Introduction
The sptotal
package was developed for predicting a
weighted sum, most commonly a mean or total, from a finite number of
sample units in a fixed geographic area. Estimating totals and means
from a finite population is an important goal for both academic research
and management of environmental data. One naturally turns to classical
sampling methods, such as simple random sampling or stratified random
sampling. Classical sampling methods depend on probability-based sample
designs and are robust. Very few assumptions are required because the
probability distribution for inference comes from the sample design,
which is known and under our control. For design-based methods, sample
plots are chosen at random, they are measured or counted, and inference
is obtained from the probability of sampling those units randomly based
on the design (e.g., Horwitz-Thompson estimation). As an alternative, we
will use model-based methods, specifically geostatistics, to accomplish
the same goals. Geostatistics does not rely on a specific sampling
design. Instead, when using geostatistics, we assume the data were
produced by a stochastic process with parameters that can be estimated.
The relevant theory is given by Ver Hoef (2008). The
sptotal
package puts much of the code and plots in Ver Hoef
(2008) in easily accessible, convenient functions.
In the sptotal
package, our goal is to estimate some
linear function of all of the sample units, call it \(\tau(\mathbf{z}) = \mathbf{b}^\prime
\mathbf{z}\), where \(\mathbf{z}\) is a vector of the realized
values for all the sample units and \(\mathbf{b}\) is a vector of weights. By
“realized,” we mean that whatever processes produced the data have
already happened, and that, if we had enough resources, we could measure
them all, obtaining a complete census. If \(\tau(\mathbf{z})\) is a population total,
then every element of \(\mathbf{b}\)
contains a \(1\). Generally, \(\mathbf{b}\) can contain any set of weights
that we would like to multiply times each value in a population, and
then these are summed, yielding a weighted sum.
The vector \(\mathbf{b}\) contains the weights that we would apply if we could measure or count every observation, but, because of cost consideration, we usually only have a sample.
Data
Prior to using the sptotal
package, the data needs to be
in R
in the proper format. For this package, we assume that
your data set is a data.frame()
object, described
below.
Data Frame Structure
Data input for the sptotal
package is a
data.frame
. The basic information required to fit a spatial
linear model, and make predictions, are the response variable,
covariates, the x- and y-coordinates, and a column of weights. You can
envision your whole population of possible samples as a
data.frame
organized as follows,
where the red rectangle represents the column of the response variable, and the top part, colored in red, are observed locations, and the lower part, colored in white, are the unobserved values. To the right, colored in blue, are possibly several columns containing covariates thought to be predictive for the response value at each location. Covariates must be known for both observed and unobserved locations, and the covariates for unobserved locations are shown as pale blue below the darker blue covariates for observed locations above. It is also possible that there are no available covariates.
The data.frame
must have x- and y-coordinates, and they
are shown as two columns colored in green, with the coordinates for the
unobserved locations shown as pale green below the darker green
coordinates for the observed locations above. The
data.frame
can have a column of weights. If one is not
provided, we assume a column of all ones so that the prediction is for
the population total. The column of weights is purple, with weights for
the observed locations a darker shade, above the lighter shade of purple
representing weights for unsampled locations. Finally, the
data.frame
may contain columns that are not relevant to
predicting the weighted sum. These columns are represented by the orange
color, with the sampled locations a darker shade, above the unsampled
locations with the lighter shade.
Of course, the data do not have to be in exactly this order, either in terms of rows or columns. Sampled and unsampled rows can be intermingled, and columns of response variable, covariates, coordinates, and weights can be also be intermingled. The figure above is an idealized graphic of the data. However, this figure helps envision how the data are used and illustrate the goal. We desire a weighted sum, where the weights (in the purple column) are multiplied with the response variable (red/white) column, and then summed. Because some of the response values are unknown (the white values in the response column), covariates and spatial information (obtained from the x- and y-coordinates) are used to predict the unobserved (white) values. The weights (purple) are then applied to both the observed response values (red), and the predicted response values (white), to obtain a weighted sum. Because we use predictions for unobserved response values, it is important to assess our uncertainty, and the software provides both an estimate of the weighted sum, mean, or total for the response variable as well as its estimated prediction variance.
