 4, M = 5 machine precision = 2.22045e-16
At X0, 0 variables are exactly at the bounds
At iterate     0  f=       78.892  |proj g|=       1.2117
At iterate     1  f =       77.219  |proj g|=        1.3524
At iterate     2  f =       67.721  |proj g|=        1.0722
ys=-1.779e+01  -gs= 7.958e+00, BFGS update SKIPPED
At iterate     3  f =       67.145  |proj g|=        1.0538
At iterate     4  f =       66.859  |proj g|=        1.0395
At iterate     5  f =       66.767  |proj g|=        1.9679
At iterate     6  f =       66.638  |proj g|=        1.9699
At iterate     7  f =       66.489  |proj g|=       0.90603
At iterate     8  f =       66.478  |proj g|=       0.91348
At iterate     9  f =       66.477  |proj g|=       0.90178
At iterate    10  f =       66.473  |proj g|=        1.4322
At iterate    11  f =       66.466  |proj g|=        1.9701
At iterate    12  f =        66.45  |proj g|=        1.9715
At iterate    13  f =       66.434  |proj g|=        1.9382
At iterate    14  f =       66.432  |proj g|=       0.29568
At iterate    15  f =       66.432  |proj g|=      0.031858
At iterate    16  f =       66.431  |proj g|=      0.029762
At iterate    17  f =       66.431  |proj g|=      0.025418
At iterate    18  f =       66.431  |proj g|=     0.0091401
At iterate    19  f =       66.431  |proj g|=     0.0030923

iterations 19
function evaluations 25
segments explored during Cauchy searches 24
BFGS updates skipped 1
active bounds at final generalized Cauchy point 1
norm of the final projected gradient 0.00309229
final function value 66.4315

F = 66.4315
final  value 66.431473 
converged

optimisation start
------------------
* estimation method   : MLE 
* optimisation method : BFGS 
* analytical gradient : used
* trend model : ~1
* covariance model : 
  - type :  powexp 
  - nugget : NO
  - parameters lower bounds :  1e-10 1e-10 1e-10 1e-10 
  - parameters upper bounds :  1.481862 1.776442 2 2 
  - best initial criterion value(s) :  -70.97419 

N = 4, M = 5 machine precision = 2.22045e-16
At X0, 0 variables are exactly at the bounds
At iterate     0  f=       70.974  |proj g|=       1.2117
At iterate     1  f =       69.829  |proj g|=       0.89517
At iterate     2  f =       68.023  |proj g|=        1.3047
At iterate     3  f =       65.605  |proj g|=             2
At iterate     4  f =       64.491  |proj g|=        1.0745
At iterate     5  f =       62.731  |proj g|=        1.0658
At iterate     6  f =       62.517  |proj g|=        1.0514
At iterate     7  f =       62.297  |proj g|=        1.9639
At iterate     8  f =       62.124  |proj g|=        1.1357
At iterate     9  f =       62.083  |proj g|=        1.1218
At iterate    10  f =       61.982  |proj g|=        1.9568
At iterate    11  f =       61.961  |proj g|=        0.8761
At iterate    12  f =       61.954  |proj g|=       0.21916
At iterate    13  f =       61.953  |proj g|=      0.045153
At iterate    14  f =       61.953  |proj g|=      0.044836
At iterate    15  f =       61.953  |proj g|=      0.006876
At iterate    16  f =       61.953  |proj g|=    0.00063468

iterations 16
function evaluations 18
segments explored during Cauchy searches 19
BFGS updates skipped 0
active bounds at final generalized Cauchy point 1
norm of the final projected gradient 0.000634677
final function value 61.9535

F = 61.9535
final  value 61.953480 
converged

optimisation start
------------------
* estimation method   : MLE 
* optimisation method : BFGS 
* analytical gradient : used
* trend model : ~1
* covariance model : 
  - type :  powexp 
  - nugget : NO
  - parameters lower bounds :  1e-10 1e-10 1e-10 1e-10 
  - parameters upper bounds :  1.481862 1.776442 2 2 
  - best initial criterion value(s) :  -70.85623 

