1 Introduction

The purpose of top-down disaggregation is to estimate population counts at a finer spatial resolution than the available population totals for administrative units. WorldPop top-down disaggregation implements a dasymetric mapping approach that uses the random forest machine learning algorithm to disaggregate projected census totals to estimate population counts for 100 m grid cells (Sorichetta et al. 2015, Stevens et al. 2015). Dasymetric mapping estimates population counts at a finer resolution than the input population totals based on relationships with high resolution geospatial covariates like building locations and road networks.

In this tutorial, we will demonstrate how to implement this method in the R statistical programming environment. We will adapt the method to estimate population counts for census enumeration areas (EAs) rather than 100 m grid cells. To demonstrate the approach, we will dissaggregate population totals from municipalities in Brazil to estimate populations in finer-scale census EAs (Fig. 1.1).

Figure 1.1: Example of enumeration area boundaries in Brazil

1.1 Pre-requisites

The following software may be needed to complete this tutorial:

R environment,
RStudio (optional), and
ArcGIS or QGIS (optional to view results).

The following pre-requisite skills may be helpful:

R programming language
- install packages, load and manipulate data
Machine learning concepts
- predictor and response variables, model diagnostics
GIS concepts
- raster and vector data, zonal statistics

2 Background

The modelling process consists of three steps (Fig. 2.1):

Estimate the relationship between population density and geospatial covariates at the municipality level,
Predict population densities at the level of enumeration areas using the covariates, and
Use predicted densities as a weighting layer to disaggregate the population totals for munipalities.

The core product is the weighting layer that should reflect the spatial distribution of the population inside the municipalities. The final goal is to predict the population count in each enumeration area correctly.

The extent of the study area is defined by the spatial coverage of the municipalities for which we have population totals. The top-down method cannot estimate populations for enumeration areas within municipalities where population totals are unknown.

Figure 2.1: Modelling Process

Random forest is a machine learning algorithm that builds an ensemble of individual regression trees to predict values of a response variable (e.g. population density) based on its relationships with geospatial covariates (Breiman 2001, Stevens et al. 2015). Each regression tree identifies threshold values of the covariates that can be used to split the training data into fairly homogeneous groups (i.e. with similar population densities).

The algorithm prevents overfitting by withholding a subset of the training data for each individual tree that can be used to calculate rates of "out-of-bag" prediction error (i.e. out-of-sample cross-validation). The algorithm also prevents overfitting by randomly selecting a small set of covariates to be considered for splitting the training data at each node of the regression tree, rather than assessing all covariates at each split. These features make random forest fairly robust to multi-collinearity among covariates allowing it to use large sets of covariates without a priori variable selection. However, a large number of covariates might impact running time, especially at the prediction stage, and it may be useful to select a subset of covariates based on the measures of covariate importance provided by the random forest algorithm (Stevens et al. 2015, Bondarenko et al. 2018). While random forest predictions are robust to multi-collinearity among covariates, the measures of covariate importance may not be (Genuer et al. 2010).

To dive deeper on the subject of random forest, here are some online materials:

A use-R 2009 conference presentation given by Adele Cutler who developed the random forest algorithm with Leo Breiman.
A 2019 blog post by Tony Yiu with clear explanations of Random Forest main features namely bagging and covariates randomness that supports the creation of uncorrelated forest of decision trees.
A 2001 article from Breiman explaining the underlying maths.

The implementation we will provide in this tutorial will follow the guidance set up by Bondarenko et al. (2018) to apply random forest for top-down disaggregation.

3 R Environment

This model will be implemented in the R programming language (R Core Team 2020). Our first step is to setup the R environment so that it contains the R packages and data that we will need. The randomForest package (Liaw & Wiener 2002) implements the random forest algorithm in R. If you have not already installed it, then you will need to install it.

install.packages('randomForest')

Once installed, you must load the randomForest package into your R environment.

library(randomForest)

We will define our working directory to be the location where we have stored the input data for the tutorial (master_train.csv and master_predict.csv described below).

setwd('c:/myfolder')

3.1 Source Data

This tutorial includes two input data sets, provided as csv spreadsheets (comma-separate-values text files):

master_train.csv Training data for the model that contains population totals and covariate values for 5568 municipalities in Brazil, and
master_predict.csv Prediction data for the model that contains covariate values for 444215 enumeration areas where population estimates are needed.