Simulated Data Creation
To demonstrate the package, we created some simulated data so they are perfectly behaved, and we know exactly how they were produced. Here, we give a brief description before using the main features of the package. To get started, install the package
install.packages("sptotal")
and then type
library(sptotal)
Type
data(simdata)
and then simdata
will be available in your workspace. To
see the first six observations of simdata
, type
head(simdata)
#> x y X1 X2 X3 X4 X5
#> 1 0.025 0.975 -0.8460525 0.11866907 -0.2123901 0.38430607 0.08154129
#> 2 0.025 0.925 -0.6583116 -0.07686491 -0.9001410 -1.24774376 1.46631630
#> 3 0.025 0.875 0.2222961 -0.22803942 0.2820468 0.20560677 0.48713665
#> 4 0.025 0.825 -0.5433925 0.56894993 -0.9839629 -0.04950434 -0.78195604
#> 5 0.025 0.775 -0.7550155 -0.72592167 -0.4217208 0.26767033 0.40493269
#> 6 0.025 0.725 -0.1786784 0.33452155 -1.2134533 2.18704575 -0.54903128
#> X6 X7 F1 F2 Z wts1 wts2
#> 1 1.0747592 -0.0252824 3 3 15.94380 0.0025 0
#> 2 0.1299263 1.4651052 2 5 15.04616 0.0025 0
#> 3 -0.2537515 0.2682010 2 3 14.52765 0.0025 0
#> 4 -0.3259937 0.7858140 2 5 12.13401 0.0025 0
#> 5 -1.2284475 1.2944342 2 2 11.75260 0.0025 0
#> 6 -1.0366099 0.7938890 1 4 11.58142 0.0025 0
simdata
is a data frame with 400 observations. The
spatial coordinates are numeric
variables in columns named
x
and y
. We created 7 continuous covariates,
X1
through X7
. The variables X1
through X5
were all created using the rnorm()
function, so they are all standard normal variates that are independent
between and within variable. Variables X6
and
X7
were independent from each other, but spatially
autocorrelated within, each with a variance parameter of 1, an
autocorrelation range parameter of 0.2 from an exponential model, and a
small nugget effect of 0.01. The variables F1
and
F2
are factor variables with 3 and 5 levels, respectively.
The variable Z
is the response. Data were simulated from
the model
\[\begin{align*} Z_i = 10 & + 0 \cdot X1_i + 0.1 \cdot X2_i + 0.2 \cdot X3_i + 0.3 \cdot X4_i + \\ & 0.4 \cdot X5_i + 0.4 \cdot X6_i + 0.1 \cdot X7_i + F1_i + F2_i + \delta_i + \varepsilon_i \end{align*}\]
where factor levels for F1
have effects \(0, 0.4, 0.8\), and factor levels for
F2
have effects \(0, 0.1, 0.2,
0.3, 0.4\). The random errors \(\{\delta_i\}\) are spatially autocorrelated
from an exponential model,
\[ \textrm{cov}(\delta_i,\delta_j) = 2*\exp(-d_{i,j}) \]
where \(d_{i,j}\) is Euclidean
distance between locations \(i\) and
\(j\). In geostatistics terminology,
this model has a partial sill of 2 and a range of 1. The random errors
\(\{\varepsilon_i\}\) are independent
with variance 0.02, and this variance is called the nugget effect. Two
columns with weights are included, wts1
contains 1/400 for
each row, so the weighted sum will yield a prediction of the overall
mean. The column wts2
contains a 1 for 25 locations, and 0
elsewhere, so the weighted sum will be a prediction of a total in the
subset of 25 locations.
The spatial locations of simdata
are in a \(20 \times 20\) grid uniformly spaced in a
box with sides of length 1,
require(ggplot2)
ggplot(data = simdata, aes(x = x, y = y)) + geom_point(size = 3) +
geom_point(data = subset(simdata, wts2 == 1), colour = "red",
size = 3)
The locations of the 25 sites where wts2
is equal to one
are shown in red.
