Variability analysis of monthly precipitation vector time series in Australia by a new spatiotemporal entropy statistic

Changing climate in Australia has significant impacts on the country’s economy, environment and social well-being. Addressing such impacts, particularly that of precipitation change, entails immediate action due to the more frequent occurrence of extreme dry or wet events in Australia in recent decades. In this paper we investigate the intra-annual Australian precipitation variability and how it changes over space and time. We quantify this variability using information entropy—a statistical tool for measuring the uncertainty of a random variable over its sample space, and propose a compositional data model to compute optimal spatiotemporal estimators of this entropy using 1/1979-to-3/2022 monthly satellite precipitation estimates from the National Oceanic and Atmospheric Administration. The results enable us to identify those locations/times where/when extreme intra-annual precipitation variation or unevenness occurred. We find this variability has been changing over time in large regions of southeastern Queensland and on the coast of South Australia, which would be difficult to find without using the proposed approach. We uncover the development of extreme entropy in the months leading up to, and in the location of, four extreme precipitation events in Australia where inter-annual precipitation amounts and/or trends proved insignificant. In marked contrast to annual precipitation, we found entropy has a weak association with the El Niño Southern Oscillation cycle.


Introduction
The total annual precipitation and its inter-annual variability have long been regarded as key descriptors of the climatic environment [1], with direct impacts on the occurrence of hazards like heavy rainfall, floods [2], landslides [3], droughts [4] and bushfires [5]. Yet, the 2019-2020 Black Summer Bushfires along with some of Australia's worst recorded flood disasters namely the 2011 Queensland floods, and the 2021 and 2022 Eastern Australia Floods have all taken place in locations with insignificant inter-annual precipitation trends over the past four decades [6]. Understanding these complex and ever changing precipitation patterns is critical to building resilience to climate change [7].
Precipitation is fundamental to many aspects of life. It determines the distribution of water resources for drinking [8], irrigation [9], industry [10][11][12][13] and the survival of ecosystems [14]. Changes in the amount, timing and variability of precipitation can disrupt many natural processes, particularly if these changes occur at rates faster than living species can adapt [15,16]. The risks posed by shifting precipitation patterns due to climate change are growing worldwide with two-thirds of the world's population likely to face water shortages by 2025 [17], impacting agriculture [10,11], health [18], energy [12], infrastructure [19] and manufacturing [13].
Climate change induced variations in the annual precipitation distribution are believed to contribute to the increased frequency [20] and intensity [21] of extreme precipitation events. This paper aims to quantify and better understand the impact of these variations on the location and timing of extreme precipitation events from the perspective of the intra-and inter-annual variabilities of precipitation distribution across Australia. Changes in precipitation distribution present a serious challenge to Australia [22]. As the driest inhabited continent, precipitation variability has significant influence on the aforementioned sectors [23]. While it has been shown that the majority of Australia has stationary annual precipitation [6], in this study we quantify the variability, e.g. concentration or unevenness in each annual precipitation distribution across years and locations. Australia has one of the most variable climates in the world [24], and here we show its precipitation distribution also has significant spatiotemporal variability.
Identifying regions where the trends of intra-annual precipitation distribution variability have had statistically significant changes may inform policy, risk management and investment on building resilience to interconnected climate hazards in a range of areas including: critical infrastructure and transport [19], energy [25], agriculture [26] and tourism [27,28]. For example, regions with high risk of seasonal bushfires often have the majority of their annual precipitation proportions in winter/wet months [29]. An increased proportion of this annual precipitation occurring in winter significantly increases the risk of fire [30]. Early adaptation to changing intra-annual precipitation concentration allows for better mitigation of such extreme precipitation hazards.
The precipitation concentration index was developed to evaluate the degree of daily precipitation concentration (in terms of proportion) over a certain number of days [31]. Other indices have been studied to measure the precipitation variability including concentration, unevenness and uncertainty. In particular, information entropy as such an index has been used to measure the uncertainty involved in monthly precipitation distribution [32][33][34]. Entropy can be explicitly calculated if the underlying probability distribution is known [35], having influential applications in many areas of study, e.g. communication [36], computational sciences [37] and ecology [38]. However, the precipitation entropy cannot be computed because the true distribution of precipitation is mostly unknown. In this study, we propose to optimally estimate the intra-annual precipitation entropy based on a validated Dirichlet distribution for the monthly precipitation proportions within a year. The resultant entropy estimator is a sampling statistic computable from the data, and thus can be used to analyse the spatiotemporal variability in annual precipitation distribution.
The remainder of this paper is organised as follows. In section 2, we introduce the data set before discussing the methodology. In section 3, we show the results of intra-and inter-annual variability across Australia where we uncover extreme precipitation events that cannot be distinguished in the inter-annual precipitation amounts and trends. In section 4 we summarise our key findings and identify future research directions.

