Monday, May 27, 2013

Poisson Petals

Today is Memorial Day, a day to remember those that have fallen during US military service. It is also the traditional beginning of Summer. But Spring has been hard pressed to give-way to Summer here in Washington. Just two days ago. the temperature struggled upward, but stayed in the 60's (F), [10-15 Celsius]. A cold front kept it cold and rainy all day, much like much of our Spring this year.

Washington's iconic sign of Spring, the cherry blossoms, have long faded and fallen. This picture of fallen blossoms was taken a few weeks ago beneath a cherry tree that stretches like an umbrella above my front walk. What the picture shows is a realization of a spatial Poisson process. Such a random process counts, in continuous time, the number of petals that fall into non-overlapping regions. As the petals randomly land, the number of petals landing in any two separate paving stones are independent of each other. This would indicate that one petal, or its method or path falling from the tree, does not affect any other. The probability distribution of the count of petals on any paving stone depends only on the area of the stone.

I counted (likely with some error) the number of petals on each whole paving stone shown in this image. The mean number of petals on the square stones is 5.58. The mean number of petals on the rectangular stones is 8.71. If these were a result of a Poisson process these means should be proportional to the areas of the stones. The rectangular, larger stones are half again larger than the square ones, that is, the ratio of the areas (rectangular/square) is 1.5. Under a Poisson process we should expect the same for the mean. And sure enough the ratio of the means is 8.71/5.58 = 1.56.


Kevin said...

A pretty picture and a nice comparison of the means, but missing the important next step: is the 1.56 ratio significantly different from the expected 1.5 ratio? Should we reject the null hypothesis that number of petals is proportional to area?

Robert W. Jernigan said...

Your answer is given in the next post. Thanks for the prompting.