## 1. Introduction

Weather forecasts are very important for many parts of social and economic life. For example, they are used for severe weather warnings, for decision making in agriculture and industry, or for planning of leisure activities. Generally these forecasts are based on numerical weather prediction (NWP) models. Unfortunately, because of uncertainties in the initial conditions and unknown or unresolved atmospheric processes these models are always subject to error. Luckily some of these errors are systematic and can be corrected with statistical postprocessing, often also referred to as model output statistics (MOS; Glahn and Lowry 1972). However, not all errors can be corrected and for many customers it is important to get additional information about the remaining forecast uncertainty. For this purpose many forecasting centers provide ensemble forecasts. These are multiple NWP forecasts with slightly perturbed initial conditions and sometimes also different model formulations. The idea is that these different forecasts should represent the range of possible outcomes (Lorenz 1996). Large ensemble spreads are then presumably associated with high forecast uncertainties and small spreads that signify low uncertainties. However, in practice the initial ensemble members do not represent initial-condition uncertainty (Hamill et al. 2003; Wang and Bishop 2003). Furthermore ensemble forecasts exhibit the same model errors as single integration forecasts. Thus, to achieve unbiased and calibrated uncertainty forecasts, statistical postprocessing is needed.

In the past decade much research has gone into finding appropriate methods to postprocess ensemble forecasts. For example, Roulston and Smith (2003) proposed dressing the ensemble members with historical model errors and Raftery et al. (2005) suggested Bayesian model averaging for this purpose. Gneiting et al. (2005) proposed to use linear regression with error variances depending on the ensemble spread, and for binary predictands Hamill et al. (2004) proposed to use logistic regression. Comparisons of these and other methods (Wilks 2006a; Wilks and Hamill 2007) showed that logistic regression is one of the better approaches. A very promising extension of logistic regression has been proposed recently (Wilks 2009). By including the predictand threshold in the regression equations this extended logistic regression allows derivation of full predictive distributions. The extended logistic regression method has been used in several studies for probabilistic precipitation forecasts (Schmeits and Kok 2010; Ruiz and Saulo 2012; Roulin and Vannitsem 2012; Hamill 2012; Ben Bouallègue 2013; Scheuerer 2013) and was shown to perform very well compared to standard logistic regression (Wilks 2009; Ruiz and Saulo 2012) and other ensemble postprocessing methods (Schmeits and Kok 2010; Ruiz and Saulo 2012; Scheuerer 2013). In all of these studies, extended logistic regression is used to postprocess ensemble forecasts, but usually the ensemble mean was used as the only predictor variable. There were also several attempts to additionally include the ensemble spread, but with the exception of Hamill (2012) it was always disregarded because it did not improve the forecasts.

In this study we show that the predictive distribution of the transformed predictand is logistic and that the predictor variables only affect the location (mean) but not the dispersion (variance) of this logistic distribution. So far the ensemble spread was always included as ordinary predictor variable in extended logistic regression so that its information was only used to predict the location but not the dispersion of the forecast distribution. However, the ensemble spread is generally expected to mainly contain information about the forecast uncertainty, which in turn should be directly related to the dispersion of the predictive distribution. Hence, the uncertainty information contained in the ensemble spread cannot be utilized properly by extended logistic regression, so that it is not surprising that no improvements could be found.

To solve this drawback of extended logistic regression, we therefore propose a simple new approach in which the ensemble spread can be directly used as predictor for the dispersion of the forecast probability distribution. To illustrate our findings and test if improvements can be achieved with this new approach, we compare different approaches to include the ensemble spread in extended logistic regression on wind speed data from 11 European locations and ensemble forecasts from the European Centre for Medium-Range Weather Forecasts (ECMWF).

The remainder of the paper is organized as follows: in section 2 we describe the extended logistic regression model and show the problems when including the ensemble spread as ordinary predictor variable. Our new approach is introduced in section 3. Results from the case study are shown in section 4 and a summary and conclusions can be found in section 5.

## 2. Extended logistic regression

*y*to fall below a certain threshold

*q*can be predicted withwhere

**x**is a vector of predictor variables [e.g., NWP forecasts;

**x**= (1,

*x*

_{1},

*x*

_{2}, …)

^{T}] and

**is a vector of regression coefficients [**

*β***= (**

*β**β*

_{0},

*β*

_{1},

*β*

_{2}, …)

^{T}] that is generally estimated with maximum likelihood estimation (see appendix A). The regression function has the same mathematical form as the cumulative distribution function of the standard logistic distribution Λ, which is indicated by the final equality in Eq. (1).

