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Jim Kompanek
The Independent Random Process
The below figures represent thirty randomly generated Cartesian coordinates (using the =RAND() command Microsoft Excel) and their associated scatter plots (Figures 1-3). Each cell was randomly generated and represents a value between 0 and 1. This is the fundamental concept behind The Independent Random Process (IRP), which follow the following criteria (Hardisty 2007):
1. Events occur with equal probability anywhere; and
2. The place of occurrence of an event is not affected by the occurrence of other events.
Each of the below figures follow the above criteria and are clearly random; nonetheless, it is human nature to look for patterns where there are none. This concept is clearly demonstrable in the casino, where many people wrongly believe a slot machine may be "hot" or "cold" and a machine that hasn't paid out in a long time is more likely to pay out (where in reality, each pull of the lever stands an equal chance of winning the jackpot). Or for example, if one were to flip a coin ten times and each time got "heads", the odds of the next flip resulting in "heads" is still fifty-fifty.
This being said, even though the below figures are random, they do not necessarily "look" random. For example, Figure 1 has a somewhat empty space in the SE quadrant but a relatively equal distribution everywhere else. Figure 2 has an empty space in the SW quadrant but a cluster of points in the NE quadrant and Figure 4 contains four clusters of points surrounded by empty space. Though some clusters and empty space are visible in each figure, they still random; where each point has an equal chance of being plotted as any other, and the location of each point is not impacted by the location of other points. Because of these two principles, each plot is random even though patterns may appear visible.
Figure 1. Scatter plot generated by Microsoft Excel demonstrating The Independent Random Process.
Figure 2. Scatter plot generated by Microsoft Excel demonstrating The Independent Random Process.
Figure 3. Scatter plot generated by Microsoft Excel demonstrating The Independent Random Process.
The Inhomogeneous Poisson Process
The following figures (Figures 4-6) demonstrate The Inhomogeneous Poisson Process. While in the previous section, each point stood an equal chance of being plotted, in the following figures each point is weighted based on the value entered within 10-x-10 matrix. Although there are no strongly apparent clustering of events in Figure 4 (the unaltered matrix), the majority of events were placed within the NE half of the scatter plot.
With Figure 5, only the 0,0 cell was modified to a value of 500. As a result, there is clearly a cluster of events in this block but the overall trend of the scatter plot still places the majority of the points in the NE half of the plot. Finally, in Figure 6 three separate cells were given the value of 500. This created three distinct clusters of events and weighted the matrix in such a way that the overall NE skewing of the scatter plot was no longer visible. In fact, all of the remaining events appeared to be somewhat clustered together.
Overall, in each of the plots the patterns appear to be random but also somewhat biased based on modifying the weights of the matrix. While in the previous section each event had an equal chance of occurring, in the following Figures the chance of any event was tied to the weight given; although, it should be noted that giving an individual cell an unusually high weight (in this case, 500) does not appear impact the surrounding cells.
Figure 4. Scatter plot generated by Microsoft Excel demonstrating The Inhomogeneous Poisson Process.
Figure 5. Scatter plot generated by Microsoft Excel demonstrating The Inhomogeneous Poisson Process.
Figure 6. Scatter plot generated by Microsoft Excel demonstrating The Inhomogeneous Poisson Process.
A Process with Interaction Between Events
The following figures (Figures 7-9) demonstrate randomly placed events which are "placed at random or placed at a random distance and direction from a randomly selected previous event" (Hardisty 2007). In each of the figures, I experimented with modifying the probability of new seed points, which ranged from 0.001 (highly clustered) to 1 (highly random). The degree of randomness seems to be based off of the probability of the new seed point. The higher the probability, the higher the chance each event will be randomly placed irregardless of the previous point.
For example, in Figure 7 the probability was set to 0.05 which created a cluster of events with a few outliers. The low probability resulted in a random initial event but each subsequent event was likely to be based on the location of the previous point. This is in contrast to Figure 8, where the probability was set to 1, where each point was randomly placed throughout the scatter plot between the defined range. On the other extreme (Figure 9), the probability was set extremely low to 0.001 and subsequently resulted in a strong cluster of points.
The degree of how random each event looks appears to be tied into the probability variable. The higher the value (up to 1), the more random each event appears, as it is less likely to be impacted by the previous result. With a value of 1, the randomness appears somewhat similar to what was generated in the first section of the lesson. In the same manner, the result of clustering/empty space is also tied to this value, with less of each with the higher probability value.
Figure 7. Scatter plot generated by Microsoft Excel demonstrating interaction effects.
Figure 8. Scatter plot generated by Microsoft Excel demonstrating interaction effects.
Figure 9. Scatter plot generated by Microsoft Excel demonstrating interaction effects.
References
Hardisty, Frank
2007 Lesson 3 Cassic Spatial Analysis. Geography 586 Penn State University World Campus Progam in GIS. Available online. Accessed 28 July 07.
This document is published in fulfillment of an assignment by a student enrolled in an educational offering of The Pennsylvania State University. The student, named above, retains all rights to the document and responsibility for its accuracy and originality.