We have simulated the data for the whole population. This is
convenient, because we know the true means and totals. In order to
compare with the prediction from the sptotal
package, let’s
find the true population total
sum(simdata[ ,'Z'])
#> [1] 4834.326
as well as the total in the subset of 25 sites
sum(simdata[ ,'wts2'] * simdata[ ,'Z'])
#> [1] 273.3751
However, we will now sample from this population to provide a more
realistic setting where we can measure only a part of the whole
population. In order to make results reproducible, we use the
set.seed
command, along with sample
. The code
below will replace some of the response values with NA
to
represent the unsampled sites.
set.seed(1)
# take a random sample of 100
<- sample(1:nrow(simdata), 100)
obsID <- simdata
simobs $Z <- NA
simobs'Z'] <- simdata[obsID, 'Z'] simobs[obsID,
We now have a data set where the whole population is known,
simdata
, and another one, simobs
, where 75% of
the response variable of the population has been replaced by
NA
. Next we show the sampled sites as solid circles, while
the missing values are shown as open circles, and we use red again to
show the sites within the small area of 25 locations.
ggplot(data = simobs, aes(x = x, y = y)) +
geom_point(shape = 1, size = 2.5, stroke = 1.5) +
geom_point(data = subset(simobs, !is.na(Z)), shape = 16, size = 3.5) +
geom_point(data = subset(simobs, !is.na(Z) & wts2 == 1), shape = 16,
colour = "red", size = 3.5) +
geom_point(data = subset(simobs, is.na(Z) & wts2 == 1), shape = 1,
colour = "red", size = 2.5, stroke = 1.5)
We will use the simobs
data to illustrate use of the
sptotal
package.
Using the sptotal
Package
After your data is in a similar format to simobs
, using
the sptotal
package occurs in two primary stages. In the
first, we fit a spatial linear model. This stage estimates spatial
regression coefficients and spatial autocorrelation parameters. In the
second stage, we predict the unsampled locations for the response value,
and create a prediction for the weighted sum (e.g. the total) of all
response variable values, both observed and predicted. To show how the
package works, we demonstrate on ideal, simulated data. Then, we give a
realistic example on moose data and a second example on lakes data to
provide further insight and documentation. The moose example also has a
section on data preparation steps.
Fitting a Spatial Linear Model: slmfit
We continue with our use of the simulated data, simobs
,
to illustrate fitting the spatial linear model. The spatial
model-fitting function is slmfit
(spatial-linear-model-fit), which uses a formula like many other
model-fitting functions in R
(e.g., the lm()
function). To fit a basic spatial linear model we use
<- slmfit(formula = Z ~ X1 + X2 + X3 + X4 + X5 +
slmfit_out1 + X7 + F1 + F2,
X6 data = simobs, xcoordcol = 'x',
ycoordcol = 'y',
CorModel = "Exponential")
The documentation describes the arguments in more detail, but as
mentioned earlier, the linear model includes a formula argument, and the
data.frame
that is being used as a data set. We also need
to include which columns contain the \(x\)- and \(y\)-coordinates, which are arguments to
xcoordcol
and ycoordcol
, respectively. In the
above example, we specify 'x'
and 'y'
as the
column coordinates arguments since the names of the coordinate columns
in our simulated data set are 'x'
and 'y'
. We
also need to specify a spatial autocorrelation model, which is given by
the CorModel
argument. As with many other linear model
fits, we can obtain a summary of the model fit,
summary(slmfit_out1)
#>
#> Call:
#> Z ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + F1 + F2
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -1.9390 -0.6271 0.3338 1.2520 2.8137
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 11.36965 1.03775 10.956 < 2e-16 ***
#> X1 -0.05596 0.06400 -0.874 0.38437
#> X2 0.02661 0.06606 0.403 0.68814
#> X3 0.18292 0.06469 2.828 0.00583 **
#> X4 0.26487 0.05741 4.613 1e-05 ***
#> X5 0.38434 0.06022 6.382 < 2e-16 ***
#> X6 0.47612 0.11198 4.252 5e-05 ***
#> X7 0.02893 0.11761 0.246 0.80625
#> F12 0.29596 0.15154 1.953 0.05407 .
#> F13 0.70853 0.13136 5.394 < 2e-16 ***
#> F22 0.15384 0.17073 0.901 0.37008
#> F23 0.19804 0.17828 1.111 0.26973
#> F24 0.25492 0.20024 1.273 0.20641
#> F25 0.39748 0.23691 1.678 0.09703 .