Often, more than one threshold is of interest and separate logistic regressions are fitted for each of these thresholds. This approach has the disadvantage that the predicted probabilities are not constrained to be mutually consistent. In other words, for two thresholds *q*_{a} and *q*_{b} with *q*_{a} < *q*_{b} it can occur that *P*(*y* < *q*_{a} | **x**) > *P*(*y* < *q*_{b} | **x**), which would imply nonsense negative probabilities for *P*(*q*_{a} ≤ *y* < *q*_{b} | **x**).

*q*

_{j}as additional predictor variable:Here

*g*(

*q*

_{j}) is a nondecreasing function of

*q*

_{j}and

*α*is an additional coefficient that has to be estimated. In addition to avoiding negative probabilities, this extended logistic regression has the advantage that fewer coefficients have to be estimated (instead of different vectors

**for each threshold,**

*β**α*and

**are the same for all thresholds), which is especially advantageous for small training datasets (Wilks 2009). Furthermore, the probability to fall below any arbitrary value**

*β**Q*can be easily computed by replacing

*q*

_{j}with

*Q*:Equation (3) can also be interpreted as continuous cumulative distribution function, which implies that full continuous probability distributions can be provided.

*g*( ) has to be a nondecreasing function, the equationis always fulfilled. With Eq. (4) and some rearrangements, Eq. (3) can also be written asand upon setting

*μ*= −

**x**

^{T}

**/**

*β**α*and

*σ*= 1/

*α*we obtain the following:This notation allows one to easily see that the conditional probability distribution of the transformed predictand

*g*(

*y*) given the predictor variables

**x**is a logistic distribution with location parameter

*μ*and scale parameter

*σ*. Cumulative distribution functions and probability density functions of this distribution with different scale parameters

*σ*are shown in Fig. 1. The shape of the logistic distribution is very similar to that of the normal distribution but with somewhat heavier tails. The mean of this distribution is

*μ*and in terms of the scale parameter the variance is

*σ*

^{2}

*π*

^{2}/3 (Johnson et al. 1995).

Note that the scale parameter *σ* = 1/*α* is constant so that the predictor variables in **x** only affect the mean, not the variance of the logistic predictive distribution. Hence, when included as additional predictor variable in **x**, the ensemble spread has no effect on the dispersion of the predictive distribution. However, usually large ensemble spreads are associated with high forecast uncertainties, which in turn should be related to wider predictive distributions. In contrast the level of uncertainty should generally have no effect on the location of the forecast probability distribution.

## 3. Heteroscedastic extended logistic regression

*dispersion*of the predictive distribution. Therefore, we simply replace

*μ*and

*σ*in Eq. (6) withandrespectively. Here

**z**is an additional vector of input variables (i.e., the ensemble spread) and

**and**

*γ***are coefficient vectors that have to be estimated. The exponential function is used here as a simple method to ensure positive values for**

*δ**σ*.

Note that with **z** =1 this model is completely equivalent to the original extended logistic regression [Eq. (2)] with *α* = 1/exp(** δ**) and

**= −**

*β***/exp(**

*γ***).**

*δ*The idea of using the ensemble spread as predictor for the dispersion is not completely new. For Gaussian linear regression models, Gneiting et al. (2005) proposed a similar approach, which has been proven to perform well in several studies (e.g., Wilks 2006a; Wilks and Hamill 2007).

## 4. Case study

In this section, we apply the findings from the previous sections on real data. We use 10-m wind speed observations (mean over last 10 min) from the following 11 European weather stations: Amsterdam Airport Schiphol in Amsterdam, Netherlands (52.3°N, 4.783°E); Berlin Tegel Airport in Berlin, Germany (52.55°N, 13.3°E); National Airport in Brussels, Belgium (50.9°N, 4.533°E); Copenhagen Airport in Copenhagen, Denmark (55.6°N, 12.633°E); Frankfurt Main in Frankfurt, Germany (50.033°N, 8.583°E); Heathrow in London, United Kingdom (51.467°N, −0.45°E); Geof in Lisbon, Portugal (38.767°N, −9.133°E); Barajas in Madrid, Spain (40.467°N, −3.55°E); Orly in Paris, France (48.717°N, 2.383°E); Fiumicino in Rome, Italy (41.8°N, 12.233°E); and Wien-Hohe-Warte in Vienna, Austria (48.249°N, 16.356°E), from April 2010 to December 2012. As NWP forecasts we use ensemble wind speed forecasts bilinearly interpolated to the instrument location from the ECMWF (Molteni et al. 1996), initialized at 0000 UTC for the lead times 24, 36, 48, and 60 h.