Australian precipitation data and an initial analysis
In this study we are using monthly satellite precipitation estimates from the National Oceanic and Atmospheric Administration (NOAA). NOAA is an American government agency responsible for providing weather forecasts, oceanic and atmospheric condition monitoring, etc. The data is provided as a grid with spatial resolution of 1 • × 1 • over Australia, starting in January of 1979 and finishing in March of 2022. An ordinary kriging model is used to interpolate these observations to a higher resolution grid over Australia [39]. Figure 1 shows the mean annual amount of precipitation across Australia. As a country with one of the most diverse climates [24,40], the average annual amount of precipitation is also spatially variable. Moreover, the mean annual precipitation gives an indicator to the climate classification of a given region [41]. For example, Darwin and Melbourne shown in Figure 1 are located in the north and south of Australia respectively, resulting in significantly different annual precipitation totals, at means of 1691 mm and 445 mm respectively.
While annual precipitation gives a good indication of which climate regime the associated location falls into, intra-and inter-annual distributions of the precipitation contain more relevant information on the occurrence, frequency and intensity of extreme precipitation events. Information on such distributions for Lismore-a city on a low flood plain in northeastern New South Wales of Australia, known for its worst ever flood in March 2022-may be retrieved from Figure 2, where time-series plots of annual precipitation and intra-annual monthly precipitation proportions (for January (mid-summer), April (mid-autumn), July (mid-winter) and October (mid-spring), respectively) over years 1979 to 2021 are displayed in the top panel. Also displayed in the bottom panel in Figure 2 are frequency histograms for the corresponding intra-annual monthly precipitation proportions.
We can glean three main points from Figure Figures 2 and 13). In particular, the sample correlations at Lismore are  0.143, 0.389, 0.234 and −0.223 for January, April, July and October, respectively (p-value 0.36, 0.07, 0.13 and 0.15, respectively). (b) Intra-annual monthly precipitation proportions distribution (which will be measured by entropy later in the paper) may vary significantly over years for some regions in Australia, cf Figure 12. (c) Inter-annual distribution of each individual monthly precipitation proportion can be well fit by a Beta distribution.
The above suggest that the intra-and inter-annual precipitation distributions should be analysed based on the 12 within-the-year monthly precipitation proportions. Collection of these 12-proportion sets across all years constitute the compositional data with each set being regarded as a point on a simplex [42]. Moreover, the intra-and inter-annual precipitation variabilities should be optimally estimated based on the entropy of the distribution of the compositional data.

Modelling monthly precipitation proportions
The monthly precipitation proportions within each year, deemed compositional data, require an appropriate multivariate distribution to model. Dirichlet distribution, a multivariate generalisation of Beta distribution [43], is a natural choice for this. Formally, let X i,j (s) be a monthly precipitation proportion for month j ∈ {1, . . . , n = 12} in year i ∈ {1, . . . , m} at location s ∈ S. We say X i (s) = (X i,1 (s), . . . , X i,n (s)) ′ follows a Dirichlet distribution, denoted as where α(s) = (α 1 (s), . . . , α n (s)) ′ is the concentration parameter vector to be estimated from data. The probability density function for X i (s) is on the domain: is the multivariate beta function [43]. The Dirichlet distribution has mean, variance and covariance as Var and Cov respectively, where α 0 (s) = ∑ n j=1 α j (s) [44]. Multiple approaches are available in literature for estimating the concentration parameter vector α(s), e.g. the maximum likelihood method [45]. Here we use a fixed-point algorithm proposed in [46] to calculate the maximum likelihood estimator for α(s). The results will be summarised in section 3.

Variabilities in monthly precipitation proportions
Three types of variabilities in hierarchy may be studied for the within-a-year monthly precipitation proportions X i,j (s)'s. The first type is that in X i (s) for fixed year i and location s, which is named the intra-annual variability. The second one is how the intra-annual variability in X i (s) changes over years for fixed location s, which is named the inter-annual variability. The third one is how the intra-and inter-annual variability changes over locations. This third type is outside the scope of this paper and details will be reported elsewhere. Statistical methods for assessing the first two types will be presented in the remainder of this section, while the real data analysis results obtained from applying these methods are to be summarised and discussed in section 3.

