**Contents:**

## Overview of Ripley’s K-Function

The **K-Function** (also called *Ripley’s K-function* or the *network K-Function*) is a way to analyze completely mapped event location data (i.e. the locations of all events from a study).

The function is generally used to find out if the phenomenon of interest (e.g. trees,) is clustered, dispersed, or randomly distributed throughout the area of interest. Other uses include:

- Estimating parameters. A parameter is the actual figure from the whole population.
- Fit a model to the data. For example, your data might fit a linear function.
- Perform a hypothesis test about the pattern. Hypothesis tests are common in calculus-based statistics, where you have to “prove” your results are valid with a test.
- Summarize a pattern of points.

In general, the K-function is a summary of **how observed distances between similar events are clustered or dispersed.** If the events are clustered, it tells you the distance where the clustering occurs. It was originally developed by Ripley in 1976.

A comparison of observed vs. expected distances give you the following information:

- If the observed value is higher than the expected value, the events tend to cluster.
- f the observed value is lower than the expected value, the events are dispersed.

## Formal Definition of Ripley’s K-Function

The K-function is denoted as:

**K(t) = λ ^{-1} E,**

Where:

“E” = the number of additional events within distance t of a random event.

Formally, the K-Function is defined as:

Where:

- R = region of interest,
- s = distance,
- d
_{ij}= distance between the*i*th and*j*th events, - I
_{s}(d_{ij}= An indicator function. If d_{ij}≤ s, it equals 1. It equals zero for all other values.

The function gives the **expected number of events** within distance of a randomly chosen event.

## Limitation

Although useful, the function has one very **limiting assumption**: points are assumed to be continuous on a Euclidean plane. This could be problematic if you’re analyzing data that *isn’t *on a Euclidean plane.

As an example, let’s say you’re analyzing traffic flow for a **busy New York intersection**. The points here might be “cars”, and their distances can all be expressed in Euclidean terms: Left, Right, Up, Down (relative to the intersection). However, let’s say your study extended to several city blocks. A car doesn’t travel straight to the next block through building: it takes the only route available: along streets. Those distances are non-Euclidean (i.e. they belong to a type of geometry called *Taxicab geometry*).

## References

Dixon, P. (2002). Ripley’s K-Function. In Encyclopedia of Environmetrics. Volume 3, pp 1796-1803. Retrieved November 28 from: https://www3.nd.edu/~mhaenggi/ee87021/Dixon-K-Function.pdf

Gelfand, A. et al. (2010). Handbook of Spatial Statistics.

Popovich, V. et al. (2015). Information Fusion and Geographic Information Systems (IF&GIS’ 2015): Deep Virtualization for Mobile GIS. Springer.

Ripley, B.D. (1976). The second-order analysis of stationary point processes, Journal of Applied Probability

13, 255–266.