N = 4, M = 5 machine precision = 2.22045e-16
At X0, 0 variables are exactly at the bounds
At iterate     0  f=       70.856  |proj g|=       1.2117
At iterate     1  f =        69.24  |proj g|=       0.83292
At iterate     2  f =       67.406  |proj g|=          1.34
At iterate     3  f =       65.585  |proj g|=             2
At iterate     4  f =       65.317  |proj g|=        1.6078
At iterate     5  f =       64.371  |proj g|=        1.1121
At iterate     6  f =       62.737  |proj g|=        1.0658
ys=-4.052e-01  -gs= 1.416e+00, BFGS update SKIPPED
At iterate     7  f =       61.893  |proj g|=        1.1211
At iterate     8  f =       61.741  |proj g|=        1.1764
At iterate     9  f =        61.65  |proj g|=        1.9699
At iterate    10  f =       61.539  |proj g|=        1.9678
At iterate    11  f =       61.449  |proj g|=       0.88992
At iterate    12  f =       61.432  |proj g|=       0.27053
At iterate    13  f =       61.431  |proj g|=      0.049971
At iterate    14  f =       61.431  |proj g|=      0.018914
At iterate    15  f =       61.431  |proj g|=      0.016332
At iterate    16  f =       61.431  |proj g|=       0.03052
At iterate    17  f =       61.431  |proj g|=      0.034397
At iterate    18  f =       61.431  |proj g|=     0.0028097
At iterate    19  f =       61.431  |proj g|=    0.00045263

iterations 19
function evaluations 23
segments explored during Cauchy searches 22
BFGS updates skipped 1
active bounds at final generalized Cauchy point 1
norm of the final projected gradient 0.000452632
final function value 61.4308

F = 61.4308
final  value 61.430801 
converged

optimisation start
------------------
* estimation method   : MLE 
* optimisation method : BFGS 
* analytical gradient : used
* trend model : ~1
* covariance model : 
  - type :  powexp 
  - nugget : NO
  - parameters lower bounds :  1e-10 1e-10 1e-10 1e-10 
  - parameters upper bounds :  1.481862 1.776442 2 2 
  - best initial criterion value(s) :  -69.24872 

N = 4, M = 5 machine precision = 2.22045e-16
At X0, 0 variables are exactly at the bounds
At iterate     0  f=       69.249  |proj g|=       1.2117
At iterate     1  f =       65.902  |proj g|=        1.0515
At iterate     2  f =       60.611  |proj g|=       0.95426
ys=-1.724e+01  -gs= 3.858e+00, BFGS update SKIPPED
At iterate     3  f =       60.321  |proj g|=        1.9628
At iterate     4  f =       60.072  |proj g|=         1.977
At iterate     5  f =       59.387  |proj g|=        1.9967
At iterate     6  f =       59.142  |proj g|=       0.65201
At iterate     7  f =       59.113  |proj g|=       0.64851
At iterate     8  f =       58.942  |proj g|=       0.94813
At iterate     9  f =       58.923  |proj g|=             2
At iterate    10  f =       58.883  |proj g|=        1.9998
At iterate    11  f =       58.866  |proj g|=        1.1067
At iterate    12  f =       58.866  |proj g|=       0.67758
At iterate    13  f =       58.866  |proj g|=       0.31985
At iterate    14  f =       58.866  |proj g|=      0.038333
At iterate    15  f =       58.866  |proj g|=    0.00081442
At iterate    16  f =       58.866  |proj g|=    0.00064075

iterations 16
function evaluations 25
segments explored during Cauchy searches 21
BFGS updates skipped 1
active bounds at final generalized Cauchy point 1
norm of the final projected gradient 0.000640753
final function value 58.8656

F = 58.8656
final  value 58.865630 
converged

optimisation start
------------------
* estimation method   : MLE 
* optimisation method : BFGS 
* analytical gradient : used
* trend model : ~1
* covariance model : 
  - type :  powexp 
  - nugget : NO
  - parameters lower bounds :  1e-10 1e-10 1e-10 1e-10 
  - parameters upper bounds :  1.481862 1.776442 2 2 
  - best initial criterion value(s) :  -69.09983 

