# Introduction

The package spatialRF facilitates fitting spatial regression models on regular or irregular data with Random Forest by generating spatial predictors that allow the model to take into account the spatial structure of the training data. The end goal is minimizing the spatial autocorrelation of the model residuals as much as possible.

Two main methods to generate spatial predictors from the distance matrix of the data points are implemented in the package:

The package is designed to minimize the amount of code required to fit a spatial model from a training dataset, the names of the response and the predictors, and a distance matrix, as the example below shows.

spatial.model <- spatialRF::rf_spatial(
data = your_dataframe,
dependent.variable.name = "your_response_variable",
predictor.variable.names = c("predictor1", "predictor2", ..., "predictorN"),
distance.matrix = your_distance_matrix
)

The package, that uses the ranger package under the hood (Wright and Ziegler 2017), also provides tools to identify potentially interesting variable interactions, tune random forest hyperparameters, assess model performance on spatially independent data folds, and examine the resulting models via importance plots, response curves, and response surfaces.

# Development

This package is reaching its final form, and big changes are not expected at this stage. However, it has many functions, and even though all them have been tested, only one dataset has been used for those tests. You will find bugs, and something will go wrong almost surely. If you have time to report bugs, please, do so in any of the following ways:

I will do my best to solve any issues ASAP!

# Applications

The goal of spatialRF is to help fitting explanatory spatial regression, where the target is to understand how a set of predictors and the spatial structure of the data influences response variable. Therefore, the spatial analyses implemented in the package can be applied to any spatial dataset, regular or irregular, with a sample size between ~100 and ~5000 cases (the higher end will depend on the RAM memory available), a quantitative or binary (values 0 and 1) response variable, and a more or less large set of predictive variables.

All functions but rf_spatial() work with non-spatial data as well if the arguments distance.matrix and distance.thresholds are ignored. In such case, the number of cases is no longer limited by the size of the distance matrix, and models can be trained with hundreds of thousands of rows.

However, when the focus is on fitting spatial models, and due to the nature of the spatial predictors used to represent the spatial structure of the training data, there are many things this package cannot do:

• Predict model results over raster data.

• Predict a model result over another region with a different spatial structure.

• Work with “big data”, whatever that means.

• Imputation or extrapolation (it can be done, but models based on spatial predictors are hardly transferable).

• Take temporal autocorrelation into account (but this is something that might be implemented later on).

If after considering these limitations you are still interested, follow me, I will show you how it works.

# Install

The package is not yet in the CRAN repositories, so at the moment it must be installed from GitHub as follows.

remotes::install_github(
repo = "blasbenito/spatialRF",
ref = "main",
force = TRUE,
quiet = TRUE
)
library(spatialRF)

There are a few other libraries that will be useful during this tutorial.

library(kableExtra)
library(rnaturalearth)
library(rnaturalearthdata)

# Data requirements

The data required to fit random forest models with spatialRF must fulfill several conditions:

• The input format is data.frame. At the moment, tibbles are not fully supported.
• The number of rows must be somewhere between 100 and ~5000, but that will depend on the RAM available in your system. However, this limitation only affects spatial analyses performed with rf_spatial(), while all other modeling and plotting functions should work without a distance matrix (if they don’t tell me, that’d be a bug!), and therefore analyses in large datasets can still be done with the package.
• The number of predictors should be larger than 3, fitting a Random Forest model is moot otherwise.
• Factors in the response or the predictors are not explicitly supported in the package, they may work, or they won’t, but in any case, I designed this package for quantitative data alone. However, binary data with values 0 and 1 in the response variable are supported.
• Must be free of NA. You can check if there are NA records with sum(apply(df, 2, is.na)). If the result is larger than 0, then just execute df <- na.omit(df) to remove rows with empty cells.
• Columns cannot have zero variance. This condition can be checked with apply(df, 2, var) == 0. Columns yielding TRUE should be removed.
• Columns must not yield NaN or Inf when scaled. You can check each condition with sum(apply(scale(df), 2, is.nan)) and sum(apply(scale(df), 2, is.infinite)). If higher than 0, you can find what columns are giving issues with sapply(as.data.frame(scale(df)), function(x)any(is.nan(x))) and sapply(as.data.frame(scale(df)), function(x)any(is.infinite(x))). Any column yielding TRUE will generate issues while trying to fit models with spatialRF.

