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Concept and benchmark results for Big Data energy forecasting based on Apache Spark
- Jorge Ángel González Ordiano†^{1}Email authorView ORCID ID profile,
- Andreas Bartschat†^{1},
- Nicole Ludwig†^{1},
- Eric Braun†^{1},
- Simon Waczowicz†^{1},
- Nicolas Renkamp†^{1},
- Nico Peter†^{1},
- Clemens Düpmeier†^{1},
- Ralf Mikut†^{1} and
- Veit Hagenmeyer†^{1}
Received: 5 December 2017
Accepted: 24 February 2018
Published: 6 March 2018
Abstract
The present article describes a concept for the creation and application of energy forecasting models in a distributed environment. Additionally, a benchmark comparing the time required for the training and application of data-driven forecasting models on a single computer and a computing cluster is presented. This comparison is based on a simulated dataset and both R and Apache Spark are used. Furthermore, the obtained results show certain points in which the utilization of distributed computing based on Spark may be advantageous.
Keywords
Introduction
The transformation of the current energy grid into a Smart Grid [1] is an ongoing challenge in the pursuit of an environmentally-friendly energy supply. This transformation is exemplified—from a data perspective—by the European Union’s decision to replace 80 percent of electricity meters with smart meters by the year 2020 [2] and by the desire to automate and monitor each of the power grid’s voltage levels [3]. The increasing installation of information and communication technologies (ICT) comes hand in hand with an increment in the volume and variety of the collected data, i.e. Big Data. The difficulties present in the analysis and utilization of this Big Data—in the context of the Smart Grid—have caught the interest of the energy research community. Possible solutions have been proposed in the literature, e.g., the use of cloud computing [4, 5]. Nonetheless, the utilization of Big Data is not the only complication in the development of the future energy grid. The continuous integration of volatile renewable power systems (e.g., photovoltaic (PV) and wind power systems) poses an additional challenge, since power generation volatility complicates the required balancing of energy supply and demand [6]. However, data-driven forecasting models trained using the available Big Data may be a possible solution.
Unlocking the hidden potential in Big Data requires distributed algorithms and data storage systems. To this end, the present contribution offers a description of a Big Data forecasting concept that utilizes the distributed computing framework Apache Spark^{1} for the creation and application of forecasting models. Spark possesses a number of Big Data processing methodologies that may be helpful when analyzing and using Smart Grid Big Data [7]. In addition, the presented Big Data forecasting concept can serve in the development of Big Data forecasting tools, for example, in the Helmholtz Association’s Energy System 2050 (ES 2050) project.^{2}
There is widespread belief that the utilization of Big Data can improve forecasting results if its underlying patterns can be analyzed [8]. However, the creation of data-driven forecasting models with Big Data proves to be challenging, since most data-driven approaches have not been designed to work on a distributed environment [8]. Therefore, the present contribution presents a benchmark to determine the possibility of training and applying data-driven forecasting models on Big Data. The goals of the benchmark are the assessment of the necessary time to obtain data-driven forecasting models when using Big Data and to determine the point at which a distributed computing framework based on, e.g., Spark becomes necessary. These goals are achieved by comparing the required time for training and applying different data-driven forecasting models on a computing cluster (using Spark) and on a single computer (using R and Spark). A similar study with a focus in the analysis of smart meter data using distributed computing can be found in [9]. It is important to mention, that the discussion and conclusion of the present work does not come from a Spark/Big Data expert point of view, but rather from a user’s (e.g., energy researcher) perspective.
The present work is structured as follows: first general information on energy related forecasting is given. Thereafter, the Big Data forecasting concept and the conducted benchmark are presented. Afterwards, the obtained results are shown and discussed. Lastly, the conclusion and outlook are offered.
Energy forecasting
Time series forecasting models are useful at predicting values that are changing over time [10]. Hence, they are commonly used to forecast energy values, as e.g., electrical load and volatile renewable power generation. Energy forecasting models can generally be divided into white-box models, data-driven models (i.e. black-box models), and their combination (i.e. gray-box models) [11]. While white-box models conduct their forecasts through the utilization of known relations and expert knowledge (e.g., physical models for volatile renewable power generation [12, 13]), data-driven models try to infer—via data mining techniques—the relation between their input values and the future time series values.
