The selection of predictors in a regression-based method for gap filling in daily temperature datasets

Transcription

1 INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. 34: (2014) Published online 20 June 2013 in Wiley Online Library (wileyonlinelibrary.com) DOI: /joc.3766 The selection of predictors in a regression-based method for gap filling in daily temperature datasets Gianmarco Tardivo* and Antonio Berti Dipartimento di Agronomia Animali Alimenti Risorse Naturali e Ambiente, Università degli Studi di Padova., Legnaro, Italy ABSTRACT: The presence of gaps in meteorological time series is a very common problem for long term studies, for example when computer activity is needed to carry out general climatological analysis. This problem can be solved through a method to reconstruct missing data; the method must be adapted to the density of the suitable stations and the climate zone which they belong to. Regression-based ones are among the most important methods used to carry out such reconstructions. A suitable search strategy for identifying the best reconstructing stations is a basic requisite for the proper implementation of this class of methods. In this article a detailed analysis of the effects of the number of predictors for a regression-based approach and their search strategy is presented. The multiple correlation between stations, related to the distance from the target station, was studied checking performances with a recently published regression model. This study was carried out for daily data of minimum, mean and maximum temperature of a dense network (111 stations within an area of a 76.5 km radius, on average). For the density of this network and comparing the system through different values of distance from target station, a better performance was achieved when the maximum radius within which to start searching for predictors was equal to or greater than 40 km. As a consequence it can be deduced that stations used to reconstruct gaps do not strictly need to be close to the target station. Setting the maximum number of predictors at four, and the maximum radius at exactly 40 km significantly reduces the number of the cases in which the reconstructed values present a reversing of the natural order: minimum < mean < maximum temperature. KEY WORDS daily temperature; predictors; regression model Received 13 November 2012; Revised 5 April 2013; Accepted 19 May Introduction Long-term time series often contain gaps due to failures of the measuring instruments or radio-software systems acquiring data from them. This issue is particularly acute in meteorological and climatological fields where monitoring station networks are frequently used to measure key variables such as temperature, precipitation, pressure, humidity, radiation, etc. Over the years, a lot of methods were tested to provide as accurate as possible climatological data reconstructions. Many climatological models require a number of surrounding stations to reconstruct missing values of a given station; for example: between-station methods (Kemp et al., 1983), kriging approaches (Jeffrey et al., 2001), thin-plate smoothing splines (Price et al., 2000), artificial neural networks (Kim and Pachepsky, 2010), and so on. The choice of the number of surrounding stations (predictors useful to reconstruct the gap) and their closeness to the target station is strictly dependent on the type of model and on the total number of available stations and their density in the study area. * Correspondence to: G. Tardivo, Dipartimento di Agronomia Animali Alimenti Risorse Naturali e Ambiente, Università degli Studi, di Padova., Viale dell Università, Legnaro, Padova, Italy. gian0812@gmail.com If the territory includes significant altitudinal gradients, a morphological dependence should be considered. However, Agusti-Panareda et al. (2000) showed that lowland stations can be effectively used for reconstructing missing values on upland locations, despite the differences in average temperatures. In this context, there are essentially three ways to differentiate and select predictors: by climatic zone, by distance and by correlation indices. For example, Steurer s (1985) method, rely on selection of stations using broad and somewhat arbitrary climate or political boundaries; DeGaetano et al. (1995) have shown that using a distance criterion versus climate boundaries significantly reduces the overall range of errors. On the other hand, in the Normal Ratio approach (Young, 1992), the choice is driven by the value of the correlation coefficient (using the three stations having the highest values of this coefficient). In most cases, however, the selection of predictors is based in a mixed distance/correlation approach (Eischeid et al., 1995; Vicente-Serrano et al., 2010). Temperature is one of the least problematic variables and reconstructing methods are often based on multilinear regression (Eischeid et al., 1995; Nalder and Wein, 1998; Kotsiantis et al., 2006; Tardivo and Berti, 2012). Eischeid et al. (1995) solved the problem of number of predictors by requiring a preliminary choice of 10 stations 2013 Royal Meteorological Society

