Statistical Modeling I

HES 505 Fall 2025: Session 23

Matt Williamson

Objectives

By the end of today you should be able to:

  • Define the likelihood function and its relationship to statistical infrerence.

  • Recognize key assumptions of statistical models and how spatial data may challenge those assumptions.

  • Simulate fake data with known relationships

  • Fit simple linear and generalized linear models to spatial data

Inference for First Order Properties

\[ \begin{equation} z(\mathbf{x}) = \mu(\mathbf{x}) + \epsilon(\mathbf{x}) \end{equation} \]

  • Often we actually care about the “drivers” of \(\mu(x)\)

  • Inference not prediction

  • “Which spatial attributes drive \(z(\mathbf{x})\) ?”

  • Does \(\mu(x)\) increase or decrease with changes in a particular variable?”

Using regression to estimate mu

\[ \begin{equation} z(\mathbf{s}) \sim Distr(\mu, \sigma)\\ \mu = w_0 + \sum_{i=1}^{m}w_iX_i(\mathbf{s}) + \epsilon \end{equation} \]

  • When \(z(s)\) is binary → logistic regression

  • When \(z(s)\) is continuous → linear (gamma) regression

  • When \(z(s)\) is discrete → Poisson regression

  • Assumptions about \(\epsilon\) matter!!

Common Regression Forms

\[ \begin{aligned} y &\sim N(\bar{y}, \bar{\sigma}) &&\quad\text{(linear regression)}\\ y &= w_0 + \sum_{i=1}^{m}w_iX_i(\mathbf{s}) + \epsilon\\ \\ y &\sim \text{Bern}(p) &&\quad\text{(logistic regression)}\\ \text{logit}(y) &= w_0 + \sum_{i=1}^{m}w_iX_i(\mathbf{s}) + \epsilon\\ \\ y &\sim \text{Poisson}(\lambda) &&\quad\text{(Poisson regression)}\\ \log(\lambda) &= w_0 + \sum_{i=1}^{m}w_iX_i(\mathbf{s}) + \epsilon \end{aligned} \]

Key components

  • Distributional assumptions - the likelihood

  • \(w_i\) is ‘spatial weight’, equivalent to \(\beta\) in typical regression

  • link function scales linear model to appropriate support

Estimating parameters

  • You measure \(y\)

  • Your model expresses your hypothesis about the rules governing \(y\)

  • We need estimates of \(w_i\) to complete our rule

Estimating parameters via the Likelihood Function

\[ p(y_i \mid w, \sigma) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\!\left(-\frac{(y_i - w_iX_i(\mathbf{s}))^2}{2\sigma^2}\right)\\ p(\mathbf{y}\mid w, \sigma) = \prod_{i=1}^n \frac{1}{\sqrt{2\pi\sigma^2}} \exp\!\left(-\frac{(y_i - w_iX_i(\mathbf{s}))^2}{2\sigma^2}\right) \]

Estimating parameters via the likelihood function

  • Maximum Likelihood: treats \(w_i\) as fixed, unknown constants. Uncertainty calculated after…

  • Bayesian = \(w_i\) is a random quantity which varies within the constraints of our prior. Uncertainty built into the estimation process.

  • MLE: “What’s the best dart throw?”

  • Bayesian: “Given where darts tend to land, what is the distribution of possible aiming points?”

Deriving model assumptions from the likelihood

  • Shape of likelihood function gives an indication of residual behavior

  • \(\prod\) signals independent observations

  • Moments of the distribution add assumptions (i.e. Poisson mean = variance)

Logistic Regression and Distribution Models

Why do we create distribution models?

  • To identify important correlations between predictors and the occurrence of an event

  • Generate maps of the ‘range’ or ‘niche’ of events

  • Understand spatial patterns of event co-occurrence

  • Forecast changes in event distributions

From Wiens et al. 2009

General analysis situation

From Long

  • Spatially referenced locations of events \((\mathbf{y})\) sampled from the study extent

  • A matrix of predictors \((\mathbf{X})\) that can be assigned to each event based on spatial location

Goal: Estimate the probability of occurrence of events across unsampled regions of the study area based on correlations with predictors

Modeling Presence-Absence Data

  • Random or systematic sample of the study region

  • The presence (or absence) of the event is recorded for each point

  • Hypothesized predictors of occurrence are measured (or extracted) at each point

From By Ragnvald - Own work, CC BY-SA 3.0

Logistic regression

  • We can model favorability as the probability of occurrence using a logistic regression

  • A link function maps the linear predictor \((\mathbf{x_i}'\beta + \alpha)\) onto the support (0-1) for probabilities

  • Estimates of \(\beta\) can then be used to generate ‘wall-to-wall’ spatial predictions

\[ \begin{equation} y_{i} \sim \text{Bern}(p_i)\\ \text{link}(p_i) = \mathbf{x_i}'\beta + \alpha \end{equation} \]

From Mendoza

Comparison with Machine Learning

  • Statistical models describe a generative process: how’d the data get here

  • Machine learning models ask something different

  • Underlying modes of prediction