Twelve-monthly values of rainfall since 1883 at Manilla NSW yield the four moments of their frequency distributions: mean, variance, skewness, and kurtosis. I plotted the history of each moment (when smoothed) in **an earlier post**.

Here, I compare the moments in pairs. Connected scatterplots reveal the trajectory of each relationship with time.

**Some linear and cyclic trends persist through decades, but none persists through the whole record.**

The first image is an index to the suite of six graphs of pair-wise relationships that I present below.

# Tag Archives: moments

# Rainfall kurtosis vs. HadCRUT4, revised

## The **kurtosis** of annual rainfall at Manilla NSW forms a time-series that matches the time-series of global surface temperature when detrended.

**[REVISED:**

**Earlier posts were based on rainfall data sets that were too small. Estimates of kurtosis and skewness were unstable. ****For details please read “Rainfall kurtosis matches HadCRUT4” and “Rainfall kurtosis vs. HadCRUT4: scatterplots”.]**

## The variables

These two climate variables have little in common. Manilla, NSW, is a single station that has a 134-year record of daily rainfall only. That yields estimates of rainfall kurtosis, an indicator of the relative frequency of extreme values.

HadCRUT4 is one of several century-long estimates of near-surface temperature for the whole world. [See Note below: “Data Sources”.]

## The visual match of the patterns

The first graph (a dual-axis line chart) shows that these two variables have similar patterns of variation over time.

I found the best visual match by:

* scaling 0.5 units of Manilla rainfall kurtosis to 0.1° of detrended HadCRUT4 temperature;

* aligning the kurtosis value of -0.3 units with the zero of detrended temperature;

* lagging the rainfall by two years.

Features that the two patterns have in common are:

* matching main peaks at 1897, 1942 and 2005, each higher than the one before;

* persistent low values in the 1910’s, 1920’s, 1950’s, 1960’s, 1970’s and early 1980’s;

*some matching minor peaks and troughs.

## The correlation chart

The second graph is a correlation chart. The linear regression of kurtosis on detrended temperature has the reasonable R-squared value of 0.67.

As I have made it a connected scatterplot, you can see how the relation has changed through time. From the first data point in 1898 (in **red**) both variables decreased together to the lowest temperature in 1910. Both peaked in 1942, having risen since 1920, later falling until 1955-56. The final rise to the highest peak (2005) was continuous from 1984 for temperature, but the rise in kurtosis was not. It fell slightly in 1990, then remained static until 1998.

All rainfall figures actually came two years earlier. **[See note below: “Manilla’s 2-year lead”.]** The assigned two-year lag not only makes peaks match on the first graph. It sharpens the reversals on the second graph. On a trial connected scatterplot without lag, these reversals had been smooth clockwise curves.

## What it means

### As evidence of extreme behaviour in climate

It is said that more extremes in climate will occur as the world becomes warmer. The evidence is not strong. Most data sets are overwhelmed by noise, and “extreme” is seldom defined with rigor.

In the present case, I believe that the definition of “extreme” that I use is sound: that is, the kurtosis of a frequency-distribution. The instability of kurtosis when based on my small samples had been an issue. In this revision I have increased the sample population size from 21 to 125.

My rainfall data set that displays more and less extreme behaviour is not general but local. It can merely suggest that data elsewhere may reveal functional relationships.

### De-trended global temperature

# Moments of Manilla’s 12-monthly Rainfall

**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)

# Rainfall kurtosis matches HadCRUT4

## The **kurtosis** of annual rainfall at Manilla NSW forms a time-series that matches the time-series of global surface temperature when de-trended.

[**SUPERSEDED**

This post had inadequate data. It is now superseded by the post **“Rainfall kurtosis vs. HadCRUT4, revised”** of 20 May 2018.]

## Features of the data

Data sources, noted on the graph, are specified below. The best match is achieved by decadal smoothing, by scaling 1.0 units of kurtosis to 0.16 degrees of temperature, and by lagging the rainfall data five years.

## Closeness of the match

Although both variables have irregular traces, their patterns are almost the same. They begin and end very high, have a broad peak near 1943, and are low in the 1910’s, 1920’s, 1950’s, 1960’s and 1970’s.

The match is very close for ninety years from 1915 to 2005, except for one decade (at 1972). In all this time, both the values and the slopes (as scaled) agree. [See the Note below “1991-1992”.]

