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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Moments of Manilla’s 12-monthly Rainfall

Manilla 12-monthly rainfall history: Four moments

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.]

Manilla 12-monthly rainfall history: Mean

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.

Manilla 12-monthly rainfall history: Variance

12-monthly rainfall variance (the second moment)

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April 2018 very warm and sunny

Gum tree

Schoolyard Lemon-scented Gum

Warm spells several degrees above normal persisted until late in the month. Then normal temperature returned.
While no day went over 35°, thirteen days went over 30°, which was a record. ANZAC Day, at 27.3°, was 3° warmer than usual. That was much the same as last year, but not nearly as warm as in 2002 (28.7°).
A record 25 nights were warmer than 10°. There were no frosts, the coldest night (the 29th) being 5.2°.
There were only three rain days, with the highest reading of 10.2 mm on the 20th. The number of cloudless mornings (16) was a new record, beating April 2001 (15).

Weather log for April 2018

Comparing April months

As in March, so in April, this very warm dry month matched the same month in 2016. The three highest mean temperatures for April months were: in 2018, 20.7°; in 2005, 20.6°; and in 2016, 20.5°. For mean daily maximum temperatures, however, 2005 was the warmest, at 29.5°. April 2018 claims the record highest mean minimum temperature of 12.5°, beating April 2014, which had 12.2°.
The rainfall total of 17.8 mm was at the 31st percentile, well below the average (40 mm). In 2018, rainfall has been below normal in January, March and April. However, serious rainfall shortages below the 10th percentile are still seen only in the medium term: the 60-month total of 2780 mm is at the 8th percentile, and the 72-month total of 3370 mm is at the 6th percentile.

Climate in April months


Data. A Bureau of Meteorology automatic rain gauge operates in the museum yard. From 17 March 2017, 9 am daily readings are published as Manilla Museum, Station 55312.  These reports use that rainfall data when it is available.  The gauge, which had last reported on 24 September 2017, came on line again on the 16th of March.

All data, including subsoil at 750 mm, are from 3 Monash Street, Manilla.

3-year trends to April 2018

Hot and sunny

3-year trends to April 2018

April raw anomaly data (orange)

Raw temperature anomaly values for April 2018 were very high for daily maxima, daily minima and subsoil. Rather low moisture was shown by the rainfall and daily temperature range anomalies, while cloudiness was very low (sunny days), but dew point was near normal.

 Fully smoothed data (red)

In the latest fully-smoothed data, for October 2017, while rainfall anomaly continued to move up its graph towards dryness, the other three moisture anomalies (cloudy days, dew point, and daily temperature range) had just begun to move down towards wetness.
For temperatures, both daily maximum anomaly and daily minimum anomaly were rising rapidly. Subsoil temperature anomaly continued the rapid fall from its peak value (normal) in June 2017.


Note:

Fully smoothed data – Gaussian smoothing with half-width 6 months – are plotted in red, partly smoothed data uncoloured, and raw data for the last data point in orange. January data points are marked by squares.
Blue diamonds and the dashed blue rectangle show the extreme values in the fully smoothed data record since September 1999.

Normal values are based on averages for the decade from March 1999.* They appear on these graphs as a turquoise (turquoise) circle at the origin (0,0). A range of anomalies called “normal” is shown by a dashed rectangle in aqua (aqua). For values in degrees, the assigned normal range is +/-0.7°; for cloudiness, +/-7%; for monthly rainfall, +/-14 mm.

 * Normal values for rainfall are based on averages for the 125 years beginning 1883.

March 2018 very warm indeed

Acacia salicina

Young Cooba

Eight days went over 35° this March, beaten only by nine days in March 2016. The 19th (38.6°) was the fifth hottest March day of the new century. Weekly mean temperatures rose to 4.6° above normal by the 18th, and remained almost as high beyond the end of the month.
There were seven rain days, with the highest reading of 16.5 mm (est.) coming early in the month.

Weather log for March 2018

Comparing March months

Average temperatures this month very nearly match those of the record-breaking March 2016. They are only 0.1° lower! Moisture variables are also similar: rather dry, in stark contrast to the sogginess of March 2017.

The rainfall total of 25.6 mm (est.) was at the 40th percentile, well below the average (53 mm). Serious rainfall shortages are seen only in the medium term: the 60-month total of 2770 mm (8th percentile) and the 72-month total of 3410 mm (9th percentile).

Climate in March months


Data. A Bureau of Meteorology automatic rain gauge operates in the museum yard. From 17 March 2017, 9 am daily readings are published as Manilla Museum, Station 55312.  These reports use that rainfall data when it is available.  The gauge, which had last reported on 24 September 2017, came on line again on the 16th of March. However, not all later days have readings reported.

All data, including subsoil at 750 mm, are from 3 Monash Street, Manilla.

3-year trends to March 2018

Hot and rather dry

3-year trends to March 2018.

March raw anomaly data (orange)

March 2018 was more like January than February. Day and night temperature anomalies were high. Moisture anomalies retreated somewhat from very low values. Subsoil temperature moved back from low to normal.

 Fully smoothed data (red)

In the latest fully-smoothed data, for September 2017, temperatures, which had been static, began to increase. On the other had, moisture anomalies, which had been moving up the graphs towards drought, became static.

Earlier, the month of February 2017 marked a sharp reversal of trend in climate anomalies. Daily maximum temperature anomaly peaked. Most anomalies (not rainfall) retraced in March and April the values of January and December. The curvature of the trace (relative to daily maximum temperature anomaly) kept the same sense after April 2107 as before December 2016. For subsoil temperature anomaly, the trend reversal came four months later, in June 2017.


Note:

Fully smoothed data – Gaussian smoothing with half-width 6 months – are plotted in red, partly smoothed data uncoloured, and raw data for the last data point in orange. January data points are marked by squares.
Blue diamonds and the dashed blue rectangle show the extreme values in the fully smoothed data record since September 1999.

Normal values are based on averages for the decade from March 1999.* They appear on these graphs as a turquoise (turquoise) circle at the origin (0,0). A range of anomalies called “normal” is shown by a dashed rectangle in aqua (aqua). For values in degrees, the assigned normal range is +/-0.7°; for cloudiness, +/-7%; for monthly rainfall, +/-14 mm.

 * Normal values for rainfall are based on averages for the 125 years beginning 1883.

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”.