Rainfall kurtosis vs. HadCRUT4, revised

Patterns of rainfall kurtosis and global temperature.

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.

Regression rainfall kurtosis on HadCRUT4.

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

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Rainfall kurtosis vs. HadCRUT4 Scatterplots

These scatterplots and Connected Scatterplots support a relationship between the kurtosis of annual rainfall at Manilla NSW and the de-trended smoothed HadCRUT4 series of global temperatures.

Scatterplot rainfall kurtosis vs. HadCRUT all data

[SUPERSEDED
This post had inadequated data. It is now superseded by a section in the post “Rainfall kurtosis vs. HadCRUT4, revised” of 20 May 2018.]

The raw data, as observed

The first scatterplot compares (y-axis) all the calculated unsmoothed values of kurtosis of annual rainfall at Manilla, NSW with (x-axis) the unsmoothed values of the HadCRUT4 series of global near-surface temperature at those dates.
[I have plotted rainfall values lagged by five years on all of the scatterplots shown. This lagging makes little difference to the first two scatterplots.]

On this first graph, the fitted linear trend barely supports a positive relation of kurtosis to temperature. The slope is low (1.05) and the R-squared only 0.16. There is an aberrant cloud of points in the top left corner.

Scatterplot rainfall kurtosis vs. HadCRUT detrended (all data)

The raw data, HadCRUT4 de-trended

This graph takes a first step towards a better model for the relationship: the secular trend of the temperature series (that is, the global warming) is removed. For comparison, I have not re-scaled the x-axis.
Although still very weak, the relation is much enhanced. The slope (2.35) is twice as steep and the R-squared (0.24) increased by 50%.

Connected Scatterplot rainfall kurtosis vs. HadCRUT all data

Smoothed data, HadCRUT4 de-trended

This third graph uses smoothed data. The HadCRUT4 series is  “decadally-smoothed” (as published) with a 21-point binomial filter to remove high frequency noise. The rainfall data, already damped by its 21-year sampling window, has been further smoothed with a 9-point Gaussian filter.
This graph is a Connected Scatterplot, that shows the trajectory of the rainfall-temperature relation with the passing of time.

Line chart rainfall kurtosis vs. HadCRUT (detrended)Smoothing both data sets has given a much closer relation. The R-squared value is almost doubled again, to 0.43, and the slope is increased to 3.70. The date labels show that the relation before 1910 was different from that at later dates. (This had also been clear in the Dual axis line chart, copied here, from the post “Rainfall Kurtosis Matches HadCRUT4”.)

Connected Scatterplot rainfall kurtosis vs. HadCRUT from 1908

Smoothed data, HadCRUT4 de-trended, from 1908 to 2002

In this final graph, I have discarded the first eleven years. The linear regression based on smoothed values from 1908 to 2002 has a steep slope of 5.21 and a respectable R-squared value of 0.84.

I had prepared similar graphs for lag values of rainfall kurtosis from zero up to nine. The lag value of five years tends to maximise the slope and the R-squared values.
Choice of a five-year lag tends to form hair-pin loops in the trace, while shorter lags give wider clockwise loops and longer lags give wider anti-clockwise loops.
The lag value of five years implies that the Manilla annual rainfall kurtosis value for a given year matches the de-trended HadCRUT value that occurs five years later.

[Back to the main post on this topic: “Rainfall kurtosis matches HadCRUT4”.]

Rainfall kurtosis matches HadCRUT4

Line chart rainfall kurtosis vs. HadCRUT (detrended)

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.

Line chart rainfall kurtosis vs. HadCRUT
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.

Connected Scatterplot rainfall kurtosis vs. HadCRUT from 1908A 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.

Manilla yearly rainfall history: four momentsI 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.

Kurtosis, Fat Tails, and Extremes

sketch demonstrating kurtosis

PLATYKURTIC left; LEPTOKURTIC right

Why must I explain “kurtosis”?

Manilla 21-year rainfall mediansThe 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.

Histogram of annual rainfall frequency at Manilla NSWThe 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.

Smoothed rainfall frequency and a platykurtic curveManilla’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

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.

A platykurtic curve

A platykurtic curve

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When is the First Frost?

This year (2017) the first frost at Manilla came on the 11th of May, close to the middle date for it: the 13th of May. In just half of the years, the first frost comes between ANZAC Day (the 25th of April) and the 19th of May.

Graphical record of first frost dates
(See the notes below: “Observing Frosts in Manilla.”)

The date of first frost from year to year

The graph shows the dates of first frosts in the last nineteen years. One feature stands out: from a very early date of the 4th of April in 2008, the dates got later each year to a very late date of the 8th of June in 2014. Otherwise, the dates simply jumped around.

Graphical log of frostsThe date of first frost hardly relates at all to the number of frosts in a season. This graph, copied from an earlier post, shows how poorly they match. The earliest first frost, in 2008, was in a year with a normal number of frosts. In the least frosty year, 2013, the first frost did not come late.

The central date and the spread

To find the central value and the spread of a climate item like this calls for readings for a number of years called a “Normal Period”. (See note below on Climate Normals.) I chose the first eleven years of my readings (1999 to 2009) as my Normal Period. For this period I found these five order statistics:

Lowest (earliest) value: 4th April;
First Quartile value: 25th April (ANZAC Day);
Median (middle) value: 13th May;
Third Quartile value: 19th May;
Highest (latest) value: 24th May.

