Capturing the time-dependence in the precipitation process for weather risk assessment

Abstract

This work investigates the seasonal, long-term and time-dependence properties of the precipitation process using daily precipitation amount records in Calgary and develops occurrence–amount models which capture the complex properties of the process. Data show that: (a) the probability of precipitation occurrence not only depends on the occurrence of the previous days, but also depends on the amount values. Moreover this dependence varies with season. (b) The expected amount and its volatility depend on the occurrence as well as the amount on the previous days—these dependencies reveal seasonal patterns. (c) There is a strong long-term dependence in Calgary’s data. The proposed models in this work satisfy these properties. A large set of covariates is built to capture the complexity of various properties of the precipitation process. A grouped sequential model selection approach is used to pick the appropriate covariates which works by assigning the predictors to various groups and sequentially exploring them. This framework is assessed by comparing the simulations from the models to the observed data. This confirms a satisfactory agreement.

Appendices

Appendix 1: Model implementation, computation and simulation

As discussed in Sect. 3.1, the proposed model in this work can be estimated using an extension of the GLMs to the time series context via the concept of partial likelihood (Kedem and Fokianos 2002; Hosseini et al. 2011b). The model discussed in this work is implemented with the free statistical software R (https://www.r-project.org/) and the estimation is fast for a single model fitted to our data (less than 0.1 s on a laptop with CPU: 2.9 GHz Intel Core i7, RAM: 8 GB). The parameter estimates for the occurrence and amount components with their standard errors are given in Tables 5 and 6. However the model selection can be computationally intensive because of the large number of covariates (lags, seasonal lags, non-linear lags, etc.). As discussed in Sect. 3.2, there are more than 80 covariates considered here and therefore more than $2^{80} \approx 10^{24}$ models could be potentially compared—which is not computationally feasible. The GSMS approach overcomes this issue by grouping the data into smaller groups and sequentially exploring each group. For example, for a group with 10 covariates, we need to compare only $2^{10}=1024$ models. The model selection task for this work was performed in less than 2 h. One way to make the computation faster is to use parallel computing as for a given group, e.g., with $2^{10}=1024$ subsets, we can partition the power set into 8 partitions of size 128 and use separate cores to fit the models in each partition. By performing parallel computing with 8 cores we reduced the model selection time to less than 20 min on the same computer.

Table 5 Occurrence component parameters’ estimates and standard errors

Table 6 Amount component parameters’ estimates and standard errors

Appendix 2: Application to agro-climate risk

An important application of modeling precipitation series concerns weather risk assessment which plays a central role in many sectors, e.g., energy, agriculture, tourism, aviation and retail industry. Attention to hedging the weather risk using weather derivatives—which are financial instruments whose payoff are based on a specified weather event—has increased significantly from the turn of the century. Temperature weather derivatives are the most commonly used derivatives in the industry and various daily models are suggested in the literature for this purpose. These models usually consider a seasonal trend. Some authors, e.g., Campbell (2005) and Hosseini et al. (2012) use Fourier series for this purpose and Alexandridis and Zapranis (2013) use wavelet networks. The autocorrelation in the temperature series is captured by AR or moving average methods (Caballero et al. 2002; Campbell 2005; Svec and Stevenson 2007; Alexandridis and Zapranis 2013). Caballero et al. (2002) observed that autocorrelation in the temperature series in some cities in the US depends on the season and they suggested fitting a separate model to each season as an ad hoc solution. Hosseini et al. (2012, 2014) also observed this property and developed models with time-varying autocorrelation. Similar issues were seen when dealing with binary weather processes such as the binary temperature series of frost occurrence (below and above zero) and Hosseini et al. (2012) suggested models which capture the seasonal variations in the first- and second-order transition probabilities. This work showed similar time-varying properties of precipitation occurrence and amount and developed time-varying models which capture them. Below we show how these models can be utilized in practice.

We can use simulations from the developed models to calculate the risk of arbitrary hazard events or find the distribution of quantities of interest. An example of a hazard event is: there is a period of 1 month without any precipitation in the coming year. An example of a quantity of interest is: the total precipitation during the coming spring; or the number of wet days during the same period.

The simulation of future precipitation is performed as follows. First we estimate the parameters $\mathbf{\beta }^{o}$ and $\mathbf{\beta }^{a}$ by applying the extension of GLMs which also estimates the dispersion parameter for the amount model as discussed in Kedem and Fokianos (2002) (the dispersion parameter for the first model is equal to 1 and does not need to be estimated). At time $t-1$ to simulate 1 day ahead, i.e., the precipitation amount at time $t{\text {:}}$ (1) we calculate the covariate processes $Z^{o}(t-1)$ and $Z^{a}(t-1)$ which depend only on past observations, (2) we obtain the conditional means for the occurrence and amount, $E\{O(t)| {\mathcal {F}}(t-1)\}$ and $E\{Y(t)=1 | Y(t)>0,\,{\mathcal {F}}(t-1)\},$ using Eqs. 2 and 3, (3) the first conditional mean is the probability of precipitation at time t. We use a Bernoulli distribution to simulate the occurrence value, O(t),  which is 0 with that probability and 1 otherwise, (4) using the dispersion parameter and the conditional mean, the Gamma distribution is specified. Using this distribution, we simulate an amount value, A(t),  (5) the final value is obtained by multiplying O(t) and $A(t){\text {:}}\,Y(t)=O(t)A(t).$ Appendix 1 provides more details about fitting the model, simulating from it and applications in agroclimate risk analysis.

Here we calculate the probability of the following event:

Event A: the total precipitation during the frost-free period during 2007 in Calgary is less than 150 mm and there are no dry periods longer than 2 weeks.

According to Agroclimatic Atlas of Alberta (Huggins 2003; Chetner 2003), the frost-free period in Calgary is considered to be between 20 May 20 and 14 September. In order to calculate the probability of Event A, we simulate 1000 series from the last available date (31 December 2006) to the end of 2007 and we calculate the proportion of the series for which the event occurs. This proportion is 30 %. However due to the parameter uncertainty in the models, this number is an estimate rather than an exact value. To obtain confidence interval for this value, we can use the extension of the GLMs methodology to time series (Kedem and Fokianos 2002) or parametric and non-parametric bootstrap methods. See Hosseini et al. (2011a, 2012) for more details on these approaches for binary series and Hosseini et al. (2012, 2014) for the continuous series of temperature. The 95 % confidence intervals obtained from these various approaches were similar. The non-parametric bootstrap confidence interval is (0.26, 0.33).

We can also use the models developed in this paper to obtain the distribution of quantities of interest, e.g., the longest dry spell in the growing season or frost-free period (Miller 1999). We define two quantities of interest:

Quantity B: longest dry spell in the frost-free period;

Quantity C: accumulated rain amount during the frost-free period.

In Fig. 13 the dashed curves are kernel density estimates fitted to the distribution of the observed quantities between the year 1900 and 2006. We also simulate 100 chains from the model for the same period (1900–2006). Then we build a density curve for each simulation (transparent grey). The figure shows good agreement between the model simulated densities and the data estimated density for both quantities.

Fig. 13

(Left) the dashed curve shows the density of the longest dry spell in the frost-free period estimated from observed data between 1900 and 2006. 100 simulations of the model for the same period are used to estimate 100 density curves denoted in grey. (Right) the dashed curve shows the density of the accumulated precipitation amount (mm) in the frost-free period estimated from observed data between 1900 and 2006 by finding each year’s accumulated rain during the frost-free period and fitting a density curve. 100 simulations of the model for the same period are used to estimate 100 density curves (grey). The grey color is transparent and looks darker where there are multiple grey curves overlapping