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BMeteorologische Zeitschrift, PrePub DOI 10.1127/metz/2018/0908 PrePub Article
© 2018 The authors
Introducing Gradient Boosting as a universal gap filling tool
for meteorological time series
Philipp Körner∗, Rico Kronenberg, Sandra Genzel and Christian Bernhofer
Technische Universität Dresden
(Manuscript received February 8, 2018; in revised form August 14, 2018; accepted August 14, 2018)
Abstract
In this article, Gradient Boosting (gb) is introduced as an easily adaptable machine learning method to fill gaps
caused by missing or erroneous data in meteorological time series. The gb routine is applied on a large data
set of hourly time series of the measurands: air temperature, wind speed and relative humidity for station-
based observations in Germany covering the period from 1951 to 2015. Our analysis shows that Gradient
Boosting produces small errors by estimating missing values with median RMSE for temperature of 0.73 °C,
wind speed of 0.82 m/s, relative humidity of 4.3 %, respectively. The comparison between the results achieved
by Gradient Boosting with results from other gap filling techniques like neural networks or multiple linear
regression shows considerably better statistics. The comparison clearly shows that the Gradient Boosting
approach outperforms the other techniques particularly in calculation time, performance and the handling of
missing predictor values.
Keywords: gap filling, missing values, hourly resolution, gradient boosting, xgboost, air temperature, wind
speed, relative humidity, Germany, xgb, imputation
1 Introduction1
Gap filling of meteorological time series of sites is a2
necessary issue for a wide range of tasks like time series3
analysis, hydrological, meteorological or climatological4
modelling and working with fluxes, where continuous5
data series are required. Therefore a broad selection of6
methods for the task of dealing with missing data is7
known.8
A linear interpolation from adjacent time steps is rea-9
sonable, in case a few data points are missing (Reich-10
stein et al., 2005). Autoregressive models were found11
suitable for longer periods of missing values without ad-12
jacent sites as predictors (Giller, 2012;Shahin et al.,13
2014).14
In the case adjacent sites are available, on the one15
hand, spatial interpolation like inverse distance weight-16
ing (Xia et al., 2001), Kriging (Stein, 2012) and thin17
plate spline interpolation (Hijmans et al., 2005)are18
commonly used. On the other hand, data-driven methods19
like nearest neighbor (Junninen et al., 2004), linear or20
multiple linear regression (Junninen et al., 2004;Kot-21
siantis et al., 2006;Ramos-Calzado et al., 2008) look22
up tables (Reichstein et al., 2005) or artificial neural23
networks (Falge et al., 2001; Reichstein et al., 2005)24
were found to be successful for the task of filling gaps25
in time series.26
In this paper, “Gradient Boosting” (gb), a method27
commonly used in the field of data mining (Lawrence28
et al., 2004; Li et al., 2007), is introduced as an alter-29
native to the aforementioned approaches. Therefore, a30
∗Corresponding author: Philipp Körner, Technische Universität Dresden,
Pienner Straße 23, 01737 Tharandt, philipp.koerner@tu-dresden.de
large data set of wind speed, air temperature and rela- 31
tive humidity in hourly resolution for up to 588 sites in 32
Germany is created. This data served as test case to fill 33
gaps of missing values in time series. The objective of 34
this paper is to demonstrate that the Gradient Boosting 35
method can be applied to fill gaps in time series to reach 36
a maximum of performance. 37
2 Material and methods 38
2.1 Case study region 39
The investigated sites cover the whole territory of the 40
federal state of Germany from the coast of the Baltic 41
Sea and North Sea with an elevation of 0 m a.s.l. and 42
the lowlands in the north to the Alps in the south and 43
the highest mountain “Zugspitze” with an elevation of 44
2962 m a.s.l. The Climate is characterized as a transition 45
region from Oceanic Climate of Western Europe to Con- 46
tinental Climate of Eastern Europe. Mean annual tem- 47
perature spreads from −4.8 °C at mountain Zugspitze 48
to 11 °C at the Upper Rhine Plain in the period of 49
1981–2010. 50
2.2 Meteorological data 51
We are analyzing hourly data of relative humidity (rH), 52
temperature (T) and wind speed (u). The period of in- 53
vestigation is 1951 to 2015. The data are provided by the 54
German Weather via public ftp Server. We downloaded 55
the data in March 2016. 56
Altogether, we collected 588 time series of observed 57
relative humidity and temperature and 397 time series 58
© 2018 The authors
DOI 10.1127/metz/2018/0908 Gebrüder Borntraeger Science Publishers, Stuttgart, www.borntraeger-cramer.com
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Figure 1: Temporal availability of sites for the parameters tempera-
ture, relative humidity and wind speed.