N = 4, M = 5 machine precision = 2.22045e-16
At X0, 0 variables are exactly at the bounds
At iterate     0  f=         69.1  |proj g|=       1.2117
At iterate     1  f =       67.245  |proj g|=       0.83749
At iterate     2  f =       65.212  |proj g|=        1.3298
At iterate     3  f =       62.975  |proj g|=        1.9787
At iterate     4  f =       61.261  |proj g|=        1.1165
At iterate     5  f =       60.571  |proj g|=        1.0912
At iterate     6  f =       59.909  |proj g|=        1.1259
At iterate     7  f =       59.718  |proj g|=        1.9646
At iterate     8  f =       59.445  |proj g|=        1.9633
At iterate     9  f =       59.315  |proj g|=        1.9686
At iterate    10  f =       59.256  |proj g|=       0.92942
At iterate    11  f =       59.224  |proj g|=        1.9753
At iterate    12  f =       59.217  |proj g|=       0.21145
At iterate    13  f =       59.213  |proj g|=        0.0814
At iterate    14  f =       59.213  |proj g|=      0.076274
At iterate    15  f =       59.213  |proj g|=      0.018228
At iterate    16  f =       59.213  |proj g|=    0.00097147
At iterate    17  f =       59.213  |proj g|=    6.7025e-05

iterations 17
function evaluations 22
segments explored during Cauchy searches 21
BFGS updates skipped 0
active bounds at final generalized Cauchy point 1
norm of the final projected gradient 6.70248e-05
final function value 59.2128

F = 59.2128
final  value 59.212793 
converged

optimisation start
------------------
* estimation method   : MLE 
* optimisation method : BFGS 
* analytical gradient : used
* trend model : ~1
* covariance model : 
  - type :  powexp 
  - nugget : NO
  - parameters lower bounds :  1e-10 1e-10 1e-10 1e-10 
  - parameters upper bounds :  1.481862 1.776442 2 2 
  - best initial criterion value(s) :  -71.21742 

N = 4, M = 5 machine precision = 2.22045e-16
At X0, 0 variables are exactly at the bounds
At iterate     0  f=       71.217  |proj g|=       1.2117
At iterate     1  f =       70.321  |proj g|=        1.3293
At iterate     2  f =       62.283  |proj g|=       0.92117
ys=-3.937e+00  -gs= 7.162e+00, BFGS update SKIPPED
At iterate     3  f =       62.053  |proj g|=        1.0785
At iterate     4  f =       62.032  |proj g|=        1.0286
At iterate     5  f =       62.028  |proj g|=       0.93077
At iterate     6  f =       62.011  |proj g|=       0.74251
At iterate     7  f =       62.001  |proj g|=        0.5191
At iterate     8  f =       61.994  |proj g|=      0.058859
At iterate     9  f =       61.994  |proj g|=      0.019752
At iterate    10  f =       61.994  |proj g|=      0.015203
At iterate    11  f =       61.994  |proj g|=    0.00087174
At iterate    12  f =       61.994  |proj g|=    6.2583e-05

iterations 12
function evaluations 18
segments explored during Cauchy searches 16
BFGS updates skipped 1
active bounds at final generalized Cauchy point 1
norm of the final projected gradient 6.2583e-05
final function value 61.9937

F = 61.9937
final  value 61.993652 
converged

optimisation start
------------------
* estimation method   : MLE 
* optimisation method : BFGS 
* analytical gradient : used
* trend model : ~1
* covariance model : 
  - type :  powexp 
  - nugget : NO
  - parameters lower bounds :  1e-10 1e-10 1e-10 1e-10 
  - parameters upper bounds :  1.481862 1.776442 2 2 
  - best initial criterion value(s) :  -70.62542 

N = 4, M = 5 machine precision = 2.22045e-16
At X0, 0 variables are exactly at the bounds
At iterate     0  f=       70.625  |proj g|=       1.2117
At iterate     1  f =       68.984  |proj g|=        1.0741
At iterate     2  f =       67.373  |proj g|=        1.3178
At iterate     3  f =       65.179  |proj g|=             2
At iterate     4  f =       64.946  |proj g|=        1.2234
At iterate     5  f =       64.602  |proj g|=        1.1966
At iterate     6  f =       63.138  |proj g|=        1.0178
At iterate     7  f =       62.808  |proj g|=        1.9615
At iterate     8  f =       62.763  |proj g|=        1.0945
At iterate     9  f =       62.737  |proj g|=        1.0781
At iterate    10  f =       62.681  |proj g|=       0.39153
At iterate    11  f =       62.665  |proj g|=       0.15083
At iterate    12  f =        62.66  |proj g|=       0.27128
At iterate    13  f =        62.66  |proj g|=       0.03915
At iterate    14  f =        62.66  |proj g|=     0.0068872
At iterate    15  f =        62.66  |proj g|=     0.0060669
At iterate    16  f =        62.66  |proj g|=    0.00077154

iterations 16
function evaluations 19
segments explored during Cauchy searches 19
BFGS updates skipped 0
active bounds at final generalized Cauchy point 1
norm of the final projected gradient 0.000771537
final function value 62.6599

