The main motivation for using a GAM is the possibility to estimate (potentially) non-linear relationships between the response and the covariates. In the following, the basic concept of GAMs is introduced for an arbitrary parameter θ of some probability density function d(⋅;θ) (PDF). A GAM aiming at modelling a spatio-temporal climatology over complex terrain would have the form, g(θ)=β0+f1(logalt)+f2(doy)+f3(lon,lat), where g(⋅) is a link function that maps the scale of the parameter θ to the real line. The right-hand side is called the additive predictor, where β0 is the intercept term and fj are unspecified (potentially) non-linear smooth functions that are modelled using regression splines . For each fj a design matrix Xj containing spline basis functions is constructed. Thus the GAM can be written as generalized linear model (GLM), g(θ)=β0+∑j=13Xjβj. The coefficients β = (β0, β1⊤, β2⊤, β3⊤) are estimated by maximizing the penalized log-likelihood, l(β)=∑i=1nlog⁡dyi;g-1β0+∑j=13Xjβj-12∑j=13λjβj⊤Sjβj, where the first term on the right-hand side is the unpenalized log-likelihood. The second term is added to prevent overfitting by penalizing too-abrupt jumps of the functional forms. λj are the smoothing parameters corresponding to the functions fj. For λj = 0 the log-likelihood is unpenalized with respect to fj. When λj → ∞ the fitting procedure will select a linear effect for fj. The selection of the smoothing parameters λj is performed by cross-validation (Sect. ).

The value of λj determines the degrees of freedom of the associated effect. Lower values of λj, e.g. small penalization, lead to an effect with more degrees of freedom, which might explain more features but is also prone to overfitting. High values of λj, e.g. strong penalization, result in an effect with fewer degrees of freedom. Thus fewer features can be explained. The balance between small and strong penalization and thus the corresponding degrees of freedom is found by performing cross-validation (Sect. ).

Sj in Eq. () are prespecified penalty matrices, which depend on the choice of spline basis for the single terms. The reader is referred to for more details.

Estimation of a GAM for such a large data set, i.e. 7 309 152 data points, is feasible, e.g. via function bam() (for big additive models) implemented in the mgcv package of the statistical software R .

The first component models the probability of lightning Pr(Y > 0) = π occurring within a 1 km × 1 km cell and a day. The Bernoulli distribution with the parameter π of the probability density function (PDF), dBe(y;π)=πy(1-π)1-y, will be fitted. Since the data are binomial, y ∈ {0, 1} indicates no lightning and lightning within the cells. The model for π takes the form of Eq. () with π replacing θ. The complementary log-log g(π) = log⁡(-log⁡(1 - π)) is implemented as link function.

The second part is the truncated count component for the expected number of flashes given lightning activity. We will refer to this component as the intensity model. It is assumed that the positive counts of flashes within a spatial cell and day follow a zero-truncated Poisson distribution with the PDF, dZTP(y;μ)=dPois(y,μ)1-dPois(0,μ), where y ∈ {1, 2, …}, and dPois(⋅;μ) is the PDF of the Poisson distribution with expectation μ. The conditional expectation is E[Y|Y > 0] = μ/(1 - e-μ). The GAM for this component has the form of Eq. () with μ replacing θ. The logarithm serves as link function g(μ) = log⁡(μ).

The family for modelling the zero-truncated Poisson distribution ztpoisson() within a GLM or GAM framework is implemented in the R-package countreg . For more information on and a formal definition of hurdle models the reader is referred to .

In this section the verification procedures are briefly introduced, namely the cross-validation, the applied scores and the block-bootstrapping.

In order to ensure the verification of the model along independent data, we applied a 6-fold cross-validation . Six years of data are available. The parameters of the model are estimated based on 5 years of the data and validated on the remaining year. This is done 6 times with every single year serving as a validation period once.

The log-likelihood is applied as scoring function, which is also called logarithmic score in the literature on proper scoring rules .

To assess confidence intervals of the estimated parameters and effects, day-wise block-bootstrapping was performed. By day-wise block-bootstrapping we mean the following: we resample the 738 dates of all available days with repetition and pick all the data observed on these days spatially. This procedure is executed 1000 times in order to assess confidence intervals.