# Example data

The package includes an example dataset that fulfills the conditions mentioned above, named plant_richness_df. It is a data frame with plant species richness and predictors for 227 ecoregions in the Americas, and a distance matrix among the ecoregion edges named, well, distance_matrix.

data(plant_richness_df)
data(distance_matrix)

#names of the response variable and the predictors
dependent.variable.name <- "richness_species_vascular"
predictor.variable.names <- colnames(plant_richness_df)[5:21]

#random seed for reproducibility
random.seed <- 100

The response variable of plant_richness_df is “richness_species_vascular”, with the total count of vascular plant species found on each ecoregion. The figure below shows the centroids of each ecoregion along with their associated value of the response variable.

The predictors (columns 5 to 21) represent diverse factors that may influence plant richness such as sampling bias, the area of the ecoregion, climatic variables, human presence and impact, topography, geographical fragmentation, and features of the neighbors of each ecoregion. The figure below shows the scatterplots of the response variable (y axis) against each predictor (x axis).

The function plot_training_df_moran() helps to check the spatial autocorrelation of the response variable and the predictors.

# Finding promising variable interactions

Random Forests already takes into account variable interactions of the form “variable a becomes important when b is higher than x”. However, Random Forest can also take advantage of variable interactions of the form a * b, as they are commonly defined in regression models.

The function rf_interactions() tests all possible interactions among predictors by using each one of them in a separate model, and suggesting the ones with the higher potential contribution to the model’s R squared and the higher relative importance (presented as a percentage of the maximum importance of a variable in the model).

interactions <- spatialRF::rf_interactions(
data = plant_richness_df,
dependent.variable.name = dependent.variable.name,
predictor.variable.names = predictor.variable.names,
seed = random.seed
)
## Testing 10 candidate interactions.

## 3 potential interactions identified.

##       ┌─────────────────────────┬───────────────────────┬────────────────┐
##       │ Interaction             │ Importance (% of max) │ R2 improvement │
##       ├─────────────────────────┼───────────────────────┼────────────────┤
##       │ human_population_X_bias │                  93.2 │          0.002 │
##       │ _area_km2               │                       │                │
##       ├─────────────────────────┼───────────────────────┼────────────────┤
##       │ human_population_X_huma │                  82.8 │          0.000 │
##       │ n_population_density    │                       │                │
##       ├─────────────────────────┼───────────────────────┼────────────────┤
##       │ climate_bio1_average_X_ │                  77.9 │          0     │
##       │ bias_area_km2           │                       │                │
##       └─────────────────────────┴───────────────────────┴────────────────┘

Here rf_interactions() suggests several candidate interactions ordered by their impact on the model. The function cannot say whether an interaction makes sense, and it is up to the user to choose wisely whether to select an interaction or not.

For the sake of the example, I will choose climate_bio1_average_X_bias_area_km2, hypothesizing that ecoregions with higher area (bias_area_km2) and energy (represented by the annual temperature, climate_bio1_average) will have more species of vascular plants (this is just an example, many other rationales are possible when choosing between candidate interactions). The data required to add the interaction to the training data is in the output of rf_interactions().

#adding interaction column to the training data
plant_richness_df[, "climate_bio1_average_X_bias_area_km2"] <- interactions$columns[, "climate_bio1_average_X_bias_area_km2"] #adding interaction name to predictor.variable.names predictor.variable.names <- c(predictor.variable.names, "climate_bio1_average_X_bias_area_km2") # Reducing multicollinearity in the predictors The functions auto_cor() and auto_vif() help reduce redundancy in the predictors by using different criteria (bivariate R squared vs. variance inflation factor), while allowing the user to define an order of preference, which can be based either on domain expertise or on a quantitative assessment. The preference order is defined as a character vector in the preference.order argument of both functions, and does not need to include the names of all predictors, but just the ones the user would like to keep in the analysis. In the example below I give preference to the interaction suggested by rf_interactions() over it’s two components, and prioritize climate over other types of predictors (any other choice would be valid, it just depends on the scope of the study). These rules are applied to both auto_cor() and auto_vif(), that are executed sequentially by using the %>% pipe from the magrittr package. Notice that I have set cor.threshold and vif.threshold to low values because the predictors in plant_richness_df already have little multicollinearity,. The default values (cor.threshold = 0.75 and vif.threshold = 5) should work well when combined together for any other set of predictors. preference.order <- c( "climate_bio1_average_X_bias_area_km2", "climate_aridity_index_average", "climate_hypervolume", "climate_bio1_average", "climate_bio15_minimum", "bias_area_km2" ) predictor.variable.names <- spatialRF::auto_cor( x = plant_richness_df[, predictor.variable.names], cor.threshold = 0.6, preference.order = preference.order ) %>% spatialRF::auto_vif( vif.threshold = 2.5, preference.order = preference.order ) ## [auto_cor()]: Removed variables: bias_area_km2, human_footprint_average ## [auto_vif()]: Removed variables: human_population The output of auto_cor() or auto_vif() is of the class “variable_selection”, that can be used as input for the argument predictor.variable.names of any modeling function within the package. An example is shown in the next section. names(predictor.variable.names) ## [1] "vif" "selected.variables" "selected.variables.df" The slot selected.variables contains the names of the selected predictors. predictor.variable.names$selected.variables
##  [1] "climate_bio1_average_X_bias_area_km2"
##  [2] "climate_aridity_index_average"
##  [3] "climate_hypervolume"
##  [4] "climate_bio1_average"
##  [5] "climate_bio15_minimum"
##  [6] "bias_species_per_record"
##  [7] "climate_velocity_lgm_average"
##  [8] "neighbors_count"
##  [9] "neighbors_percent_shared_edge"
## [10] "human_population_density"
## [11] "topography_elevation_average"
## [12] "landcover_herbs_percent_average"
## [13] "fragmentation_cohesion"
## [14] "fragmentation_division"
## [15] "neighbors_area"