Entropy for describing intra-annual variability
Uncertainty or variability of a probability distribution can be measured by information entropy. Also known as Shannon's entropy, the measure was first developed as a method to quantify the information value of a communicated message [35]. It has since developed beyond communicated information to many areas of study, and is being used for compositional data [47,48].
For a discrete random variable X having n possible outcomes x 1 , . . . , x n and probability mass function (pmf) P(X), the entropy of P(X) is defined as It is easy to see H(P(X)) achieves the maximum log n when X is uniformly or evenly distributed, and achieves the minimum 0 when X becomes degenerate. Hence H(P(X)) provides a good indication of the unevenness of X's distribution, with smaller H(P(X)) values meaning more uneven.
Since the intra-year monthly precipitation proportion set x i (s) as a compositional data vector forms a pmf by itself, the entropy of x i (s) isĤ which measures the intra-annual variability or unevenness of x i (s).
Regarding x i (s) as the observation sampled from year i and location s, the mean and variance for the and is the trigamma function, and e j is a vector of 0's except its jth element equal to 1. Their estimates can be computed using the maximum likelihood estimate of α(s).

Inter-annual trend analysis of intra-annual entropy
Once we have calculated the sample entropyĤ(x i (s)) of the monthly precipitation proportions in each year, at each location, we will be able to analyse how this entropy changes across years, i.e. the inter-annual trend of the intra-annual entropy at each location. This is different from the inter-annual trend analysis of annual precipitation. Linear regression can be used to fitĤ(x i (s)) versus the time (year i) at each location. Significant inter-annual trend ofĤ(x i (s)) will be manifested on those locations where the slope coefficient of the linear regression is statistically significantly positive or negative. The above trend analysis can be slightly modified to test whether the mean intra-annual entropy E[Ĥ(x i (s))] significantly changes over two non-overlapping periods of years, e.g. 1979-2000 and 2001-2021. This is essentially a statistical 2-sample comparison t-or Z-test, which can be performed based on test statistic [H 1 (x(s)) −H 2 (x(s))]/s.e[H 1 (x(s)) −H 2 (x(s))], whereH k (x(s)) is the sample mean ofĤ(x i (s)) over period k with k = 1, 2, and the 's.e' refers to standard error. This test statistic can be shown to asymptotically follow a standard normal distribution N (0, 1) under the null hypothesis of no change in E[Ĥ(x i (s))] across the two periods, whereby the test's p-value can be computed and a conclusion on entropy change can be drawn. Some numeric results obtained from implementing this test procedure for the Australian precipitation data are summarised into Figure 12 in Appendix B.