Figure 2 shows a clear positive correlation between ensemble spread and forecast error for Wien-Hohe-Warte (similar for most other locations). This positive spread–skill relationship suggests that the ensemble spread contains potentially useful uncertainty information. To investigate how this information might be used most effectively, we compare different extended logistic regression models.

For all models we use the square root function for *M* and standard deviation *S* of the square root–transformed ensemble wind speed forecasts. Furthermore, we selected *J* = 9 climatological quantiles with probabilities 1/10, 2/10, …, 9/10 as thresholds *q*_{j} for each location separately.

Table 1 lists the models that are used in the following. In addition to the extended logistic regression model with the ensemble mean as single predictor variable (XLR) there are four models that use the ensemble standard deviation. The models XLR:S and XLR:SM are standard extended logistic regression models with the ensemble standard deviation as additional predictor variable, either alone (XLR:S) or multiplied with the ensemble mean (XLR:SM). In the heteroscedastic extended logistic regression model (HXLR) the ensemble standard deviation is only included as predictor variable for the scale and in HXLR:S it is additionally also used as predictor variable for the location of the predictive distribution.

List of different extended logistic regression models. The **x** and **z** are vectors of predictor variables for the location and scale of the predictive distribution, respectively. The *M* and *S* are the mean and standard deviation of square root transformed wind speed ensemble forecasts, respectively.

Before reporting the forecast quality of these different models it is interesting to investigate the effect of the ensemble spread on the predicted probability distributions. Figure 3 shows predicted probability density functions of the XLR:S and HXLR models for different ensemble standard deviations. For the XLR:S model it can be seen that contrary to the desired effect, larger ensemble standard deviations are related to slightly sharper distributions. In contrast, the HXLR model uses the ensemble standard deviation more appropriately and larger ensemble standard deviations are clearly related to wider distributions.

*J*= 9 is the number of thresholds and

*I*(·) = 1 if the argument in brackets is true and 0 if it is not. To get independent training and test datasets we estimate and verify the models with tenfold cross validation. With this cross validation we get one RPS value for each event in the dataset. From these individual RPS values, 250 estimates of the mean

Figure 4 shows the RPSS of the different models and lead times aggregated over the 11 locations. It can be seen that including the ensemble standard deviation simply as ordinary predictor variable (XLR:S, XLR:SM) does not improve forecast quality of extended logistic regression. However, the reason is not the absence of predictive information in the ensemble standard deviation since using it with our new approach (HXLR) clearly improves the forecast quality, especially for day time forecasts (36- and 60-h lead time). Since the ensemble standard deviation seems not to contain any predictive information on the location it is also not advantageous to include it additionally as predictor variable for the location (HXLR:S). The effect of the lead time on the RPSS is only weak but for daytime forecasts (12 and 36 h) the superiority of HXLR is more pronounced. Note that we also tested longer lead times (up to 96 h) and shorter training data lengths (down to 6 months), but results were similar and are therefore not shown.

Figure 5 shows the RPSS for selected locations aggregated over lead times 24–60 h. While most of the locations show similar patterns as in Fig. 4 (e.g., Amsterdam, Wien) there are also some locations (e.g., Berlin) where including the ensemble spread as ordinary predictor variable (XLR:S, XLR:SM) is superior to heteroscedastic extended logistic regression (HXLR). This suggests that for these locations the ensemble spread also contains predictive information on the location. For nonnegative predictands like wind speed, large observed values are generally related to large ensemble spreads. Therefore, it is indeed conceivable that the ensemble spread contains some predictive information on the location that is not yet covered by the ensemble mean. However, additional improvements can be achieved when including the ensemble spread as a predictor for both location and scale of the predictive distribution (HXLR:S).

