Kurtosis, Fat Tails, and Extremes

sketch demonstrating kurtosis


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 Westfal 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 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.


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.”

Manilla’s Hot Days

I have used my 13-year weather record to find the number of hot days in each year and in each month. Earlier I did the same for frosty mornings. 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”.

Total hot days

The first graph shows the number of hot and very hot days in each year. The most hot days were in the year ’09-’10, which had 44; the fewest were in the year ’11-’12, which had only 4. The 13-year average is 26, but the number of hot days is quite different from year to year.

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.
Hot days each year, and seasonal distribution.

Months with hot days

The second graph shows how the number of hot and very hot days peaks strongly in January, with very few earlier than November or later than March. On average, Manilla’s summer has about 22 days warmer than 35°, while spring has 3, and autumn only 1.

The other graphs show how each year had a different pattern of hot days. The highest monthly peaks, each 19 hot days, came in January 2003 (following drought) and January 2007. Annual peaks also came in January in 2008 and 2012, but these peaks were extremely low: only 4 and 3 hot days. Continue reading

Manilla’s Frosts

I have a 13 year record that shows how frosty Manilla is, and how some years are frostier than others. My thermometer is on high ground, so people living where cold air collects will have had more frosts. However, my readings show changes from one time to another. As I do not have a thermometer in the grass, I have recorded a frost when my screen reading is below +2.2°.

There is a matching post for Manilla’s hot days.

Total frosts

The first graph shows the number of frosty mornings in each year. The most frosts were in the years 2004 (68) and 2006 (70); the fewest were in the years 2007 (43) and 2010 (44). The 13-year average is 54.
The graph also shows the number of mornings colder than zero, minus two, and minus four degrees. On the average, these occurred on 26, 7, and 1 mornings per year. For those colder than zero degrees, 2006 was again the frostiest, but 2002 was also very frosty. Counting only the most severe frosts (below minus two or minus four degrees) 2002 was the frostiest year. It had the coldest mornings: -5.1° on both the 2nd and 11th of July.Frosts each year, and seasonal distribution.

Frosty months

The second graph shows how most frosts come in the winter months, especially July, with some frosts in autumn, but few in spring and none in summer. Few come before Anzac Day (25th April) or after Labour Day (first Monday in October in NSW).

The graphs below show that each year was different. The drought year 2002 had the highest number of frosts in a single month: 27 in July – half of all frosts in that year. By contrast, the 70 frosts of 2006 were spread through the months of winter and autumn: there were more frosts in June and August than in July. Continue reading