Xvalid
Meryem Meziane and Xavier Freulon
2023-11-30
loading libraries
knitr::opts_chunk$set(echo = TRUE)
rm(list=ls())
library(gstlearn)##
## Attaching package: 'gstlearn'## The following objects are masked from 'package:base':
##
## message, toStringlibrary(ggplot2)
library(ggpubr)
OptCst_defineByKey("ASP",0)## NULLset.seed(43243)
opers = EStatOption_fromKeys(c("NUM","MINI","MAXI","MEAN","STDV"))This script demonstrates the capabilities of the cross-validation feature in gstlearn
A fictitious data set is generated by sampling a given simulation. Then it is used to demonstrate the cross-validation tool.
A data set is first generated, composed of a series of a few samples scattered randomly within a 100 by 100 square. The number of samples is set to a small number, so as to make the results more legible. However, it could be raised to a higher value to see better results (say, 100 - but note that large numbers should be avoided since the test is performed for a unique neighborhood). In that case, the variable turnPrintON below may be switched off to avoid generating tedious printouts.
generating the data set
nech = 10
turnPrintON = TRUE
data = Db_createFromBox(nech, # number of samples
coormin = c(0,0),
coormax = c(100,100))
# Plotting the sampling locations generated
ggplot()+
plot.point(db = data,
color = "red",
size = 0.5) +
plot.decoration(title = "Sampling locations")
A spherical model is defined, with the universality condition and a non conditional simulation is performed at the sampling locations. The values generated, renamed as “data”, will from now on constitute the data set.
model = Model_createFromParam(type = ECov_SPHERICAL(), range = 30, sill = 4)
model$setDriftIRF(0)## NULLerr = simtub(NULL,
data,
model = model)
data$setName("Simu","data")## NULLdata##
## Data Base Characteristics
## =========================
##
## Data Base Summary
## -----------------
## File is organized as a set of isolated points
## Space dimension = 2
## Number of Columns = 4
## Total number of samples = 10
##
## Variables
## ---------
## Column = 0 - Name = rank - Locator = NA
## Column = 1 - Name = x-1 - Locator = x1
## Column = 2 - Name = x-2 - Locator = x2
## Column = 3 - Name = data - Locator = z1ggplot()+
plot.point(data, nameSize="data",
color = "red")+
plot.decoration(title = "Measurement values")
Now cross-validation is performed. This requires the definition of a neighborhood, chosen to be unique due to the small number of samples. If necessary, a moving neighborhood could be used.
neigh = NeighUnique()
err = xvalid(db = data,
model = model,
neigh = neigh,
flag_xvalid_est=1,
flag_xvalid_std=1,
namconv=NamingConvention("Xvalid",TRUE,TRUE,FALSE))The cross-validation feature offers several types of outputs, according to the flags: * flag_xvalid_est indicates if the function must return the estimation error Z-Z (flag_xvalid_est=1) or the estimation Z per se (flag_xvalid_est=-1) * flag_xvalid_std indicates if the function must return the normalized error (Z*-Z)/S (flag_xvalid_std=1) or the standard deviation S (flag_xvalid_std=-1)
For a complete demonstration, all options are used. Note the use of NamingConvention which explicitly maintains the Z locator on the input variable (= “data”).
The cross-validation is performed again, but the storing option is changed, as well as the radix given to the output variables.
err = xvalid(db = data,
model = model,
neigh = neigh,
flag_xvalid_est=-1,
flag_xvalid_std=-1,
namconv=NamingConvention("Xvalid2",TRUE, TRUE, FALSE))
data##
## Data Base Characteristics
## =========================
##
## Data Base Summary
## -----------------
## File is organized as a set of isolated points
## Space dimension = 2
## Number of Columns = 8
## Total number of samples = 10
##
## Variables
## ---------
## Column = 0 - Name = rank - Locator = NA
## Column = 1 - Name = x-1 - Locator = x1
## Column = 2 - Name = x-2 - Locator = x2
## Column = 3 - Name = data - Locator = z1
## Column = 4 - Name = Xvalid.data.esterr - Locator = NA
## Column = 5 - Name = Xvalid.data.stderr - Locator = NA
## Column = 6 - Name = Xvalid2.data.estim - Locator = NA
## Column = 7 - Name = Xvalid2.data.stdev - Locator = NAChecking the results gathered on the first sample
data[1,0:8]## [1] 1.000000 22.701345 83.641175 2.501800 -1.951677 -1.094985 0.550123
## [8] 1.782378The printed values correspond to the following information: * the sample rank : 1 * the sample abscissa X : 22.7 * the sample y-coordinate Y : 83.6 * the data value Z : 2.5 * the cross-validation error Z-Z : -1.952 the standardized cross-validation error Z∗−Z/S : -1.095 * the cross-validation estimated value Z∗ : 0.550 * the standard deviation of the cross-validation error S : 1.781
These results can also be double-checked by requesting a full dump of all the information when processing the first sample. The next chunk doesn’t store any result : it simply produces an output on the terminal to better understand the process.