Before 1915, the patterns do not match well, but they remain similar. Both traces descend rapidly together from 1903 to 1910. The initial peak in the rainfall trace at 1903 (actually 1898) is similar in height (as scaled) to a peak of the de-trended temperature trace just off the graph at 1879.

## Discovering the pattern match

I was seeking a robust measure of the occurrence of extreme values in annual rainfall at Manilla, NSW. As kurtosis is just such a measure, I calculated it. I then plotted out the time-series, as shown here. It reminded me of the well-established time-series of smoothed HadCRUT4 global near-surface temperature. In particular, I recalled a locally-dominant peak near 1940.

Simply reconciling the vertical scales of the two time-series gave me the second graph.

While not matching in details, the two curves remain very close from 1940 to 1995. Matching over the whole rainfall record is prevented by a difference in trend. While the rainfall kurtosis has no trend, the HadCRUT4 curve has a secular trend rising at half a degree per century (known as “global warming”).

To extend and improve the match, I subtracted the linear trend from the global temperature curve, and lagged the rainfall points by five years. The first graph is the closely-matching result.

## What it means

### As evidence of extreme behaviour in climate

It is said that more extremes in climate will occur as the world becomes warmer. The evidence is not strong. Most data sets are overwhelmed by noise, and “extreme” is seldom defined with rigor.

In the present case, I believe that the definition of “extreme” that I use is sound: that is, the kurtosis of a frequency-distribution. Only the instability of kurtosis when based on small samples is an issue.

My rainfall data set that displays more and less extreme behaviour is not general but local. It can merely suggest that data elsewhere may reveal functional relationships.

A very strong and persistent empirical relationship is shown by the graphical logs above. In another post, **“Rainfall Kurtosis vs. HadCRUT4 Scatterplots”**, I show scatterplots like this in support of it.

### De-trended global temperature

This strong link between local annual rainfall kurtosis and global climate has a surprising feature. Although this extreme behaviour seems to relate to global temperature, it does not relate to **global warming**! It relates to a temperature trace from which the global warming trend has been removed. Times of high kurtosis, denoting enhanced extremes, correspond to times when the global temperature was highest **above trend**. Such times occurred not only in the twenty-first century, but equally in the nineteenth century. There was another (widely-known), lower peak in de-trended global temperature near 1940: at that time also kurtosis was above normal.

Should global temperature remain static for a time, it would be falling rapidly below its rising trend. According to this data set, that should bring reduced extreme behaviour in annual rainfall at Manilla.

## Data Sources

### (i) Global temperature time-series

From the three available century-long time series of global near-surface temperature I have chosen to use HadCRUT4, published by the British Met Office Hadley Centre. The link is **here**.

I selected from the section: “HadCRUT4 time series: ensemble medians and uncertainties”.

From this, I downloaded two files:

(i) “Global (NH+SH)/2, annual”;

(ii) “Global (NH+SH)/2, decadally smoothed”.

[The “Decadally smoothed” data supplied is annual data smoothed with a 21-point binomial filter.]

From each data file, I used only the first column: the year date, and the second column: the median value.

I established the secular trend of global warming using the linear trend function in Charts for “Excel”. I found the linear trend of the whole HadCRUT4 annual series data (1850 to 2016) to be:

y = 0.005x – 0.52.

I then subtracted the annual value at the trend line from the decadally smoothed HadCRUT4 value to get the de-trended smoothed value shown on the first graph.

### (ii) Kurtosis of Manilla annual rainfall

The rainfall data is that for Manilla Post Office, **Station 055031** of the Australian Bureau of Meteorology. Station 055031 functioned without gaps from 1883 to March 2015. Since then, the official record is fragmentary.

I found kurtosis values for annual rainfall by using the (excess) kurtosis function in “Excel”. I used sub-populations of 21 successive years, referred to the median year. I found values for the years 1893 to 2006. I smoothed these values with a 9-point gaussian filter (yielding similar smoothing to that of HadCRUT4). Smoothing reduced the plottable years to those from 1897 to 2002.

I posted a line graph of this kurtosis data earlier, in **“Moments of Manilla’s Yearly Rainfall History”**.