We will load the datasets into our R environment using the read.csv function and display their top five rows with the head function:

# training data from municipalities
master_train <- read.csv("master_train.csv")
# covariates from enumeration areas
master_predict <- read.csv("master_predict.csv")

head(master_train[,1:5]) # only showing first five columns

##        name_muni geo_code     pop       area mean.bra_viirs_100m_2016
## 1       Brasília  5300108 3055149 5779998938               8.51690292
## 2  Vila Propício  5222302    5882 2181582535               0.08895954
## 3       Vila Boa  5222203    6312 1060171349               0.07032987
## 4 Vicentinópolis  5222054    8873  737256139               0.13859431
## 5     Vianópolis  5222005   13977  954283662               0.17625734
## 6         Varjão  5221908    3838  519194233               0.09681950

head(master_predict[,1:4]) # only showing first four columns

##   EA_id geo_code mean.bra_viirs_100m_2016 mean.bra_srtm_slope_100m
## 1     1  1100015                21.700214                 1.954565
## 2     2  1100015                12.255162                 2.374444
## 3     3  1100015                18.794325                 1.983213
## 4     4  1100015                11.781239                 1.631729
## 5     5  1100015                 3.152316                 2.861502
## 6     6  1100015                 7.395526                 1.814885

Notice that only the municipality-level dataset master_train.csv contains a column for population. The EA-level dataset does not contain a column for population, and it is the goal of this tutorial to estimate those EA-level populations based on the EA-level covariates in master_predict.csv.

Also, notice that the column names for the covariates are exactly the same in the municipality-level dataset and the EA-level dataset. All of the municipality-level covariates used to train the model must also be available at the EA level for the model predictions.

The geo_code column contains a numeric identifier for each municipality and will be used in later steps to identify the municipality that each EA belongs to. Boundaries of enumeration areas in Brazil roughly follow municipality boundaries (i.e. enumeration areas are nested within municipalities), but the boundaries are not harmonized exactly. We assigned each enumeration area to the municipality that contained the majority of its area.

Note: We derived the two spreadsheets (master_train.csv and master_predict.csv) from the following sources:

Brazil's municipality boundaries (IBGE 2019),
Brazil's 2020 census projections for municipalities (IBGE 2020 a),
Brazil's census EA boundaries (IBGE 2020 b),
WorldPop's mastergrid for Brazil (WorldPop & CIESIN 2018 a), and
WorldPop's geospatial covariate rasters for Brazil (Lloyd et al. 2017, 2019).

See the section Zonal Statistics for more information about how the input data were created.

4 Random Forest

We will reformat the source data into the correct format for the random forest model before we fit the model and apply it to estimate populations in each enumeration area (EA). We will start by preparing the response and predictor variables from master_train.csv for training the model.

4.1 Response Variable

The response variable y_data for our random forest model must be a vector with a population value for each municipality. We will define our response variable as the log of population density:

y_data <- master_train$pop / master_train$area

y_data <- log(y_data)

We use log population density as the response variable rather than population counts for two main reasons. First, population densities are more comparable than counts among spatial units of varying sizes (e.g. municipalities and EAs). Second, the logarithm transformation reshapes the response variable as a Gaussian distribution, better inline with the distributions of covariates (Stevens et al. 2015) (Fig. 4.1). Note that master_train$area is measured in square meters in this case but any unit of area will work.

Figure 4.1: Histogram of the log-transformed population density.

We now have y_data, a vector with the log population density for every municipality in Brazil.

4.2 Predictor Variables

The predictor data x_data for our random forest model must be a data.frame with a row for each municipality and a column for each covariate. The row order must match that of y_data (i.e. row 1 from x_data must represent the same enumeration area as element 1 from y_data).

We build the predictor data x_data by subsetting the covariates from master_train.csv that we would like to include in our model. Remember, we want to select covariates that will be good predictors of log population density.

# list all covariate names (i.e. column names)
cols <- colnames(master_train)
print(cols)

##  [1] "name_muni"                          "geo_code"                          
##  [3] "pop"                                "area"                              
##  [5] "mean.bra_viirs_100m_2016"           "mean.bra_srtm_slope_100m"          
##  [7] "mean.bra_osm_dst_road_100m_2016"    "mean.bra_esaccilc_dst200_100m_2015"
##  [9] "mean.bra_esaccilc_dst190_100m_2015" "mean.bra_esaccilc_dst150_100m_2015"
## [11] "mean.bra_esaccilc_dst140_100m_2015" "mean.bra_esaccilc_dst130_100m_2015"
## [13] "mean.bra_esaccilc_dst040_100m_2015" "mean.bra_esaccilc_dst011_100m_2015"
## [15] "code_state"                         "code_muni"                         
## [17] "abbrev_state"

Our example data master_train.csv includes a few columns that are not covariates (e.g. geo_code) and so we do not want to include them in x_data. Our covariates all include the word "mean" in their column names because they were calculated as the mean value of underlying covariate rasters within each municipality. So, we will use "mean" as a search term to identify the correct columns. This provides an example of selecting a subset of the predictors, but you could use other subsets, search terms, or methods of subsetting.