Extreme entropy
A small value of entropyĤ(x i (s)) indicates large unevenness of monthly precipitation distribution in year i, thereby implying high likelihood of an extreme precipitation event in year i. A parametric bootstrapping method can be used to decide whether or not an intra-annual entropyĤ(x i (s)) value is extremely small. Specifically, for each location s the maximum likelihood estimate of the Dirichlet concentration parameter vector α(s) determines a Dirichlet distribution, from which replications of the monthly precipitation proportions x i (s) are generated. Bootstrap replications ofĤ(x i (s)) are accordingly obtained, wherefrom a level 5% quantile ofĤ(x i (s)), denoted asĤ (0.05) (x i (s)), can be computed. We say the intra-annual precipitation proportions entropy in year i and at location s is extreme at 5% significance level if H(x i (s)) ⩽Ĥ (0.05) (x i (s)). The mathematical detail is skipped here. The data analysis results are to be presented in section 3. Figure 3 shows how the estimates for α 1 (s) and α 7 (s) (January and July respectively) behave for all of Australia. The estimates for the concentration parameters of individual months are not scaled thus not much informative when they are separately looked into. Hence, the mean and variance of the monthly proportions of precipitation have been included for these two months which, as can be seen in equations (3) and (4), combine all concentration parameters and are properly scaled. There is a very high degree of spatial dependency for the values of concentration parameters, the means and the variances. In the northern regions of Australia, the January concentration and mean proportion are significantly higher than what they are in July, due to the tropical nature of the region. By contrast, there is no significant difference in the estimates and proportion means between these two months for the southern more temperate regions of the country. The estimated concentration parameter and the estimated mean of January depend somewhat on the latitude of the location. For July, the estimates for the concentration parameter and the mean also appear to depend on the latitude as well as the location's proximity to the southwest facing coast line. Interestingly, the estimated variance in January is at its largest around the middle of Australia and the variance decreases toward the northern, southern and eastern coasts. The variance is most likely larger in January in this region than in July, due to the likely occurrence of tropical cyclones in January. After hitting the coastal regions of northern Australia, tropical cyclones can continue and affect more inland regions, which in turn can significantly affect the amount of precipitation an arid region may receive.   Figure 10, and how the entropy mean and variance change with location longitude and latitude is displayed in Figure 15, both in Appendix B. From Figure 4's first row one can find similar pattern as shown in Figure 3. That is, there is a lot spatial dependence in both the mean and variance of the entropy. The entropy variance in Figure 4 does not have as strong of relationship with the latitude of the location as the entropy mean, but is similar to the January variance in Figure 3, where it is maximised around the centre of Australia and decreases when moving towards the coastline. Also shown in Figure 4 is a time series comparing the inter-annual precipitation amount to the intra-annual precipitation entropy for Lismore (second row). The inter-annual precipitation amount and the intra-annual precipitation entropy value are not strongly correlated, validating the perception these are two different attributes each providing unique information to the precipitation system of a given region. Figure 4 also shows the intra-annual monthly precipitation proportions across time (third row). Years where the precipitations are lower, yet more evenly distributed yield higher entropy values, whereas years in which one or two months account for large proportion of the annual precipitation yield lower entropy values. The coloured horizontal line indicates the phase of the El Niño-Southern Oscillation (ENSO) for that year (La Niña:red, El Niño:blue, neutral:cyan). It is known the ENSO has an influence on the annual precipitation amount that a location on the eastern coast of Australia receives [49]. In performing a t-test testing difference of two means of annual precipitation between the years of El Niño and La Niña at Lismore, a p-value of 0.009 is obtained. However, when performing this same test but for the entropy at Lismore, we obtain a p-value of 0.7. This indicates intra-annual precipitation proportions entropy is not significantly influenced by the same factors as annual precipitation.  Figure 5 shows the correlation coefficient between the intra-annual precipitation proportions entropy and the annual precipitation amount. Similar results can be found in Figure 14 in Appendix B. Overall, the correlation coefficient is at most moderately positive or moderately negative for almost all regions of Australia. This confirms the premise that annual precipitation amount and the intra-annual precipitation proportions entropy are two different attributes each providing its unique information to the precipitation system. Note that negatively correlated entropy and annual precipitation indicates when the annual precipitation increases, the increase is typically concentrated on the months which already had larger proportions of annual precipitation, rather than uniformly across the year or in the months with less proportion. Conversely, for when there is a decrease in annual precipitation, the decrease is only across a few months. The opposite is true for the locations with positive correlation between the variables. An increase in the annual precipitation typically results in the usually lower proportion months receiving more precipitation. There is no obvious consistent spatial trend for the correlation, and the spatial dependency is not as strong as what was seen in Figure 4. Figure 6 shows a comparison between the locations which have a significant linear temporal trend (significance level of α = 0.05) for the inter-annual precipitation amount (left), and the intra-annual precipitation proportions entropy (right). A location can be classified as one of the following four types.