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Covariance Parameters:
#> Exponential Model
#> Nugget 1.009265e-06
#> Partial Sill 2.930385e+00
#> Range 5.891474e-01
#>
#> Generalized R-squared: 0.5996812
The output looks similar to the summary
of a standard
lm
object, but there is some extra output at the end that
gives our fitted covariance parameters. Plotting
slmfit_out1
gives a semi-variogram of the residuals along
with the fitted model:
plot(slmfit_out1)
Note that the fitted curve may not appear to fit the empirical variogram perfectly for a couple of reasons. First, only pairs of points that have a distance between 0 and one-half the maximum distance are shown. Second, the fitted model is estimated using REML, which may give different results than using weighted least squares.
We can also examine a histogram of the residuals as well as a histogram of the cross-validation (leave-one-out) residuals:
<- residuals(slmfit_out1)
residraw qplot(residraw, bins = 20) + xlab("Residuals")
#> Warning: `qplot()` was deprecated in ggplot2 3.4.0.
<- residuals(slmfit_out1, cross.validation = TRUE)
residcv qplot(residcv, bins = 20) + xlab("CV Residuals")
There is still one somewhat large cross-validation residual for an observed count that is larger than what would be predicted from a model without that particular count. The cause of this somewhat large residual can be attributed to random chance because we know that the data was simulated to follow all assumptions.
Prediction: predict
After we have obtained a fitted spatial linear model, we can use the
predict()
function to construct a data frame of predictions
for the unsampled sites. By default, the predict()
function
assumes that we are predicting the population total and outputs this
predicted total, the prediction variance for the total, a 90% prediction
interval for the total, and some basic summary information about the
number of sites sampled, the total number of units counted, etc. We name
this object pred_obj
in the chunk below and also construct
a 90% confidence interval for the total.
<- predict(slmfit_out1, conf_level = 0.90)
pred_obj pred_obj
We predict a total of 4817 units in this simulated region with 90% confidence bounds of (4779, 4856). The prediction interval is fairly small because we simulated data that were highly correlated, increasing precision in prediction for unobserved sites. We can see that the prediction of the total is close to the true value of 4834.326, and the true value is within the prediction interval.
To access the data.frame
that was input into
slmfit
, but is now appended with site-by-site predictions
and site-by-site prediction variances, we can use
pred_obj$Pred_df
. This data set might be particularly
useful if you would like to generate your own map with site-by-site
predictions using other tools. The site-by-site predictions for density
are given by the variable name_of_response_pred_density
while the site-by-site predictions for counts are given by
name_of_response_pred_count
. These two columns will only
differ if you have provided a column for areas of each site.
<- pred_obj$Pred_df
prediction_df head(prediction_df[ ,c("x", "y", "Z", "Z_pred_density")])
Examining results: plot()
Finally, to get a basic plot of the predictions, we can use the
plot()
function.
plot(pred_obj)
The map shows the distribution of the response across sampled and
unsampled sites. Its purpose is simply to give the user a very quick
idea of the distribution of the response. For example, we see from the
plot that the predicted response is low in the upper-right region of the
graph, is high in the middle of the region and in the upper-left corner
of the region, and is low again at the lower portion of the area of
interest. However, using the prediction data frame generated from the
predict()
function, you can use ggplot2
or any
other plotting package to construct your own map that may be more useful
in your context.
Prediction for a Small Area of Interest
Spatial prediction can be used to estimate means and totals over finite populations from geographic regions, but can also be used for the special case of estimating a mean or total in a small area of interest. The term small area estimation refers to making an inference on a smaller geographic area within the overall study area. There may be few or no samples within that small area, so that estimation by classical sampling methods may not be possible or variances become exceedingly large.
If we want to predict a quantity other than the population total,
then we need to specify the column in our data set that has the
appropriate prediction weights in a wtscol
argument. For
example, we might want to predict the total for a small area of
interest. if we want to predict the total for the 25 sites in coloured
in red, then we can use
<- predict(slmfit_out1, wtscol = "wts2")
pred_obj2 print(pred_obj2)
#> Prediction Info:
#> Prediction SE 90% LB 90% UB
#> Z 282.2 7.342 270.1 294.3
#> Numb. Sites Sampled Total Numb. Sites Total Observed Average Density
#> Z 100 400 1220 12.2
Recall that the true total for this small area was 273.4. We see that this is close to our prediction of 282.2 and is also within the bounds of our prediction interval.