Data-driven models (e.g., artificial neural networks, regression models, support vector regressions) have become quite popular in the energy forecasting community [11]. These models have the advantage of not requiring an explicit description of system specific properties (e.g., wind power curves, PV modules’ tilt, power line losses, customer behaviour) to conduct their forecast, as this information is implicitly contained in the measured data. Examples of data-driven energy forecasting models found in literature are given in [14–16].
As already mentioned, forecasting models that are able to predict the future power generation and/or load are of major importance in assuring the power grid’s stability. Therefore, both load and renewable power forecasting have been thoroughly discussed in literature. Several reviews outlining the state of the art of energy forecasting are presented in [17] (PV power forecasting), in [18] (load forecasting), and in [19] (wind power forecasting).
Big Data forecasting concept
Preliminary work regarding the Big Data concept described in the present section, including a first concept and benchmarks, can be found in [20]. The new widened concept for the Big Data forecasting infrastructure—i.e. an extended view of the infrastructure presented in [21] and [22]—is depicted in Fig. 1.
The foundation of the infrastructure shown in Fig. 1 is a series of Linux nodes with many CPUs and large data storage arrays. To create a computing cluster using the available Linux nodes, a distributed cluster computing system needs to be installed. Two systems that are currently being considered for the Big Data forecasting infrastructure are MapR^{3} and DC/OS.^{4} These systems allow the installation of a large range of Big Data components, as e.g., Apache Hadoop^{5} [23–25] and Spark [26], and different databases for the storage of time series data.
As shown in Fig. 1, different services are situated—in a microservice-based architecture—on top of the Big Data software stack. The data analysis service is the one that allows frontend applications to start, monitor, and manage forecasting computations on the cluster. Such applications can access the service through a REST API. Additionally, with the help of a Web UI, i.e. a more convenient frontend application, the data analyst can operate the services using a highly customizable and dynamic user interface. These frontend applications spare the data analyst any interaction with the actual cluster and hide the complexity of the different Big Data tools. Furthermore, to develop different algorithms using Spark the data analyst can use the Apache Zeppelin^{6} tool to implement and document new computation jobs dynamically.
A usage scenario for the Big Data forecasting concept is illustrated in Fig. 2. First, the data analyst implements a forecasting model using the Zeppelin software. After thorough tests, the source code can be uploaded to the cluster using the Web UI. Thereafter, the data analysis service triggers the compilation of the source code and the creation of a new Spark job that is then persisted in the Hadoop Distributed File System (HDFS). In the next step, the data analyst selects a time series training set and starts the training algorithm. The resulting model and parameters are stored in the cluster. Using these results, the forecasting model can be applied to new data. Afterwards, the forecast results are stored in the time series database with a link to the forecasting model for later retrieval. The Spark job itself has to be able to access the databases in order to load the data in a distributed way. This is an important criterion in terms of performance.
Methods (benchmark)
The benchmark conducted in the present paper has two main goals: (i) to assess the necessary time to obtain data-driven forecasting models on a distributed environment and (ii) to determine the point at which a Big Data computing framework based on Spark becomes necessary. To achieve these goals, a test scenario is conducted in which the times needed for training and evaluating data-driven forecasting models on a single computer and in a distributed environment are calculated and compared. The specifics of the tested scenario, as well as of the data and data mining techniques used, are described below. Moreover, only the computation times for training and evaluating the forecasting models are of interest in the present contribution.
Data
Data mining techniques
In the present paper three different data mining techniques are used to obtain the various data-driven forecasting models: a multiple linear regression (MLR) [32], a least absolute shrinkage and selection operator (LASSO) [32], and a random forest [33]. All created models use only five previously selected past load values as input.
While the models created from the multiple linear regression approach and LASSO are linear combinations of their used features, the models obtained from the random forest are not. A random forest is what is called an ensemble learning method, meaning that its computed forecasting model is a combination of several different underlying models created using the same data mining technique. It is important to mention that all techniques used in Spark were taken from its machine learning libraries.
Test scenario
In the test scenario, data-driven forecasting models are trained on a single computer or on a computing cluster using the previously described techniques and an amount of training data corresponding either to 1 day (1D), 1 week (1W), 1 month (1M), 6 months (6M), 1 year (1Y), 5 years (5Y), or 10 years (10Y) of the load time series. R and Spark – with Spark using eight (SC8) computing cores—are used in the case of the single computer, while on the computing cluster only Spark is utilized (SCl). The combination of the different amounts of training data and the different approaches for training the models results in 21 different tests. The abbreviations used to refer to the conducted tests are contained in Table 1. The forecasting models are, thereafter, evaluated on their training data. This evaluation consists of applying the forecasting model and calculating a corresponding evaluation value (e.g., the mean absolute error). The application and evaluation procedures are coupled since Spark does not apply a forecasting model unless it is necessary (i.e. lazy evaluation). The necessary computation time for both the models’ training and evaluation in each test is measured and compared. The time needed for loading the data in-memory is not measured in the present contribution. Additionally, since only the computation times and not the forecasting accuracy are relevant in the present article, the data set is not separated in a training and a test set.