of observed wind speed data. Since several sites were59
closed down, the maximum of available sites per year is60
less than the sum of the available sites over time. We61
analyzed only hourly time series with a minimum of62
at least 20,000 observations. Fig. 1shows the temporal63
availability of temperature, relative humidity and wind64
speed sites, respectively.65
The data provided by DWD was quality checked by66
the DWD (Behrendt, 1992, quoted in Isotta et al.,67
2014), but still we could find suspicious values in the68
data. Our suspicion was raised by temporal drifts, erro-69
neous values or trends in the series. Since the data and70
methods presented are freely available, we did not cor-71
rect or exclude erroneous values to keep the results of72
our study reproducible.73
2.3 Gradient Boosting74
Gradient Boosting was introduced by Friedmann75
(1999) and Friedmann et al. (2000). We refer to both76
papers for a detailed mathematical description of the ap-77
proach. However, the approach is about the building of78
an ensemble of regression trees with an output depend-79
ing on one or more predictors. For instance, a single tree80
model makes a single value prediction, but has a low per-81
formance. The advantage of the approach is that it uses82
not only one, but many of these trees. The number of re-83
gression trees depends on the investigated data and the84
number of leafs of a single tree. Due to the divergence85
within the data set and its size, the approach establishes86
hundreds to thousands of trees. The result is calculated87
as the sum of the output of all the regression trees the88
model is built of (see Fig. 2).89
Gradient Boosting is classified as machine learning90
method, which is why the approach needs a learning91
phase with a training data set. In this training period92
Figure 2: gb-Regression tree ensemble; the predicted temperature
value is the sum of the predicted values of the respective leafs
(Modified after Chen and Guestrin, 2016).
either a predefined number of trees or the ‘best’ number 93
of trees is built, which is decided by a cross validation in 94
runtime. The orange boxes (Fig. 2) with the probabilities 95
are called leafs, and, additionally, every prediction (blue 96
boxes) and every leaf is a node. 97
Fig. 2gives an example how the results are calcu- 98
lated. The goal of the gb-approach is to predict the tem- 99
perature value at an arbitrary site X. Let us assume we 100
have two predictors, Temperatures T1 and T2 measured 101
at two neighboring sites. The model is trained for sev- 102
eral time steps, which optimizes thresholds (circles in 103
Fig. 2) and predictions (rectangles in Fig. 2) for each tree 104
(Friedmann, 1999;Chen and Guestrin, 2016) with a 105
greedy algorithm. Target figure is a minimum RMSE. 106
After the training period, the model runs through all the 107
regression trees and sums up the predictions of the re- 108
spective trees to predict the value for a certain time step. 109
In Case 1, Tree 1 results in −4 °C, Tree 2 adds up 1 °C, 110
whichsumsuptoatemperatureof−3 °C at the site X. 111
Case 2 we achieve a temperature value for site X of 1 °C. 112
However, the tree architecture is not limited to our ex- 113
ample. One tree may contain several leafs, typically 2 to 114
few hundred; the nodes in one tree may recall different 115
predictors and a predictor can occur multiple times in 116
different trees. 117
The example aside, for every respective site and each 118
parameter, a single model is built with the “R”-package 119
“xgboost” (xgb) (Chen and Guestrin, 2016). The ad- 120
ditional “x” is for “extreme” and refers to a faster opti- 121
mization algorithm. The data set is split into two parts, 122
training and validation period. In the training period, the 123
regression trees are built providing a minimum RMSE 124
between model prediction and measured values in the 125
training period. Afterwards, the model calculates values 126
for the validation period, which are used to calculate the 127
statistical measures. For each parameter of a respective 128
station, the validation time steps are different and inde- 129
pendent from each other. The time series of the predic- 130
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Meteorol. Z., PrePub Article, 2018 P. Körner etal.: Introducing Gradient Boosting as a universal gap filling tool 3
tors are selected through the example of the temperature
131
at the arbitrary site “X” as follows:132