# select column names that contain the word 'mean'
cov_names <- cols[grepl('mean', cols)] 
print(cov_names)

##  [1] "mean.bra_viirs_100m_2016"           "mean.bra_srtm_slope_100m"          
##  [3] "mean.bra_osm_dst_road_100m_2016"    "mean.bra_esaccilc_dst200_100m_2015"
##  [5] "mean.bra_esaccilc_dst190_100m_2015" "mean.bra_esaccilc_dst150_100m_2015"
##  [7] "mean.bra_esaccilc_dst140_100m_2015" "mean.bra_esaccilc_dst130_100m_2015"
##  [9] "mean.bra_esaccilc_dst040_100m_2015" "mean.bra_esaccilc_dst011_100m_2015"

# subset the data.frame to only these columns
x_data <- master_train[,cov_names]
head(x_data[,1:2]) # only showing first two columns

##   mean.bra_viirs_100m_2016 mean.bra_srtm_slope_100m
## 1               8.51690292                 3.547058
## 2               0.08895954                 4.198463
## 3               0.07032987                 3.522752
## 4               0.13859431                 1.620756
## 5               0.17625734                 3.577609
## 6               0.09681950                 5.659529

We now have a data.frame x_data with a row for each municipality and a column for each covariate.

4.3 Model Fitting

The model fitting process will identify relationships between the response variable (i.e. population) and predictor variables at the municipality level.

4.3.1 Settings

Before running the random forest model, we need to identify appropriate values for each argument of the randomForest() and/or tuneRF() functions. See ?randomForest and ?tuneRF for detailed descriptions of all arguments to the functions.

A critical argument in randomForest() function is mtry, the number of randomly selected variables for evaluating the best split at each node in a regression tree within the random forest model. This process is at the root of the randomness in random forest models. To find the optimal mtry value, we use the function tuneRF() that compares different models based on their "out-of-bag" error rates.

Note: Out-of-bag error is a form of cross-validation that is calculated by comparing model predictions to observed data from EAs that were randomly selected to be witheld from model fitting (i.e. held "out-of-bag") for an iteration of the model (i.e. a single regression tree in the random forest model).

During each iteration of a random forest model, a regression tree is fit to a random sub-sample of the data. This prevents overfitting and also provides a built-in mechanism for cross-validation. The sampsize argument sets the sample size, and the replace argument defines the sampling strategy. In our case, we will define sampsize as the total count of observations (i.e. 5568 municipalities) and replace will be TRUE to sample with replacement from the training data.

Our specifications for other parameters follow Bondarenko et al. (2018).

ntree: Number of trees to grow. There is no issue of overfitting when adding additional trees. We opt for 500, the default value of the randomForest() function.
importance: If TRUE, the model will calculate the covariate importance for further analysis. We opt for TRUE.
nodesize: Minimum size of the terminal node of the trees controls the tree complexity. We make it about 0.1% of the training data sample size (i.e. 5).

4.3.2 Run the Model

Once we have decided on appropriate settings, we will input them as arguments to the tuneRF() function (note: this may take several minutes to run).

# model fitting
popfit <- tuneRF(x=x_data, 
                 y=y_data, 
                 plot=TRUE, 
                 mtryStart=length(x_data)/3, 
                 ntreeTry=500, 
                 improve=0.0001, # threshold on the OOB error to continue the search
                 stepFactor=1.20, # incremental improvement of mtry
                 trace=TRUE, 
                 doBest=TRUE, # last model trained with the best mtry
                 nodesize=length(y_data)/1000, 
                 na.action=na.omit, 
                 importance=TRUE, # calculate variable importance
                 sampsize=length(y_data), # size of the sample to draw for OOB
                 replace=TRUE) # sample with replacement

The popfit object contains the fitted random forest model. We can see the items within the object:

names(popfit)

##  [1] "call"            "type"            "predicted"       "mse"            
##  [5] "rsq"             "oob.times"       "importance"      "importanceSD"   
##  [9] "localImportance" "proximity"       "ntree"           "mtry"           
## [13] "forest"          "coefs"           "y"               "test"           
## [17] "inbag"

We can retrieve individual values, such as the value of the mtry setting that was selected by the tuneRF function.

popfit$mtry

## [1] 4

See the "Value" section in the help for the randomForest function ?randomForest::randomForest for an explanation of each value.

We will now save the popfit object to our hard drive.

save(popfit, file='popfit.Rdata')

We can load the fitted model back into our R environment later using:

load('popfit.Rdata')

4.4 Weighting Layer

We will use the fitted model to create a weighting layer that will be used to redistribute the population totals for municipalities into the enumeration areas (EA) within them. The EA-level covariate data master_predict.csv contains 444215 rows, one for each EA in Brazil.