# Fitting a non-spatial Random Forest model

To fit basic Random Forest models spatialRF provides the rf() function. It takes the training data, the names of the response and the predictors, and optionally (to assess the spatial autocorrelation of the residuals), the distance matrix, and a vector of distance thresholds (in the same units as the distances in distance_matrix).

These distance thresholds are the neighborhoods at which the model will check the spatial autocorrelation of the residuals. Their values may depend on the spatial scale of the data, and the ecological system under study.

Notice that here I plug the object predictor.variable.names, output of auto_cor() and auto_vif(), directly into the predictor.variable.names argument.

model.non.spatial <- spatialRF::rf(
data = plant_richness_df,
dependent.variable.name = dependent.variable.name,
predictor.variable.names = predictor.variable.names,
distance.matrix = distance_matrix,
distance.thresholds = c(0, 1500, 3000),
seed = random.seed,
verbose = FALSE
)

The model output can be printed or plotted with a plethora of functions such as print(), print_importance(), print_performance(), plot_importance(), print_moran(), plot_moran(), plot_response_curves(), or plot_response_surfaces), among many others.

spatialRF::plot_response_curves(model.non.spatial)

In the response curves above, the other predictors are set to their quantiles 0.1, 0.5, and 0.8, but the user can change this behavior by modifying the values of the quantiles argument.

spatialRF::plot_response_surfaces(
model.non.spatial,
a = "climate_bio1_average",
b = "neighbors_count"
)

In this response surface, the predictors that are not shown are set to their medians (but other quantiles are possible).

spatialRF::plot_importance(model.non.spatial, verbose = FALSE)

Predicting onto new data

Models fitted with rf() and other rf_X() functions within the package can be predicted onto new data just as it is done with ranger() models:

