Development and validation of a clinical risk score to predict the risk of SARS-CoV-2 infection

6 Andrea Nigri and Federico Crescenzi differently by the causes of death, specific causes may need to be targeted to reduce inequalities.

References

1. Preston S.H. (1974). Evaluation of Postwar Mortality Projections in the United States, Canada, Australia, New Zealand. World Health Statistics Report, 27(1): 719-745.

2. Mackenbach, JP., Looman CWN. Life expectancy and national income in Europe, 1900-2008: an update of Preston’s analysis. International Journal of Epidemiology 2013;42:1100–1110 3. Mehta, N.K., Abrams, L.R., Myrskylä, M. (2020). US life expectancy stalls due to

car-diovascular disease, not drug deaths. Proceedings of the National Academy of Sciences, 117(13):6998–7000.

4. Aburto, J. M., van Raalte, A. (2018). Lifespan Dispersion in Times of Life Expectancy Fluctuation: The Case of Central and Eastern Europe. Demography, 55: 2071-2096. 5. Vallin, J., Meslé, F. (2009). The Segmented Trend Line of Highest Life Expectancies.

Popula-tion and development review, 35 (1): 159–187.

6. White, Kevin M. 2002. Longevity advances in high-income countries, 1955-1996. Population and Development Review, 28: 59-76.

7. Caussinus, H. and Courgeau, D. (2010). Estimating Age without Measuring it: A New Method in Paleodemography. Population (English Edition, 2002), 65(1):117–144.

8. Łukasik, S., Bijak, J., Krenz-Niedbała, M., Liczbinska, G., Sinika, V., and Piontek, J. (2017). Warriors Die Young: Increased Mortality in Early Adulthood of Scythians from Glinoe, Moldova, Fourth through Second Centuries BC. Journal of Anthropological Research, 73(4):584–616.

9. Human Mortality Causes of Death Database 2021. University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). Data downloaded on 01/01/2021.

10. Oeppen, J., Vaupel, J.W.: Broken Limits to Life Expectancy. Science296 (5570): 1029–1031 (2002)

11. Riley J. Rising Life Expectancy: A Global History. Cambridge. Cambridge University Press (2001)

12. Shkolnikov V.M., Andreev E.M. Tursun-zade R., Leon D.A. (2019). Patterns in the relationship between life expectancy and gross domestic product in Russia in 2005–15: a cross-sectional analysis. The Lancet Public Health.

13. World Bank. GDP (current US).2021.htt ps : //data.worldbank.org/

Locally sparse functional regression with an

application to mortality data

Regressione funzionale localmente sparsa con

un’applicazione a dati di mortalit`a

Mauro Bernardi, Antonio Canale, Marco Stefanucci

Abstract A novel method for functional regression with functional response and functional covariate is discussed. The method is particularly useful when the regres-sion surface is non-zero only on a subset of its bivariate domain, allowing for a local relation between the response and predictor variable. By means of a tensor product splines representation of the unknown functional coefficient and an overlap group lasso penalty we are able to effectively estimate the regression function. The model performance is illustrated through its application to the well-known Swedish Mor-tality dataset, clearly showing the local nature of the relation between the morMor-tality at consecutive years.

Abstract Viene discusso un nuovo metodo per implementare la regressione fun-zionale penalizzata per una risposta funfun-zionale ed una covariata funfun-zionale. Il metodo `e particolarmente utile quando la superficie di regressione `e diversa da zero solo in un sottoinsieme del suo dominio bivariato, permettendo una relazione locale tra la variabile risposta e la variabile di previsione. Grazie ad una rappre-sentazione del coefficiente funzionale incognito con splines prodotto–tensoriali e ad una penalit`a lasso a gruppi sovrapposti siamo in grado di stimare efficacemente la funzione di regressione. Tale procedura di stima viene quindi applicata al noto set di dati Swedish Mortality, mostrando chiaramente la natura locale della relazione tra la mortalit`a in anni consecutivi.

