Using Analyze of Average Nearest Neighbor prove that Hotelling model of spatial competition is…


Average of Nearest Neighbor (ANN) is a measurement of series of point data which tell us that series of point data is clustered or dispersed. In this article, I will briefly talk about how ANN works and the application of ANN. Lastly, I will prove the Hotelling model of spatial competition by means of ANN.

Introduction of Average of Nearest Neighbor

Before we describe the data is clustered or dispersed, we have to realize how ANN works, the figure shown below is the formula of ANN. It is clear to see that the ANN is DO divided by DE, the formula of DO,DE will be discuss respectively.

Formula of ANN

The D bar of O is observed mean distance between each feature and its nearest neighbor. In formula, D bar of O is :

Formula of D bar of observed mean

The D bar of E is the expected mean features for the given in a random pattern. NOTE: The “A” stand for area of research.

Formula of D bar of expected mean

As a result, ANN is the relation ship of observed and expected mean, and expect mean is random pattern. We could inference that data is completely random scattered if ANN = 1. Plus, if observe mean less than expected mean, it means that the distance in reality is less than random pattern, which means clustered, vice versa.

Dispersed to Clustered

The important thing is: the parameter “A”. The A is stand for area of research. Based on the formula of expected mean, the A play an important role of whole ANN, so it should be pay more attention on it. the figure show the different A cause different ANN.

prone to dispersed/prone to clustered

As we know that the ANN >1 is dispersed; ANN <1 is clustered; ANN =1 is random pattern. We don’t know if the data has enough evidence to describe the data is clustered or dispersed. In the situation, we can use the basic concept of statistic, the null hypothesis.

Base on the null hypothesis, we know that the random is the thing we don’t want to happen, so the null hypothesis is random pattern. In the contrast, the alternative hypothesis is clusteredand dispersed (left and right tailed test respectively). As the figure shown below.

the null hypothesis

Also, we all know that if the the H0 is rejected depends on the Z value, the Z value should be enough big or small. If the Z value bigger than critical value, we describe data as Dispersed, vice versa. The formula of Z is

(x-x bar)/standard error, which is:

Z score formula in ANN

In brief:

If ANN>1 data is dispersed

If ANN=1 data is random pattern

If ANN<1 data is clustered

The higher Z score is, the data is more extremely.

The area has a large impact of ANN.

Application of ANN — Convenience store in Taipei

Now we want to know the ANN of convenience store in Taipei urban area. Taiwan is known for high density of convenience store. The figure shown is the map of convenience in Taipei.

map of convenience store in Taipei city

As a result, before we do the ANN analysis, we can estimate that the ANN would prone to less than 1 and in what degree are we don’t know. Nevertheless, I only want to know the urban area in Taipei city as the mountain region (Beitou) may disturb my ANN .

Now we select the convenience store in the core area, known as CBD, and open the ANN analysis sheet.

Note: leave the “Area” blank and the default area is selected features enclosing rectangle.

ANN analysis in ArcGIS Pro

Now see the report that ArcGIS generate for us :

Report of ANN

We can see that the ANN is small and there is enough statistic evidence to prove that convenience is clustered.

Review of Hotelling model of spatial competition

Imagine that there are two convenience store one road and both of them are locate at the end of road. Both of them acquire half customer on the road.

Both of them acquire half customer on the road

However, the Convenience store A wants more customer, so A move their store closer to B. and acquire more customer.

The B profit decrease, so B move closer to A. To make them acquire equal profit.

After a lot of turns. We can predict that A will near to B at the end of game.

Using concept of ANN to determine if Hotelling model of spatial competition applied

In Taiwan, there are 5 company dominated the market of convenience. I use different color to represent them respectively. the figure are shown below.

Different company of convenience store

Now we know that the Z score of every convenience store in the research area is -4.7 . We want to know is, if we only apply one company’s convenience store, according to Hotelling model, the ANN prone to dispersed or Z score more than -4.7. Take the green company for example:

Green company

The result is, the green convenience store is dispersed. That is make sense because each of convenience store will be cooperate relationship, those convenience store scatter evenly with an eye to maximize profit.

The table below show all of the company’s z-score :

Table : z=score

Hence, according to the table above, we could find out that take five company into consideration may be clustered just like we predict before. However, for every single company, there is no any sign of clustered. We can inference that Hotelling model of spatial competition can apply in Taipei's CBD via the analysis of ANN.

Conclusion and Discussion

In this article, we briefly talk about how the ANN works and the important thing of ANN — research area. Plus, review the concept of Hotelling model of spatial competition, knowing that two firm or store will open very close even near to each other. Finally, we combine Hotelling model and ANN, proving that Hotelling model works in Taipei CBD because that for a single company data is dispersed or random, and take all company into consideration, there is very clustered.

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