• All time series of temperature are selected; except the133
time series of site “X”.134
• All time series with less than 8760 overlapping135
hourly time steps (approx. one year) with site “X“136
in calibration period are excluded.137
• All time series with less than 8760 overlapping138
hourly time steps (approx. one year) with site “X”139
in validation period are excluded.140
• The 100 sites, that are closest to the site “X” by Eu-141
clidian distance and fulfill the previous requirements,142
are selected as predictors.143
The aforementioned steps are performed for every144
single site and meteorological parameter, respectively.145
The following error corrections were necessary: in all146
cases, generated negative values for wind speed and147
for relative humidity are set zero, values above 100 %148
humidity are set to 100 %.149
Our results display only the validation period (Ta-150
ble 4to Table 5).151
2.4 Multiple linear regression152
Two other commonly used methods, multiple linear re-153
gression and neural networks, are used for comparison154
with gb-results. Due to calculation time issues, this com-155
parison was done with a reduced data set (Section 4).156
The best possible multiple linear regression model157
(mlr) with the other sites for a considered time step is158
applied to generate values as a replacement for miss-159
ing values. The procedure is performed using R soft-160
ware (RCoreTeam, 2015). The R package leaps (Lum-161
ley, 2017) offers an efficient branch-and-bound algo-162
rithm (Land and Doig, 1960) to perform an exhaustive163
search for the best subset. The regression model with the164
highest adjusted R2is selected and subsequently com-165
puted with all available pairs of variates and applied for166
the gap filling.167
2.5 Neural network168
The applied neural networks (nn) are designed as169
feed-forward neural networks (Dibike and Coulibaly,170
2006). The hidden layer was fitted to five neurons with171
sigmoid activation functions. The output neurons have172
a linear activation function. The weights are adjusted173
by the Levenberg-Marquardt algorithm in the training174
(More, 1977). Target figure is a minimum RMSE. The175
input consists of all the available station data of the176
reduced data set (see Section 4) without the station,177
which is filled by the approach. The nn output is vali-178
dated against the observed values of the respective sta-179
tion filled by the network. Due to a random initialization,180
the neural network routines are applied for each site ten181
times, the best net with minimum RMSE for a site is182
selected for the comparison.183
2.6 Statistical evaluation 184
The following measures of performance are used to 185
evaluate the results: root mean square error (RMSE), 186
mean absolute error (MAE) and squared coefficient of 187
correlation (R2). These measured are defined by: 188
RMSE =n
i=1(xi−yi)2
n(2.1)
with xias the measured value at time i,yias the predicted 189
value at time ifor npredictions. 190
MAE is defined as 191
MAE =n
i=1|xi−yi|
n(2.2)
R2equals the square of the Pearson correlation coeffi- 192
cient. 193
The lower the RMSE, MAE and the higher the R2194
value, the closer the results are to the measured values. 195
3 Results and discussion 196
The presented hourly gap filling approach was per- 197
formed for observed hourly temperature, relative humid- 198
ity and wind speed for the available DWD-data set from 199
1951 to 2015. Statistical measures RMSE, MAE, R2are 200
given for minimum, 25th percentile, median, 75th Per- 201
centile and Maximum, respectively. All of our results are 202
based of the 50 % validation data, which was excluded 203
from the training. 204
3.1 Temperature 205
Table 1shows the analysis based on the quantiles of the 206
resulting measures. Besides the minimum and maximum 207
values, the 25th quantile, median and 75th quantile (Q25, 208
Q50, and Q75, respectively) are 209
The measure performance for temperature at the sites 210
spreads from 0.15 to 0.5 °C (DIN IEC 751, 1990). Con- 211
sidering the median MAE of 0.52 °C achieved by gb is 212
sligthly out of this range. Combined with the high R2-213
values Gradient Boosting shows actually a strong per- 214
formance. 215
Fig. 3a) shows the scatter plot for the site “Hahn”. As 216
can be seen, there are values of −20 °C, 0 °C and 30 °C, 217
Table 1: Measures of performance for modeling hourly values for
air temperature.
