In the smart manufacturing sector, analyzing time series data is essential for monitoring plants and machinery to prevent costly failures or shutdowns. In order to gain new insights and make better control decisions, new methods are needed for extracting information and interpreting sensor data from hundreds of systems.
Due to advances in digitalization and the growing market in the smart manufacturing sector, most plants and machinery are now equipped with new sensor technology. The possible use cases of sensors can vary greatly: in the aluminum-production industry, sensors are used to monitor the state of reduction cells 
or to measure individual anode currents 
while in the food-refrigeration industry sensor data are analyzed to predict over-temperature disturbances 
. To gain new insights about a system, the sensor data have to be preprocessed, analyzed and visualized in a compact way. However, manually analyzing the data from each sensor is a time consuming process. To tackle this problem new methods have to be developed that are capable of quickly extracting information and interpreting the sensor data from hundreds of systems.
Calculating certain features is a promising solution for quantifying the characteristics of a time series 
which reduces the amount of data to a specific number of meta-features and can be understood as a dimensionality reduction technique 
. According to 
, a representation of time series and similarity metric can be used in different fields, i.e., query by content, anomaly detection, motif discovery, clustering and classification. Furthermore, calculating meta-features is an essential part in the field of meta-learning to improve learning systems by incorporating already collected knowledge 
Meta-features should describe the “global picture” of a time series 
. Specific time series meta-features, such as the trend (or trend-cycle) and seasonal strength, can be computed just like the mean or median. Examples of these meta-features can be seen in Figure 1
, which shows different time series from the Makridakis Competitions, i.e., M1- and M3-Competition 
. Strength values close to 1 indicate a strong trend or seasonal component in the corresponding time series. A strength value close to 0 indicates a weak trend or seasonal component in the corresponding time series.
Several raw time series from the M1- and M3-Competition 
, which have different characteristics regarding the trend and seasonal strength. Values were rounded to two decimal places. Trend strength and seasonal strength were computed using feasts 0.1.6 
. Further examples can be found in 
Beside the trend and seasonal strength, more time series meta-features, e.g., the spectral entropy, seasonal period or autocorrelation coefficients, can be computed. Time series meta-features should be selected according to the type of the time series and the problem in question 
. Expert knowledge about a specific domain, in the case the aluminum electrolysis, can help to find suitable meta-features that describe the different characteristics of time series.
The core business of TRIMET Aluminium SE is the development and production of aluminum products. Figure 2
shows one of the three pot rooms containing 120 reductions in an end-to-end configuration at the Essen site in Germany. In total, TRIMET Aluminium SE Essen (TAE) operates 360 reduction cells that produce liquid aluminum. The industrial production of aluminum is based on the Hall–Héroult process, in which alumina (Al2
) is dissolved in liquid cryolite (Na3
) with an excess of aluminum fluoride (AlF3
. The Hall–Héroult process is performed with an almost constant energy supply to ensure stable aluminum production 
Figure 2. One of the three pot rooms operated at the Essen site in Germany by TRIMET Aluminium SE. One pot room contains 120 reduction cells in an end-to-end configuration (©TRIMET Aluminium SE).
Due to the energy transition, the energy supply in Germany is increasingly reliant on renewable energy sources, which means that the energy supply for the industrial aluminum production is no longer constant but variable 
. In order to ensure a stable operation of the reduction cells with a variable energy supply, TAE has started to equip the reduction cells with magnetic field compensation and shell heat exchangers 
. The magnetic field compensation is used to prevent bulging of the liquid aluminum at high current levels, while the shell heat exchangers are designed to control the heat loss from the side walls of the cells, maintaining a protective side ledge. Additional thermocouples are mounted on some cells to monitor the heat balance. Due to the corrosive environment, loose or faulty thermocouples are unavoidable and can result in incorrect heat balance calculations. Abnormal thermocouples are identified by manual evaluation of the recorded temperature signals (time series). However since hundreds of temperature signals have to be checked, this procedure is time consuming.
2. Literature Overview: Time Series Meta-Features
Grabowski et al. 
calculated meta-features, i.e., the mean for numeric data and the sum for binary data, of several process variables in a rolling window to predict the bath temperature of reduction cells using a random forest. Kremser et al. 
used several time series meta-features from 
in a rolling window to predict anode effects in reduction cells using logistic regression, linear support vector machine, random forest and eXtreme Gradient Boosting.
Nanopoulos et al. 
used a multilayer perceptron (MLP) and eight time series meta-features to classify patterns in control charts. The authors compared the performance of the MLP with another MLP that is solely based on the values of the time series. In further experiments, the performance of both MLPs was analyzed with noise corrupted time series and time series at varying length. Horvath et al. 
used hierarchical clustering and seven parameters (meta-features) to divide reduction cells with similar behavior into control groups.
Hyndman et al. 
computed 18 meta-features from time series representing the performance of servers at the internet company Yahoo. With the help of a principal component analysis (PCA), 𝛼
-hulls and highest density regions, the authors used the calculated meta-features to identify entire time series that differed from other time series in the data set. In 
, the extreme value theory was used to identify anomalous streaming time series data. The presented framework is based on the calculation of 14 time series meta-features, which are used in an offline and an online phase. The offline phase is used to train a model on the typical behavior of the system. Afterward, the model was deployed in the online phase to identify anomalous time series using a rolling window. Furthermore, the authors present an algorithm that updates the model if a significant change in the distribution of the typical behavior of the system is detected. In a practical example, the authors used the framework to identify anomalous sensor data.
Kang et al. 
computed six time series meta-features to visualize and analyze the M3-Competition data set 
. The authors performed a PCA using the calculated meta-features, then visualized the first and second principal component (PC) to obtain an overview of the time series characteristics of the M3-Competition data set. Furthermore, the two-dimensional feature space and a genetic algorithm were used to generate new time series that extend the original data set. The authors then compared the feature space with the performance of selected forecasting methods. Talagala et al. present in 
the FFORMS framework which includes the training of a classifier with time series meta-features to predict an algorithm that might be appropriate in forecasting a time series.
The authors in 
applied the meta-learning approach to automate the process of selecting forecasting models for individual time series using meta-features. In their first case study decision trees with 10 meta-features calculated from stationary time series were trained to choose between two forecasting models. The second case study considers NOEMON 
and five meta-features, which were calculated for each time series in the yearly M3-Competition data set 
, to rank forecasting models, i.e., random walk, Holt’s linear exponential smoothing, and auto-regressive model.