Real Data Examples
Moose Abundance from Aerial Surveys
The simulated data example assumes that the coordinates are a
Transverse Mercator projection (TM), that the vector of the response is
numeric and has NA
values for sites that were not sampled,
and that the areas of each site sampled are all the same. For this
example, we consider a data set on moose abundance in Alaska obtained
from Alaska
Department of Fish and Game, Division of Wildlife Conservation. Each
observation corresponds to a moose counted at a particular site, but
operational constraints do not permit all sites to be counted. We begin
by loading the data into R
.
data(AKmoose_df)
AKmoose_df#> elev_mean strat surveyed total x y lon lat
#> 0 560.3333 L 0 NA 38.98384825 1.301806e+02 -147.8750 63.71667
#> 1 620.4167 L 0 NA 34.86652773 1.302284e+02 -147.9583 63.71667
#> 2 468.9167 L 1 0 30.74963291 1.302815e+02 -148.0417 63.71667
#> 3 492.7500 L 0 NA 26.63241710 1.303400e+02 -148.1250 63.71667
#> 4 379.5833 L 0 NA 22.51526115 1.304038e+02 -148.2083 63.71667
#> 5 463.7500 L 0 NA 38.94319469 1.264665e+02 -147.8750 63.68333
#> 6 456.4375 L 0 NA 34.82103091 1.265143e+02 -147.9583 63.68333
#> 7 358.9375 L 0 NA 30.69929374 1.265674e+02 -148.0417 63.68333
#> 8 333.1875 L 0 NA 26.57723416 1.266260e+02 -148.1250 63.68333
#> 9 257.8750 L 0 NA 22.45523537 1.266899e+02 -148.2083 63.68333
#> 10 417.6250 L 0 NA 38.90255070 1.227521e+02 -147.8750 63.65000
#> 11 362.3125 L 1 0 34.77554479 1.228000e+02 -147.9583 63.65000
#> 12 265.7500 L 0 NA 30.64896544 1.228532e+02 -148.0417 63.65000
#> 13 269.3125 L 0 NA 26.52206419 1.229118e+02 -148.1250 63.65000
#> 14 225.0000 L 0 NA 22.39522272 1.229758e+02 -148.2083 63.65000
#> 15 172.7500 M 1 0 18.26882464 1.230451e+02 -148.2917 63.65000
#> 16 398.6250 L 0 NA 38.86191861 1.190378e+02 -147.8750 63.61666
#> 17 279.6250 L 1 0 34.73007200 1.190857e+02 -147.9583 63.61666
#> 18 227.8750 L 0 NA 30.59865236 1.191390e+02 -148.0417 63.61666
#> 19 197.8750 L 0 NA 26.46691037 1.191976e+02 -148.1250 63.61666
#> 20 194.7500 L 0 NA 22.33522813 1.192616e+02 -148.2083 63.61666
#> 21 167.9375 M 0 NA 18.20398971 1.193311e+02 -148.2917 63.61666
#> 22 204.0000 L 0 NA 14.07244432 1.194059e+02 -148.3750 63.61666
#> 23 619.6000 L 1 0 47.09442203 1.152440e+02 -147.7083 63.58333
#> 24 479.7500 L 0 NA 42.95803072 1.152812e+02 -147.7917 63.58333
#> 25 350.5000 L 0 NA 38.82130074 1.153237e+02 -147.8750 63.58333
#> 26 286.4167 L 0 NA 34.68461462 1.153716e+02 -147.9583 63.58333
#> 27 207.5833 L 0 NA 30.54835739 1.154250e+02 -148.0417 63.58333
#> 28 181.5833 L 0 NA 26.41177586 1.154837e+02 -148.1250 63.58333
#> 29 174.3333 L 0 NA 22.27525505 1.155478e+02 -148.2083 63.58333
#> 30 164.5833 M 0 NA 18.13917747 1.156172e+02 -148.2917 63.58333
#> 31 188.4000 L 0 NA 14.00279297 1.156921e+02 -148.3750 63.58333
#> 32 515.0500 L 0 NA 47.06349078 1.115299e+02 -147.7083 63.55000
