Gamma distribution

18 3.2. PARAMETRIC MODELS

curve becomes more symmetrical. This is simply illustrated on the graph3.1ausing a variety of shape parametersαwith given scale parameterβ=1.

In contrast, changing the value of scale parameterβdoes not affect the shape of the distri-bution. Instead, the scale parameterβ describes how spread out the distributions are. The larger the scale parameterβ, the more spread out the distribution is. The smaller the scale parameterβ, the more compressed the distribution. This is illustrated on the graph 3.1b using a variety of scale parametersβwith given shape parameterα= 1.

For the shape parameterα≤1, the density function has a maximum atx= 0and is strongly skewed, and then it reduces to the exponential distribution atα= 1, which is in fact a special kind of Gamma distribution in whichα= 1. For larger values of the shape parameterα, the density function exhibits a single maximum atx = β(α−1)and still continues to exhibit skewness until the shape parameterαis quite large (Wilks,1990).

(a) Parameter of scaleβ=1 (b) Parameter of shapeα=1

Figure 3.1: Probability of density function of Gamma distribution with a variety of parame-ters combinations.

The mean and the variance of the Gamma distribution can be calculated using the following formulas, respectively.

E(X) =αβ (3.3)

V ar(X) =αβ² (3.4)

The Gamma distribution is an attractive model, especially for precipitation applications, due to several reasons (Husak et al.,2007;Wilks,1990). First, the Gamma distribution is rel-atively straightforward utilizing only two parameters (namely, the parameters of shape and scale), but it offers a flexible representation of a variety of distribution shapes. The Gamma

20 3.2. PARAMETRIC MODELS distribution ranges from exponential-decay forms for shape values near one to nearly nor-mal forms for shape values beyond 20. This flexibility allows for the Gamma distribution to be fit to any number of precipitation regimes with reasonable accuracy, while other distri-butions may fit only a single rainfall regime. Second, the Gamma distribution is bounded on the left at zero, and it is capable of mimicking non-negative precipitation amounts in a reasonable way. This is an important point to avoid negative values, especially for precipi-tation events with high variability and low mean values. Finally, the Gamma distribution is mainly characterized by its positively skewed characteristic because it has a long tail to the right of the distribution. Thus, non-zero probability of extremely high precipitation amounts can still be captured by using Gamma distribution. Besides, the Gamma distribution param-eters can be scaled to describe precipitation amounts at a variety of time-scales, from hourly to monthlyIson et al.(1971).

Furthermore, parameters of the Gamma distribution can be used to describe precipitation amounts in a variety of applications, such as the mean and the standard deviation (Husak et al.,2007). However, interpretation of the distribution parameters of the Gamma tion requires some understanding of the distribution properties. Unlike the normal distribu-tion, where a parameter, such as the mean or the standard deviadistribu-tion, can directly provide an intuitive understanding of some aspect of the distribution, the Gamma distribution requires both shape and scale parameters to be interpreted together. For example, precipitation oc-currences with similar shape values, but different scale values, might have very different probability density functions than other precipitation events. Understanding of the distri-bution of the parameters of shape and scale is, therefore, of importance when describing precipitation events reasonably.

In many practical applications, the simultaneous interpretation of the parameters of shape and scale is challenging. The linear relationship between both parameters is not obviously found and not easy to be interpreted together. In this study, however, both parameters, which are obtained by fitting the Gamma distribution to the empirical distribution of pre-cipitation amounts at a given time step for all selected prepre-cipitation occurrences, are scaled in the same unit ranging [0,1] on the uniform distribution using the rank transformation.

This procedure will be discussed in detail in the section of copulas as given in section 4.2.

As mentioned, the selected precipitation occurrences used in this study are the precipitation values on a given time step at which more than 0.7 of all gauge stations in the region of interest are wet with a precipitation depth of more than 0.1 mm.

Interestingly, overall, there is a reasonably good visual agreement of linear dependencies be-tween the parameters of shape and scale describing precipitation occurrences as presented in Figure3.2. Figure3.2shows multiple scatter plots between the parameters of shape and scale from the Gamma distribution at a variety of time scales from all time periods of differ-ent seasons in Singapore and Bavaria. The units are ranked variables scaled on the range (0, 1) using a rank transformation. In Singapore, (Figure3.2a), there is a strong linear depen-dency between shape parameters and scale parameters across temporal scales and seasons, but it is a negative linear relationship, which is an consequence of the fact that the product of both shape and scale must equal the mean. If the scale parameter decreases then the shape parameter must increase, in order to preserve a constant mean (Husak et al.,2007).

(a) Singapore

(b) Bavaria

Figure 3.2: Scatter Plots of parameters of Gamma distribution (shape and scale) at a variety of time scales in different seasons in the regions of Singapore (Fig a) and Bavaria (Fig b).

22 3.2. PARAMETRIC MODELS In this case, a high shape value means that the precipitation occurs more spatially symmet-rically distributed, implying that the probability of ”drier” areas than the average is approx-imately equal to the probability of ”wetter” areas than the average. In general, all spatial observation points are typically received consistent precipitation amounts. In addition, it also results in less variance in the distribution function. In contrast, a large-scale value means that the variance is quite large in comparison to the mean resulting in a more posi-tively skewed distribution function (Husak et al.,2007). Similar to Singapore, there is also a negative linear dependency between the parameters of shape and the parameters of scale across temporal scales and seasons in Bavaria but more scattered as shown in Figure3.2b.