We will first use the fitted random forest model to generate raw model predictions for each EA and save them as a column in the master_predict data.frame:

# random forest predictions
master_predict$predicted <- predict(popfit, 
                                    newdata = master_predict)

The weights are then calculated from the model predictions using the following formula:

\[\begin{equation} weight_{i} = \frac{exp(predicted_{i})}{\sum_{i=1}^{I_j}{exp(predicted_{i})}} \tag{4.1} \end{equation}\]

where $predicted_i$ is the model prediction for enumeration area $i$, and $I_j$ is the total number of enumeration areas in municipality $j$. These weights represent the proportion of the population from municipality $j$ that lives in enumeration area $i$.

4.5 Redistribution to EA-level

We will now use the weights to estimate the population for each enumeration area $i$ using the following formula:

\[\begin{equation} population_{i} = total_j \times weight_{i} \tag{4.2} \end{equation}\]

where $total_j$ is the population total for municipality $j$ and $weight_i$ is the proportion of the population that lives in enumeration area $i$ (Eq. (4.1)). We will now work through this formula step-by-step in R.

First, we will exponentiate the predictions (numerator from Eq. (4.1)) and save them as a column in master_predict:

# back-transform predictions to natural scale
master_predict$predicted_exp <- exp(master_predict$predicted)

We need to exponentiate the predictions because we log-transformed the population densities that we used as the response variable for training the model. Therefore, the model predictions are population densities on the log-scale. We exponentiate them to derive the predicted population densities that we will use as weighting factors to disaggregate the municipality-level population totals.

Next, we will sum the predicted population densities among EAs in each municipality (denominator from Eq. (4.1)) and merge the results into master_predict using the geo_code column to match municipalities:

# sum exponentiated predictions among EAs in each municipality
predicted_exp_sum <- aggregate(list(predicted_exp_sum=master_predict$predicted_exp), 
                               by = list(geo_code=master_predict$geo_code), 
                               FUN = sum)

# merge predicted_exp_sum into master_train based on geo_code
master_predict <- merge(master_predict, 
                        predicted_exp_sum, 
                        by='geo_code')

These sums (i.e. the denominator from Eq. (4.1)) are used to scale the exponentiated model predictions to sum to one among EAs in each municipality.

Then, we will merge the total populations for each municipality into master_predict, again using the geo_code column to match EAs to the correct municipalities:

# merge municipality total populations from master_train into master_predict
master_predict <- merge(master_predict,
                        master_train[,c('geo_code','pop')],
                        by = 'geo_code')

# modify column name
names(master_predict)[ncol(master_predict)] <- 'pop_municipality'

We have now added all of the required information from Eqs. (4.2) and (4.1) into the master_predict data.frame. We will calculate the EA-level population estimates and add them to master_predict as a column:

# calculate EA-level population estimates
master_predict$predicted_pop <- with(master_predict, predicted_exp / predicted_exp_sum * pop_municipality)

Notice that this line of code is equivalent to Eqs. (4.2) and (4.1).

We have now disaggregated the population totals from 5568 municipalities in Brazil into population estimates for 444215 enumeration areas (Fig. 4.2).

Census-based population totals for municipalities (left) that have been disaggregated into population estimates for enumeration areas (right) east of Sao Paulo, Brazil. Darker red corresponds to more people.

Figure 4.2: Census-based population totals for municipalities (left) that have been disaggregated into population estimates for enumeration areas (right) east of Sao Paulo, Brazil. Darker red corresponds to more people.

4.6 Diagnostics

4.6.1 Summing Enumeration Areas

We first want to check that the EA-level population estimates sum to the municipality-level population totals. To do this, we will aggregate the EA-level predictions using the geo_code column to identify the municipalities that each EA belongs to:

# sum EA population estimates within each municipality
test <- aggregate(master_predict$predicted_pop, 
                  by = list(geo_code=master_predict$geo_code), 
                  FUN = sum)

# modify column names
names(test) <- c('geo_code','predicted_pop')

# merge municipality population totals
test <- merge(test,
              master_train[,c('geo_code','pop')],
              by = 'geo_code')

# test if estimates match muncipality population totals
all(test$pop == round(test$predicted_pop))

## [1] TRUE

If we fail this test (i.e. if the last line of code returns a value of FALSE), it could mean that there was an error in our implementation of Eq. (4.2) or Eq. (4.1) because those equations should re-scale the EA-level population estimates so that they always sum to the population totals for each municipality. This test does not assess the goodness-of-fit for the random forest model.

4.6.2 Goodness-of-Fit

The print function from the randomForest package displays two metrics that describe the out-of-bag prediction residuals:

the mean squared residuals (MSE)
the % of variance explained which corresponds to a pseudo $R^2$, (1 - MSE/ Var(y_data)).

The first metric (MSE) is dependent on the scale of the input response variable (e.g. the unit of area used to calculate population density), whereas the second metric ($R^2$) can be compared across models with different response variables.