F = 62.6599
final  value 62.659940 
converged

optimisation start
------------------
* estimation method   : MLE 
* optimisation method : BFGS 
* analytical gradient : used
* trend model : ~1
* covariance model : 
  - type :  powexp 
  - nugget : NO
  - parameters lower bounds :  1e-10 1e-10 1e-10 1e-10 
  - parameters upper bounds :  1.481862 1.776442 2 2 
  - best initial criterion value(s) :  -74.97118 

N = 4, M = 5 machine precision = 2.22045e-16
At X0, 0 variables are exactly at the bounds
At iterate     0  f=       74.971  |proj g|=       1.2117
At iterate     1  f =       73.207  |proj g|=        1.2512
At iterate     2  f =       71.197  |proj g|=        1.3219
At iterate     3  f =       68.662  |proj g|=             2
At iterate     4  f =        68.26  |proj g|=        1.2027
At iterate     5  f =       67.696  |proj g|=        1.0705
At iterate     6  f =       65.454  |proj g|=        1.0431
ys=-2.646e+00  -gs= 1.790e+00, BFGS update SKIPPED
At iterate     7  f =       64.949  |proj g|=         1.119
At iterate     8  f =       64.883  |proj g|=        1.9613
At iterate     9  f =       64.834  |proj g|=        1.9647
At iterate    10  f =       64.716  |proj g|=        1.9669
At iterate    11  f =       64.658  |proj g|=       0.79549
At iterate    12  f =       64.649  |proj g|=       0.33978
At iterate    13  f =       64.647  |proj g|=       0.19202
At iterate    14  f =       64.647  |proj g|=      0.036033
At iterate    15  f =       64.646  |proj g|=      0.020934
At iterate    16  f =       64.646  |proj g|=     0.0033787
At iterate    17  f =       64.646  |proj g|=    0.00082071

iterations 17
function evaluations 20
segments explored during Cauchy searches 20
BFGS updates skipped 1
active bounds at final generalized Cauchy point 1
norm of the final projected gradient 0.000820706
final function value 64.6465

F = 64.6465
final  value 64.646481 
converged

optimisation start
------------------
* estimation method   : MLE 
* optimisation method : BFGS 
* analytical gradient : used
* trend model : ~1
* covariance model : 
  - type :  powexp 
  - nugget : NO
  - parameters lower bounds :  1e-10 1e-10 1e-10 1e-10 
  - parameters upper bounds :  1.481862 1.776442 2 2 
  - best initial criterion value(s) :  -74.85036 

N = 4, M = 5 machine precision = 2.22045e-16
At X0, 0 variables are exactly at the bounds
At iterate     0  f=        74.85  |proj g|=       1.2117
At iterate     1  f =       73.106  |proj g|=        1.3249
At iterate     2  f =       67.516  |proj g|=        1.9743
At iterate     3  f =       65.248  |proj g|=        1.1745
At iterate     4  f =       65.119  |proj g|=        1.1297
At iterate     5  f =       64.777  |proj g|=        1.0951
At iterate     6  f =       64.456  |proj g|=        1.0379
At iterate     7  f =       64.364  |proj g|=        1.9701
At iterate     8  f =       64.291  |proj g|=        1.3754
At iterate     9  f =       64.263  |proj g|=        1.8927
At iterate    10  f =       64.183  |proj g|=        1.9662
At iterate    11  f =       64.173  |proj g|=       0.17514
At iterate    12  f =       64.172  |proj g|=      0.033342
At iterate    13  f =       64.172  |proj g|=       0.15002
At iterate    14  f =       64.172  |proj g|=     0.0010331
At iterate    15  f =       64.172  |proj g|=    7.7481e-05

iterations 15
function evaluations 21
segments explored during Cauchy searches 18
BFGS updates skipped 0
active bounds at final generalized Cauchy point 1
norm of the final projected gradient 7.7481e-05
final function value 64.1721

