Create an 'aurelhy' object that contains required data to perform AURELHY interpolation
Description
An 'aurelhy' object contains principal components calculated after the various variables describing the landscape,
as well as other useful descriptors. Use the predict()
method to interpolate some data with the AURELHY method.
Usage
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  aurelhy(geotm, geomask, landmask = auremask(), x0 = 30, y0 = 30, step = 12,
nbr.pc = 10, scale = FALSE, model = "data ~ .", vgmodel = vgm(1, "Sph", 10, 1),
add.vars = NULL, var.name = NULL, resample.geomask = TRUE)
## S3 method for class 'aurelhy'
print(x, ...)
## S3 method for class 'aurelhy'
plot(x, y, main = "PCA on land descriptors", ...)
## S3 method for class 'aurelhy'
points(x, pch = ".", ...)
## S3 method for class 'aurelhy'
summary(object, ...)
## S3 method for class 'aurelhy'
update(object, nbr.pc, scale, model, vgmodel, ...)
## S3 method for class 'aurelhy'
predict(object, geopoints, variable, v.fit = NULL, ...)
## S3 method for class 'predict.aurelhy'
print(x, ...)
## S3 method for class 'predict.aurelhy'
summary(object, ...)
## S3 method for class 'predict.aurelhy'
plot(x, y, which = 1, ...)
## S3 method for class 'aurelhy'
as.geomat(x, what = "PC1", nodata = NA, ...)
## S3 method for class 'predict.aurelhy'
as.geomat(x,
what = c("Interpolated", "Predicted", "KrigedResiduals", "KrigeVariance"),
nodata = NA,...)

Arguments
geotm 
a terrain model ('geotm' object) with enough resolution to be
able to calculate all landscape descriptors (use 
geomask 
a 'geomask' object with same resolution and coverage of the
'geotm' object or the final interpolation grid (with 
landmask 
an 'auremask' object that defines the window of analysis used around each point ot calculate its landscape descriptors. 
x0 
shift in X direction (longitude) where to consider the first point of the interpolation grid (note that interpolation grid must be less dense or equal to the terrain model grid, depending on the mask used). 
y0 
shift in Y direction (latitude) for the first point of the interpolation grid. 
step 
resolution of the interpolation grid, i.e., we keep one point
every 
nbr.pc 
number of PCA's principal components to keep in the interpolation. This is the initial value; the example show you how you can change this after the 'aurelhy' object is calculated. 
scale 
should we scale the landscape descriptors (variance = 1) before
performing the PCA? If 
model 
a formula describing the model used to predict the data. The
lefthand side of the formula must always be 'data' and the righthand
considers all predictors spearated by a plus sign. To use all predictors,
specify 
vgmodel 
the variogram model to fit, as defined by the 
add.vars 
additional variable(s) measured at the same points as the
geotm object, or the final interpolation grid. They will be used as
additional predictors. The example show you how you can add or remove such
variables after the 'aurelhy' object is calculated. If 
var.name 
if 
resample.geomask 
do we resample the geomask using 
x 
an 'aurelhy' or 'predict.aurelhy' object, depending on the method invoked 
y 
a 'geopoints' object to create a plot best depicting the interpolation process, or nothing to just plot the interpolation grid. 
main 
the main title of the graph 
pch 
the symbol to use for plotting points. The default value,