predicted <- stats::predict(
object = model.non.spatial,
data = plant_richness_df,
type = "response"
)$predictions Repeating a model execution Random Forest is an stochastic algorithm that yields slightly different results on each run unless a random seed is set. This particularity has implications for the interpretation of variable importance scores. For example, in the plot above, the difference in importance between the predictors climate_hypervolume and climate_bio1_average_X_bias_area_km2 could be just the result of chance. The function rf_repeat() repeats a model execution and yields the distribution of importance scores of the predictors across executions. model.non.spatial.repeat <- spatialRF::rf_repeat( model = model.non.spatial, repetitions = 30, seed = random.seed, verbose = FALSE ) plot_importance(model.non.spatial.repeat, verbose = FALSE) After 30 model repetitions it is clear that the difference in importance between climate_hypervolume and climate_bio1_average_X_bias_area_km2 is not the result of chance. The response curves of models fitted with rf_repeat() can be plotted with plot_response_curves() as well. The median prediction is shown with a thicker line. spatialRF::plot_response_curves( model.non.spatial.repeat, quantiles = 0.5 ) The function get_response_curves() returns a data frame with the data required to make custom plots of the response curves. plot.df <- spatialRF::get_response_curves(model.non.spatial.repeat) response predictor quantile model predictor.name response.name 1410.021 -183.8091 0.1 1 climate_bio1_average richness_species_vascular 1410.021 -181.5008 0.1 1 climate_bio1_average richness_species_vascular 1410.021 -179.1924 0.1 1 climate_bio1_average richness_species_vascular 1410.021 -176.8841 0.1 1 climate_bio1_average richness_species_vascular 1410.021 -174.5758 0.1 1 climate_bio1_average richness_species_vascular 1410.021 -172.2675 0.1 1 climate_bio1_average richness_species_vascular 1410.021 -169.9592 0.1 1 climate_bio1_average richness_species_vascular 1410.021 -167.6509 0.1 1 climate_bio1_average richness_species_vascular 1410.021 -165.3426 0.1 1 climate_bio1_average richness_species_vascular 1410.021 -163.0343 0.1 1 climate_bio1_average richness_species_vascular # Tuning Random Forest hyperparameters The model fitted above was based on the default hyperparameter values provided by ranger(), and those might not be the most adequate ones for a given dataset. The function rf_tuning() helps the user to choose sensible values for three Random Forest hyperparameters that are critical to model performance: • num.trees: number of regression trees in the forest. • mtry: number of variables to choose from on each tree split. • min.node.size: minimum number of cases on a terminal node. Model tuning is done via spatial cross-validation, to ensure that the selected combination of hyperparameters maximizes the ability of the model to predict data not used to train it. model.non.spatial.tuned <- spatialRF::rf_tuning( model = model.non.spatial, xy = plant_richness_df[, c("x", "y")], repetitions = 30, training.fraction = 0.75, num.trees = c(500, 1000), mtry = seq( 2, 14, #equal or lower than the number of predictors by = 3 ), min.node.size = c(5, 10, 20), seed = random.seed ) ## Exploring 30 combinations of hyperparameters. ## Best hyperparameters: ## - num.trees: 1000 ## - mtry: 14 ## - min.node.size: 5 ## gain in r.squared: 0.033 The function rf_tuning() returns a model fitted with the same data as the original model, but using the best hyperparameters found during tuning. Model tuning has helped to a very small improvement in performance measures (+ 0.033 R squared), so from here, we can keep working with model.non.spatial.tuned. # Fitting a spatial model The spatial autocorrelation of the residuals of model.non.spatial, measured with Moran’s I, can be plotted with plot_moran(). spatialRF::plot_moran( model.non.spatial.tuned, verbose = FALSE ) According to the plot, the spatial autocorrelation of the residuals is highly positive for a neighborhood of 0 km, while it becomes non-significant (p-value > 0.05, whatever that means) at 1500 and 3000 km. To reduce the spatial autocorrelation of the residuals as much as possible, the non-spatial tuned model fitted above can be converted into a spatial model easily with rf_spatial(), that by default uses the Moran’s Eigenvector Maps method. model.spatial <- spatialRF::rf_spatial( model = model.non.spatial.tuned, method = "mem.moran.sequential", #default method verbose = FALSE, seed = random.seed ) The plot below shows the Moran’s I of the residuals of the spatial model. It shows that rf_spatial() has managed to remove the spatial autocorrelation (p-values of the Moran’s I estimates for each neighborhood distance are higher than 0.05) of the model residuals for every neighborhood distance. spatialRF::plot_moran( model.spatial, verbose = FALSE ) If we compare the variable importance plots of both models, we can see that the spatial model has an additional set of dots under the name “spatial_predictors”, and that the maximum importance of a few of these spatial predictors matches the importance of the most relevant non-spatial predictors. p1 <- spatialRF::plot_importance( model.non.spatial, verbose = FALSE) + ggplot2::ggtitle("Non-spatial model") p2 <- spatialRF::plot_importance( model.spatial, verbose = FALSE) + ggplot2::ggtitle("Spatial model") p1 | p2 If we take a look to the ten most important variables in model.spatial we will see that a few of them are spatial predictors. variable importance climate_bio1_average_X_bias_area_km2 0.151 spatial_predictor_0_2 0.147 climate_hypervolume 0.140 climate_bio1_average 0.132 bias_species_per_record 0.080 spatial_predictor_0_1 0.064 spatial_predictor_3000_1 0.057 spatial_predictor_0_6 0.053 spatial_predictor_0_5 0.045 human_population_density 0.041 Spatial predictors are named spatial_predictor_X_Y, where X is the neighborhood distance at which the predictor has been generated, and Y is the index of the predictor. Spatial predictors, as shown below, are smooth surfaces representing neighborhood among records at different spatial scales. The spatial predictors in the spatial model have been generated using the method “mem.moran.sequential” (function’s default), that mimics the Moran’s Eigenvector Maps method described in (Dray, Legendre, and Peres-Neto 2006). In brief, the method consist on transforming the distance matrix into a double-centered matrix of normalized weights, to then compute the positive eigenvectors of the weights matrix (a.k.a, Moran’s Eigenvector Maps, or MEMs). The MEMs are included in the model one by one in the order of their Moran’s I, and the subset of MEMs maximizing the model’s R squared and minimizing the Moran’s I of the residuals and the number of MEMs added to the model are selected, as shown in the optimization plot below (dots linked by lines represent the selected spatial predictors). The selection procedure is performed by the function select_spatial_predictors_sequential(). Tuning spatial models Spatial models fitted with rf_spatial() can be tuned as well with rf_tuning(). However, tuning may in some cases increase the spatial autocorrelation of the model residuals. In that case, the function will return a message explaining the situation, and the original model without any sort of tuning applied model.spatial.tuned <- rf_tuning( model = model.spatial, xy = plant_richness_df[, c("x", "y")], repetitions = 30, num.trees = c(500, 1000), mtry = seq( 2, length(model.spatial$ranger.arguments$predictor.variable.names), by = 9), min.node.size = c(5, 20), seed = random.seed ) ## Exploring 24 combinations of hyperparameters. ## Best hyperparameters: ## - num.trees: 1000 ## - mtry: 47 ## - min.node.size: 5 ## gain in r.squared: 0.024 # Assessing model performance on spatially independent folds The function rf_evaluate() separates the training data into a number of spatially independent training and testing folds, fits a model on each training fold, predicts over each testing fold, and computes performance measures, to finally aggregate them across model repetitions. Let’s see how it works. model.spatial.tuned <- spatialRF::rf_evaluate( model = model.spatial.tuned, xy = plant_richness_df[, c("x", "y")], #data coordinates repetitions = 30, #number of folds training.fraction = 0.8, #training data fraction metrics = c("r.squared", "rmse"), seed = random.seed, verbose = FALSE ) The function generates a new slot in the model named “evaluation” with several objects that summarize the spatial cross-validation results. names(model.spatial.tuned$evaluation)
## [1] "metrics"           "training.fraction" "spatial.folds"
## [4] "per.fold"          "per.fold.long"     "per.model"
## [7] "aggregated"