Finally, Fig. 6 shows reliability diagrams for 36-h forecasts of the first climatological decile [*P*(*y* < *q*_{1} | **x**)] and the climatological median *P*(*y* < *q*_{5} | **x**) for the models XLR:S and HXLR. Both models are fairly reliable with only little differences between each other. For the lower decile both models are slightly overforecasting (points below diagonal). The logistic predictive distribution of extended logistic regression involves a point mass at zero (i.e., positive predictive density for negative wind speeds; Schefzik et al. 2013). Because zero wind speeds occur relatively rarely, this might be the reason for the overestimated probabilities to fall below the lower decile.

## 5. Summary and conclusions

The inclusion of the ensemble spread in extended logistic regression has been shown in several studies not to improve the forecast skill. As we have shown in this paper this is not surprising because when the ensemble spread is included as an ordinary predictor variable it modifies only the location but not the dispersion of the forecast distribution. Uncertainty information contained in the ensemble spread is therefore not used appropriately. To solve this problem we proposed a new approach called heteroscedastic extended logistic regression where the ensemble spread is directly used as predictor for the *scale* of the predictive distribution.

To illustrate the advantages of this new approach we used wind speed observations from 11 European locations and ensemble forecasts from ECMWF. Consistent with our findings and with results from previous studies, the inclusion of the ensemble standard deviation as an ordinary predictor variable has no clear positive effects on forecast quality. In contrast, with our new approach the uncertainty information in the ensemble standard deviation is used effectively to achieve clear improvements.

An additional single case study with precipitation data showed similar results. We therefore expect that our results can be transferred to other variables and/or locations. However, this still has to be tested.

Hamill (2012) got better forecasts when using the ensemble variance multiplied with the ensemble mean as an additional predictor variable. This suggests that in his data, the ensemble spread also contained predictive information on the location of the predictive distribution. Consistent with these findings, we also found individual weather stations where including the ensemble spread as ordinary predictor variable is even superior to heteroscedastic extended logistic regression. However, further improvements could be achieved when including the ensemble spread as predictor variable for both location and spread of the predictive distribution.

To enhance the flexibility of extended logistic regression, Ben Bouallègue (2013) proposed the use of interaction terms between the threshold and the predictor variables. An interaction term between threshold and ensemble spread could also be used to control the dispersion of the predictive distribution. Contrary to heteroscedastic extended logistic regression such a model can be easily implemented with standard binary logistic regression software. However, with interaction terms the ensemble spread also has some undesired effects on the distribution location.

Extended logistic regression has been shown in several studies to perform well compared to other ensemble postprocessing algorithms (e.g., Schmeits and Kok 2010; Ruiz and Saulo 2012; Scheuerer 2013). However, a major drawback of this method was that uncertainty information contained in the ensemble spread could not be utilized effectively. Heteroscedastic extended logistic regression is therefore a very attractive extension of extended logistic regression to further enhance its competitiveness.

## Acknowledgments

We thank Tom Hamill, Tilmann Gneiting, Constantin Junk, and an anonymous reviewer for their valuable comments that helped to improve this manuscript. This study was supported by the Austrian Science Fund (FWF): L615-N10. The first author was also supported by a Ph.D. scholarship from the University of Innsbruck, Vizerektorat für Forschung. Data from the ECMWF forecasting system were obtained from the ECMWF Data Server.

## APPENDIX A

### Likelihood Function

**and**

*α***(extended logistic regression) or**

*β***and**

*γ***(heteroscedastic extended logistic regression) maximum likelihood estimation is used. The general log-likelihood function for logistic regression models iswhere**

*δ**N*is the length of the training dataset and

*π*

_{i}is the predicted probability for the

*i*th observed outcome. For binary logistic regression there are two possible outcomes, so thatIn previous studies the sum of this binary log-likelihood over all thresholds is used as objective function that is maximized to estimate the regression coefficients. However, the predicted probability of the

*i*th outcome actually isso that the correct maximum likelihood estimator is given by the maximization of Eqs. (A1) and (A3). In this study we employ this maximum likelihood estimator to take advantage of all standard asymptotic inference in the maximum likelihood framework. However, the concepts presented in this paper do not depend on the objective function and results should also not differ significantly when using the sum of binary log-likelihoods [Eq. (A2)] to estimate the coefficients.

## APPENDIX B

### Computational Details

Our results were obtained on Ubuntu using R 2.15.2 (R Core Team 2012). A function to fit (heteroscedastic) extended logistic regression models is included in the package crch 0.1-0 (Messner and Zeileis 2013).

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