#> 33 358.8750 L 0 NA 42.92226236 1.115671e+02 -147.7917 63.55000
#> 34 364.1250 L 0 NA 38.78069481 1.116096e+02 -147.8750 63.55000
#> 35 217.5625 L 0 NA 34.63917159 1.116576e+02 -147.9583 63.55000
#> 36 191.5625 L 0 NA 30.49807620 1.117110e+02 -148.0417 63.55000
#> 37 172.6250 L 0 NA 26.35665755 1.117697e+02 -148.1250 63.55000
#> 38 162.0625 M 0 NA 22.21529857 1.118339e+02 -148.2083 63.55000
#> 39 164.5625 L 0 NA 18.07438426 1.119034e+02 -148.2917 63.55000
#> 40 167.5500 L 0 NA 13.93316208 1.119784e+02 -148.3750 63.55000
#> 41 543.8750 L 0 NA 67.76480465 1.077107e+02 -147.2917 63.51667
#> 42 577.3125 L 0 NA 63.61824587 1.077209e+02 -147.3750 63.51667
#> 43 593.1875 L 0 NA 59.47169973 1.077365e+02 -147.4583 63.51667
#> 44 612.0000 L 0 NA 55.32555017 1.077575e+02 -147.5417 63.51667
#> 45 630.5625 L 0 NA 51.17904516 1.077839e+02 -147.6250 63.51667
#> 46 455.8500 L 0 NA 47.03256864 1.078158e+02 -147.7083 63.51667
#> 47 319.5625 L 0 NA 42.88650452 1.078530e+02 -147.7917 63.51667
#> 48 289.2500 L 0 NA 38.74010081 1.078956e+02 -147.8750 63.51667
#> 49 211.6875 L 0 NA 34.59374144 1.079436e+02 -147.9583 63.51667
#> 50 181.8125 L 0 NA 30.44781080 1.079970e+02 -148.0417 63.51667
#> 51 164.1250 M 1 0 26.30155543 1.080558e+02 -148.1250 63.51667
#> 52 163.5000 M 1 0 22.15536074 1.081200e+02 -148.2083 63.51667
#> 53 174.9375 L 0 NA 18.00961012 1.081896e+02 -148.2917 63.51667
#> 54 186.3500 L 1 0 13.86355163 1.082646e+02 -148.3750 63.51667
#> 55 503.6923 L 0 NA 76.06048918 1.039922e+02 -147.1250 63.48333
#> 56 620.5000 L 0 NA 71.90908579 1.039916e+02 -147.2083 63.48333
#> 57 466.6667 L 0 NA 67.75806458 1.039965e+02 -147.2917 63.48333
#> 58 447.9375 L 0 NA 63.60667113 1.040067e+02 -147.3750 63.48333
#> 59 597.9375 L 0 NA 59.45529032 1.040223e+02 -147.4583 63.48333
#> 60 655.8750 L 0 NA 55.30430652 1.040434e+02 -147.5417 63.48333
#> 61 566.3750 L 0 NA 51.15296682 1.040698e+02 -147.6250 63.48333
#> 62 507.7000 L 0 NA 47.00165560 1.041017e+02 -147.7083 63.48333
#> 63 375.7500 L 0 NA 42.85075722 1.041389e+02 -147.7917 63.48333
#> 64 261.4375 L 0 NA 38.69951878 1.041815e+02 -147.8750 63.48333
#> 65 198.4375 L 0 NA 34.54832466 1.042296e+02 -147.9583 63.48333
#> 66 173.5000 M 0 NA 30.39755970 1.042831e+02 -148.0417 63.48333
#> 67 158.6250 M 0 NA 26.24646955 1.043419e+02 -148.1250 63.48333
#> 68 170.1667 L 0 NA 22.09544005 1.044062e+02 -148.2083 63.48333
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#> 70 219.0769 L 0 NA 13.79396170 1.045509e+02 -148.3750 63.48333
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#> 782 152.0588 L 0 NA 144.47635663 1.190124e+01 -145.7917 62.65000
#> 783 134.9231 L 0 NA 140.20522995 1.181235e+01 -145.8750 62.65000
#> 784 103.0000 M 0 NA 135.93402127 1.172899e+01 -145.9583 62.65000
#> 785 108.2500 M 1 2 131.66312741 1.165115e+01 -146.0417 62.65000