The slot “spatial.folds”, produced by make_spatial_folds(), contains the indices of the training and testing cases for each cross-validation repetition. The maps below show two sets of training and testing spatial folds.

The functions plot_evaluation() and print_evaluation() allow to check the evaluation results as a plot or as a table.

spatialRF::print_evaluation(model.spatial.tuned)
##
## Spatial evaluation
##   - Training fraction:             0.8
##   - Spatial folds:                 25
##
##     Metric     Mean Standard deviation  Minimum  Maximum
##  r.squared    0.250              0.159    0.080    0.601
##       rmse 3223.499            810.886 2285.215 4748.118

The low R squared yielded by the model evaluation shows that the spatial model is hard to transfer outside of the training space. Models based on a spatial structure like the ones fitted with rf_spatial() do not work well when transferred to a different place (that is what rf_compare() does), because spatial structures are not transferable when the data is irregularly distributed, as it is the case with plant_richness_df. The comparison below shows how non-spatial models may show better (not bad, not great) evaluation scores on independent spatial folds.

# Comparing several models

The function rf_evaluate() only assesses the predictive performance on unseen data of one model at a time. If the goal is to compare two models, rf_evaluate() can be indeed ran twice, but spatialRF offers a more convenient option named rf_compare(). It takes as input a named list with as many models as the user needs to compare.

comparison <- spatialRF::rf_compare(
models = list(
Non-spatial = model.non.spatial,
Non-spatial tuned = model.non.spatial.tuned,
Spatial = model.spatial,
Spatial tuned = model.spatial.tuned
),
xy = plant_richness_df[, c("x", "y")],
repetitions = 30,
training.fraction = 0.8,
metrics = c("r.squared", "rmse"),
notch = TRUE,
seed = random.seed
)

Model Metric Mean
Non-spatial r.squared 0.336
Non-spatial tuned r.squared 0.412
Spatial r.squared 0.167
Spatial tuned r.squared 0.217
Non-spatial rmse 2817.225
Non-spatial tuned rmse 2329.933
Spatial rmse 3086.170
Spatial tuned rmse 2930.469

# Generating spatial predictors for other models

You might not love Random Forest, but spatialRF loves you, and as such, it gives you tools to generate spatial predictors for other models anyway.