Conducted tests
Data amount | R | SC8 | SCl |
---|---|---|---|
1D (\(8.64\cdot 10^{4}\) values) | \(R_\text {1D}\) | \(SC8_\text {1D}\) | \(SCl_\text {1D}\) |
1W (\(6.05\cdot 10^{5}\) values) | \(R_\text {1W}\) | \(SC8_\text {1W}\) | \(SCl_\text {1W}\) |
1M (\(2.68\cdot 10^{6}\) values) | \(R_\text {1M}\) | \(SC8_\text {1M}\) | \(SCl_\text {1M}\) |
6M (\(1.57\cdot 10^{7}\) values) | \(R_\text {6M}\) | \(SC8_\text {6M}\) | \(SCl_\text {6M}\) |
1Y (\(3.14\cdot 10^{7}\) values) | \(R_\text {1Y}\) | \(SC8_\text {1Y}\) | \(SCl_\text {1Y}\) |
5Y (\(1.58\cdot 10^{8}\) values) | \(R_\text {5Y}\) | \(SC8_\text {5Y}\) | \(SCl_\text {5Y}\) |
10Y (\(3.16\cdot 10^{8}\) values) | \(R_\text {10Y}\) | \(SC8_\text {10Y}\) | \(SCl_\text {10Y}\) |
Results and discussion
Mean computation times (seconds) for forecasting models training and evaluation
Tests | Training | Evaluation | ||||
---|---|---|---|---|---|---|
MLR | LASSO | Random forest | MLR | LASSO | Random forest | |
\(R_\text {1D}\) | 0.014 (0.009) | 0.407 (0.074) | 1773.078 (18.274) | 0.01 (0.008) | 0.073 (0.026) | 583.438 (16.824) |
\(SC8_\text {1D}\) | 0.208 (0.015) | 0.223 (0.026) | 22.122 (1.639) | 0.192 (0.027) | 0.188 (0.015) | 20.905 (1.755) |
\(SCl_\text {1D}\) | 0.399 (0.022) | 0.404 (0.040) | 10.342 (0.202) | 0.462 (0.029) | 0.440 (0.020) | 7.247 (0.199) |
\(R_\text {1W}\) | 0.07 (0.006) | 2.7 (0.075) | OoM | 0.072 (0.007) | 0.27 (0.021) | OoM |
\(SC8_\text {1W}\) | 0.271 (0.013) | 0.281 (0.013) | 108.993 (6.285) | 0.348 (0.022) | 0.353 (0.012) | 71.187 (7.316) |
\(SCl_\text {1W}\) | 0.390 (0.022) | 0.409 (0.025) | 28.418 (0.491) | 0.475 (0.025) | 0.476 (0.018) | 14.416 (0.169) |
\(R_\text {1M}\) | 0.324 (0.056) | 13.438 (0.247) | OoM | 0.267 (0.074) | 1.346 (0.223) | OoM |
\(SC8_\text {1M}\) | 0.627 (0.015) | 0.638 (0.018) | 454.759 (3.311) | 1.058 (0.15) | 1.074 (0.031) | 243.008 (6.786) |
\(SCl_\text {1M}\) | 0.463 (0.021) | 0.482 (0.032) | 67.469 (2.125) | 0.644 (0.042) | 0.652 (0.036) | 33.546 (1.908) |
\(R_\text {6M}\) | 1.862 (0.146) | 115.960 (17.589) | OoM | 3.206 (0.311) | 13.610 (11.116) | OoM |
\(SC8_\text {6M}\) | 2.822 (0.060) | 2.857 (0.077) | 3101.202 (14.726) | 5.420 (0.101) | 5.457 (0.150) | 1722.941 (36.603) |
\(SCl_\text {6M}\) | 1.133 (0.055) | 1.115 (0.028) | 357.388 (5.037) | 1.747 (0.050) | 1.757 (0.061) | 161.561 (2.156) |
\(R_\text {1Y}\) | 4.061 (0.165) | OoM | OoM | 6.903 (0.718) | OoM | OoM |
\(SC8_\text {1Y}\) | 5.588 (0.050) | 5.604 (0.041) | 6291.860 (43.074) | 10.809 (0.049) | 10.800 (0.053) | 3246.918 (35.248) |
\(SCl_\text {1Y}\) | 1.934 (0.061) | 1.903 (0.055) | 784.018 (11.487) | 1.934 (0.061) | 1.903 (0.055) | 304.891 (3.768) |
\(R_\text {5Y}\) | OoM | OoM | OoM | OoM | OoM | OoM |
\(SC8_\text {5Y}\) | 41.464 (0.575) | 42.520 (1.937) | NT | 72.940 (1.726) | 73.728 (2.644) | NT |
\(SCl_\text {5Y}\) | 16.104 (0.711) | 15.474 (0.683) | IOF | 26.528 (1.346) | 26.380 (1.000) | IOF |
\(R_\text {10Y}\) | OoM | OoM | OoM | OoM | OoM | OoM |
\(SC8_\text {10Y}\) | NT | NT | NT | NT | NT | NT |
\(SCl_\text {10Y}\) | 38.997 (1.869) | 39.300 (1.755) | IOF | 63.044 (2.157) | 63.127 (2.301) | IOF |
The computation times required for the training and evaluation of forecasting models for the different tests and the three different data mining techniques are shown in Table 2. The presented results are mean and standard deviation values obtained by training and evaluating the different models ten separate times. Note that the tests for a random forest with Spark on the single computer using 5 and 10 years of training data were not conducted.