# goodness-of-fit metrics
print(popfit)

## 
## Call:
##  randomForest(x = x, y = y, mtry = res[which.min(res[, 2]), 1],      replace = TRUE, sampsize = ..4, nodesize = ..1, importance = TRUE,      na.action = ..2) 
##                Type of random forest: regression
##                      Number of trees: 500
## No. of variables tried at each split: 4
## 
##           Mean of squared residuals: 0.2089229
##                     % Var explained: 89.97

We can visualize the out-of-bag predictions for municipalities.

First we plot the observed vs predicted values. This enables us to spot any municipalities where the random forest predictions are not accurate.

# plot observed vs predicted (out-of-bag)
plot(x = y_data,
     y = predict(popfit), 
     main = 'Observed vs Predicted log-Densities')

# 1:1 line
abline(a=0, b=1, col='red')

Note: The predict function in this code will return "out-of-bag" predictions (i.e. weights) for every municipality because we did not give it a new prediction data set, like we did in the section Weighting Layer (see?predict.randomForest for more info). The "out-of-bag" prediction for a municipality is generated from the regression trees of the model that did not include that municipality in their randomly sampled training data. This provides a cross-validation (a.k.a. out-of-bag, or out-of-sample) assessment.

Plotting the residuals versus the predictions allows us to detect any pattern towards over or underestimation.

# plot residuals (out-of-bag)
plot(x = predict(popfit), 
     y = y_data - predict(popfit), 
     main = 'Residuals vs Predicted',
     ylab = 'Out-of-bag residuals',
     xlab = 'Out-of-bag prediction')

# horizontal line at zero
abline(h=0, col='red')

Important Note: The different goodness-of-fit diagnostics presented here are focused on the training dataset and thus on predicting at the municipality level, but remember that the final goal is to predict at the enumeration area level.

4.6.3 EA-level Assessment

There are two ways to evaluate the disaggregation results at the EA-level:

Comparing to an independent sample of enumeration areas or any other fine-scale spatial units for which we know the population counts, or
Build a test model using aggregated population counts at a coarser resolution (e.g. state-level) and evaluating its predictions at the municipality level.

The first option relies on the availability of an additional dataset. Disaggregating population estimates into 100 m grids rather than enumeration area polygons (as in this tutorial) would provide greater flexibility for aggregating units to match the spatial units of the independent validation data.

The second option is discussed by Stevens et al. (2015). Its effectiveness depends on the size of the aggregated training dataset (e.g. the number of states). Indeed a coarser scale means a decreasing number of input observations, which undermines the quantity of information the random forest can extract.

4.6.4 Covariate Importance

We can look at the influence of individual covariates on model predictions. There are two measures of variable importance available. "Type 1" covariate importance is computed as the difference between out-of-bag mean squared error (MSE) from the full model compared to a model in which the variable is randomly permuted for each tree. If MSE is not affected by random permutations of the variable, then it has low importance according to this metric.

# for covariate importance
varImpPlot(popfit, type=1)

"Type 2" covariate importance is computed as the total decrease in node purity when the variable is used for a split in the regression trees. Node purity is defined as the residual sum of squares among samples that are classified into the same terminal node of a regression tree. If splits in the tree based on this covariate do not decrease the residual sum of squares, then it has low importance according to this metric.

# for covariate importance
varImpPlot(popfit, type=2)

See ?randomForest::importance for more details.

Note: Use caution when interpreting covariate importance. A high importance value indicates that the covariate improves predictive performance, but does not indicate a causal relationship with spatial patterns of population density. Covariate importance scores may also be underestimated for large groups of correlated predictor variables.

5 Limitations

Random Forest can be time-intensive for large datasets.

If running time becomes an issue, there are several solutions that are outlined in the random forest set-up of Bondarenko et al. (2018).

If the model takes too long to be fitted, one can play with the parameter sampsize, which has a great impact on modelling time or ntreeSize. Be careful though as a small number of trees can lead to overfitting. This issue can be clearly visible in any out-of-bag goodness-of-fit metrics.

If the prediction takes too much time, Stevens et al. (2015) proposes to prune the number of covariates based on their importance metric. We have also provided an example of how to generate predictions more efficiently using parallel processing (see Parallel Processing section below).
Random Forest models are not good at extrapolating.

They can only predict values that are in the range of observed values. This issue becomes a problem when the training and prediction inputs differ in range and distribution, which is known as covariate shift. The following plot can help detect a mismatch in covariates distribution between the two scales.
```
# two panel layout
layout(matrix(1:2, nrow=1))

# loop through two covariates
for(cov_name in c('mean.bra_srtm_slope_100m', 'mean.bra_viirs_100m_2016')){

  # combine EA-level and municipality-level values into a single vector
  y <- c(master_predict[,cov_name], master_train[,cov_name])

  # create corresponding vector identifying spatial scale
  x <- c(rep('EA', nrow(master_predict)),
     rep('Municipality', nrow(master_train)))

  # create boxplot
  boxplot(y~x, xlab=NA, ylab=cov_name)
}
```
Figure 5.1: The distributions of two covariates at each spatial scale: municipalities and enumeration areas. The left panel shows similar distributions across scales, but the right panel shows a covariate with significantly different distributions when measured for municipalities compared to enumeration areas.

