REVISED, WITH MORE PRECISE DATA
Supersedes the post “Moments of Manilla’s Annual Rainfall Frequency” (15 November 2017). This post includes twelve times as much data.[See Note below: “Data handling”]
Comparing all four moments of the frequency-distributions
Yearly rainfall for Manilla, NSW, has varied widely from decade to decade, but it is not only the mean amounts that have varied. Three other measures have varied, all in different ways.
I based the graph on 125-month (decadal) sub-populations of the 134-year record. I plotted data for every month, at the middle month of each sub-population.
I analysed each sub-population as a frequency-distribution, to give values of the four moments: mean (drawn in indigo), variance (drawn in orange), skewness (drawn in green) and kurtosis (drawn in blue).
[For more about the moments of frequency-distributions, see the post: “Kurtosis, Fat Tails, and Extremes”.]
Each trace of a moment measure seems to have a pattern: they are not like random “noise”. Yet each trace is quite unlike the others.
Twenty-first century values are on the right. They are remarkable in three of the four moments. First, the mean rainfall (indigo) stays near the long-term mean, which has seldom happened before. By contrast, two moments are now near historical extremes: variance (orange) is very low and kurtosis (blue) very positive. Skewness (green) is rather negative.
To my knowledge, such a result has not been observed or predicted, or even suspected, anywhere.
[Note. The main difference from the earlier 4-moment graph based on more sparse data is that skewness does not trend downward.]
The mean 12-monthly rainfall (the first moment)
The first moment of the frequency-distribution of 12-monthly rainfall is the mean, or average. It measures of the amount of rain.
As I have shown before, the rainfall was low in the first half of the 20th century, and high in the 1890’s, 1950’s and 1970’s. Rainfall crashed in 1900 and again in 1980.
12-monthly rainfall variance (the second moment)
The second moment of the frequency-distribution, the variance, measures the scatter or spread of the 12-monthly rainfall values. The pattern has some features in common with that of the mean rainfall: values of the variance fell rapidly at 1900, were low at 1902 and 1912, and very low at 1940. Otherwise, the pattern was different: variance values were extremely high in the 1950’s, and below normal from 1970 onward.
One example of strong disagreement between the first and second moments is that, while the rainfall was very high around both 1953 and 1973, the scatter was extremely high around 1953 but very low around 1973.
12-monthly rainfall skewness (the third moment)
The third moment of the frequency-distribution, the skewness, shows how the curve is lop-sided one way or the other. With positive skewness, the tail on the right side (high values) is heavier, being longer or fatter or both.
Skewness oscillated about the mean value of +0.2, with seven peaks and seven troughs. Negative skewness occurred only near 1907 and from 1967 to 1978.
12-monthly rainfall kurtosis (the fourth moment)
The fourth moment of the frequency-distribution is the kurtosis. As explained in the post “Kurtosis, Fat Tails, and Extremes”, kurtosis measures how frequent (or how rare) extreme values are in the data. With positive values of “Excess kurtosis”, extremes are more frequent than in a normal distribution that has the same variance.
The pattern is stark: there were three peaks, increasing in height, one very early (1895), one in the middle (1941), and one very late (2004). Strongly platykurtic troughs came in 1910, 1920, 1950, and 1980.
[The latest post before this move from annual data to 12-monthly data was “Annual Rainfall Extremes at Manilla NSW: V. Extremes marked by high kurtosis” (14 March 2018).]
Data used are the monthly rainfall totals in millimetres at Manilla NSW for 1621 months of record from March 1883 to March 2018 inclusive.
Twelve-monthly rainfall totals are calculated for every possible month. Each total is plotted against the sixth month of the twelve. Because of the end effect, there are only 1610 12-monthly totals.
Sub-populations of 125 months
Each of the four moments is calculated in the same way. Each uses overlapping sub-populations of 125 12-monthly totals that are only one month apart. Results are plotted at the 63rd month of each sub-population. Because of the end effect, there are only 1486 sub-populations.
Smoothing function 181 months wide
The smoothing function is a Gaussian function with Standard Deviation 30 months (and width at half height of 72 months). It uses 181 monthly data points extending for 90 monthly data points before and after the point plotted. Because of the end effect, there are only 1305 smoothed monthly data points. The first smoothed data point is at April 1896, and the last at January 2005.
Choice of sub-population size and smoothing function width
In the original study that used annual rainfall figures, the sub-population size had been set at 21 years. A larger sub-population would have better stabilised the moment estimates. However, the size was limited by the need for time-resolution and to limit end losses.
Adequate smoothing was achieved using a 9-point Gaussian function (Standard Deviation = 1.65 years), at the cost of further end losses.
Replacing annual rainfall data by 12-monthly rainfall data increased the data density by a factor of twelve. By experiment, I found that the benefit could be shared between stability and resolution.
Rather than keeping the scope of each sub-population at 21 years (now including 252 12-monthly totals), I could reduce each sub-population to 125 12-monthly totals, spanning only 10.4 years. This doubled the time-resolution, and reduced end losses. At the same time, it increased the data density within each sub-population by a factor of six, tending to stabilise the moment estimates.
Adequate smoothing was achieved using a 181-point Gaussian function (Standard Deviation = 30 months (= 2.5 years)), at the cost of further end losses.