RMSE [°C] MAE [°C] R2[–]
Minimum 0.43 0.31 0.943
Q25 0.66 0.47 0.989
Q50 0.73 0.52 0.991
Q75 0.82 0.59 0.993
Maximum 1.7 1.24 0.997
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(a) (b)
Figure 3: Model validation results for hourly air temperature compared with the measured values for a) site Hahn (DWD ID: 5871,
RMSE = 0.84 °C, R2= 0.988) and b) site “Münster” (DWD ID: 3404) as an example for median skills at RMSE (RMSE = 0.74°C,
R2= 0.990).
Table 2: Measures of performance for modeling hourly values of
relative humidity.
RMSE [%] MAE [%] R2[–]
Minimum 2.7 1.9 0.53
Q25 3.8 2.7 0.91
Q50 4.3 3.0 0.94
Q75 5.0 3.5 0.95
Maximum 16.3 11.5 0.98
respectively, in the site data that are not out of the cli-218
matic range, but still erroneous. For instance, a sudden219
jump from 10 °C to −20 °C and back to 10 °C occurs.220
This causes a relatively high RMSE of 0.84 °C. Never-221
theless a high correlation of 0.988 could be achieved.222
Fig. 3b) shows validation results for the site “Münster”223
as a representative sample of the data set without erro-224
neous measurements.225
3.2 Relative humidity226
The statistics of the measures of performance for the gb227
results of relative humidity are presented as minimum,228
maximum, Q25, Q50 and Q75, respectively in Table 2.229
The R2-value, as well as the error-values, show a230
very high gap filling performance. Compared to the231
measuring performance at the sites, which scatters from232
1% to 3% (HMP45A/D User Guide, 2006), the me-233
dian MAE-value is within this range. However, there234
are single stations with a significant lower perfor-235
mance. All these sites have in common that they ei-236
ther are exposed single mountain sites like “Zugspitze”237
(2964 m a.s.l., RMSE: 16.3 %, R2: 0.53) or “Feldberg/ 238
Schwarzwald” (1490 m, 13.4 %, 0.62) or coastal sites 239
on islands like “Arkona” (43 m, 6.2 %, 0.70) or “Helgo- 240
land” (4 m, 5.3 %, 0.71). These exposed sites are often 241
influenced by air masses, which are not represented in 242
the measured values at the other stations. 243
Therefore, an additional model run with a changed 244
configuration was performed, to show the potential of a 245
possible improvement: 246
First, the data set is divided into 5-year sub data 247
sets. This time step increases the amount of available 248
predictor sites, especially during the first decades with 249
lower observation density, see Fig. 1.250
Second, the temperature values of the closest 100 251
stations per time step to a respective site, are added as 252
predictors. Generally, correlations between temperature 253
and relative humidity over a year are quite poor; there- 254
fore, a significant increase of performance might not be 255
expected. However, the aforementioned structure in the 256
gradient boosting model can also be interpreted as some 257
kind of weather pattern classification. Hence, the addi- 258
tional temperature information is mostly used to decide 259
which weather pattern is predominant at a certain time 260
step. 261
If the temperature of the gap-filled station itself is in- 262
cluded, the improvement is much higher than without, 263
but to get results that are more realistic, these tempera- 264
ture time series were excluded as predictors. The statis- 265
tics of the measures are shown in Table 3.266
Both Table 3and Fig. 4show considerably better re- 267
sults in case that temperature is taken into account as 268
additional predictor. The improvement is obvious in all 269
statistical parameters and sites. It affects especially the 270
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Meteorol. Z., PrePub Article, 2018 P. Körner et al.: Introducing Gradient Boosting as a universal gap filling tool 5
(a) (b)
Figure 4: Histogram of R2(a) and RMSE (b) for hourly rH gap filling. Red bars show distribution with rH as predictor, blue bar with rH
and T as predictors and the subdivision in 5-year sub data sets.
