# How Useful Are Strain Rates for Estimating the Long-Term Spatial Distribution of Earthquakes?
## 2 Method and Data  
### Multiplicative Hybrid Model  
We follow the method used by [PERSON] et al. in [6; 10; 11] to produce hybrid earthquake likelihood models from a baseline model and a set of covariates. As in [10; 11], the stationary uniform Poisson (SUP) model is used as the baseline model, and subsets of the available spatially varying covariates are assimilated into it to form a range of hybrid models. The hybrid and baseline models are regional earthquake likelihood models [8] defined by a set of expected numbers of earthquakes \(\{\lambda(j,k)\}\) in cells of location, indexed by \(j\), and magnitude, indexed by \(k\), where \(j=1,\,\ldots,\,n_{j}\), and \(k=1,\,\ldots,\,n_{k}\). The baseline model is denoted by \(\{\lambda_{0}(j,k)\}\), and the hybrid model by \(\{\lambda_{H}(j,k)\}\). The \(n_{i}\) covariates take values in the location cells only and are denoted by \(\{p_{i}(j),j=1,\,\ldots,\,n_{j};i=1,\,\ldots,\,n_{i}\}\). The covariates may be real-valued or categorical (e.g., binary) variables. The multiplicative hybrid model is of the form  
\[\lambda_{H}(j,k)=\lambda_{0}(j,k)\exp(a)\prod_{i=1}^{n_{i}}f_{i}[\rho_{i}(j)]. \tag{1}\]  
Here, \(a\) is a normalising parameter that ensures the total expected numbers of earthquakes in the hybrid model match with the actual number in the catalogue to which it is fitted. The function \(f_{i}\) converts a cell value of the \(i\)th covariate into a multiplier to be applied to the corresponding cells of the baseline model. For a real-valued covariate, the multiplier \(f_{i}\) is a non-negative, monotone, non-decreasing and nonlinear function of the form  
\[f_{i}[\rho]=\exp(b_{i}[\ln(1+\rho)]^{c_{i}}),\,\,(b_{i}\geq 0;c_{i}\geq 0). \tag{2}\]It thus preserves the ordering of values in cells but not the ratio of values between cells. For a binary covariate, where \(\rho(j)\) takes only the values \(0\) and \(1,f_{i}\) is of the form  
\[f_{i}[\rho]=\exp(d_{i}\rho), \tag{3}\]  
where adjustable parameters \(d_{i}\) are allowed to take positive or negative values. The adjustable parameters in the hybrid model are the normalising parameter \(a\), the shape parameters \(b_{i}\) and \(c_{i}\) for each real-valued covariate, and \(d_{i}\) for each binary covariate. In fitting a hybrid, the parameters are chosen to optimize the log-likelihood, In \(L\), of the target earthquakes, given by [PERSON] et al. [15] as  
\[\ln L=\sum_{n=1}^{N}\ln\lambda_{H}(j_{n},k_{n})-\hat{N}_{H}, \tag{4}\]  
in which the target earthquakes occur in cells \(\{(j_{n},k_{n}),n=1,\cdots,N\}\), and \(\hat{N}_{H}\) denotes the total expected number of target earthquakes, i.e.,  
\[\hat{N}_{H}=\Sigma_{j=1}^{n_{j}}\Sigma_{k=1}^{n_{k}}\lambda_{H}(j,k). \tag{5}\]  
Following [PERSON] et al. [6], the corrected information gain per earthquake (IGPEc) of a fitted hybrid model \(\{\lambda_{H}(j_{k})\}\) over the baseline model \(\{\lambda_{0}(j_{k})\}\) is calculated using  
\[\text{IGPEc}=\frac{-\Delta}{2N}, \tag{6}\]  
in which \(\Delta\) is the change in the corrected Akaike Information Criterion (AICc) between the hybrid model and the baseline model. AICc is defined by  
\[\text{AICc}=-2\ln L+2p+\frac{p+1}{N-p-1}, \tag{7}\]  
in which \(p\) is the number of fitted parameters in the hybrid model, and \(N\) is the number of target earthquakes [16]. Confidence limits on IGPEc are estimated using an adaptation of the \(T\)-test [15] for comparing one earthquake likelihood model to another based on independent data, as described by [PERSON] et al. [6]. For testing of the fitted models on independent data, in which there are no adjustable parameters, the information gain per earthquake (IGPE) is simply the change in log-likelihood between the hybrid and baseline models divided by the number of target earthquakes, and the \(T\)-test given in [15] applies. The covariates are defined on the NZ Collaboratory for the Study of Earthquake Predictability (CSEP) testing grid of 0.1-degree rectangular cells [17].