For the most complex approach in this contribution, the random forest, the Spark cluster shows its full potential and is clearly the fastest. Neither the single computer Spark variant nor the R variant comes close to the cluster’s computation times. Additionally, the single computer using R runs out of memory quickly. The computer can only train a random forest with the data corresponding to one day. However, the computing cluster also fails to train a random forest with data larger than 1 year. This failure most likely stems from an integer overflow. As it can be seen in Table 2, finding a single winner for all models and amounts of data is not possible. Looking at the results for the LASSO model, R exhibits the worst computation times for training, regardless of the data amount. Yet, it is not clear which of the remaining two variants is best. While the Spark computing cluster is faster for data larger than 1 month, its single machine variant is the best choice for smaller amounts of data. With respect to the evaluation computation times, the Spark cluster is again the fastest for data larger than a week. This time however, R is the fastest for the smallest two amounts of data.
- 1.
CS1: No caching
- 2.
CS2: Caching of input data, but not of intermediate results
- 3.
CS3: Caching of input data and of intermediate results
As seen by the results obtained for CS1, no caching results in poor computation performance; this can be explained by the fact that Spark is unable to keep the necessary data in-memory. Hence, Spark is forced to read the data from the provided source using slow read operations every time the data is required. Interestingly, caching the necessary input data and all the intermediate results (CS3) is not the optimal solution either; since the used workers may run out of memory. Nonetheless, CS3 still results in the fastest training of an MLR using Spark on the cluster. CS2, i.e. caching only the necessary input data, resulted in the lowest evaluation computation times. It is important to note that CS2 is the caching strategy used to obtain the results shown in Table 2.
Mean computation times (seconds) for MLR forecasting models training and evaluation using different caching strategies
Tests | Training | Evaluation | ||||
---|---|---|---|---|---|---|
CS1 | CS2 | CS3 | CS1 | CS2 | CS3 | |
\(SC8_{6M}\) | 356.695 (4.019) | 2.822 (0.060) | 2.925 (0.047) | 233.737 (2.003) | 5.420 (0.101) | 9.467 (0.364) |
\(SCl_{6M}\) | 37.562 (0.290) | 1.133 (0.055) | 1.128 (0.041) | 47.532 (0.374) | 1.747 (0.050) | 1.777 (0.055) |
Conclusion and outlook
In the present contribution, a Big Data forecasting concept is described. Afterwards, a benchmark comparing the training and evaluation of data-driven forecasting models using different amounts of data, as well as R and Spark on a single computer and Spark on a computing cluster is presented. The obtained results show the points at which a Big Data computing framework based on Spark may be advantageous, for instance, when using a complex data mining technique or when surpassing a specific amount of data. The former is shown by the fact that Spark on the cluster has–—for the conducted benchmark—the lowest computation times for training and evaluating a complex data-driven model, i.e. a random forest; the latter is shown by Spark on the computing cluster outpacing both single computer approaches independently of the utilized technique once a data amount threshold is surpassed (i.e. in the presented benchmark a training set comprised of \(1.57\cdot 10^{7}\) values). The results also show that Spark is sensitive towards certain factors like caching and the repartitioning of data; factors that when disregarded may reduce the computational advantages provided by Spark.