Key words: Functional Data Analysis; Function-on-function regression; Sparsity; Swedish Mortality

Mauro Bernardi, Antonio Canale, Marco Stefanucci,

mbernardi@stat.unipd.it, canale@stat.unipd.it, stefanucci@stat.unipd.it, Universit`a di Padova, Via Cesare Battisti, 241, 35121 Padova

1

2 Bernardi et al.

1 Introduction

Functional linear model investigates the indirect relation between two or more vari-ables, where at least one is functional in nature. The function-on-scalar regression model focuses scalar response variable and functional explanatory variables [1]. The converse applies to the scalar-on-function model [3]. In this short paper we study the function-on-function regression model, i.e.

y(t) =$ β(s,t)x(s)ds + ε(t), (1)

where both the response and explanatory variable are defined over a continuum, s ∈ S ,t ∈ T and ε(t) is functional noise. This framework has increased in popu-larity thanks to the famous book by Ramsay and Silverman [5], where a penalized estimation approach based on B-splines is discussed. Real data applications involv-ing the function-on-function regression model include —but are not limited to— chemometrics, pharmacology, neuroscience, demography and meteorology. In the aforementioned book, Ramsay and Silverman discussed the interesting special case of the Historical Functional Model (HFM) where the influence of the explanatory variable x(s) on the response y(t) is confined to a specific interval of the domain, namely s < t, and the resulting model is

y(t) =$

s<tβ(s,t)x(s)ds + ε(t).

With this formulation only values of s that are lower than t are used in the prediction of y(t) and the estimatedβ(s,t) is an upper–triangular matrix. Considering the well-known Canadian Weather dataset discussed in [5] as illustrative application, x(s) is the temperature observed from January 1st to December 31st and y(t) is the level of precipitation observed in the same period (S = T ). Of course only the temper-ature before the time t can be predictive of the precipitation at time t, and the HFM results in an elegant way to force this constraint. In this application the interval of integration is motivated by the phenomenon under study and hence is chosen by the user. However this aspect poses some limitations:

• often the choice of the interval of integration is not obvious. Beyond simple cases, the user could be unable to make this crucial choice.

• the choice is not data-dependent. Specifically, integration on a restricted domain acts like a model selection, similarly to an a priori subset selection not derived by using the data.

From these reasons we propose a methodology able to automatically detect the re-gions of sparsity in the unknown operator, without any restriction on the domain of integration. The method is based on a tensor product splines representation of the regression function and a penalized approach that makes use of a modified lasso penalty [4]: the Overlap Group Lasso (OGL).

Locally sparse functional regression with an application to mortality data 3

2 Model

We define {ηj(s)}Lj=1 and {θk(t)}M_k=1 as B-splines bases over S and T . Then we

have xi(s) =∑Lj=1ai jηj(s) and yi(t) =∑Mk=1cikθk(t) for each i = 1,...,n. The

bi-variate functionβ(s,t) can be expressed as β(s,t) = ∑L

j=1∑Mk=1bjkηj(s)θk(t). If L

and M, the dimensions of the two spline bases, are small enough, model (1) can be fitted computing the

argmin

B ||CΘΘΘ −ANN T_BΘΘΘ||2

F. (2)

whereΘΘΘ and N are matrices of basis functions and A,B,C are matrices of coeffi-cients. However, a penalized estimation is often preferable. Consistently with this the dimensions of the spline bases are much higher and to regularize the solution a penalty is added to (2). A popular choice for the penalty term is the ridge penalty defined as"β2_{(s,t)dsdt. In this case the coefficient function can be obtained by}

argmin

B ||CΘΘΘ −ANN T_BΘΘΘ||2

F+λpen(B), (3)

where the parameterλ controls the balance between the two terms. The main limi-tation of this particular penalty function and, in general, of all smooth penalties, is that they are not suited for identify sparsity and the %β(s,t) obtained in (2) and (3) will result in a overall smooth function. A possible way to detect sparsity is the use of the lasso penalty [7] on the coefficients of the expansion ofβ(s,t) but this comes at a price: sacrifice smoothness. The main idea behind our locally sparse model is to borrow strength from the two approaches in order to construct an estimator able to identify both the smooth and sparse regions of the unknown function.

We achieve this exploiting the OGL penalty and a feature of the B-splines represen-tation ofβ(s,t) that allows β(s,t) to be exactly zero on a region of its domain if a block of adjacent coefficients bjk of suitable dimension depending on the B-spline

order is jointly zero.

3 Application

We apply the method described in previous section to the well–known Swedish Mortality dataset, where log-hazard rates are observed for the Swedish female pop-ulation between years 1751 and 1894. The aim is to predict the log-hazard function yi(t) at a specific calendar year i by using the log-hazard function at previous year

xi(s) = yi−1(t). Existing studies [6, 2] show that the hazard function at year i and

age t is mainly influenced by the hazard function at the previous year i − 1 at age s = t − 1, resembling a quasi–concurrent relation, following the terminology of [5]. However, none of these studies report the total absence of relation when s and t are far away. Figure 2 shows the %β(s,t) obtained by the proposed methodology. It

4 Bernardi et al.

s t

Fig. 1 Estimated β(s,t) with the proposed locally sparse approach for the Swedish mortality database

is easy to see that smoothness is preserved along the main diagonal and that the function is flat when s and t are far enough.