2 1312 G. TARDIVO AND A. BERTI Figure 1. Distribution of meteorological stations across Veneto Region (from Tardivo and Berti, 2012). Figure 2. Cumulative distribution of the extent of missing data intervals (log scale). For the 114 stations. The time span considered is from 1 January 1993 to 31 December The total number of intervals of missing days is 1480 (from Tardivo and Berti, 2012). closest to the target station stating that up to four stations are needed as predictors, while above this value could degrade the estimate. Another regression-based but adaptive method is the Tardivo and Berti (2012) one, consisting of a statisticalcomputational approach that tackles each gap separately. Comparing both methods and setting the maximum number of reconstructing stations at four or ten, it was observed that the best behaviour was achieved with four stations, especially from the point of view of inversions (cases when reconstructed data present maximum temperature less than mean or mean less than minimum), roughly agreeing with Eischeid s assertion. Similar analyses conducted in the weather forecasting field (Carr, 1988), referring in particular to the Model Output Statistics (MOS) forecasting technique, confirmed the findings of Lorenz (1956), who states that the chance of false correlations between some of the predictors and the predictand arising from observational errors increases with the number of predictors. Figure 3. Progress of CV trials for every sampling size in the case of the selection of the period before the real gap. D = (U 1) + T + I ; I = maximum sampling size allowed; i = tested sampling size ; U = number of CV trials ; T = length of the gap (days). Each u represents a CV trial (from Tardivo and Berti, 2012) It is worth noting that generally, the parameters defining the type of predictors search are chosen by experience, without a precise definition of their effects on the reconstruction of data. Thus, this article aims to evaluate the effects of (1) the variation of the maximum distance from the target station within which to start the search of predictors and (2) the number of predictors used on the quality of reconstruction. 2. Materials and methods 2.1. Data This work was carried out over 111 stations of the Meteorological Centre of the Veneto Region Environmental Protection Agency (Centro Meteorologico di

3 SELECTION OF PREDICTORS FOR GAP FILLING IN TEMPERATURE DATASETS 1313 Figure 4. Mode-values of histograms of the whole set of sizes of selected subgroups for each subgroups-set, with mn = 1 (this graph refers to Tmax data, the same behaviour was found for Tmean and Tmin). Table 1. Mean distances (in m) of the stations with suitable and available D-period (from the 1st to the 12th nearest station). Station order Mean distance (m) Teolo ARPA Veneto; Figure 1). Daily data span a period of 15 years; more than 4400 gaps can be counted, 86% of which are less than 5 d, Figure 2. A more detailed description of this data was given in Tardivo and Berti (2012) Reference method Tardivo and Berti (2012) describe a dynamic method to reconstruct gaps of daily temperature datasets, via the multiregression model; the key-points are here described (for each gap): (1) Analysis of the target station to identify a period without gaps of D days contiguous to the gap to be filled preceding and/or following the gap: D = (U 1) + T + I, U = number of CV-trials (cross-validation trials, user defined), T = length of the gap (days), I = maximum sampling size allowed; (2) Identification of two groups of stations with a continuous period of D days + gap-size available (one considering data preceding the gap to be filled, the other group following the gap), with a maximum of mx and a minimum of mn stations per group, starting searching for those within a radius of Sr kilometres. If there is not at least mn stations with available data within Sr kilometres of the target station, the search radius is increased by 10 km steps until the minimum of mn suitable stations is reached. If more than mx stations are found, the mx with the higher coefficient of determination (R 2 ) over the period of D days are selected. (3) If a target gap has sets of stations available on both sides, a choice is made ranking the two sets by R 2 and picking the set having the station with the best R 2 ; (4) From this last set the subgroup of stations reporting the smaller MAE (mean absolute error) with the target station (on the coupling period of D days) is selected; the search is done considering all the possible subsets with a maximum of mx stations; (5) Identification of the best sampling size (length of the period used for data coupling for the final reconstruction) that minimizes the error deduced from a series of trials. The length of this period (i) can vary between regressors + 2toamaximumof I days. For each sampling size i a number of CV trials equal to U is done, starting from the farthest position from the gap and moving the period of T + i days towards the real gap with a 1-d step within the interval of D days (Figure 3). For each CV trial, a gap of T days is simulated and the reconstruction of this simulated gap is done with the multiple regression obtained for the set of considered reconstructing stations in the period of i days contiguous to the simulated gap (Figure 3). For each simulated gap, a MAE is computed and, over the U CV trials, a mean MAE (MAE i ) can be computed for each i. The final sampling size (n) that will be used for the reconstruction is equal to the i value giving the minimum MAE i. (6) Reconstruction of the gap with the selected subgroup and sampling size. In Tardivo and Berti (2012) mn was setted at one, mx at four and Sr at 40 km, without special inquiries about these values Selection of predictors In this article these three parameters (mn, mx and Sr) are varied, ranging mn and mx from 1 station to 12, with mn mx and Sr from 10 to 60 km with a 10 km step. Setting Sr, mn and mx for each target station and each of their gaps, a subgroup of predictors can be found; a set of subgroups (hereafter referred as subgroups-set ) is obtained (matched with the previously set Sr, mn, mx) scanning the whole network of target stations with their gaps.