object 
an 'aurelhy' object 
geopoints 
a 'geopoints' object with data to be interpolated. 
variable 
the name of the variable in the 'geopoints' object to interpolate 
v.fit 
the fitted variogram model used to krige residuals. If 
which 
which graph to plot 
what 
what is extracted as a 'geomat' object 
nodata 
the code used to represent missing data in the 'geomat' object 
... 
further arguments passed to the function 
Details
aurelhy()
creates a new 'aurelhy' object. The object has print()
and plot()
methods for further diagnostics. You should use the
predict()
method to perform the AURELHY interpolation on some data. The
'aurelhy' object is also easy to save for further reuse (it is designed so
that the most timeconsumming operations are done during its creation; so, it
is supposed to be generated only once and reused for different interpolations
on the same terrain model).
Value
An 'aurelhy' object with all information required to perform an AURELHY interpolation with any 'geopoints' data.
Author(s)
Philippe Grosjean <phgrosjean@sciviews.org>
Source
Benichou P, Le Breton O (1987). Prise en compte de la topographie pour la cartographie des champs pluviometriques statistiques. La Meteorologie, 7:2334.
See Also
geotm
, auremask
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 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129  # Create an aurelhy object for the Morocco terrain data
data(morocco) # The terrain model with a grid of about 924x924m
data(mbord) # A shape with the area around Morocco to analyze
data(mmask) # A 924x924m grid with a mask covering territory to analyze
data(mseadist) # The distance to the sea for territory to analyze
data(mrain) # Rain data measured at 43 stations to be interpolated
# Create a map with these data
image(morocco) # Plot the terrain model
grid()
lines(mbord, col = "red") # Add borders of territory to analyze in red
# Make sure we use all the stations from mrain in the geomask
mmask2 < add.points(mmask, mrain)
# Now, create an aurelhy object with landscape description, using the
# first ten PCs, plus the distance to the sea (mseadist) for prediction
# Use a default radial window of analysis of 26km as maximum distance
# and an interpolation grid of 0.1x0.1degrees (roughly 11x11km)
# The variogram model is kept simple here, see ?vgm for other choices
# Be patient... this takes a little time to calculate!
maurelhy < aurelhy(morocco, mmask2, auremask(), x0 = 30, y0 = 54, step = 12,
scale = TRUE, nbr.pc = 10, vgmodel = vgm(100, "Sph", 1, 1),
add.vars = mseadist, var.name = "seadist")
maurelhy
points(maurelhy) # Add the interpolated points on the map
points(mrain, col = "red") # Add location of weather stations in red
# Diagnostic of the PCA on land descriptors
summary(maurelhy)
plot(maurelhy)
# Interpolate 'rain' variable on the considered territory around Morocco
# Since we do not want negative values for this variable and it is lognormally
# distributed, we will interpolate log10(rain) instead
mrain$logRain < log10(mrain$rain)
pmrain < predict(maurelhy, mrain, "logRain")
pmrain
# Diagnostic of regression model
summary(pmrain) # Significant predictors at alpha = 0.01 are x, y, PC3, PC6 and PC7
# one could simplify the model as data ~ x + y + PC3 + PC6 + PC7
# but it is faster to keep the full model for final interpolation
# when we are only interested by the final interpolation or when processing
# is automated...
# Any of the predictors can be extracted from maurelhy as a geomat object
# for further inspection. For instance, let's look at PC3, PC6 and PC7 components
persp(as.geomat(maurelhy, "PC3"), expand = 50)
persp(as.geomat(maurelhy, "PC6"), expand = 50)
persp(as.geomat(maurelhy, "PC7"), expand = 50)
plot(pmrain, which = 1) # Residuals versus fitted (how residuals spread?)
plot(pmrain, which = 2) # Normal QQ plot of residuals (residuals distribution)
plot(pmrain, which = 3) # Best graph to look at residuals homoscedasticity
plot(pmrain, which = 4) # Cook's distance of residuals versus observation
plot(pmrain, which = 5) # Residuals leverage: are there influencial points?
# Map of predicted values
filled.contour(as.geomat(pmrain, "Predicted"), asp = 1,
color.palette = terrain.colors, main = "Values predicted by the linear model")
# Residuals kriging diagnostic
plot(pmrain, which = 6) # Semivariogram and adjusted model
filled.contour(as.geomat(pmrain, "KrigedResiduals"), asp = 1,
color.palette = terrain.colors, main = "Kriged residuals")
filled.contour(as.geomat(pmrain, "KrigeVariance"), asp = 1,
color.palette = terrain.colors, main = "Kriged residuals variance")
# As we can expect, kriging variance is larger in the south/southwest part
# where density of stations is low
# AURELHY interpolation diagnostic plots
# Graph showing the importance of predicted versus kriged residuals for
# all observations
plot(pmrain, which = 7) # Model prediction and kriged residuals
# Extract interpolated log(rain) and transform back into rain (mm)
geomrain < as.geomat(pmrain)
geomrain < 10^geomrain
# How is interpolated rain distributed?
range(geomrain, na.rm = TRUE)
range(mrain$rain)
# Ooops! We have some very high values! How many?
sum(geomrain > 1000, na.rm = TRUE)
# This is probably due to a lack of data at high altitudes
# Let's truncate them to 1000 for a better graph
geomrain[geomrain > 1000] < 1000
# ... and plot the result
image(geomrain, col = topo.colors(12))
contour(geomrain, add = TRUE)
lines(mbord, col = "red")
points(mrain, col = "red")
# A better plot for these interpolated rain data
filled.contour(coords(geomrain, "x"), coords(geomrain, "y"), geomrain,
asp = 1, xlab = "Longitude", ylab = "Latitude",
main = "AURELHY interpolated rain data", key.title = title(sub = "Rain (mm)\n\n"),
color.palette = colorRampPalette(c("red", "gray", "blue"), bias = 2))
# One can experiment different interpolation parameters using update()
# Suppose we (1) don't want to scale PCs, (2) to keep only first 7 PCs,
# (3) we want an upgraded linear model like this:
# data < a.x + b.y + c.z + d.PC3 + e.PC6 + f.PC7 + g.seadist + h.seadist^2 + i
# and (4) we want a Gaussian model for the semivariogram
# (note that one can also regress against seadist2 < seadist^2), just do:
# maurelhy2$seadist2 < maurelhy$seadist^2
# Even with all these changes, you don't have to recompute maurelhy,
# just update() it and the costly steps of calculating landscape descriptors
# are reused (not the use of I() to protect calcs inside a formula)!
maurelhy2 < update(maurelhy, scale = FALSE, nbr.pc = 7,
model = data ~ x + y + z + PC3 + PC6 + PC7 + seadist + I(seadist^2),
vgmodel = vgm(100, "Gau", 1, 1))
maurelhy2
# Diagnostic of the new PCA on land descriptors without scaling
summary(maurelhy2)
plot(maurelhy2)
# Interpolate with the new parameters
pmrain2 < predict(maurelhy2, mrain, "logRain")
summary(pmrain2)
# A couple of graphs
plot(pmrain2, which = 1) # Residuals versus fitted (how residuals spread?)
plot(pmrain, which = 6) # Semivariogram and adjusted model
plot(pmrain2, which = 7) # Model prediction and kriged residuals
#... Explore as much as you like until you find the set of parameters that suits you!

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