References

1. Cardot, H., Ferraty, F., Sarda, P. (1999), Functional linear model. Statistics and Probability Letters45, 11–22.

2. Chiou, J.M. and Muller, H.G. (2009), Modeling hazard rates as functional data for the analysis of cohort lifetables and mortality forecasting. Journal of the American Statistical Association 104 (486), 572–585.

3. Faraway, J.J. (1997), Regression analysis for a functional response. Technometrics39, 254– 261.

4. Jenatton, R., Audibert, J.-Y., and Bach, F. (2011), Structured variable selection with sparsity– inducing norms. Journal of Machine Learning Research,12, 2777–2824.

5. Ramsay, J.O. and Silverman B.W. (2005), Functional Data Analysis, 2nd edition. Springer, New York

6. Ramsay, J.O., Hooker, G., and Graves, S. (2009), Functional data analysis with R and MAT-LAB. Springer Science & Business Media.

Locally sparse functional regression with an application to mortality data 5 7. Tibshirani, R. (1996), Regression shrinkage and selection via the lasso. Journal of the Royal

Statistical Society: Series B,58, 267–288.

8. Wu, T.T. and Lange, K. (2010), The MM alternative to EM. Statistical Science,25, 492–505.

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883

Effect of ties on the empirical copula methods

for weather forecasting

L’effetto di dati ripetuti nei metodi di meteorologia basati

su copule empiriche

Elisa Perrone, Fabrizio Durante, and Irene Schicker

Abstract Weather forecasts are typically in the form of an ensemble of forecasts obtained through different runs of physical models. Ensemble forecasts are often biased and affected by errors and need to be statistically post-processed to be cor-rected. Here, we focus on the empirical copula based techniques for the statistical post-processing of multivariate forecasts. We present the methodology and discuss its pros and cons, especially when ties appear in the ensemble. We consider a case study of joint temperature forecasts for three locations in Austria. We analyze var-ious ways of dealing with ties and show that, in general, the current practice of breaking them at random may not be the optimal solution for forecasting purposes. Abstract Le previsioni meteorologiche sono spesso espresse sotto forma di un in-sieme di scenari ottenuti tramite diversi modelli fisici. Queste sono di solito distorte e viziate da errori e devono essere corrette mediante opportune tecniche statistiche. Qui si analizzano le tecniche di correzione basate su copule empiriche. Si presen-tano i principali aspetti di tale approccio e si discute il caso in cui l’insieme di previsioni presenti dei dati ripetuti (ties). In particolare, si considera un caso di studio riguardante le previsioni di temperatura in tre stazioni meteo localizzate in Austria. Analizzando vari modi di trattare i dati ripetuti si mostra che, in generale, la loro randomizzazione non sempre fornisce la migliore previsione.

Key words: Empirical copulas, Weather forecasting, Statistical post-processing, Ensemble Copula Coupling, Ties.

Elisa Perrone

Eindhoven University of Technology, Groene Loper 3, 5612 AE Eindhoven (The Netherlands) e-mail: e.perrone@tue.nl

Fabrizio Durante

Università del Salento, Centro Ecotekne, 73100 Lecce (Italy) e-mail: fabrizio.durante@unisalento.it

Irene Schicker

Zentralanstalt für Meteorologie und Geodynamik, Hohe Warte 38, 1190 Vienna (Austria) e-mail: irene.schicker@zamg.ac.at

1

2 Elisa Perrone, Fabrizio Durante, and Irene Schicker

1 Introduction

Accounting for the right amount of uncertainty is key to the quality of weather pre-dictions. The complexity of the physical atmospheric phenomena makes it hard to achieve so through a single deterministic forecast. As a consequence, meteorolo-gists often use probabilistic numerical weather prediction (NWP) models, which represent the forecast probability distribution as an ensemble of possible forecasts obtained by multiple runs of deterministic physical models. Still, the uncertain ini-tial and boundary model conditions are reflected in the ensemble forecasts, making them biased and affected by errors. As a consequence, raw ensemble forecasts are often statistically post-processed to account for such errors and gain accuracy. In this respect, commonly used approaches consist of two steps. First, we obtain a univari-ate corrected (parametric) distribution for every single variable of the forecasting problem. Then, we reconstruct the dependence structure from the rank structure of a reference sample. The most popular approaches of this type are Schaake Shuffle [2], Ensemble Copula Coupling (ECC) [21], and Sim Schaake [20]. Each method differs for the chosen reference sample: ECC obtains a reference sample from the raw ensemble forecasts, while Schaake Shuffle and Sim Schaake use past observa-tions. We notice that these methods are all based on empirical copulas, which are mathematical tools to describe the dependence structure of a multivariate sample through their associated ranks [15]. As such, the procedure is implicitly requiring that data can be uniquely ranked and no ties, i.e. repeated observations, appear in the reference sample. Since this is often not the case in practice, how do we handle situations when ties appear in the reference sample?

