Histogram binwidth

  • Binwidth has nothing to do with bandwidth in Kernel density.
  • Binwidth simply controls for the size of breaks in the histogram.

\(\hat F\)-function

Empty space distances:

\[d(u, \mathbf x) = \min\{\Vert u - x_i \Vert : x_i \in \mathbf x\}\] from a fixed location \(u \in \mathbb R^2\) to the nearest point in a point pattern \(\mathbf x\) is called the empty space distance or void distance.

The empty space function \(F\):

\[ \hat F(r)= \frac{1}{m}\sum_{j}\mathbf{1}\{d(u_j, \mathbf x) \le r\} \]

The \(F\)-function measures the distribution of all distances from an arbitrary reference location \(u\) (random or evenly distributed) in the plane to the nearest observed event \(u\).

\(\hat K\)-function

\[ \hat K(r) = \frac{1}{\hat\lambda \text{ area}(W)}\sum_i\sum_{i \ne j}\mathbf 1\{\Vert x_i-x_j \Vert \le r\} \]

where \(\hat\lambda\) is the estimated intensity of the point patterns.

Of the distance-based techniques that you have seen so far, \(\hat G\) and \(\hat F\) are often used as complements. The \(\hat K\) is useful when exploring multi-scale patterns.

Simulation to obtain a confidence interval

Given the challenge of solving for the standard deviations of test statistics, we employ simulations to derive a confidence interval.

Activities for today

  • We will work on the following chapter from the textbook:
    • Chapter 16: Activity 7: Point Pattern Analysis IV
    • Chapter 18: Activity 8: Point Pattern Analysis V
  • The hard deadline is Friday, February 14.

Reference