14 Isotropy
In the previous section, we assumed stationarity, which removes dependence on absolute location.
We now introduce a further structural assumption: isotropy.
14.1 Definition
A stationary point process is said to be isotropic if its distribution is invariant under rotations.
Formally, for any rotation \(R \in SO(d)\),
\[ X \overset{d}{=} R X, \]
where \[ R X := \{ R u : u \in X \}. \]
14.2 Intuition
Isotropy means:
The process has no preferred direction.
- Patterns look the same in all orientations
- There is no directional bias or anisotropy
- Only distance, not direction, matters
14.3 Consequences for Second-Order Structure
From stationarity, we already have:
\[ \lambda^{(2)}(u,v) = \lambda^{(2)}(u - v). \]
Let \(h = u - v\).
Under isotropy, this must be invariant under rotations of \(h\):
\[ \lambda^{(2)}(h) = \lambda^{(2)}(R h). \]
Therefore, \(\lambda^{(2)}\) depends only on the length of \(h\):
\[ \lambda^{(2)}(u,v) = \lambda^{(2)}(\|u - v\|). \]
14.4 Pair Correlation Function
Recall:
\[ g(u,v) = \frac{\lambda^{(2)}(u,v)}{\lambda^2} \]
Under stationarity and isotropy:
\[ g(u,v) = g(\|u - v\|) = g(r), \]
where
\[ r = \|u - v\| \]
14.5 Why This Matters
This is the key simplification:
| Assumption | PCF form |
|---|---|
| None | \(g(u,v)\) |
| Stationary | \(g(u - v)\) |
| Stationary + isotropic | \(g(r)\) |
We have reduced:
- A function of two locations to a function of one scalar distance
This is what makes:
- PCF plots possible
- Minimum contrast feasible
- Your entire CSCP vs LGCP comparison tractable
14.6 Interpretation of \(g(r)\)
Under isotropy, \(g(r)\) has a clean interpretation:
- \(g(r) = 1\) → no interaction at distance \(r\)
- \(g(r) > 1\) → clustering at distance \(r\)
- \(g(r) < 1\) → inhibition at distance \(r\)
Crucially:
These interpretations now depend only on distance, not direction.
14.7 Connection to \(K\)-Function
Under isotropy, the \(K\)-function can be written as:
\[ K(r) = \int_0^r 2\pi s \, g(s) \, ds \quad \text{(in 2D)}. \]
Thus:
- \(g(r)\) is the derivative of \(K(r)\) (up to scaling)
- Both summaries are now 1D functions of distance
14.8 Limitations
Isotropy can fail in many real datasets:
- Directional clustering (e.g. along roads, rivers)
- Geological or environmental gradients
- Wind or flow-driven processes
In such cases, we may need:
- Anisotropic models
- Direction-dependent summaries (e.g. directional PCFs)
14.9 Summary
Isotropy assumes:
- No preferred direction in the process
- Dependence depends only on distance
Combined with stationarity, this yields:
\[ g(u,v) \rightarrow g(r), \]
which is the foundation for nearly all practical second-order analysis in spatial statistics.