The first step requires generating Moran’s Eigenvector Maps (MEMs) from the distance matrix. Here there are two options, computing MEMs for a single neighborhood distance with mem(), and computing MEMs for several neighborhood distances at once with mem_multithreshold().

#single distance (0km by default)
mems <- mem(x = distance_matrix)

#several distances
mems <- spatialRF::mem_multithreshold(
x = distance_matrix,
distance.thresholds = c(0, 1000, 2000)
)

In either case the result is a data frame with Moran’s Eigenvector Maps (“just” the positive eigenvectors of the double-centered distance matrix).

spatial_predictor_0_1 spatial_predictor_0_2 spatial_predictor_0_3 spatial_predictor_0_4
0.0259217 0.0052203 0.0416969 -0.0363324
0.0996679 0.0539713 0.1324480 0.3826928
0.0010477 -0.0143046 -0.0443602 -0.0031386
0.0165695 0.0047991 0.0307457 0.0005170
0.0225761 0.0019595 0.0230368 -0.0524239
0.0155252 0.0023742 0.0197953 -0.0338956
0.0229197 0.0039860 0.0312561 -0.0416697
-0.2436009 -0.1155295 0.0791452 0.0189996
0.0150725 -0.0158684 -0.1010284 0.0095590
-0.1187381 -0.0471879 0.0359881 0.0065211

But not all MEMs are made equal, and you will need to rank them by their Moran’s I. The function rank_spatial_predictors() will help you do so.

mem.rank <- spatialRF::rank_spatial_predictors(
distance.matrix = distance_matrix,
spatial.predictors.df = mems
)

The output of rank_spatial_predictors() is a list with three slots: “method”, a character string with the name of the ranking method; “criteria”, an ordered data frame with the criteria used to rank the spatial predictors; and “ranking”, a character vector with the names of the spatial predictors in the order of their ranking (it is just the first column of the “criteria” data frame). We can use this “ranking” object to reorder or mems data frame.

mems <- mems[, mem.rank$ranking] #also: #mems <- mem.rank$spatial.predictors.df

From here, spatial predictors can be included in any model one by one, in the order of the ranking, until the spatial autocorrelation of the residuals is gone, or our model gets totally defaced. A little example with a linear model follows.

#model definition
predictors <- c(
"climate_aridity_index_average ",
"climate_bio1_average",
"bias_species_per_record",
"human_population_density",
"topography_elevation_average",
"fragmentation_division"
)
model.formula <- as.formula(
paste(
dependent.variable.name,
" ~ ",
paste(
predictors,
collapse = " + "
)
)
)

#scaling the data
model.data <- scale(plant_richness_df) %>%
as.data.frame()

#fitting the model
m <- lm(model.formula, data = plant_richness_df)

#Moran's I test of the residuals
moran.test <- spatialRF::moran(
x = residuals(m),
distance.matrix = distance_matrix,
)
moran.test
##   moran.i p.value               interpretation
## 1    0.21       0 Positive spatial correlation

According to the Moran’s I test, the model residuals show spatial autocorrelation. Let’s introduce MEMs one by one until the problem is solved.

#add mems to the data and applies scale()
model.data <- data.frame(
plant_richness_df,
mems
) %>%
scale() %>%
as.data.frame()

#initialize predictors.i
predictors.i <- predictors

#iterating through MEMs
for(mem.i in colnames(mems)){

#add mem name to model definintion
predictors.i <- c(predictors.i, mem.i)

#generate model formula with the new spatial predictor
model.formula.i <- as.formula(
paste(
dependent.variable.name,
" ~ ",
paste(
predictors.i,
collapse = " + "
)
)
)

#fit model
m.i <- lm(model.formula.i, data = model.data)

#Moran's I test
moran.test.i <- moran(
x = residuals(m.i),
distance.matrix = distance_matrix,
)

#stop if no autocorrelation
if(moran.test.i\$interpretation != "Positive spatial correlation"){
break
}

}#end of loop

Now we can compare the model without spatial predictors m and the model with spatial predictors m.i.

Model Predictors R_squared AIC BIC Moran.I
Non-spatial 6 0.38 4238 4266 0.21
Spatial 21 0.50 530 608 0.06

According to the model comparison, it can be concluded that the addition of spatial predictors, in spite of the increase in complexity, has improved the model. In any case, this is just a simple demonstration of how spatial predictors generated with functions of the spatialRF package can still help you fit spatial models with other modeling methods.