#### Note: 1991-1992

The most striking match in the graph is that both traces pause at 1991-1992 within a two-decade-long steady rapid rise. That pause in the global temperature series has been attributed with little doubt to the injection into the atmosphere of **seventeen million tonnes of sulphur dioxide** by the eruption of Mount Pinatubo in the Philippines. That eruption cannot have affected the rainfall kurtosis five years earlier.

# Moments of Manilla’s Yearly Rainfall History

## 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.

[**SEE A REVISED VERSION OF THIS WORK**

A revised version of this post uses twelve times as much data. It is **“Moments of Manilla’s 12-monthly Rainfall”** posted on 15 May 2018.]

### 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.

## Yearly rainfall variance (the second moment)

# Kurtosis, Fat Tails, and Extremes

## Why must I explain “kurtosis”?

The annual rainfall at Manilla, NSW has changed dramatically decade by decade since the record began in 1883. One way that it has changed is in the amount of rain each year, as shown in this graph that **I posted earlier**.

**Another way, unrelated to the amount of rain, is in its kurtosis. Higher kurtosis brings more rainfall values that are extreme; lower kurtosis brings fewer.** We would do well to learn more about rainfall kurtosis.

**[A comment on the meaning of kurtosis by Peter Westfall is posted below.]**

## Describing Frequency Distributions

### The Normal Distribution

Many things vary in a way that seems random: pure chance causes values to spread above and below the average.

If the values are counted into “bins” of equal width, the pattern is called a frequency-distribution. Randomness makes this pattern form the unique bell-shaped curve of Normal Distribution.

The values of annual total rainfall measured each year at Manilla have a frequency-distribution that is rather like that. This graph compares the actual distribution with a curve of Normal Distribution.

### Moments of a Normal Distribution: (i) Mean, and (ii) Variance

The **shape** of any frequency-distribution is described in a simple way by a set of four numbers called **moments**. A Normal Distribution is described by just the first two of them.

The first moment is the **Mean** (or average), which says where the middle line of the values is. For Manilla annual rainfall, the Mean is 652 mm.

The second moment is the **Variance**, which is also the square of the Standard Deviation. This second moment says how widely spread or scattered the values are. For Manilla annual rainfall, the Standard Deviation is 156 mm.

### Moments of other (non-normal) distributions: (iii) Skewness, and (iv) Kurtosis

The third moment, **Skewness**, describes how a frequency-distribution may have one tail longer than the other. When the tail on the right is longer, that is called right-skewness, and the skewness value is positive in that case. For the actual frequency-distribution of Manilla annual rainfall, the Skewness is slightly positive: +0.268. (That is mainly due to one extremely high rainfall value: 1192 mm in 1890.)

**Kurtosis** is the fourth moment of the distribution. It describes how the distribution differs from Normal by being higher or lower in its peak or its tails, as compared to its shoulders.

As it was defined at first, a Normal Distribution had the kurtosis value of 3, but I (and Excel) use the convention “excess kurtosis” from which 3 has been subtracted. Then the excess kurtosis value for a Normal Distribution is zero, while the kurtosis of other, non-normal distributions is either positive or negative.

Manilla’s total frequency distribution of annual rainfall has a Kurtosis of -0.427. As shown here (copied from **an earlier post**), I fitted a curve with suitably negative kurtosis to Manilla’s (smoothed) annual rainfall distribution.

## Platykurtic, Mesokurtic, and Leptokurtic distributions

Karl Pearson invented the terms: platykurtic for (excess) kurtosis well below zero, mesokurtic for kurtosis near zero, and leptokurtic for kurtosis well above zero.

The sketch by W S Gosset at the top of this page shows the typical shapes of platykurtic and leptokurtic curves.

(See the Note below: ‘The sketch by “Student”‘.)

In the two graphs that follow, I show how a curve of Normal Distribution can be modified to be leptokurtic or platykurtic while remaining near-normal in shape. (See the note “Constructing the kurtosis adjuster”)

In both of these graphs, I have drawn the curve of Normal Distribution in grey, with call-outs to locate the mean point and the two “shoulder” points that are one Standard Deviation each side of the mean.

### A leptokurtic curve

By adding the “adjuster curve” **(red)** to the Normal curve, I get the classical leptokurtic shape **(green)** as was sketched by Gosset. It has a higher peak, lowered shoulders, and fat tails. The shape is like that of a volcanic cone: the peak is narrow, and the upper slopes steep. The slopes get gentler as they get lower, but not as gentle as on the Normal Curve.