This problem has a long history, as it can also be noticed from the seminal con-tribution [10]. Nowadays, the influence of ties in copula methods and models has largely been recognized in the literature, especially for their effects on the statisti-cal estimation and goodness-of-fit tests; see, for instance, [4, 11, 14]. Usually, the practitioner’s solution of jittering the data (i.e. add a small error term) to guarantee a unique ranking of the observations should be done with extreme care (see [17]). However, to the best of our knowledge, the influence of ties has not been addressed in the weather forecasting literature, where ties are simply solved at random.

In this work, we study this aspect by focusing on ECC. Starting with a case study of temperature forecasts in Austria originally presented in [18], we construct some realistic simulation scenarios to evaluate the effect of ties on the corrected multivariate forecasts. Several ways of breaking ties are hence considered.

2 Statistical post-processing in weather forecasting

In this section, we introduce the general setting and methodology. For a detailed description of the methods and the data, we refer the reader to [18] and references in there. As mentioned in the introduction, we focus on ECC, which is a statistical

Effect of ties on the empirical copula methods for weather forecasting 3 post-processing technique based on the ranks of original raw ensemble forecasts. In particular, it consists of the following main steps.

Step 1 For each variable, we obtain a univariate corrected distribution by apply-ing any univariate statistical postprocessapply-ing method (e.g., the Ensemble Model Output Statistics (EMOS) [6]).

Step 2 We construct a new sample from the corrected univariate distributions (e.g., by taking uniform quantiles).

Step 3 We reorder each univariate corrected sample according to the rank struc-ture of the raw ensemble forecasts.

We now give a simple illustration of these steps. In general, we assume an ensemble system of M ∈ N members, with d = J, and J ∈ N, univariate raw margins of the form (x(₁j), . . . ,x(mj)), where j ∈ {1,...,J} is a location. We also denote by d∗ the

number of variables of our interest. As an example, we consider an ensemble sys-tem of size M = 6, and we aim to correct the sys-temperature forecasts of three stations at a fixed lead-time. Thus, the multivariate forecast of our interest has dimension d∗_{=3. In this scenario, the raw ensemble forecast is a (6 × 3)-matrix: each matrix}

column represents the raw forecasts of the temperature of a particular station, and each matrix row represents a (raw) three-dimensional multivariate forecast. For ex-ample, the raw forecastsR and its corresponding reference dependence structure D might be as follows: R = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 262.34 263.80 266.82 263.14 263.88 267.73 262.55 263.03 269.31 263.15 263.62 267.39 264.92 261.22 267.10 260.18 265.57 265.13 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , D = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ (2) (4) (2) (4) (5) (5) (3) (2) (6) (5) (3) (4) (6) (1) (3) (1) (6) (1) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (1)

where (·) indicates a rank, and D is obtained by transforming the raw forecasts into their corresponding ranks. The first step of ECC is to correct the forecasts of each station. For this aim, we use EMOS, which is a Gaussian-based approach originally presented in [6]. Specifically, EMOS is a regression method that uses the ensemble forecasts as covariates and optimizes the parameters of a Gaussian response distri-bution to adapt for errors in the mean and uncertainty of the forecasts.

We apply EMOS to each column ofR and obtain the (corrected) univariate dis-tributions F1∼ N (259.5,1.05), F2∼ N (270,1.13), and F3∼ N (273,0.8). Then,

we construct three samples of size M = 6 from F1, F2, and F3. Namely, the vectors:

s1= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 258.38 258.91 259.31 259.69 260.09 260.62 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ s2= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 268.79 269.36 269.80 270.20 270.64 271.21 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ s3= ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 272.15 272.55 272.86 273.14 273.45 273.85 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

886

4 Elisa Perrone, Fabrizio Durante, and Irene Schicker A multivariate sample with corrected margins is given by the following matrixC.

C = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 258.38 268.79 272.15 258.91 269.36 272.55 259.31 269.80 272.86 259.69 270.20 273.14 260.09 270.64 273.45 260.62 271.21 273.85 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (2)

We notice thatC does not account for any special dependence structure of the ran-dom vector (F1,F2,F3), and it might not be representative of the actual distribution

of (F1,F2,F3). This issue is fixed in the last step of ECC by reordering the entries

of each column ofC according to the ranks in D. In our illustrative example, the reordering step results in the matrix ˜C given below.