Notice that the covariate mean.bra_viirs_100m_2016 has a different range of values for EAs compared to municipalities (Fig. 5.1). This covariate measures the intensity of nighttime lights and we would expect its average values to be less in larger municipalities with large unsettled sections compared to small EAs, many of which are densely populated. This covariate was also the most influential on model predictions (see Covariate Importance). We may want to consider re-calculating this covariate using a mask to exclude unsettled areas within municipalities and EAs in an attempt to have a more comparable measure between spatial scales.
Random Forest models are not ideal for inferring covariate effects.

The purpose of the modelling is to have a great predictive performance. We are not aiming at explaining the drivers of human spatial distribution. Therefore no causal effect should be derived from the modelling results.

6 Tips and Tricks

The previous sections demonstrated the basic components of random forest top-down disaggregation, but there are additional tips and tricks that may be useful.

This section utilizes several source files that we did not include with the tutorial because they are large files that are publicly available from their original sources. These source files are:

Enumeration area boundaries for Brazil (IBGE 2020 b),
VIIRS nighttime lights from 2016 (WorldPop & CIESIN 2018 b), and
WorldPop mastergrid for Brazil (WorldPop & CIESIN 2018 a).

Note: The code below will assume that these source files have been downloaded and unzipped (if applicable) into your working directory.

6.1 Map Results

If you want to view the EA-level population estimates on a map (e.g. Fig. 4.2), you will need to join the model results (from section Redistribution to EA-level above) to polygon features for use with GIS software (e.g. Esri polygon shapefile).

For this step, we will need to install and load the sf package (Pebesma 2018) to read and manipulate GIS vector data (census EAs, in this case).

install.packages('sf')
library('sf')

Then, we need to load a vector GIS data set containing the polygons that we would like to join our model results to. For this example, we are using Brazilian census enumeration area polygons which are openly available online as a polygon shapefile (IBGE 2020 b).

sf_polygons <- st_read('BR_Setores_2019.shp')

Note: The st_read function can read most GIS file formats (e.g. Esri shapefile, geopackage, geojson). See ?sf::st_read for more information.

We will add an ID column that matches the enumeration area IDs from the EA_id column of the master_predict data.frame object (continuing from previous sections of this tutorial).

sf_polygons$EA_id <- 1:nrow(sf_polygons)

Then, we can merge the master_predict data.frame which contains our EA-level population estimates into sf_polygons. We will use the EA_id column to match values to the correct enumeration area polygons.

sf_polygons <- merge(sf_polygons, 
                     master_predict,
                     by='EA_id')

Finally, we save the results to disk as an Esri shapefile.

st_write(sf_polygons,
         'master_predict_polygons.shp')

This will save the result as an Esri shapefile because it uses the .shp file extension. If we want to save to a different format, we can simply modify the file extension. For example, we can save the result as a GeoPackage by adding the.gpkg file extension.

st_write(sf_polygons,
         'master_predict_polygons.gpkg')

6.2 Zonal Statistics

This tutorial began with two pre-processed spreadsheets master_train.csv and master_predict.csv. These input data were derived from raster-based covariates using zonal statistics to summarize raster values within each municipality and enumeration area using mean values or other summary statistics (e.g. median, mode, standard deviation, min, max). We will give a brief demonstration of how to do this.

We will rely on the raster and exactextractr packages (Daniel Baston 2020, Hijmans 2020) for this step and so we will first install and load those packages:

install.packages(c('raster','exactextractr'))
library('raster')
library('exactextractr')

We already have polygon features sf_polygons loaded from the previous section (Map Results). We will also need to load a covariate raster. We will use a nighttime lights raster for Brazil that is openly available (WorldPop & CIESIN 2018 b).

raster_covariate <- raster('bra_viirs_100m_2016.tif')

Next, we will calculate a zonal mean within each enumeration area using the object sf_polygons (from section Map Results).

sf_polygons$mean.bra_viirs_100m_2016 <- exact_extract(x = raster_covariate,
                                                      y = sf_polygons,
                                                      fun = 'mean')

Here, we saved the results into a new column of the sf_polygons data.frame called mean.bra_viirs_100m_2016.

Note: You may receive a warning message: Polygons transformed to raster CRS (EPSG:NA). This indicates that the projection for sf_polygons was different than raster_covariate, and that a reprojection was performed internally by exact_extract. This does not affect our results, but we could have avoided the warning message by first reprojecting the covariate raster to match the polygons (see ?raster::projectRaster) or by reprojecting the polygons to match the covariate raster (see ?sf::st_transform).

