Spatial Statistics

Give data a location and something subtle happens: the standard statistical toolkit quietly stops being valid. Most methods assume your observations are independent — but places near each other are emphatically not independent. Adjacent suburbs have similar incomes, neighbouring regions similar weather, nearby areas similar crime rates. Spatial statistics is the branch built for data where geography matters, and its whole purpose is to take that spatial dependence seriously rather than pretend it away.

It's distinct from the GIS page — that one is the tools for handling spatial data; this is the inferential statistics of location: how to measure spatial pattern, test whether a cluster is real, and predict across space. This page is the core ideas, and the recurring theme is that ignoring "where" produces confident, wrong answers.

Why geography breaks the rules

Nearly every method in classical statistics rests on an assumption of independent observations — that knowing one data point tells you nothing about the next. Spatial data violates this flagrantly: knowing one suburb's value tells you a lot about its neighbours'. This spatial autocorrelation means your effective sample size is smaller than it looks (nearby points carry redundant information), so ordinary analyses report over-confident results — significance that isn't there, correlations inflated by shared location.

So spatial statistics does two things: it measures the spatial dependence (is there a pattern, and where?), and it accounts for it in models so the conclusions stay honest. Both start from one foundational idea.

The first law of geography

Tobler's first law of geography states it plainly: "everything is related to everything else, but near things are more related than distant things." That's the engine of the whole field. Spatial autocorrelation can be positive (the usual case — similar values cluster together, like wealth or temperature) or, more rarely, negative (high values systematically next to low ones, like a checkerboard). The goal of the measures below is to detect and quantify which is happening, and where — turning a vague impression of "that looks clustered" into a testable claim.

Defining 'neighbours'

Before you can measure spatial relationships, you must formalise what counts as "near." That's the spatial weights matrix — for every pair of locations, a weight saying how connected they are. The common choices: contiguity (regions that share a border are neighbours), distance (everything within k kilometres), or k-nearest (each area's closest k others).

Measuring clustering: Moran's I

The standard global measure of spatial autocorrelation is Moran's I. It's essentially a correlation coefficient for space: it asks whether a location's value tends to match its neighbours' values, across the whole map.

I = \frac{N}{\sum_{i}\sum_{j} w_{ij}} \cdot \frac{\sum_{i}\sum_{j} w_{ij}\,(x_i - \bar{x})(x_j - \bar{x})}{\sum_{i} (x_i - \bar{x})^2}

You don't need to memorise the formula — read its behaviour. Moran's I runs roughly from −1 to +1: a value well above 0 means positive autocorrelation (clustering — similar values near each other), near 0 means a random spatial pattern, and below 0 means dispersion. Crucially you test it for significance (against the null of a random arrangement), so you can say whether an apparent cluster is real or could easily be chance — the difference between spotting a genuine pattern and seeing faces in clouds.

What Moran's I detects. Left: clustered — high values (filled) sit together, high I, positive autocorrelation. Right: random — no spatial structure, I near zero. The statistic, with a significance test, tells these apart instead of relying on the eye.

Finding hotspots: LISA

Moran's I gives one number for the whole map — but usually the interesting question is where the clusters are. LISA (Local Indicators of Spatial Association) decomposes the global statistic into a value for each location, revealing the local structure. It classifies each area as part of a:

High-High cluster — a high value surrounded by high values: a genuine hotspot.
Low-Low cluster — a coldspot (low among low).
High-Low / Low-High — spatial outliers: a value that bucks its surroundings (a high-crime pocket in a safe region), often the most interesting cases of all.

This is the workhorse of hotspot analysis — finding the statistically significant concentrations, not just the eye-catching ones, which is exactly what you need before acting on "where is the problem worst?"

Spatial regression: honest models

When you model a spatial outcome (does income predict health across regions?), ordinary regression is invalid if the residuals are spatially autocorrelated — the independence assumption is broken, and the standard errors lie. Spatial regression fixes this by building the spatial structure into the model:

Spatial lag models — include neighbours' outcome values as a predictor, capturing genuine spillover (a region's value is shaped by its neighbours').
Spatial error models — account for spatially correlated unobserved factors in the error term, so the inference stays honest.

Either way, the point is the same as the causal discipline elsewhere: respect the data's structure or your confidence is fake.

Predicting between points: kriging

The last big tool is spatial prediction. You've measured a value at scattered locations (rainfall at weather stations, a pollutant at sample sites) and want to estimate it everywhere in between. Kriging does this, and it's smarter than naive interpolation: it uses the measured spatial autocorrelation structure — how quickly similarity decays with distance — to make the statistically optimal prediction at each unsampled point, complete with an uncertainty estimate.

It's the spatial sibling of the forecasting ideas on the time-series page (interpolating across space rather than extrapolating through time), and it's how you turn a handful of sample points into a continuous, honest surface — with the crucial caveat that the uncertainty grows the further you are from any real measurement.

Where it shows up in my work

Real hotspots, not mirages

Plenty of government analysis is inescapably spatial — incidents, regional patterns, where to direct resources — and the single most valuable thing this discipline gives me is the difference between a statistically significant hotspot and an eye-catching cluster that's just noise. Moran's I tests whether there's real spatial structure at all, and LISA pins down where the genuine hotspots (and the telling outliers) are — which is what you need before acting on "where is it worst?"

It also keeps the modelling honest: treating spatially-dependent data as if it were independent produces over-confident conclusions, so spatial regression is what stops a regional analysis from overstating its certainty — the same respect-the-structure lesson as causal inference. It complements the GIS page (tools vs inference) and shares the MAUP caution about how the choice of areal units shapes everything.

Refresh in 60 seconds

Spatial data breaks the independence assumption — nearby places are related (spatial autocorrelation), so naive stats are over-confident.
Tobler's first law: near things are more related than distant things. Positive autocorrelation (clustering) is the common case.
Define neighbours via a spatial weights matrix (contiguity / distance / k-nearest) — a real modelling choice.
Moran's I = a correlation for space (well above 0 = clustering; ≈0 = random), tested for significance. LISA finds where — High-High hotspots, Low-Low coldspots, and outliers.
Spatial regression (lag/error models) builds the structure in so inference stays honest. Kriging predicts between sampled points using the autocorrelation, with uncertainty.
The throughline: respect "where" or get confident, wrong answers (and mind the MAUP).

Tobler's law, the Moran's I / LISA hotspot framing, spatial weights, and kriging reflect current spatial-statistics references alongside statistics coursework.