if (turnPrintON){
OptDbg_setReference(1)
err = xvalid(data,
model = model,
neigh = neigh,
flag_xvalid_est=1,
flag_xvalid_std=1,
namconv=NamingConvention("Xvalid3",TRUE, TRUE, FALSE))
OptDbg_setReference(0)
}##
## Target location
## ---------------
## Sample #1 (from 10)
## Coordinate #1 = 22.701345
## Coordinate #2 = 83.641175
##
## Data selected in neighborhood
## -----------------------------
## Rank Sample x1 x2
## 1 1 22.701 83.641
## 2 2 82.323 43.955
## 3 3 15.270 3.385
## 4 4 55.428 19.935
## 5 5 93.142 79.864
## 6 6 85.694 97.845
## 7 7 73.690 37.455
## 8 8 32.792 43.164
## 9 9 32.178 78.710
## 10 10 64.591 82.042
##
## LHS of Kriging matrix (compressed)
## ==================================
## Number of active samples = 10
## Total number of equations = 11
## Reduced number of equations = 11
##
## Rank 1 2 3 4 5
## Flag 1 2 3 4 5
## 1 1 4.000 0.000 0.000 0.000 0.000
## 2 2 0.000 4.000 0.000 0.000 0.000
## 3 3 0.000 0.000 4.000 0.000 0.000
## 4 4 0.000 0.000 0.000 4.000 0.000
## 5 5 0.000 0.000 0.000 0.000 4.000
## 6 6 0.000 0.000 0.000 0.000 0.654
## 7 7 0.000 1.932 0.000 0.139 0.000
## 8 8 0.000 0.000 0.000 0.000 0.000
## 9 9 1.954 0.000 0.000 0.000 0.000
## 10 10 0.000 0.000 0.000 0.000 0.012
## 11 11 1.000 1.000 1.000 1.000 1.000
##
## Rank 6 7 8 9 10
## Flag 6 7 8 9 10
## 1 1 0.000 0.000 0.000 1.954 0.000
## 2 2 0.000 1.932 0.000 0.000 0.000
## 3 3 0.000 0.000 0.000 0.000 0.000
## 4 4 0.000 0.139 0.000 0.000 0.000
## 5 5 0.654 0.000 0.000 0.000 0.012
## 6 6 4.000 0.000 0.000 0.000 0.085
## 7 7 0.000 4.000 0.000 0.000 0.000
## 8 8 0.000 0.000 4.000 0.000 0.000
## 9 9 0.000 0.000 0.000 4.000 0.000
## 10 10 0.085 0.000 0.000 0.000 4.000
## 11 11 1.000 1.000 1.000 1.000 1.000
##
## Rank 11
## Flag 11
## 1 1 1.000
## 2 2 1.000
## 3 3 1.000
## 4 4 1.000
## 5 5 1.000
## 6 6 1.000
## 7 7 1.000
## 8 8 1.000
## 9 9 1.000
## 10 10 1.000
## 11 11 0.000
##
## Cross-validation results
## ========================
## Target Sample = 1
## Variable Z1
## - True value = 2.502
## - Estimated value = 0.550
## - Estimation Error = -1.952
## - Std. deviation = 1.782
## - Normalized Error = -1.095## NULLIt appears that : * the cross-validation system is dimensionned to 40. Actually, for a unique neighborhood, the kriging system is established with the whole data set. The one suppressing one datum in turn is derived using the Schur theorem. * the result reminds the value of the measurement (2.502), the estimated value (0.550) the standard deviation of the estimation error (1.782) and the cross-validation estimation error (-1.952).
