Windowing, not quadrats

The single idea that recurs everywhere here: a hard threshold is a boxcar window, and the boxcar is the worst window there is.

The quadrat problem

Tile space into cells and count points per cell, and your answer swings with the arbitrary size, origin, and phase of the grid — the Modifiable Areal Unit Problem. Two points 1 µm apart but across a cell edge are scored as unrelated; two points in opposite corners of one cell are scored as together. The relationships you care about are exactly what the grid destroys.

In signal terms, assigning each point fully to one cell is a rectangular (boxcar) window, which has the worst sidelobes of any window (≈ −13 dB) — maximal spectral leakage, maximal edge artefacts.

The fix: a tapered, overlapping window

Replace the boxcar with a smooth, overlapping kernel — Gaussian, Hann, Epanechnikov. Overlap recovers the cross-boundary relationships; the taper removes the discontinuity; averaging over overlapping windows makes the result nearly insensitive to grid phase. This is the move from a histogram to a kernel density estimate:

$$\hat{f}(x) = \frac{1}{n}\sum_{i=1}^{n} K_h\!\left(x - x_i\right), \qquad K_h(u) = \frac{1}{h}\,K\!\left(\frac{u}{h}\right)$$

where the boxcar is the degenerate case $K(u) = \tfrac{1}{2}\,\mathbb{1}_{|u|\le 1}$.

Why it unifies the toolbox

A window is the shared operator across both fronts:

• on the image side it is a convolution kernel;
• on the cell side it is the KDE / local-statistic kernel;
• a quadrat is just window(shape = box, overlap = 0) — one (bad) setting of it.

So we make the window a parameter, not a separate family of methods. The same principle reappears in connectivity: a hard “edge if within radius R” graph is a boxcar in distance, and fuzzy adjacency is its tapered version — which is what makes fuzzy TDA tick.

What we have actually measured

This page is an argument, and it is worth being explicit about how much of it has been put to the test, because two different windowing questions live close together here.

Measured. On the neighbourhood weighting inside a statistic — the kernel $K_r$ of a topographical correlation map — replacing the hard disk with a smooth kernel at matched scale barely improves discrimination (all kernels within 0.0005 AUC on an easy scene), but removes almost all of the estimator’s discretisation: 93–99.8 % of cells share a mark value under the disk against ~0 % under any smooth kernel, and the disk is 2–5× more sensitive to the radius. So the boxcar’s cost there is resolution and parameter sensitivity, not power.

Not yet measured. Whether computing a statistic over a grid of quadrat ROIs differs materially from sweeping a window over the domain. That is the claim this page opens with, and it is a different axis — how the domain is sampled, rather than how a neighbourhood is weighted. Settling it needs a per-quadrat baseline, a swept counterpart, and a grid-phase sweep: slide the quadrat origin across its own unit cell and see whether any conclusion flips, not merely whether the spread is wider. Until that is run against published data, treat the opening claim as a position we hold, not a result we have.

Two honest caveats

• The null must follow the window. Significance tests defined on hard counts have to be re-derived as weighted permutation nulls once counts become smooth.
• For pure pairwise co-occurrence, skip the grid. The cross-PCF and Ripley’s K are already windowed in the distance domain; reach for a windowed grid when you want a spatial map, not a function of distance.