These five values divide the dates of first frost into four equal groups. For example, the first frost comes before ANZAC Day in one year out of four. This could confirm what Manilla gardeners know already!

Is the first frost getting later?

Talk of global warming leads us to expect the date of first frost to get later. By how much?
Dates on the graph after 2009 seem to be later in the season than during the Normal Period. As shown, a linear trend line fitted to the data points slopes steeply down towards later dates in later years. A curved trend line (a parabola) slopes down even more steeply. However, with so few data points, these trend lines are wild guesses, not to be relied on for forecasting future frosts.
Data for NSW from 1910 shows that daily minimum temperatures have been rising at 0.11° per decade.  (That is much faster than the rate for daily maximum temperatures, which is 0.07° per decade.) To work out how this might affect the date of first frost in Manilla, one needs to know that the daily minimum temperature in this season gets lower each day by 0.15°. One day of seasonal cooling will more than cover a decade of climate warming. The effect of global warming is to make the date of first frost only one day later in fourteen years. If the middle date of first frost was the 13th of May in the Normal Period, centred on 2004, the forecast middle date of first frost next year (2018) would be the 14th of May. This is shown by the flattest of the three trend lines on the graph.

Looking ahead, it seems unlikely that the date of first frost will get later by as much as a week within a lifetime.


Notes

1. Observing Frost in Manilla

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Manilla’s Hot Days to June 2015

Log of annual hot days in 16 years This post updates a similar one by including three more years to make a total of sixteen. It is in the same format as a recent post on Manilla’s frosts. Because the summer, which has the most hot days, crosses from one calendar year to the next, I have begun each year at July. I have called days warmer than 35° “hot days”, and days warmer than 40° “very hot days”.

Note added.

I have analysed the pattern of hot days in more detail in a later post “Hot days and ENSO”. By finding the relative frequency of hot days in all of the hotter months, I show that there is a cyclic variation related to ENSO. The cycle period is near 1.5 years, not 3 years as the log of annual frequency of hot days (above) suggests.

Graphical log of hot and very hot days

The first graph is a log of the number of hot and very hot days in each year. The three years with the most hot days had almost the same number: the year ’02-’03 had 41, the year ’09-’10 had 44, and the year ’13-’14 had 43. The two years with the fewest were ’07-’08 which had 5, and ’11-’12, which had only 4. The 13-year average is 26. Counting only the very hot days, ’03-’04 had the most (6), and four years had none at all. On the average, two days exceeded 40° in a year. (These are thirteen-year averages, not updated.) The number of hot days per year seems to have a cyclic pattern, with a period that increases from two years to four years during this short record. This is just a curiosity. The pattern of hot days has a lot in common with the pattern of smoothed monthly temperature anomalies for all months. These are plotted here, on a graph that relates them to ENSO. The relation of Manilla daily maximum temperature to ENSO was quite close from 1999 to 2011, but failed almost completely since mid-2011. In the earlier post on frosts, no cyclic pattern can be seen, nor any relation to ENSO.

New Record hottest days

In the sixteen years, there have now been 37 days hotter than 40 degrees: that is, 2.4 days per year. It remains true that December has fewer very hot days than November or February. A new record was set on 12/1/2013 by a daily maximum temperature of 43.2 degrees, beating the 42.6 degrees of 20/11/2009. This record was broken again on 3/1/2014, with 43.7 degrees. In the latest year, the hottest day (41.1 degrees) ranked only 12th, and it was not in summer, but in November.

Three new annual graphs

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Manilla’s Frosts to 2014

Graphical log of frosts

This post updates a similar one by including three more years to make a total of sixteen.

The Number of Frosts in Each Year

The first graph is a log of the number of frosts in each year. The pattern is different when counting all frosts or only severe frosts.
The log for all frosts had two periods of stable, medium numbers of frosts: from 1999 to 2003, and from 2008 to 2011. Three years had many frosts: 2004 (68), 2006 (70), and 2012 (69). The year 2007 had the fewest frosts (43) until beaten by 2013 (34).
In the logs for severe frosts below minus 2° or minus 4° in the thermometer screen, the drought year 2002 stands out as the most frosty by far. It had the coldest mornings: -5.1° on both the 2nd and 11th of July.

The Last Three Years

Monthly frosts in 2012, 2013, and 2014.

The second graph compares the mean seasonal pattern of frosts with the patterns for the three latest years: 2012, 2013, and 2014.
The frost season of 2012, which almost matched the record 70 frosts of 2006, began early and ended late. May had 13 frosts (like the 14 of 2006) and September had 10 (like the 8 of 2003).
The curve for the season of 2013 ( the new record fewest) was like that of a normal frost season, but lower.
The year 2014 was not very frosty, because the season began late, with no frosts in May and only nine in June.

Monthly frosts each year

Graphs showing the seasonal frost patterns for earlier years are copied here.

 

 

 

 

 

 


There is 2013 reserch on frost in NSW titled “Understanding frost risk in a variable and changing climate” reported here.
It is in GRDC Update Papers (Grains Research and development Corporation). The research is done by Steven Crimp and co-workers at CSIRO Climate Adaption Flagship.
Referring to the period 1960 to 2010,
“Over many parts of NSW the frost season length has broadened by as much as 40 days and the mean number of consecutive frost days has increased to 5 days.”