The result can also be checked using a moving neighborhood which has been tuned to cover a pseudo-unique neighborhood.
neighM = NeighMoving_create()
if (turnPrintON){
OptDbg_setReference(1)
err = xvalid(db = data,
model = model,
neigh = neighM,
flag_xvalid_est = 1,
flag_xvalid_std = 1,
namconv = NamingConvention("Xvalid4, TRUE, TRUE, FALSE"))
OptDbg_setReference(0)
}##
## Target location
## ---------------
## Sample #1 (from 10)
## Coordinate #1 = 22.701345
## Coordinate #2 = 83.641175
##
## Data selected in neighborhood
## -----------------------------
## Rank Sample x1 x2 Sector
## 1 2 82.323 43.955 1
## 2 3 15.270 3.385 1
## 3 4 55.428 19.935 1
## 4 5 93.142 79.864 1
## 5 6 85.694 97.845 1
## 6 7 73.690 37.455 1
## 7 8 32.792 43.164 1
## 8 9 32.178 78.710 1
## 9 10 64.591 82.042 1
##
## LHS of Kriging matrix (compressed)
## ==================================
## Number of active samples = 9
## Total number of equations = 10
## Reduced number of equations = 10
##
## Rank 1 2 3 4 5
## Flag 1 2 3 4 5
## 1 1 4.000 0.000 0.000 0.000 0.000
## 2 2 0.000 4.000 0.000 0.000 0.000
## 3 3 0.000 0.000 4.000 0.000 0.000
## 4 4 0.000 0.000 0.000 4.000 0.654
## 5 5 0.000 0.000 0.000 0.654 4.000
## 6 6 1.932 0.000 0.139 0.000 0.000
## 7 7 0.000 0.000 0.000 0.000 0.000
## 8 8 0.000 0.000 0.000 0.000 0.000
## 9 9 0.000 0.000 0.000 0.012 0.085
## 10 10 1.000 1.000 1.000 1.000 1.000
##
## Rank 6 7 8 9 10
## Flag 6 7 8 9 10
## 1 1 1.932 0.000 0.000 0.000 1.000
## 2 2 0.000 0.000 0.000 0.000 1.000
## 3 3 0.139 0.000 0.000 0.000 1.000
## 4 4 0.000 0.000 0.000 0.012 1.000
## 5 5 0.000 0.000 0.000 0.085 1.000
## 6 6 4.000 0.000 0.000 0.000 1.000
## 7 7 0.000 4.000 0.000 0.000 1.000
## 8 8 0.000 0.000 4.000 0.000 1.000
## 9 9 0.000 0.000 0.000 4.000 1.000
## 10 10 1.000 1.000 1.000 1.000 0.000
##
## RHS of Kriging matrix (compressed)
## ==================================
## Number of active samples = 9
## Total number of equations = 10
## Reduced number of equations = 10
## Number of right-hand sides = 1
## Punctual Estimation
##
## Rank Flag 1
## 1 1 0.000
## 2 2 0.000
## 3 3 0.000
## 4 4 0.000
## 5 5 0.000
## 6 6 0.000
## 7 7 0.000
## 8 8 1.954
## 9 9 0.000
## 10 10 1.000
##
## (Co-) Kriging weights
## =====================
## Rank x1 x2 Data Z1*
## 1 82.323 43.955 1.266 0.045
## 2 15.270 3.385 2.184 0.064
## 3 55.428 19.935 -2.917 0.063
## 4 93.142 79.864 0.870 0.055
## 5 85.694 97.845 -0.730 0.054
## 6 73.690 37.455 2.955 0.040
## 7 32.792 43.164 -0.573 0.064
## 8 32.178 78.710 0.824 0.553
## 9 64.591 82.042 -0.157 0.063
## Sum of weights 1.000
##
## Drift coefficients
## ==================
## Rank Lagrange Coeff
## 1 -0.256 0.289
##
## Cross-validation results
## ========================
## Target Sample = 1
## Variable Z1
## - True value = 2.502
## - Estimated value = 0.550
## - Estimation Error = -1.952
## - Std. deviation = 1.782
## - Normalized Error = -1.095## NULLThe next chunk displays the graphic outputs expected following a cross-validation process. They are provided by the function draw.xvalid, which produces : * the base map of the absolute value of the standardized cross-validation error * the histogram of the standardized cross-validation error * the scatter plot of the standardized error as a function of the estimate * the scatter plot of the real value as a function of the estimate.