#> 786 136.7000 L 0 NA 127.39177197 1.157883e+01 -146.1250 62.65000
#> 787 195.1250 L 0 NA 123.12035020 1.151202e+01 -146.2083 62.65000
#> 788 203.6875 L 1 0 118.84925895 1.145074e+01 -146.2917 62.65000
#> 789 159.6875 L 0 NA 114.57772176 1.139496e+01 -146.3750 62.65000
#> 790 153.5000 L 0 NA 110.30613442 1.134471e+01 -146.4583 62.65000
#> 791 168.1875 L 0 NA 106.03489225 1.129998e+01 -146.5417 62.65000
#> 792 161.2667 L 0 NA 101.76322030 1.126077e+01 -146.6250 62.65000
#> 793 155.0000 L 0 NA 97.49151387 1.122708e+01 -146.7083 62.65000
#> 794 152.5333 M 1 0 93.22016827 1.119890e+01 -146.7917 62.65000
#> 795 151.3333 L 0 NA 88.94840854 1.117624e+01 -146.8750 62.65000
#> 796 149.6667 L 0 NA 84.67663000 1.115910e+01 -146.9583 62.65000
#> 797 155.0000 M 1 0 80.40522845 1.114748e+01 -147.0417 62.65000
#> 798 169.3000 L 0 NA 76.13342741 1.114138e+01 -147.1250 62.65000
#> 799 190.0625 M 0 NA 71.86162373 1.114080e+01 -147.2083 62.65000
#> 800 199.8750 M 0 NA 67.59021218 1.114573e+01 -147.2917 62.65000
#> 801 203.0625 M 1 6 63.31841783 1.115618e+01 -147.3750 62.65000
#> 802 225.1500 M 1 2 59.04663597 1.117216e+01 -147.4583 62.65000
#> 803 170.1875 L 1 0 144.55601998 8.187910e+00 -145.7917 62.61666
#> 804 142.4000 L 1 8 140.28009561 8.098953e+00 -145.8750 62.61666
#> 805 165.9333 M 1 8 136.00408980 8.015521e+00 -145.9583 62.61666
#> 806 99.5000 M 0 NA 131.72839826 7.937618e+00 -146.0417 62.61666
#> 807 180.6667 L 0 NA 127.45224526 7.865232e+00 -146.1250 62.61666
#> 808 252.2778 L 0 NA 123.17602597 7.798370e+00 -146.2083 62.61666
#> 809 285.7500 M 0 NA 118.90013765 7.737036e+00 -146.2917 62.61666
#> 810 259.1875 L 0 NA 114.62380299 7.681220e+00 -146.3750 62.61666
#> 811 169.2105 M 0 NA 110.34741822 7.630928e+00 -146.4583 62.61666
#> 812 178.4706 L 0 NA 106.07137906 7.586162e+00 -146.5417 62.61666
#> 813 170.0000 L 0 NA 101.79490972 7.546916e+00 -146.6250 62.61666
#> 814 164.6250 M 0 NA 97.51840540 7.513193e+00 -146.7083 62.61666
#> 815 159.7000 L 0 NA 93.24226338 7.484995e+00 -146.7917 62.61666
#> 816 159.5000 L 0 NA 88.96570631 7.462318e+00 -146.8750 62.61666
#> 817 173.3750 L 0 NA 84.68912991 7.445165e+00 -146.9583 62.61666
#> 818 190.5714 L 0 NA 80.41293147 7.433535e+00 -147.0417 62.61666
#> 819 179.9333 M 0 NA 76.13633361 7.427427e+00 -147.1250 62.61666
#> 820 200.7500 L 0 NA 71.85973208 7.426843e+00 -147.2083 62.61666
#> 821 218.0000 M 0 NA 67.58352416 7.431781e+00 -147.2917 62.61666
#> 822 226.0000 M 1 31 63.30693248 7.442243e+00 -147.3750 62.61666
#> 823 154.4375 L 0 NA 144.63565441 4.474810e+00 -145.7917 62.58333
#> 824 198.7000 L 1 5 140.35493410 4.385780e+00 -145.8750 62.58333
#> 825 216.2500 L 0 NA 136.07413239 4.302279e+00 -145.9583 62.58333
#> 826 101.9375 M 0 NA 131.79364543 4.224313e+00 -146.0417 62.58333
#> 827 166.0000 M 1 3 127.51269660 4.151868e+00 -146.1250 62.58333
#> 828 270.2353 M 1 2 123.23168152 4.084951e+00 -146.2083 62.58333