Even though the present contribution showed certain benefits of utilizing Spark on a computer cluster for the creation and application of energy forecasting models, there is still a number of questions that need to be answered in future works. For example, how does the behaviour and computation times of Spark change if the necessary data is loaded from and saved into a database or if a forecasting model is implemented as an online streaming service.
Notes
Declarations
Authors’ contributions
All authors contributed equally. All authors read and approved the final manuscript.
Acknowledgments
The authors acknowledge the support given by the “Deutsche Forschungsgemeinschaft” and by the Open Access Publishing Fund of the Karlsruhe Institute of Technology. The authors also acknowledge careful English language editing by Alexander Murray (KIT).
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
The data will not be shared.
Consent for publication
Not applicable.
Ethics approval and consent to participate
Not applicable.
Funding
The present contribution is supported by the Helmholtz Association under the Joint Initiative “Energy System 2050 - A Contribution of the Research Field Energy”. This work was also partially funded by the DFG Research Training Group 2153: “Energy Status Data - Informatics Methods for its Collection, Analysis and Exploitation”.
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Authors’ Affiliations
References
- Fang X, Misra S, Xue G, Yang D. Smart grid—the new and improved power grid: a survey. IEEE Commun Surv Tutor. 2012;14(4):944–80.View ArticleGoogle Scholar
- Zhou S, Brown MA. Smart meter deployment in Europe: a comparative case study on the impacts of national policy schemes. J Clean Prod. 2017;144:22–32.View ArticleGoogle Scholar
- Monti A, Ponci F, Ferdowsi M, McKeever P, Löwen A. Towards a new approach for electrical grid management: the role of the cloud. In: Proc., IEEE international workshop on measurements & networking (M&N). 2015. p. 1– 6.Google Scholar
- Rusitschka S, Eger K, Gerdes C. Smart grid data cloud: a model for utilizing cloud computing in the smart grid domain. In: Proc., 1st IEEE international conference on smart grid communications (SmartGridComm). 2010. p. 483– 88.Google Scholar
- Simmhan Y, Aman S, Kumbhare A, Liu R, Stevens S, Zhou Q, Prasanna V. Cloud-based software platform for Big Data analytics in smart grids. Comput Sci Eng. 2013;15(4):38–47.View ArticleGoogle Scholar
- Gottwalt S, Gärttner J, Schmeck H, Weinhardt C. Modeling and valuation of residential demand flexibility for renewable energy integration. IEEE Trans Smart Grid. 2017;8(6):2565–74.View ArticleGoogle Scholar
- Shyam R, Kumar S, Poornachandran P, Soman K. Apache Spark a Big Data analytics platform for smart grid. Proced Technol. 2015;21:171–8.View ArticleGoogle Scholar
- Hassani H, Silva ES. Forecasting with Big Data: a review. Ann Data Sci. 2015;2(1):5–19.View ArticleGoogle Scholar
- Liu X, Golab L, Golab WM, Ilyas IF. Benchmarking smart meter data analytics. In: EDBT. 2015. p. 385– 396.Google Scholar
- Hyndman RJ, Athanasopoulos G. Forecasting: principles and practice. Lexington: OTexts; 2014.Google Scholar
- González Ordiano JÁ, Waczowicz S, Hagenmeyer V, Mikut R. Energy forecasting tools and services. Wiley Interdiscip Rev Data Min Knowl Discov. 2018;8(2):1235.View ArticleGoogle Scholar
- Monteiro C, Bessa R, Miranda V, Botterud A, Wang J, Conzelmann G. Wind power forecasting: state of the art 2009. Argonne National Laboratory (ANL): Technical report; 2009.Google Scholar
- Pelland S, Galanis G, Kallos G. Solar and photovoltaic forecasting through post-processing of the global environmental multiscale numerical weather prediction model. Prog Photovolt Res Appl. 2013;21(3):284–96.View ArticleGoogle Scholar