# plotting absolute value of the standardized error
ggplot()+
plot.point(data,
nameSize = "Xvalid.data.stderr",
color = "red",
flagAbsSize = TRUE)+
plot.decoration(title="Standardized error (absolute value)")
# plotting standardized error
ggplot()+
plot.point(data,
nameSize = "Xvalid.data.stderr",
color = "blue",
flagAbsSize = FALSE)+
plot.decoration(title = "Standardized error")
# plotting the histogram of the standardized error
ggplot()+
plot.hist(data,
name = "Xvalid.data.stderr",
bins=30,
color = "cornflowerblue",
fill="cyan")+
plot.decoration(xlab="Estimation Error", title="Cross-Validation")
# plotting the correlation between the estimate and the standardized error
ggplot()+
plot.correlation(data,
namex = "Xvalid2.data.estim",
namey = "Xvalid.data.stderr",
asPoint = TRUE)+
plot.decoration(title = "Cross-Validation",
xlab = "kriging estimate",
ylab = "kriging error") 
# plotting the correlation between the estimate and the standardized error
ggplot()+
plot.correlation(data,
namex = "Xvalid2.data.estim",
namey = "data",
flagRegr = TRUE,
regrColor = "purple",
asPoint = TRUE,
flagSameAxes = TRUE)+
plot.decoration(title = "Cross-Validation",
xlab = "kriging estimate",
ylab = "measured values") 
K-Fold cross-validation
Data set
‘dbin’ contains scattered points.
Note: the K-fold cross validation is not available for a unique neighborhood.
N = 300
K = 10
dbin = Db_createFillRandom(ndat = N, nvar = 0, seed = 125)
dbin["folds"] = sample(1:K, size = dbin$getSampleNumber(TRUE), replace = TRUE)
p <- ggDefaultGeographic() +
plot.point(dbin, nameColor = "folds", palette = "Spectral",
flagLegendColor = TRUE, flagLegendLabel = "Folds") +
plot.decoration(xlab = "X", ylab = "Z", title = "Simulated Folds")## Warning: Using size for a discrete variable is not advised.ggPrint(p)
We define a Model and simulate the variable at sample locations.
model = Model()
err = model$addCovFromParam(type = ECov_EXPONENTIAL(), range = 0.2, sill = 1)
err = model$setDriftIRF(0,0)
err = simtub(dbin = NULL, dbout = dbin, model = model, nbsimu = 1, seed = 235, nbtuba = 1000,
namconv = NamingConvention("Y"))We define a (Moving) neighborhood
neigh = NeighMoving_create(nmini = 10, nmaxi = 10)We perform the cross-validation (using Ordinary Kriging) without the folds: it is based on the Leave-One-Out algorithm..
err = dbin$deleteColumns(names = "*.xval.*")
err = dbin$setLocators("Y", locatorType = ELoc_Z(), cleanSameLocator = TRUE)
err = xvalid(db = dbin, model = model, neigh = neigh, flag_kfold = FALSE,
flag_xvalid_est = 1, flag_xvalid_std = 1, namconv = NamingConvention("OK.xval.LOO"))We now perform the same task but with K-Fold option
err = dbin$setLocators("Y", locatorType = ELoc_Z(), cleanSameLocator = TRUE)
err = dbin$setLocator("folds", locatorType = ELoc_C(), cleanSameLocator = TRUE)
err = xvalid(db = dbin, model = model, neigh = neigh, flag_kfold = TRUE,
flag_xvalid_est = 1, flag_xvalid_std = 1, namconv = NamingConvention("OK.xval.kfold"))knitr::kable(dbStatisticsMono(dbin, names = c("Y", "OK.xval.LOO.Y.*", "OK.xval.kfold.Y.*"),
opers = opers)$toTL(),
digits = 3, caption = paste0("Cross validation scores (K = ", K, ")"))| Number | Minimum | Maximum | Mean | St. Dev. | |
|---|---|---|---|---|---|
| Y | 300 | -2.859 | 2.595 | 0.081 | 0.965 |
| OK.xval.LOO.Y.esterr | 300 | -1.897 | 1.778 | -0.011 | 0.703 |
| OK.xval.LOO.Y.stderr | 300 | -2.498 | 2.560 | -0.007 | 1.033 |
| OK.xval.kfold.Y.esterr | 300 | -1.897 | 1.890 | -0.009 | 0.713 |
| OK.xval.kfold.Y.stderr | 300 | -2.498 | 2.590 | -0.003 | 1.038 |
p1 <- ggDefault() +
plot.hist(dbin, name = "OK.xval.LOO.Y.stderr", bins = 20, fill = "orange") +
plot.decoration(xlab = "Normalised estimation error (Z*-Z)/S",
title = "Leave-One-Out cross validation")
p2 <- ggDefault() +
plot.hist(dbin, name = "OK.xval.kfold.Y.stderr", bins = 20, fill = "skyblue") +
plot.decoration(xlab = "Normalised estimation error (Z*-Z)/S",
title = paste0("K-fold cross validation (K = ", K, ")"))
ggarrange(p1, p2, nrow = 1, ncol = 2)