#> 829 367.2500 M 0 NA 118.95099789 4.023567e+00 -146.2917 62.58333
#> 830 422.3333 L 0 NA 114.66986750 3.967705e+00 -146.3750 62.58333
#> 831 217.0714 L 0 NA 110.38868703 3.917371e+00 -146.4583 62.58333
#> 832 182.2000 L 0 NA 106.10785263 3.872569e+00 -146.5417 62.58333
#> 833 176.2500 L 0 NA 101.82658763 3.833290e+00 -146.6250 62.58333
#> 834 171.0625 M 0 NA 97.54528819 3.799540e+00 -146.7083 62.58333
#> 835 171.7500 L 1 0 93.26435047 3.771319e+00 -146.7917 62.58333
#> 836 214.8750 L 1 0 88.98299779 3.748624e+00 -146.8750 62.58333
#> 837 236.8750 L 0 NA 84.70162579 3.731456e+00 -146.9583 62.58333
#> 838 228.5625 L 0 NA 80.42063271 3.719817e+00 -147.0417 62.58333
#> 839 191.8500 L 0 NA 76.13923876 3.713704e+00 -147.1250 62.58333
#> 840 211.8125 L 0 NA 71.85784215 3.713119e+00 -147.2083 62.58333
#> 841 236.5625 L 0 NA 67.57683857 3.718062e+00 -147.2917 62.58333
#> 842 255.2500 L 1 9 63.29545130 3.728532e+00 -147.3750 62.58333
#> 843 252.1667 L 0 NA 144.71526444 7.617278e-01 -145.7917 62.55000
#> 844 256.8000 L 0 NA 140.42974964 6.726257e-01 -145.8750 62.55000
#> 845 129.6316 L 0 NA 136.14415350 5.890564e-01 -145.9583 62.55000
#> 846 128.0625 M 0 NA 131.85887259 5.110266e-01 -146.0417 62.55000
#> 847 128.3750 L 0 NA 127.57312940 4.385220e-01 -146.1250 62.55000
#> 848 216.6500 L 0 NA 123.28732000 3.715500e-01 -146.2083 62.55000
#> 849 310.3750 M 1 0 119.00184252 3.101159e-01 -146.2917 62.55000
#> 850 357.6250 M 1 6 114.71591788 2.542084e-01 -146.3750 62.55000
#> 851 235.3571 L 0 NA 110.42994317 2.038334e-01 -146.4583 62.55000
#> 852 191.2667 L 0 NA 106.14431501 1.589946e-01 -146.5417 62.55000
#> 853 185.5000 L 0 NA 101.85825582 1.196839e-01 -146.6250 62.55000
#> 854 181.0833 M 1 4 97.57216170 8.590565e-02 -146.7083 62.55000
#> 855 187.9333 L 0 NA 93.28643079 5.766205e-02 -146.7917 62.55000
#> 856 267.5500 L 0 NA 89.00028397 3.494797e-02 -146.8750 62.55000
#> 857 303.9375 L 0 NA 84.71411784 1.776622e-02 -146.9583 62.55000
#> 858 235.0000 M 1 0 80.42833056 6.117608e-03 -147.0417 62.55000
#> 859 209.5000 L 0 NA 76.14214301 0.000000e+00 -147.1250 62.55000
Some of the variables of interest include
total
, which has counts of moose (and isNA
for all sites that were not surveyed).strat
, a covariate that is eitherL
for Low orM
for medium.surveyed
, which is a0
if the site wasn’t sampled and a1
if the site was sampled.x
andy
, the spatial coordinates for the centroids of the sites (in a user-defined Trans-Mercator projection).
Fitting the Model and Obtaining Predictions
We can now proceed to use the functions in sptotal
in a
similar way to how the functions were used for the simulated data. To
get a sense of the data, we first give a plot of the raw observed
counts:
ggplot(data = AKmoose_df, aes(x = x, y = y)) +
geom_point(aes(colour = total), size = 4) +
scale_colour_viridis_c() +
theme_bw()