4 1314 G. TARDIVO AND A. BERTI Figure 5. Average distances of subgroups varying Sr from 10 to 60 km and setting as constant first the mn parameter (left-side) and then the mx parameter (right-side). For each subgroups-set, the mean values of sizes of all its subgroups and their mean distance (calculated over all the average distances of subgroups, of the subgroups-set, from their respective target station) were considered. Carrying out this work for each temperature: maximum temperature (Tmax), mean temperature (Tmean) and minimum temperature (Tmin). For these tests and this network of stations the best performing values of I and U discussed in Tardivo and Berti (2012) were maintained, setting I = 150 and U = 450 d for the sum of 600 d: togliamo); the same was done for the value of maximum searching distance for predictors, which was set at 100 km Evaluation of reconstruction performances During the sampling size selection (Reference method 5), a number U of gaps are simulated and reconstructed over the period of T days close to the real gap, with a sampling size equal to n (the final sampling size of Reference method 5); these simulated reconstructions (CV-trials) can then be used for assessing the performances of the reconstruction method. In this case, the

5 SELECTION OF PREDICTORS FOR GAP FILLING IN TEMPERATURE DATASETS 1315 Table 2. Differences between maximum and minimum values of distances, 95% and SD 95%, for each Sr and each temperature (Tmax, Tmean, and Tmin), varying mn and mx. Sr Distance 95% SD 95% Tmax Tmean Tmin Tmax Tmean Tmin Tmax Tmean Tmin Figure 6. Boxplot of 95% SD matched to collections varying Sr from 10 to 60 km (left-graph) and varying mn = mx (right-graph); for Tmax. The same behaviour was found for Tmean and Tmin. absolute value of the mean of the ME (AME) and the SD (over the U CV-trials) are considered as reconstruction error of a gap. For each subgroups-set the 95th percentile of the AME of all its gaps (95%) and the respective SD (SD 95%) are considered to assess the method (...to evaluate the matched Sr, mx and mn values). 3. Results and discussion 3.1. Station selection Stations that can be used as predictors in gap reconstruction must have a sufficient number of days available. If a dynamic selection method is used (Tardivo and Berti, 2012), a further condition is that the required number of days must be continuous and contiguous to the missing period in the target station. This period permits statistical (and therefore climatological) relationships with the target station to be analysed in a localized way. Generally, a dynamic and/or regression-based model deals with station-selection using correlation parameters (r and R 2 ) or with error-related statistical indexes (ME, MAE,...). In this article, R 2 is used to rank stations when the number of searched stations is equal to one, while when more than one station is searched, the possible subgroups of stations are selected via combinatorial calculus, ranking them via MAE. The selection of stations appeared to be strongly affected by mx, mn and Sr parameters, with effects involving both the number of stations selected and their distance from the target one. The mean of the number of predictors is reported in Figure 4 for each subgroupsset with mn = 1 (this graph refers to Tmax data, the same behaviour was found for Tmean and Tmin) and Table 1 representing mean distance of the closest stations, from all stations, having an available period of D days contiguous to the gaps. Setting mn = 1 permits the system to be independent from mn values and to study just the relationship of mx and Sr. If Sr 30 km, the system can find seven suitable stations at most on an average (Table 1); instead, if Sr> 30 km, the system can find 12 or more suitable stations on an average. Increasing mx the system increases the mean of the sizes (of subgroups), when Sr > 30 km (Figure 4); instead, when Sr 30 km, this mean remains constant when the corresponding number of stations (see Table 1: station-order column) is approximately reached. When Sr is greater than 30 km, the system can find 12 or more stations (Table 1), and the subgroups-set size is increased with mx (Figure 4). This suggests that the distance between the target station and the reconstructing one is not the main criterion of selection and that it is possible to find stations (and subgroups) with a high correlation even at very long distances from the target. Furthermore, it was observed (Figure 5) that the average distance of the selected stations seems to be independent from mx and varies according to mn at the smaller Sr values, while the opposite is evident for the larger Sr values. This last dependence was acquired gradually, changing Sr from 10 to 60 km. The mean value of distances reported in each graph increase with Sr (Figure 5), referring to Tmax as an example: mean value is m for Sr = 10 km, rising to , , , , up to for Sr = 60 km. When the search is restricted to a relatively reduced radius, the optimum number of predictors is mainly dictated by the parameter mn; when a number of stations equal to the minimum allowed is available, the search stops. With a wider search radius it is possible to identify many possible predictors, generally above the maximum