Temporal trend analysis results
(a) Both the annual precipitation amount, and the intra-annual precipitation proportions entropy across the year are not significantly changing with time (white in both maps). (b) The annual precipitation is not significantly changing with time, but the intra-annual precipitation proportions are becoming more evenly distributed across the year or more concentrated to fewer months (white in the inter-annual precipitation trend map, but coloured in the intra-annual precipitation entropy trend map). (c) The annual precipitation is significantly changing with time, but the intra-annual precipitation proportions entropy across the year is not (coloured in the inter-annual precipitation trend map, but white in the intra-annual precipitation entropy trend map). (d) The annual precipitation is significantly changing with time, as well as the intra-annual precipitation proportions entropy across the year (coloured in both maps).
Approximately 8.5% of the observed locations in Australia have a statistically significant regression trend for the inter-annual precipitation amount, 35% of which is decreasing. This is comparable to the work done in [6] which found approximately 9.5% of Australia had non-stationary trends in annual precipitation amounts using gauge data from 1900 to 2018. However, for intra-annual precipitation proportions entropy, roughly 9.5% of Australia have a statistically significant regression trend, 73% of which is decreasing. We find large regions of south eastern Queensland, and the southern coast of South Australia have significant negative intra-annual precipitation proportions entropy trends, but non-significant inter-annual  precipitation trends. Namely, South eastern Queensland has a decreasing intra-annual precipitation proportions entropy trend, indicating while there is no significant changes in the inter-annual precipitation amount, the annual amount is becoming more concentrated on fewer months. Same can be said to the southern coast regions of South Australia. However, a large region in northern Western Australia has a positive intra-annual precipitation proportions entropy trend, but insignificant inter-annual precipitation trend, indicating that the annual precipitation not changing in total amount is becoming more evenly distributed across the months of the year. Results presented in Figure 6 conform to that in [6], showing there are only a few locations where both the inter-annual precipitation amount and the intra-annual precipitation proportions entropy have significant temporal trends. Climate change may thus be affecting the precipitation dynamics differently for different locations [7,50]. Finally, more detailed results corresponding to that of Figure 6 left and right are plotted in Figures 11 and 10, respectively, in Appendix B. Figure 7 shows the locations where the intra-annual precipitation was shown to have extreme entropy (left), and the locations which were affected by bushfire during 2019/2020 summer (right). New South Wales was one of the most affected states, with a total of 5.68 million ha of burnt land [51,52]. While not all bushfires occurred in locations having small extreme entropy, there are some locations where they do coincide. The Gospers Mountain fire for example, was started by lightning which then moved towards the coastline. In the right panel of Figure 7, the area surrounding the Gospers Mountain experienced small extreme intra-annual precipitation entropy over this time period. This is not to say small extreme entropy caused the bushfires in those locations, but it could be a contributing factor, along with temperature, overall precipitation amount, environmental conditions and human behaviour [53]. Figure 8 shows the locations where the intra-annual precipitation entropy was considered small extreme. Note, in applying the entropy measure to the intra-annual precipitation for a location, we do not have to have the time period to be from January to December in a given year. Using a stretch of any twelve months is acceptable as the full seasonal cycle is captured. Three periods are examined:   Figure 1) and its surrounding areas received significant amounts of rain from November 2010 to January 2011. These regions are considered to have extreme intra-annual precipitation entropy. The 2021 Eastern Australian Floods saw the regions of south eastern Queensland and north eastern New South Wales receive extreme precipitation amounts from March 16 to March 23. These regions again have been considered to have extreme intra-annual precipitation entropy but not always annual precipitation amount. The 2022 Eastern Australian Floods again saw the regions of south eastern Queensland and north eastern New South Wales receive extreme precipitation amounts in late February and early March. The town of Lismore was particularly badly affected by this flooding. Regions further south such as Sydney also experienced extreme precipitation and resulting flooding, but not always extreme annual precipitation amount.

Discussions
Results obtained so far in section 3 suggest the spatiotemporal dependence and dynamics of the Australian precipitation system are associated with not only the inter-annual precipitation amount, but also the intra-annual precipitation distribution entropy, as well as other climate variables such as ENSO. Particularly, use of the intra-annual monthly precipitation proportions entropy provides new insight into the occurrence of extreme precipitation events.
The reason that the entropy of the monthly precipitation proportion set effectively helps on identifying an extreme precipitation event is that precipitation entropy in general gives a measure of total variation, in terms of unevenness, of the distribution of precipitation over a given time period (a year here), and this total variation is clearly related to the occurrence of an extreme precipitation event, e.g. flood or draught. However, this total variation cannot be computed because the true probability distribution of precipitation is unknown to us. Thus, we use the observed monthly precipitation proportions in a year to characterise the annual precipitation distribution, and calculate the entropy of the monthly precipitation proportion set, which is then used to represent the intra-annual precipitation entropy.
A technical aspect troubling understanding the entropy and its relationship with the variation, concentration, or evenness of the underlying random quantity (i.e. precipitation here) is that the entropy achieves the maximum when the precipitation is most variable, or equivalently least concentrated, or most even; and achieves the minimum at the end of the other direction. In other words, the intra-annual precipitation proportions entropy achieves the maximum when all the proportions equal with each other, and achieves the minimum (i.e. 0) when a single one proportion equals 100% and the rest equal 0. It can be readily shown that the maximum precipitation entropy value is log 12 = 2.485 and the minimum is 0 for any 12 monthly precipitation amount data observed in a year. Also the variance of this precipitation entropy has an asymptotic upper bound of 0.25 · log 2 12 = 1.544, asymptotically achieved when half of the observed entropy values equal 0 and the other half equal log 12.
Note that neither the large inter-annual precipitation amount nor the small intra-annual monthly precipitations entropy alone can ascertain the occurrence of an extreme precipitation event. The comprehensive connection between the occurrence of an extreme precipitation event and the intra-and inter-annual precipitation information, as well as with other climate and meteorological variables, is complex and remains to be further explored and determined. In this paper, we contribute to unravelling this connection by examining the development of extreme entropy in the months leading up to, and in the location of, four extreme precipitation events in Australia where inter-annual precipitation amounts and/or trends proved contrastingly insignificant. Consistent with prior findings, we found a strong association between the irregular ENSO cycle and annual precipitation. Intriguingly, we found the converse for entropy which has a weak association with the ENSO cycle. In closing, our findings suggest that the entropy of intra-annual precipitation proportions merits future research attention, especially the extent to which its spatiotemporal dynamics can be used to explain and forewarn of areas of high risk of extreme events.