The estimates of the effect of the occurrence model (Sect. ) are depicted in Fig. . The values are on the scale of the additive predictor, i.e. the right-hand side of Eq. (). The additive predictor takes the value of the intercept term β0 if the sum of all other effects is equal to zero. Its estimate is β0 = -3.97 (-4.15, -3.80) on the complementary log-log scale, which is 1.87 % (1.56, 2.21 %) in terms of the probability of lightning. The numbers in parentheses are the 95 % confidence intervals computed from 1000 day-wise block-bootstrapping estimates.

How the effects in Fig.  can be interpreted to obtain the probability of lightning at a particular location and day is shown for the location E (Rosennock in Fig. ) and 20 July. The location is at an altitude of 2440 m (Table ), for which the altitude effect is roughly 0.34 (Fig. a). The contribution of the seasonal effect (Fig. b) for 20 July is about 0.64. The spatial effect (Fig. c) has a value of -0.03 at this geographical location (13.71∘ E, 46.88∘ N). Adding these values to the intercept β0 = -3.97 yields -3.02, which is on the scale of the additive predictor. It needs to be transferred with the inverse of the complementary log-log function to obtain the value in probability space, which is 4.76 %.

The second term f1(logalt) of the additive predictor models the effect of the logarithm of the altitude. f1(logalt) varies from roughly -0.2 for low altitudes to values greater than 0.5 for altitudes above 2800 m (Fig. a). This function takes 7.9 degrees of freedom. Its shape is close to exponential, which suggests that a linear term for the altitude might be sufficient. For altitudes above 2000 m, however, the non-linear term leads to larger values than a linear term β1alt would do.

The temporal or seasonal effect f2(doy), i.e. the dependence of the target on the day of the year (doy), shows a steep increase during May, reaches its maximum in mid-July and decreases slowly during August (Fig. b). This result indicates that the main lightning season in Carinthia lasts from mid-June until end of August. The estimated degrees of freedom are 2.5, which leads to the simple shape of the seasonal effect with one clear maximum.

The spatial effect f3(lon,lat), which explains the spatial variations of the linear predictor that cannot be explained by the altitude term f1(logalt), requires 138 degrees of freedom. (Fig. c). Most prominent features are the minimum near the north-west and the maximum in the middle to eastern part of Carinthia. In the north-west the highest mountains, the High Tauern, of Carinthia are located. The minimum with values less than -0.3 on the complementary log-log scale suggests that lightning activity is less pronounced in this region. This finding is in line with former analyses of the lightning activity in Austria , which stated that the main alpine crest is an area with a minimum in flash density. The maximum zone with values exceeding 0.3 in the middle to eastern part of Carinthia covers the so-called Gurktal Alps. In comparison with the High Tauern, the Gurktal Alps have a lower average elevation and the mountains are not as steep. Such maxima at moderate or low altitude are mostly modelled by the spatial effect, not by the altitude effect.

As the altitude is a function of longitude and latitude, one could ask whether it would be sufficient to only take spatial effect into account that implicitly contains the altitude and skip the explicit altitude effect. In general the presented method would be capable of modelling the influence of the altitude within the spatial effect implicitly. However, the shape of the altitude in the region of interest is very complex. Thus, a spatial effect with a large degree of freedom would be required in order to account for the complex altitude shape. As we know the shape of the altitude we can pass it to the GAM as an isolated effect. The altitude effect contains only information associated with the altitude while the remaining effects are captured by the spatial term.

The introduced model (Eq. 1) could also be extended by potentially non-linear functions of other covariates that are meaningful for a climatological assessment, e.g. surface roughness, slope and aspect of topography. However, in the present case, adding these covariates was not improving the model.

The non-linear effects of the intensity model (Sect. ) are depicted in Fig. . The estimate of the intercept term takes the value β0 = -0.01 (-0.19, 0.14) which leads to a expected number of flashes given lightning activity of 1.57 (1.47, 1.68) when the sum of all other effects is equal to zero.

The altitude effect f1(logalt) (Fig. a), with 5.4 degrees of freedom, reveals a similar functional form as the altitude effect of the occurrence model (Fig. a). However, it has a flatter shape for the terrain between 600 and 1200 m and a steeper increase for high altitudes above 2000 m a.m.s.l.

The seasonal effect f2(doy) is -0.5 in early May, reaches a maximum of 0.3 in early July and decreases to values around -0.3 until the end of August (Fig. b). Thus the amplitude of this effect is not as strong as the seasonal effect of the occurrence model (Fig. b) and the location of the maximum is earlier. It has 2.1 degrees of freedom.

