Building the model

In Supervised machine learning, the response variable is modeled as a function of predictors. To build the model you will need to construct a dataset made of predictors (e.g. elevation, latitude, longitude, soil occupation, aspect) and response variable (e.g. temperature, humidity, rainfall). You will most likely also need to inspect and clean your dataset (e.g. removing outliers, check for errors, treat any missing values) before building the model :

Garbage in, garbage out

Once your dataset is prepared you can build your regression model. Each data-analysis software provides a set of functions to build such a kind of model (in R you do it with the lm() function).

Evaluating the model - Linear Regression Diagnostics

Once the linear model is fitted, the mathematical formula to predict the response variable is obtained. However it is not enough to actually use this model ! Before using a regression model, you have to ensure that it is statistically significant. There are many indicators that you can use to evaluate the validity of a regression model, among which :

• Residual plot
• Goodness of fit
• Standard error of the regression
• p-value

To get a deep insight of these model diagnostic indicators, check these 2 excellent R-oriented posts about evaluating regression model outputs by Felipe Rego and Selva Prabhakaran. You can also have a quick look at this page. You may also have a look at the Minitab’s blog posts (1, 2 concerning the interpretation of the R² values

Using the model and measuring its success (i.e. validation process)

Now that you have tested the validity of your model (i.e. your model is statistically significant), you can use it to make some predictions. But an important question then arises : how well your model performs at predicting the data at unknown locations ? To answer this question, you need to rigorously test your model performance as much as possible. This is done using a cross-validation (CV) process.

From Robin Lovelace’s Geocomputation with R book :

CV determines a model’s ability to predict new data or differently put its ability to generalize. To achieve this, CV splits a dataset (repeatedly) into test and training sets. It uses the training data to fit the model, and checks if the trained model is able to predict the correct results for the test data. Basically, cross-validation helps to detect over-fitting since a model that fits too closely the training data and its specific peculiarities (noise, random fluctuations) will have a bad prediction performance on the test data. However, the basic requirement for this is, that the test data is independent of the training data. CV achieves this by splitting the data randomly into test and training sets. However, randomly splitting spatial data results in the fact that training points are frequently located next to test points. Since points close to each other are more similar compared to points further away, test and training datasets might not be independent. The consequence is that cross-validation would fail to detect over-fitting in the presence of spatial autocorrelation

(To understand the importance of the autocorrelation concept, you could read the Spatial regression models paragraph of the Spatial Data Analysis and Modeling with R website and watch this short video).

To assess how well a model performs at making its predictions actually good predictions, 2 CV methods are often presented :
* (spatial) k-fold cross validation * (spatial) leave-one-out (refs : K. Le Rest & D.R. Roberts)

How to know which one best fits your needs (k-fold or leave-one-out) ? The short answer is to use the leave-one-out method when you have a small amount of samples.

To check if the trained model is able to predict the correct results for the test data, calculating the accuracy measures and error rates allows to find out the prediction accuracy of the model. Paper about spatial predictions errors

In your analysis, you might try many variants of the same kind of modeling technique, for example, by adding or removing extra independent variables. In this case, you will need to establish a diagnostic of the measure of success of each of the variants investigated. There is no universal technique to compare these. You can grab some ideas to implement in your own work from the Aertsen et al. 2010 paper where they describe a multi-criteria decision analysis for model evaluation.

Assessing the uncertainty on the predicted values

If you are totally new to programming with R…

Learning and mastering a new programming language might scare you as it seems as a very difficult goal to achieve. However, with the help of the Internet and the R community, you can quickly start to write your first R programs. You can learn through tutorials (like at Datacamp), blog posts (like the blog aggregator R-Bloggers), package documentation (on CRAN) and of course help forums like Stackoverflow which is where I spend a lot of time searching answers to my questions (most of the time already posted by others).

Why R ?

• R is open-source (why it is important)
• R is gaining more and more popularity. Mastering it can opens you many job opportunities

A good starting point to work with multiple regression analysis with R is this tutorial by Felipe Rego. On his excellent blog you will also find a detailed post about regression model output interpretation.

(A special version of spatial regression modeling is the Geographically weighted regression which is described in this R tutorial written by Pr. Chris Brundson).

Getting started with R for spatial data analysis

In the Geocomputation with R book by Robin Lovelace you will get all you need to get started with spatial data manipulation and analysis with R. The book tutorials make a heavy use of these libraries, and especially the new sf package for spatial data analysis.

library(sf)            # classes and functions for vector data
library(raster)        # classes and functions for raster data
library(spData)        # load geographic data
library(spDataLarge)   # load larger geographic data

Useful R cheatsheets

• dplyr - Data manipulation
• ggplot2 - Data Visualization
• Coordiante reference systems

Worth reading R spatial oriented blog

https://www.r-spatial.org/

Geographic vs projected coordinate reference system (CRS)

• Geographic CRS’s identify any location on the Earth’s surface using two values — longitude and latitude
• Projected CRS’s are based on Cartesian coordinates on an implicitly flat surface. They have an origin, x and y axes, and a linear unit of measurement such as meters. All projected CRSs are based on a geographic CRS, and rely on map projections to convert the three-dimensional surface of the Earth into Easting and Northing (x and y) values in a projected CRS.

Notations systems of CRSs

• EPSG vs. proj4string notations