6 1316 G. TARDIVO AND A. BERTI Table 3. Number of inversions found, for each reconstruction procedure (for each Sr value and each mx = mn). Sr mx = mn Table 4. The maximum absolute error of inversions found ( C), for each reconstruction procedure (for each Sr value and each mx = mn). Sr mx = mn allowed mx; in these conditions the number of predictors selected is mainly dictated by mx and, frequently, stations as far away as 84 km but, nevertheless, highly correlated with the target one can be found Estimation of performance To evaluate the performance of this selection procedure the CV-trial method of Tardivo and Berti (2012) dynamic model has been used, where calculations of 95%, SD 95% and inversion errors were done for each subgroups-set. For each Sr value the values of 95th percentile of the mean error (varying mn and mx parameters) are presented in Table 2, together with their standard errors (SD 95%); the absolute values of these errors are very low and are only marginally affected by Sr. The SD 95% values tend to reach a minimum when mn = mx in comparison with the cases when mn was less than mx. Considering mx = mn in Figure 6 the standard deviation of Tmax decreases rapidly with both Sr and the number of predictors, becoming roughly stable for Sr 40 km and for more than six predictors. With a proper selection of both Sr and number of predictors it is anyway possible to obtain SD 95% values very close to the best values: when mx = mn = 4 stations and Sr 40 km, the values are only C higher than the best one; the same happens with mx = mn = 10 stations, independently of Sr. The same behaviour was found for Tmean and Tmin with minimum SD 95% values of 1.254, and C for Tmax, Tmean and Tmin, respectively. In the case of the number of inversions (cases when reconstructed data present Tmax values less than Tmean or Tmean values less than Tmin; Table 3) it can be observed that the best results were found when mx = mn was equal to or greater than four and Sr 40 km; while, considering the associated errors of inversions (Table 4), the best entries are found when mx = mn was equal to or greater than three. 4. Conclusions The results presented highlight that it is preferable to search for suitable reconstructing stations over a wide search radius. This may seem counter-intuitive, being the closeness of the predicting station being a widely used and accepted criterion for station selection. In effect in most cases the closer stations are highly correlated with the target one but it is anyway possible to identify other stations that, despite their distance from the target one, present high correlations due to some specific local trait, at least for a period of sufficient length to permit data reconstruction. A further advantage of a wide search radius is that it is possible to identify a large number of possible reconstructors and the increase in the number of predictors allows a reduction of the reconstruction errors. In our case, setting Sr 40 km it is already possible to obtain a saturated selection system, i.e. a system that almost always finds at least mx stations independently from mn. Considering both SD 95% and inversion errors, the best results are obtained for Sr = 40 km and mn = mx = 4, reaching one inversion with an error of C. Extrapolating from the specific situation, it seems to be appropriate to use a search radius allowing the identification of a number of stations roughly three times the required number of predictors. This generally permits an optimal subgroup of predictors to be identified, limiting the reconstruction error to a minimum.

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