Nearest Neighbor Analysis

The 2 Distance-based Approaches¶

In this section, we focus on the first approach.

4 examples of point distribution: Clustered or Random?¶

Nearest Neighbor Analysis Calculation Process: In A Nut Shell¶

• Identifying nearest neighbors for every point

• Record the nearest neighbor distances

• Calculate the mean value

  • observed average nearest neighbor distance ($D_O$)

• The average nearest neighbor distance:

  • short: the points are close to each other

  • long: the points are far from each other

How short is short enough to be considered as “clustered”?

Let’s use the Monte Carlo Simulation approach to test the difference between the observed point pattern and a large number of random patterns generated under the CSR assumption.

Option 1: Monte Carlo Simulation¶

• Area size has to match the study area.

  • Area size will affect the distance between points.

• The number of points has to match the dataset.

  • Within the same area, more points (higher density) will lead to lower nearest neighbor distances.

Statistical Testing¶

• After simulating random for M times, we will get $M$ measurement of mean nearest neighbor distances.

• The frequency distribution of the N values would form a normal distribution.

• The range where the normal distribution is located represent the “random (CSR)” under the specific area and # point setting.

(1) If the observed value is far from the random, we found a clear evidence that the observed pattern is different from random.

• either clustered or dispersed

(2) If the observed value fall within the random range, then, we did not find any evidence to prove that the observed pattern is different from random.

• cannot reject the null hypothesis

• we have no choice but to believe it is random

(3) If the observed fall at the edge of the normal distribution

• check p-value and your confidence level setting.

p-value: the probability of being wrong to reject null hypothesis

See [ESRI “What is a z-score? What is a p-value?”(https://pro.arcgis.com/en/pro-app/latest/tool-reference/spatial-statistics/what-is-a-z-score-what-is-a-p-value.htm) for more details]

Let’s say we set the confidence level to 99%, meaning we are willing to accept a 1% risk of being wrong (incorrectly rejecting the null hypothesis, H0). If the obtained p-value is 3%, which is higher than our acceptable risk, then we cannot reject H0.

The grey area indicates the mean nearest neighbor distance of the 10k random pattern.

Option 2: Z-test approach¶

Z-test is another option for testing the significant levels of how observed nearest neighbor distance ($\bar{D}_O$) is different from the ‘expected’ average nearest neighbor distance ($\bar{D}_E$) under CSR.

See ESRI for detail.

The nearest neighbor distance ratio:

If the index ($\text{ANN}$) is less than 1, the pattern exhibits clustering. If the index is greater than 1, the trend is toward dispersion.

The expected mean distance could be calculated using the number of point (n) and the study area (A):

The z-score can then be calculated using $\bar{D}_O$ and $\bar{D}_E$:

The z-score and p-value for this statistic are sensitive to changes in the study area or changes to the Area parameter. For this reason, only compare z-score and p-value results from this statistic when the study area is fixed.

Summary: Nearest Neighbor Analysis (NNA)¶

• This approach calculates the average distance of the nearest neighbor for every point.

• The observed average nearest neighbor distance can be tested using a Monte Carlo Simulation or z-test approaches against the null hypothesis of CSR.

• The NNA has an assumption that the points being measured are free to locate anywhere within the study area (for example, there are no barriers, and all cases or features are located independently of one another).