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ (2) (4) (2) (4) (5) (5) (3) (2) (6) (5) (3) (4) (6) (1) (3) (1) (6) (1) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ → ˜C = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 258.91 270.64 272.55 259.69 271.21 273.45 259.31 269.36 273.85 260.09 269.80 273.14 260.62 268.79 272.86 258.38 271.21 272.15 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

The post-processed three-dimensional sample ˜C of size 6 has the same empirical dependence structure of the raw forecasts. Thus, ˜C represents a corrected multivari-ate sample of (F1,F2,F3)with a reconstructed inter-variable and spatial dependence.

We notice that any ties in the columns of the raw forecastsR impact the methodol-ogy since the assigned ranks to tied values are arbitrary. In the next section, we focus on this point and compare various ways of breaking ties in a case study setting.

3 A case study of temperature forecasts in Austria

We now consider a case study of joint temperature forecasts for three stations in Austria, namely, Sonnblick, Kolm Saigurn, and Rauris. Our setup is similar to the one discussed in [18], where the authors also provide a thorough description of the seventeen-member Austrian ensemble system ALADIN-LAEF. In this work, we consider a temporal period from January 2014 through May 2018, when we have both the raw forecasts and the corresponding true observations.

As the scope of this paper is to analyze the effect of ties, we examine a simulated scenario from this real data situation. Specifically, we introduce ties artificially by simply rounding the values of the raw ensemble forecasts to the closest integer. Table 1 reports five simple tie-breaking rules, which suffice to show the impact of solving ties in our case study. Our choice here is arbitrary and motivated by the exploratory goal of this work. Other criteria that apply to more than three stations are, of course, possible but beyond the scope of this work.

Effect of ties on the empirical copula methods for weather forecasting 5 For each day of the testing period, we perform our analysis as follows.

1. We apply EMOS to obtain a corrected distribution from the raw forecasts of each station and construct a sample of size M = 17 by taking uniform quantiles. 2. We reorder the corrected univariate samples according to Table 1.

3. We compute standard scoring rules, namely, the Energy Score [5] and the Vari-ogram Score [22], to quantify the quality of the corrected multivariate forecasts. We report the results in Table 2. The scores are averaged over the entire testing period, and a better forecast corresponds to a lower score. The comparison also in-cludes the sample of uniform quantiles with no reordering, corresponding to matrix C in the example discussed in Section 2. Such a sample, named EMOS-Q, is unre-alistic and can be used as a baseline for the other methods. Looking at the scores, we notice that ECC generally improves the forecasts if compared with EMOS-Q. This indicates that, despite the presence of ties, the partial rank structure of the raw ensemble forecasts is still useful to obtain a more accurate multivariate prediction. We now analyze the performance of each ECC method, corresponding to a different way of solving ties. From Table 2, we conclude that ECC 4 shows the best perfor-mance both in terms of Energy score and Variogram score. The difference between ECC 4 and ECC 3, and ECC 1 and ECC 2, respectively, reflects the effect of the non-tied values on the final empirical structure of the multivariate corrected fore-cast. This suggests the importance of choosing the most effective tie-breaking rule for a specific partial rank structure. Overall, we notice that only ECC 1 has higher scores than ECC 5, which suggests that, in this case, randomization does not result in the best multivariate forecast.

Conclusions. In this work, we discuss the impact of tied raw forecasts on the per-formance of ECC. In the future, we plan to provide a more general and effective way of solving ties in this context by accounting for the partial rank structure of the untied raw forecasts.

Table 1 The five ways of breaking ties for each corresponding station.

Station 1 Station 2 Station 3

ECC 1 Ascending order Ascending order Ascending order ECC 2 Descending order Descending order Descending order ECC 3 Ascending order Descending order Ascending order ECC 4 Descending order Ascending order Descending order

ECC 5 Random order Random order Random order

Table 2 Variogram score and energy score averaged over the period Jan 2014 – May 2018.

ECC 1 ECC 2 ECC 3 ECC 4 ECC 5 EMOS-Q

Variogram Score 0.381 0.368 0.362 0.350 0.379 0.429

Energy Score 2.710 2.686 2.689 2.668 2.705 2.787

Acknowledgements FD has been partially supported by MIUR-PRIN 2017, Project “Stochastic Models for Complex Systems” (No. 2017JFFHSH).