Table 3: Measures of performance for modeling hourly values of
relative humidity with additionally temperature as predictor, divided
in 5-year sub data sets.
RMSE [%] MAE [%] R2[–]
Minimum 2.5 1.8 0.75
Q25 3.2 2.3 0.93
Q50 3.5 2.5 0.96
Q75 4.3 3.1 0.97
Maximum 11.9 7.9 0.98
sites with lower performance. For instance, RMSE at271
medium performance decreases from 4.3 to 3.5 %, thus272
temperature as an additional predictor explains approxi-273
mately 20 % of the residual variation. It is expected that,274
adding wind speed, wind direction, air pressure or radi-275
ation information as predictors would lead to a further276
improvement of the gap filling performance.277
Fig. 5shows scatterplots for site “Fritzlar (Flug-278
platz)” as an example for median skills at RMSE for279
(a) relative humidity as predictor and (b) relative hu-280
midity and Temperature as predictors. Direct compari-281
son and statistical measures show the increased perfor-282
mance at the example site.283
3.3 Wind speed284
The measures of performance of the results achieved by285
gb are summarized in Table 4. The table shows an analy-286
sis based on the quantiles of the resulting measures. Be-287
sides the minimum and maximum values, the 25th quan-288
tile, median and 75th quantile (Q25, Q50, and Q75, re-289
spectively) are shown.290
Table 4: Measures of performance for modeling hourly values of
wind speed.
RMSE [m/s] MAE [m/s] R2[–]
Minimum 0.44 0.31 0.47
Q25 0.70 0.52 0.79
Q50 0.82 0.61 0.84
Q75 0.96 0.72 0.88
Maximum 3.04 2.29 0.94
The data resolution at the sites spreads from 0.1 291
to 1.0 m/s, since measure devices changes in time 292
(Deutscher Wetterdienst, 2017). The median MAE 293
of 0.61 m/s over all sites lies with this range, combined 294
with the high R2-values, Gradient Boosting shows a 295
strong gap filling performance. 296
An example of outliers can be seen in Fig. 6a), where 297
wind speed above 30 m/s is considered as a suspicious 298
observation. The site Hohn provides wind speed data 299
from 1969 to 2015. Approximately 30 probably erro- 300
neous dates in this period did not influence the high 301
correlation (R2=0.89). In comparison Fig. 6b) shows 302
the validation for the site Manschow, without suspicious 303
observations in the time series and an even smaller R2-304
value. 305
Fig. 7presents the spatial distribution of the gap 306
filling performance (RMSE) at the sites for wind speed 307
(a), relative humidity (b, 5-year steps, rH and T as 308
predictors) and air temperature (c), respectively. The 309
aforementioned weaker performance at higher altitudes 310
and close to the border is visible as well as the overall 311
high performance for all the measurands. 312
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6P. Körner etal.: Introducing Gradient Boosting as a universal gap filling tool Meteorol. Z., PrePub Article, 2018
(a) (b)
Figure 5: Measured and predicted values for hourly relative humidity at site “Fritzlar (Flugplatz)” (DWD ID: 1504) as an example for
median skills at RMSE with a) rH as predictor (RMSE = 4.27%, R2= 0.934) and b) rH and Temperature as predictors (RMSE = 3.48 %,
R2= 0.957).
(a) (b)
Figure 6: Model results for hourly wind speed compared with the measured values at a)‘site Hohn (DWD ID: 2303, RMSE = 0.85 m/s,
R2= 0.89) and b) site “Manschow” (DWD ID: 3158) as an example for median skills at RMSE (RMSE = 0.82 m/s, R2= 0.86).