- Duran MJ, Cros D, Riquelme J. Short-term wind power forecast based on ARX models. J Energy Eng. 2007;133(3):172–80.View ArticleGoogle Scholar
- Fan S, Chen L. Short-term load forecasting based on an adaptive hybrid method. IEEE Trans Power Syst. 2006;21(1):392–401.MathSciNetView ArticleGoogle Scholar
- González Ordiano JÁ, Waczowicz S, Reischl M, Mikut R, Hagenmeyer V. Photovoltaic power forecasting using simple data-driven models without weather data. Comput Sci Res Dev. 2017;32:237–46.View ArticleGoogle Scholar
- Antonanzas J, Osorio N, Escobar R, Urraca R, Martinez-de-Pison F, Antonanzas-Torres F. Review of photovoltaic power forecasting. Sol Energy. 2016;136:78–111.View ArticleGoogle Scholar
- Hong T, Pinson P, Fan S, Zareipour H, Troccoli A, Hyndman RJ. Probabilistic energy forecasting: global energy forecasting competition 2014 and beyond. Int J Forecast. 2016;32(3):896–913.View ArticleGoogle Scholar
- Jung J, Broadwater RP. Current status and future advances for wind speed and power forecasting. Renew Sustain Energy Rev. 2014;31:762–77.View ArticleGoogle Scholar
- Renkamp N. Short-term load forecasting in district heating networks. Master’s thesis, Karlsruher Institut für Technologie. 2016.Google Scholar
- Düpmeier C, Stucky K-U, Mikut R, Hagenmeyer V. A concept for the control, monitoring and visualization center in Energy Lab 2.0. In: Proc., of the 4th D-A-CH energy informatics conference. Berlin; Springer, Cham: 2015. p. 83– 94.Google Scholar
- Hagenmeyer V, Cakmak HK, Düpmeier C, Faulwasser T, Isele J, Keller HB, Kohlhepp P, Kühnapfel U, Stucky U, Waczowicz S, Mikut R. Information and communication technology in Energy Lab 2.0: Smart energies system simulation and control center with an Open-Street-Map-based power flow simulation example. Energy Technol. 2016;4:145–62.View ArticleGoogle Scholar
- Manikandan SG, Ravi S. Big Data analysis using Apache Hadoop. In: Proc., of the 2014 international conference on IT convergence and security (ICITCS). 2014. p. 1– 4.Google Scholar
- McAfee A, Brynjolfsson E, Davenport TH, Patil D, Barton D. Big Data. The management revolution. Harvard Bus Rev. 2012;90(10):61–7.Google Scholar
- Shvachko K, Kuang H, Radia S, Chansler R. The Hadoop distributed file system. In: Proc., of the IEEE 26th symposium on mass storage systems and technologies (MSST). 2010. p. 1– 10.Google Scholar
- Zaharia M, Chowdhury M, Franklin MJ, Shenker S, Stoica I. Spark: cluster computing with working sets. In: Proc., 2Nd USENIX conference on hot topics in cloud computing. HotCloud’10. Berkeley: USENIX Association: 2010. p. 10– 10.Google Scholar
- Taieb SB, Taylor JW, Hyndman RJ. Coherent probabilistic forecasts for hierarchical time series. In: Proc., 34th international conference on machine learning. Sydney: PMLR, International Convention Centre; 2017. p. 3348– 3357.Google Scholar
- Gneiting T, Raftery AE. Weather forecasting with ensemble methods. Science. 2005;310(5746):248–9.View ArticleGoogle Scholar
- Gneiting T, Katzfuss M. Probabilistic forecasting. Annu Rev Stat Appl. 2014;1:125–51.View ArticleGoogle Scholar
- Klaiber S, Waczowicz S, Konotop I, Westermann D, Mikut R, Bretschneider P. Prognose für preisbeeinflusstes Verbrauchsverhalten. at-Automatisierungstechnik. 2017;65(3):179–88.View ArticleGoogle Scholar
- Maaß H, Cakmak HK, Bach F, Mikut R, Harrabi A, Süß W, Jakob W, Stucky K-U, Kühnapfel UG, Hagenmeyer V. Data processing of high rate low voltage distribution grid recordings for smart grid monitoring and analysis. EURASIP J Adv Signal Process. 2015;1:1–21.Google Scholar
- Fahrmeir L, Kneib T, Lang S, Marx B. Regression: models, methods and applications. Berlin: Springer; 2013.View ArticleMATHGoogle Scholar
- Breiman L. Random forests. Mach Learn. 2001;45(1):5–32.View ArticleMATHGoogle Scholar