After we have calculated all of the zonal statistics that we need, we can save the sf_polygons data.frame to disk as a csv spreadsheet.

write.csv(st_drop_geometry(sf_polygons), 
          file = 'EA_covariates.csv',
          row.names = FALSE)

This would be comparable to master_predict.csv that we used as input data for this tutorial. Note that the st_drop_geometry() function removes the polygon geometries from the sf object so that it is a simple data.frame with no spatial geometries (i.e. polygons) associated with each row. We could also save the results as an Esri shapefile to maintain the feature geometries.

st_write(sf_polygons, 'EA_covariates.shp')

Note: The exactextractr README also demonstrates a simple workflow to calculate zonal statistics for Brazilian municipalities.

6.3 Gridded Population Estimates

In this tutorial, we disaggregated municipality-level population totals into finer resolution EA-level population estimates. We can also disaggregate population totals into grid cells (e.g. 100 m, 1 km). It is often easier to work with gridded population estimates because they can be aggregated to any areal units (e.g. census enumeration areas, administrative units, health facility catchment areas) and they may also be easier to compare to gridded data from other sources.

To do this, we need to create a new version of master_predict.csv where each row represents a grid cell rather than an enumeration area. We will start by loading a raster to use as a mastergrid that defines the locations and sizes of grid cells that we want to use.

mastergrid <- raster('bra_level0_100m_2000_2020.tif')

This mastergrid raster for Brazil is publicly available from WorldPop (2018 a). In this mastergrid, any cells that contain a value of NA are outside of Brazil.

Cell IDs for a raster are a numerical sequence starting from 1 in the top-left corner of the raster and increasing left-to-right. There are more than 2.5 billion grid cells in Brazil's mastergrid and more than one billion contain non-NA values. We want to identify the cell IDs for all cells with a non-NA value. Because Brazil contains so many grid cells, we will subset the mastergrid for processing.

cells <- which(!is.na(mastergrid[1:1e7]))

The new vector object cells will contain the cell IDs from the mastergrid that contain non-NA values. We are only processing the first 10 million grid cells (i.e. cells 1 to 1e7) in the mastergrid. We will continue the example using only this subset of the data to demonstrate how to generate gridded population estimates. In practice, we may want to process all of the grid cells at once (if computing power allows) or loop through one subset of cell IDs at a time (e.g. using a for loop) until all cell IDs are processed.

We will now setup a data.frame with a column for the cell IDs.

mastergrid_predict <- data.frame(row.names = cells)

Next, we will use the same covariate raster raster_covariate from the previous section (Zonal Statistics) (WorldPop & CIESIN 2018 b). This raster has the exact same extent and cell size as the mastergrid. Because the two rasters have identical extent and cell size, their cell IDs are comparable allowing us to use our vector of cell IDs cells to extract values very quickly and save them as a new column in the data.frame master_predict.

mastergrid_predict[cells, 'bra_viirs_100m_2016'] <- raster_covariate[cells]

If the extent and cell size of a covariate raster do not match, then we can extract values from the covariate raster using the point coordinates associated with each grid cell rather than using cell IDs, but this is a slower process.

# XY coordinates of each grid cell
xy <- xyFromCell(mastergrid, cells)

# extract raster values at those coordinates  
mastergrid_predict[cells, 'bra_viirs_100m_2016_alt'] <- extract(raster_covariate, xy)

Note: Another option would be to resample the covariate raster to match the mastergrid, but exploring the different options for resampling is beyond the scope of this tutorial (see ?raster::resample for more information).

We can then save the results as a csv spreadsheet.

write.csv(mastergrid_predict,
          file = 'mastergrid_predict.csv',
          row.names = FALSE)

This spreadsheet could be used in exactly the same way as master_predict.csv was used above to generate model predictions (see section Weighting Layer). Because there are so many grid cells in Brazil where predictions are needed we would likely want to use parallel processing to speed up the calculations (see Parallel Processing section below). With either approach, we would generate a new column called predicted_pop in the master_predict data.frame that would contain the population estimates for each grid cell.

After we have these model predictions in tabular form in master_predict, we would then want to rasterize those results. First, we create an empty raster.

raster_predict <- raster(mastergrid)

Then, we insert the population estimates into the raster based on the cell IDs of the predicted values.

raster_predict[cells] <- mastergrid_predict[cells, 'predicted_pop']

If we are processing the cells in different subsets (i.e. using a for loop), we can repeat this step on the same raster using different sets of cell IDs until the raster is fully populated.

Finally, we can save the rasterized results to disk as a geotiff raster.

writeRaster(raster_predict, 
            file = 'raster_predict.tif')

6.4 Parallel Processing

Parallel processing can improve processing speed when the study area contains a large number of spatial units where predictions are required (i.e. grid cells or polygons). For parallel processing, we will split the data into chunks that will be processed simultaneously using all of the processing cores available in our computer. The process that we described above to generate model predictions (see section Weighting Layer) used only a single processing core even if your computer had four or more cores available.

