Annual Rainfall Extremes at Manilla NSW: V

V. Extremes marked by high kurtosis

Manilla annual rainfall kurtosis

This graph shows how the extreme values of annual rainfall at Manilla, NSW have varied, becoming rarer or more frequent with passing time.
The graph quantifies the occurrence of extreme values by the kurtosis of 21-year samples centred on successive years.

The main features of the pattern are:
* Two highly leptokurtic peaks, showing times with strong extremes in annual rainfall values. One is very early (1897) and one very late (1998).
* One broad mesokurtic peak, in 1938, showing a time with somewhat weaker extremes.
* Broad platykurtic troughs through the 1910’s, 1920’s, 1950’s, 1960’s and 1970’s, decades in which extremes were rare.
All these features were evident in the cruder attempts to recognise times of more and less occurrence of extremes in Parts I, II, III and IV of this series of posts. This graph is more precise, both in quantity and in timing.

However, kurtosis (the fourth moment of the distribution) does not distinguish extremes above normal from those below normal. It is known that some early dates at Manilla had extremes that were above normal, and some late dates had extremes that were below normal.

Use of skewness

Extremes above normal are distinguished from those below normal by the third moment of the distribution, that is, the skewness.
Manilla Annual rainfall history: SkewnessThe post “Moments of Manilla’s Yearly Rainfall History” shows graphs of the time sequence of each of the four moments, including the skewness (copied here) and the kurtosis ( the main graph, copied above). The skewness function, like the kurtosis function, relates to the most extreme values of the frequency distribution, but to a lesser extent (by the third power, not the fourth).

I have shown the combined effect of kurtosis and skewness on the occurrence of positive and negative extremes in this data set in the connected scatterplot below.

Manilla rain skew vs.kurt

The early and late times of strong extremes were times of strongly positive and strongly negative skewness respectively. As kurtosis fell rapidly from the initial peak (+0.9) in 1897 to slightly platykurtic (-0.4) in 1902, the skewness also fell rapidly, from +0.7 to +0.3.
Much later, in mirror image, values were almost the same in 1983 as in 1902, then kurtosis rapidly rose while skewness rapidly fell, until kurtosis reached +0.9 and skewness -0.3 by 1998.
Between 1902 and 1983, while kurtosis remained below -0.2, the pattern was complex. In the decades of strong platykurtosis (below -0.9) there were extremes of skewness: +0.7 in 1919 and -0.3 in 1968.
Note that the skewness range was as high in times of low kurtosis as in times of high kurtosis, and the same applies to kurtosis range in relation to skewness. Conversely, when either moment was near its mean, the range of the other was not high.

See also:
“Rainfall kurtosis matches HadCRUT4” and “Rainfall kurtosis vs. HadCRUT4 Scatterplots”.


Moments of Manilla’s Yearly Rainfall History

Manilla Annual rainfall history: Four moments

Comparing all four moments of the frequency-distributions

Annual rainfall for Manilla, NSW, has varied widely from decade to decade, but it is not only the mean amounts that have varied. Three others measures have varied, all in different ways.

I based the graph on 21-year sub-populations of the 134-year record, centred on consecutive years. I analysed each sub-population as a frequency-distribution, to give values of the four moments: mean (drawn in black), variance (drawn in red), skewness (drawn in blue) and kurtosis (drawn in magenta).

[For more about the moments of frequency-distributions, see the recent post: “Kurtosis, Fat Tails, and Extremes”. See also the Note below: “Instability in the third and fourth moments.”]

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.

The latest values are on the right. They show that the annual rainfall is now remarkable in all four respects. First, the mean rainfall (black) closely matches the long-term mean, which has seldom happened before. By contrast, the other three moments are now near historical extremes: variance (red) is very low, skewness (blue) very negative, and kurtosis (magenta) very positive.

To my knowledge, such a result has not been observed or predicted, or even suspected, anywhere.

Manilla Annual rainfall history: Mean

The mean yearly rainfall (the first moment)

As I have shown before, the mean annual rainfall was low in the first half of the 20th century, and high in the 1890’s, 1960’s and 1970’s. Rainfall crashed in 1900 and again in 1980.

Manilla Annual rainfall history: Variance

Yearly rainfall variance (the second moment)

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