Conclusion
We set out to quantify the impact of climate change across different parts of the Australian continent by analysing precipitation variability at two temporal scales, viz., inter-annual and intra-annual, using monthly precipitation data from NOAA satellites for the period 1979-2021. Of key interest is whether intra-annual precipitation dynamics can deliver new insights into the emergence of extreme events, which may not be evident from inter-annual precipitation patterns alone. To this end, information entropy was used to quantify the variability in intra-annual monthly precipitation proportions and how it evolves over years. By this approach, interesting results have been obtained in this paper.
We found the impacts of climate change to vary considerably across the continent, as evident in the spatial variability of the entropy mean and variance. The tropical northern regions exhibited low values of entropy mean, while the more temperate southern and eastern regions saw higher values in entropy.
Relatively high values of entropy variance can be observed in the arid central regions of Australia, albeit this generally decreased towards the coast.
A significant temporal trend for annual precipitation amount and intra-annual precipitation distribution entropy can be observed for around 8.5% and 9.5% of Australia, respectively, but these locations are vastly different from each other. Some parts of Australia have seen marked shifts in the entropy of intra-annual precipitation proportions which are masked by the amount and/or trend of the inter-annual precipitation. Some other parts of Australia have seen significant changes of the annual precipitation amount which may or may not masked by the changes of the intra-annual precipitation proportions entropy. Our results suggest extreme entropy of intra-annual precipitation proportions contain useful additional insights to that obtained from inter-annual precipitation, which may shape extreme events such as droughts, bushfires and floods.
Next, we found a strong association between the irregular ENSO cycle and annual precipitation amount, while the intra-annual precipitation distribution entropy has a weak association with the ENSO cycle. More detailed connection between the ENSO cycle and the precipitation system leading to extreme precipitation events will be investigated in our other projects.
Finally, the spatiotemporal entropy statistic developed in this paper has prospects to be used in other areas of atomospheric science, climate change, environmental science, and earth science etc where extreme events under investigation are spatiotemporal and multifaceted, and vary with the probability distribution of the underpinning predictors. Some examples include tropical cyclone genesis study, geohazard events forecasting, and soil carbon farming. Research on this is beyond the scope of this paper.

Data availability statement
The data that support the findings of this study are openly available at the following URL/DOI: www.ncei. noaa.gov/data/global-precipitation-climatology-project-gpcp-monthly/.

Appendix A. Derivation of Dirichlet entropy mean and variance
The expected value of the entropy of a Dirichlet distribution is given by where e j is a vector of 0's except with its jth element equalling 1, and the forth equality comes from finding the expectation of the log of a Dirichlet marginal where ψ (0) (·) is the digamma function. The variance of the entropy of a Dirichlet distribution is given by and and further where ψ (1) (·) is the trigamma function. Thus Figure 9 shows the spatial distribution for theoretical Dirichlet entropy mean and variance from equations (8) and (9) respectively, using the optimally estimated values of the concentration parameters. Figure 10 shows the spatial distribution of the linear regression trend coefficient estimates for the intra-annual precipitation entropy values. Figure 11 shows the locations in which each month has a significant linear model trend coefficient when regressed against the year for the monthly precipitation amount from 1979-2021. Figure 12 displays the locations having significant difference of two entropy means t-test (left) and ratio of variance (right), for the sample entropy between 1979-2000 and 2001-2021. Figure 13 displays the sample correlation coefficients between the location's annual precipitation, and the month's relative proportion of the annual precipitation. Figure 14 shows the sample correlation coefficients between the location's intra-annual precipitation entropy, and the month's relative proportion of the annual precipitation. Figure 15 shows the relationships between the estimated annual entropy mean (first row) and entropy variance (second row) between 1979 and 2021, and the longitude (left) and latitude (right) of the location.