The spatial effect f3(lon,lat) varies strongly and requires 166 degrees of freedom which is more than the corresponding effect of the occurrence model. However, there are some features in common for both effects. For instance, the prominent maximum visible in Fig. c in the Gurktal Alps also appears in the spatial effect of the count model (Fig. c). The strong minimum in the western part of the domain is also common. The most pronounced new feature is the strong local maximum with values exceeding 0.9 in the south of Carinthia. A 165 m radio tower is installed on the peak of the Dobratsch mountain (location C in Fig. ), which triggers lightning strokes under suitable conditions, i.e. occurrence of a thunderstorm. Other maxima of this effect could also be attributed to sites of radio towers, which suggests that the number of flashes is more sensitive to local constructions than the probability of lightning.

In order to illustrate how climatological information can be drawn from the GAMs, two different kinds of applications are presented. First, maps show spatial climatologies (Figs.  and ). Here, the occurrence model and/or the intensity model are evaluated for one specific day. Second, the seasonal climatology for selected 1 km × 1 km grid cells are discussed (Fig. ). Here, the models are evaluated with respect to the geographical location of the point of interest and its altitude.

The spatial distribution of climatological probabilities of lightning occurring in a cell for 20 July (close to the seasonal peak) varies from 1.8 to 6.5 % (Fig. ). In the western part of the domain, local valleys and mountain ridges become visible through the altitude effect (Fig. a). However, the highest probabilities do not occur over the highest terrain in the north-west, where the spatial effect counteracts the altitude effect, leading to moderate probabilities around 2 to 3 %. The spatial effect (Fig. c) is responsible for the maximum over the moderate altitude region of the Gurktal Alps. Such a map can also serve as thunderstorm climatology when lightning is taken as a proxy for thunderstorms.

A comparison of Figs.  and  illustrates some of the benefits of using GAMs instead of taking averages in each grid cell for computing expected values of lightning occurrence. Harnessing the information from the complete data set instead of using only information contained in each grid cell removes the noise and makes the overall pattern visible, e.g. the difference between lightning over valleys and ridges.

For the same day, 20 July, the expected number of flashes is depicted in Fig. . This is the product of probabilities of lightning π from the occurrence model and the expected number of flashes given lightning activity, which is derived from the intensity model. Values range from 0.028 to 0.166. The lowest values can be found in the north-western part of Carinthia where the spatial effects of both models reveal a minimum. Next to the maximum in the Gurktal Alps, where maxima in the spatial effects of both models can also be found, a second peak appears at the Dobratsch mountain (location C in Fig. ), which is due to the local maximum in the spatial effect of the intensity model (Fig. c).

Next to the spatial information one can extract seasonal climatologies for different locations (Fig. ). These are computed exemplary for five sites (Table ). Figure a shows the climatologies of lightning probability. Differences between the annual cycles of the probabilities are due to the altitude effect and the spatial effect of the occurrence model (Fig. ). The highest probabilities between 4 and 5 % are modelled in July for location B (dashed line), which is located at the south-western border of Carinthia in the vicinity of a local maximum of the spatial effect (Fig. c). This climatology exhibits a strong seasonality, as probabilities fall below the 1 % level. Though located at a similar altitude, the climatology of location A (solid line) reveals maximum values less than 2 %. This difference is due to the spatial effect, which exhibits a clear minimum in north-western Carinthia. The climatology of location D (dashed dotted line) in the lower plains in the eastern part of Carinthia shows a moderate chance of lightning with values around 3 % during the peak of the season.

The climatologies of the expected number of flashes are depicted in Fig. b. The order of location has changed. In particular, the highest number of flashes are expected for location C which is the Dobratsch mountain. This is caused by the strong local maximum in the spatial effect of the intensity model (Fig. c). The legend shows expected number of flashes accumulated over the lightning season, which leads to values between 2.1 for location A and 7.6 for location C. These values are in good agreement with the analysis by Fig. 5 therein.

Finally, it is also possible to derive relative frequencies of the number of flashes of a specific location and day of the year from the GAM. The relative frequencies have been derived for the five sample locations (Table ). The first column of the table with the probabilities of no flashes occurring only contains information from the occurrence model (cf. Fig. a). All other probabilities for one or more flashes are derived from both the occurrence and the intensity model. The influence of the intensity model is especially dominant in the relative frequencies for location C, where the probability of having four or more flashes on 20 July is 1.54 %.