F = 64.1721
final  value 64.172131 
converged

optimisation start
------------------
* estimation method   : MLE 
* optimisation method : BFGS 
* analytical gradient : used
* trend model : ~1
* covariance model : 
  - type :  powexp 
  - nugget : NO
  - parameters lower bounds :  1e-10 1e-10 1e-10 1e-10 
  - parameters upper bounds :  1.481862 1.776442 2 2 
  - best initial criterion value(s) :  -74.72783 

N = 4, M = 5 machine precision = 2.22045e-16
At X0, 0 variables are exactly at the bounds
At iterate     0  f=       74.728  |proj g|=       1.2117
At iterate     1  f =       72.951  |proj g|=        1.2482
At iterate     2  f =       70.904  |proj g|=        1.3162
At iterate     3  f =       68.299  |proj g|=        1.9775
At iterate     4  f =       66.184  |proj g|=        1.1237
At iterate     5  f =       65.287  |proj g|=        1.0815
At iterate     6  f =        64.96  |proj g|=        1.2039
At iterate     7  f =       64.555  |proj g|=        1.1417
At iterate     8  f =       64.494  |proj g|=        1.9723
At iterate     9  f =        64.35  |proj g|=        1.9647
At iterate    10  f =       64.233  |proj g|=       0.89444
At iterate    11  f =         64.2  |proj g|=        0.1873
At iterate    12  f =       64.193  |proj g|=       0.18535
At iterate    13  f =       64.193  |proj g|=       0.12591
At iterate    14  f =       64.193  |proj g|=      0.032715
At iterate    15  f =       64.193  |proj g|=      0.076216
At iterate    16  f =       64.193  |proj g|=       0.01954
At iterate    17  f =       64.193  |proj g|=     0.0022613
At iterate    18  f =       64.193  |proj g|=    2.7094e-06

iterations 18
function evaluations 24
segments explored during Cauchy searches 22
BFGS updates skipped 0
active bounds at final generalized Cauchy point 1
norm of the final projected gradient 2.70939e-06
final function value 64.1929

F = 64.1929
final  value 64.192908 
converged

optimisation start
------------------
* estimation method   : MLE 
* optimisation method : BFGS 
* analytical gradient : used
* trend model : ~1
* covariance model : 
  - type :  powexp 
  - nugget : NO
  - parameters lower bounds :  1e-10 1e-10 1e-10 1e-10 
  - parameters upper bounds :  1.481862 1.776442 2 2 
  - best initial criterion value(s) :  -69.72732 

N = 4, M = 5 machine precision = 2.22045e-16
At X0, 0 variables are exactly at the bounds
At iterate     0  f=       69.727  |proj g|=       1.2117
At iterate     1  f =       64.982  |proj g|=       0.99107
At iterate     2  f =       59.128  |proj g|=       0.94297
ys=-4.160e+01  -gs= 4.281e+00, BFGS update SKIPPED
At iterate     3  f =       59.085  |proj g|=        1.9432
At iterate     4  f =       59.019  |proj g|=        1.9526
At iterate     5  f =       58.954  |proj g|=        1.9516
At iterate     6  f =       58.947  |proj g|=       0.24411
At iterate     7  f =       58.947  |proj g|=       0.14265
At iterate     8  f =       58.947  |proj g|=       0.14425
At iterate     9  f =       58.947  |proj g|=       0.14523
At iterate    10  f =       58.947  |proj g|=       0.13875
At iterate    11  f =       58.946  |proj g|=       0.11261
At iterate    12  f =       58.946  |proj g|=       0.05713
At iterate    13  f =       58.946  |proj g|=      0.055076
At iterate    14  f =       58.945  |proj g|=      0.054965
At iterate    15  f =       58.945  |proj g|=     0.0098254
At iterate    16  f =       58.945  |proj g|=    0.00017735
At iterate    17  f =       58.945  |proj g|=    1.0832e-05

iterations 17
function evaluations 21
segments explored during Cauchy searches 20
BFGS updates skipped 1
active bounds at final generalized Cauchy point 1
norm of the final projected gradient 1.08323e-05
final function value 58.9453

F = 58.9453
final  value 58.945297 
converged

DiceEval documentation built on May 2, 2019, 2:09 a.m.