1- and 2-dimensional Poisson processes

We introduced the idea of a Poisson process as a model for events that happen at random over time. It is also applicable to "events" that arise on other 1-dimensional continua such as flaws in a length of fabric.

We can even generalise to events that happen in a 2-dimensional surface or 3-dimensional volume. For example, 2-dimensional Poisson processes are of great interest in biology where the spatial distribution of animals and plants are examined; events then correspond to the occurrence of an animal or plant at a particular position in the study area. Such processes are also important in geography.

In all such applications, a Poisson process assumes that

A homogeneous Poisson process also assumes that

Poisson distribution

In any homogeneous Poisson process with rate \(\lambda\) events per unit size, the number of events in a period of time (or length or area) of size \(t\) has a \(\PoissonDistn(\lambda)\) distribution.

Location of houses

West of Tokyo lies a large alluvial plain, dotted by a network of farming villages. A researcher analysed the positioning of the 911 houses making up one of those villages. The area studied was a rectangle, 3km by 4km. A grid was superimposed over a map of the village, dividing its 12 square kilometres into 1200 plots, each 100 metres on a side. The number of houses located on each of those plots is displayed in the 30 × 40 matrix shown below.

2221010012000012010122011201111211201202
0201201112201100010102201221210010102012
1011001011101011012020013012102112001022
0111020120002200010012000100010900011111
1200000000102022012101110301201111001031
1310101000002202001001000012111210213111
0100010101201311413101100000002220120301
0010100100130010010010220200121220011001
0110110113113010201000133200001010100010
0000011200152000020021010020001001000120
0200111011102142101220112100001220000000
0001101000012220001013120000021200020111
0100120000000110111121113010110141120102
0001111011000012011113021000020003020112
0110001120010010020001100011100002002100
3411031000200010121001410022000101110440
0010011111100102032022310011013001101110
1101010021002200215200000000100122002101
0301000200020000010200001100200000013001
0110201000001100101000112110000110100212
1000110011100210000130221401001030011010
0211011000110031100101025211012001101200
0000020111200112101003214502111120200101
0011200010011000002001221003311010000010
1011001122110000010021100000110011002002
0011111000221200021000001103001207102002
0111122200203101010010001113101021210001
0210002120000010301100010010002211010110
0000100200000001100110100110111201021011
2001200000100112132000000000100011110210

If houses are located in this village as a homogeneous Poisson process, then these 1,200 counts will be a random sample from a \(\PoissonDistn(\lambda)\) distribution in which \(\lambda\) is the rate of houses per \(10,000\text{ m}^2\).