In the example from the previous section (Gridded Population Estimates), we saw that Brazil contained more than one billion grid cells where we would need model predictions. This would likely require parallel processing to generate model predictions efficiently. For a simple and efficient example, we will demonstrate parallel processing using the EA-level data in master_predict that will allow you to explore the process easily. When you are comfortable with the process, it can also be applied to the much larger data set in mastergrid_predict.

For this task we will need the doParallel R package (Corporation & Weston 2020) so we will start by installing and loading that package.

install.packages('doParallel')
library('doParallel')

We will need to create a new R function that will generate model predictions. In the parallel processing steps that will follow, this function will be passed to a processing core along with a chunk of the data to generate predictions.

predict_pop <- function(df, model=popfit){
  
  # EA-level predictions
  prediction <- predict(model, newdata = df)
  
  # back-transform to population density
  density <- exp(prediction)
  
  # calculate weights
  weights <- density / sum(density) 
  
  # disaggregate municipality total to EA-level
  pop <- weights * df$pop_municipality[1]
  
  # result to data.frame
  result <- data.frame(id=df$id, predicted_pop_parallel=pop)
  
  # return result
  return(result)
}

The df argument of the function will be a subset of rows from master_predict and the model argument is the fitted random forest object popfit. This function generates model predictions using the predict function from the randomForest package and then it back-transforms the predicted log-densities to the natural scale. It then calculates weights from those densities and multiplies the weights by the total population for the municipality that each grid cell belongs to (note: these are the same steps as in section Redistribution to EA-level). The results are returned as a data.frame with an "id" column and a column containing the predictions.

Now, we will add an "id" column into master_predict that we will later use to merge the predictions back into this data.frame.

master_predict$id <- 1:nrow(master_predict)

Next, we will break the master_predict data.frame into chunks that will be stored as elements in a list object.

list_master_predict <- split(x = master_predict,
                             f = master_predict$geo_code)

Notice that we have used the geo_code column in master_predict to define the chunks. Remember, that geo_code identifies each municipality, so we will be processing each municipality separately. This simplifies the steps that are required in the predict function (above) because all units processed in a chunk use the same municipality-level total population (i.e. df$pop_municipality[1]). The new object list_master_predict is a list with an element for each municipality that contains a data.frame with a row for each spatial unit within that municipality.

Now, we must setup parallel processing by identifying the number of cores available, assigning them to a processing cluster, and activating the cluster.

cores <- detectCores()
  
cluster <- makeCluster(cores)
  
registerDoParallel(cluster)

Then, we can run the predictions in parallel using this cluster.

predicted <- foreach(i = 1:length(list_master_predict), 
                     .combine = 'rbind',
                     .packages = c("randomForest")) %dopar% 
  predict_pop(df = list_master_predict[[i]])

stopCluster(cluster)

This code will pass each element i from list_master_predict to our predict function predict_pop() and then combine the resulting data.frames using the rbind() function. The last line of code simply deactivates our cluster of processing cores.

The last step is to merge the results back into our original data.frame master_predict using the id column that we setup earlier.

master_predict <- merge(master_predict,
                        predicted,
                        by = 'id')

Note: We have demonstrated parallel processing using the EA-level data in master_predict. It is important to note that EA-level predictions do not take too long to compute using the simpler approach (see section Weighting Layer). We recommend parallel processing only for larger tasks like generating gridded predictions from mastergrid_predict. We did not use this data in our example because it would potentially introduce complications that would distract from the demonstration of parallel processing, but the workflow that we demonstrated should be easily adaptable to accommodate any larger data sets.

Contributions

This tutorial was written by Doug Leasure and Edith Darin from the WorldPop Research Group at the University of Southampton with oversight from Andy Tatem. This work was part of a WorldPop collaboration with the Brazilian Institute of Geography and Statistics (IBGE), United Nations Population Fund (UNFPA), and the Environmental Systems Research Institute (Esri). Attila Lazar and Chris Jochem provided internal reviews that helped improve the tutorial.

Suggested Citation

Leasure DR, Darin E, Tatem AJ. 2020. Small area population estimates using random forest top-down disaggregation: An R tutorial. WorldPop, University of Southampton. doi:10.5258/SOTON/WP00697.

License

You are free to redistribute this tutorial under the terms of a CC BY 4.0 license.

WorldPop, University of Southampton, and their sponsors offer these data on a "where is, as is" basis; do not offer an express or implied warranty of any kind; do not guarantee the quality, applicability, accuracy, reliability or completeness of any data provided; and shall not be liable for incidental, consequential, or special damages arising out of the use of any data that they offer. If users encounter apparent errors or misstatements, they should contact WorldPop at release@worldpop.org.

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Small area population estimates using random forest top-down disaggregation: An R tutorial

WorldPop, University of Southampton

2021-01-20