4 Comparison of different gap filling313 approaches314
Due to the out of scale calculation time for nn and315
mlr the dataset was reduced to a test data set with316
lower amount of stations, 16, and a shorter time period,317
1990–2015, for the comparison between gradient boost-318
ing, nn and mlr. Methods gb, nn and mlr run the same319
tasks with the same predictors. For instance, temperature320
at site 1 is calculated with the temperature values at sites321
2 to 16 as predictors, temperature at site 2 is calculated 322
with temperature at sites 1 and 3 to 16 as predictors. 323
50 % of the time series was used for calibration, the 324
other 50 % for validation. The results are shown for the 325
validation period. 326
Table 5summarizes the performance of the gradient 327
boosting method, neural networks and multiple linear 328
regression, respectively. Due to the reduced number of 329
predictors, it does not show the accessible performance 330
of the methods (see Table 1to Table 4). 331
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Meteorol. Z., PrePub Article, 2018 P. Körner et al.: Introducing Gradient Boosting as a universal gap filling tool 7
Figure 7: spatial distribution of RMSE for hourly wind speed (a), relative humidity (b) and temperature (c) at the sites.
Table 5: Mean statistical measures for temperature, relative humid-
ity and wind for gradient boosting (xgb), neural networks (nn) and
multiple linear regression (mlr).
Temperature RMSE [°C] MAE [°C] R2[–] Calculation
time [min]
xgb 1.42 1.05 0.965 1.4
nn 1.57 1.16 0.956 900
mlr 2.09 1.54 0.926 470
Relative
Humidity
RMSE [%] MAE [%] R2[–] Calculation
time [min]
xgb 8.8 6.4 0.71 1.2
nn 9.3 6.9 0.67 470
mlr 12.1 8.9 0.51 360
Wind Speed RMSE [m/s] MAE [m/s] R2[–] Calculation
time [min]
xgb 1.52 1.16 0.55 0.9
nn 1.56 1.20 0.52 360
mlr 1.97 1.45 0.40 255
The comparison shows that xgb performs best ac-332
cording to all statistical measures for meteorological pa-333
rameters temperature, relative humidity and wind speed,334
followedbynnandmlr.335
More precisely, xgb performs best for every single336
site for all the three meteorological measured values337
concerning RMSE, MAE and R2. Additionally, the cal-338
culation speed is in the worst case still 255 times faster339
if compared to mlr.340
Running xgb for the comparison task in this section341
lasted 210 seconds in total, while nn lasted 29 hours and342
mlr 18 hours on the same machine. Moreover, xgb and343
mlr are, more or less, linear scalable, while calculation 344
time at nn increases in an exponential way. 345
5Conclusion 346
Gradient boosting proved to be a very suitable tool to 347
fill gaps in meteorological time series. There are several 348
advantages compared to other methods like multiple 349
linear regression or neural networks. 350
• Computation time: The calculations can be done with 351
an ordinary desktop PC in 1/500 to 1/300 of the time 352
for the test data set compared with neural networks 353
and multiple linear regression, respectively. 354
• Handling of missing values in the predictor data set: 355
There is no need to fill missing predictor data; the 356
procedure learns how to handle missing data the best 357
way in the training period. 358
• Performance: Gap filling performance shows very 359
good results which can and should be further im- 360
proved by preprocessing the data, like excluding out- 361
liers and applying detrending for wind speed time se- 362
ries as well as calculating the wind speed at the stan- 363
dard height of 10 meters. 364
Additionally, the results can be improved by splitting 365
the data set into decades or even smaller periods and 366
use the 100 closest stations as predictors per time step. 367
Moreover, using 90 % of the data instead of 50 % for the 368
training period will improve the results. Improvements 369
also can be achieved, if other parameters even with 370
low correlations, like temperature and relative humidity, 371
are used as combined predictors. However, due to the 372
additive calculation, the performance for extremes, are 373
expected to be relatively weak. 374
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8P. Körner etal.: Introducing Gradient Boosting as a universal gap filling tool Meteorol. Z., PrePub Article, 2018
Finally, as pointed out, xgb outperformed other ap-375
proaches in the achieved gap filling performance and in376
the computational time. First tests with further measures377
like air pressure, precipitation and fluxes (not shown)378
suggest the method would be applicable for a wide range379
of measurands.380
Acknowledgments381
We want to thank Klemens Barfus for proofreading,382
the reviewers for many helpful hints and the German383
Weather Service for the public data access.384
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