1 Introduction

Time-series data classification is an active research area for the past decade [1]. Time series is a set of data points taken consecutively through time. It is mathematically denoted as

$$\begin{aligned} D_{e}(f);[e=1,2, \ldots , g; f=1,2, \ldots , t]; \end{aligned}$$
(1)

  • e = index of various measurement at each point of time—f,

  • t = number of observed variables, and

  • g = number of observations.

In time-series data analysis, we have one variable called time. We can scrutinize this time-series data with the objective to extract meaningful statistics and other characteristics. The main goal of this time-series data is to predict the subsequent values on the basis of previous observation values. If the data has one variable, i.e., g = 1, it is referred as uni-variate. Uni-variate analysis is a very simplest form of statistical analysis. It is essentially the descriptive analysis of a single variable used to describe characteristics of a sample. It is used to get the picture of how the sample looks like rather than examining their relationships and causes. If the data has one variable, i.e., \( g >1 \), it is referred as multivariate.

Fig. 1
figure 1

Physical model of a hydraulic system used to collect training data

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Hydraulic systems must be contained at any rate for essential segments. There should be a compartment that stores oil and liquids, a pump that impels the liquids through the system, a valve to control the pressure and flow of the liquids inside the system, and a cylinder to change over the development of liquids into actual work. There are different segments in the middle, yet all systems must have these four [2]. Figure 1 illustrates a physical model of hydraulic test rig. The test system is outfitted with a few sensors measuring process values such as flow (HFS1, HFS2), electrical power (HEPS1), pressure (HPS1–HPS6), vibration (HVS1), and temperature (HTS1–HTS5) with standard industrial 20 mA current loop interfaces connected to a data acquisition system. Sampling rates range from 1 Hz (flow sensor) to 100 Hz (electric power/pressure) to contingent upon the dynamics of the underlying physical values.

Figure 2 illustrates an example of multivariate time series and is a hydraulic test rig data, where numerous parameters such as pressure sensors, hydraulic-(HPS1–3), motor power sensor-HEPS1, and volume flow sensor-HFS, are continually measured and stored by test rig in real time. The test rig system then executed various thousand working cycles during which distinct fault conditions were simulated in all combinations. Figure 2 shows hydraulic test rig multivariate time-series data consisting of five parameters [2].

Fig. 2
figure 2

A hydraulic test rig multivariate time-series data consisting of five parameters (pressure sensors hydraulic-(HPS1–3), motor power sensor-HEPS1, and volume flow sensor-HFS

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Multivariate data classification is devised as a supervised machine learning problem mainly intended for labeling data of varying length. Each parameter is a time series, sequence of pairs (timestamp, value) [1]. Hydraulic system consists of a set of parameters like HPS1–3, HEPS1, and HFS1, i.e., a multivariate time series (MTS). This system requires to be classified as Healthy or Unhealthy, in accordance with the values of the parameters [3, 4]. The time-series classification is divided into two broad categories: [1] conventional or weak classification [3] and contemporary or strong classification. Figure 3 shows weak classification, in which each succession is affiliated with only one class label and the entire succession is available to a classifier in prior to the classification. Figure 3 shows strong classification, in which each set of time-series data is organized into a different succession of classes.

Fig. 3
figure 3

Example of weak and strong classification

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The major contributions of this paper are as follows:

  • In feature extraction phase, statistical features are explored. The features are extracted using a single-window approach as well as the segmented-window approach from raw sensor data with a given state of health.

  • For classification phase, we come up with a window-feature-based classifier which takes the window feature of the current window into account.

The rest of the paper is catalogued as follows: Sect. 2, presents related work of multivariate classification. In Sect. 3, we introduce the proposed feature extraction and classification model. Section 4 presents the experimental results of the proposed approach and Sect. 5 concludes the work.

2 Related work

Multivariate classification problem falls under two major categories: [1] distance/instance-based approach [3] and a feature-based approach. In instance-based approach, the distance between two time-series data is computed. There is an extensive survey on categories of distance measure employed for time-series classification. On the other side, feature-based approach mainly aims to represent data using a set of features or acquired properties and thus the temporal time-series problem is transformed to a static problem. For example, if we represent a time-series data using its maximum, minimum, variance, and mean, thus transforming varied length data into short length vectors which encapsulate the above four properties. Feature-based classification is most widely employed across various domains including science. This approach is mainly applied to longer time-series data like medical, electrical, mechanical, or recordings of speech signals than a short time-series pattern. Table 1 shows the list of features extracted in multivariate classification and Table 2 shows the list of algorithms used in various conditional monitoring systems. Nikolai Helwig detected sensor faults using feature-based approach and linear discriminant analysis using hydraulic dataset. A. D. Bykov applied machine learning classification approach for hydraulic system using gradient boosting, K-nearest neighbor, and SVM using hydraulic dataset. Frank L monitored health for gas turbine engine with artificial neural networks and rule-based algorithms. The data is collected by TEDANN. Yu Chen detected and diagnosed the fault of HVDC systems using extreme learning machines and bagged trees. Pallanti Srinivasa Rao et al. detected and diagnosed health of aircraft engine using ANN method. Mustagime monitored the health of aircraft engine (gas turbine) using multiple regression analysis.

Table 1 Review of features extracted in multivariate classification
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Table 2 Review of algorithms used in conditional monitoring
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3 Feature-Based Time-Series Classification Model

The modeling of the proposed framework involves two key phases: [1] feature extraction and [3] feature classification. Each of the phases is performed in two approaches: [1] single-window approach and [3] segment-window approach. Feature Extraction: In the current section, we explore the extraction of various statistical features from time-series data and their use in health monitoring (classification). Classification is accomplished on the basis of features extracted for each time series and not on real values. The extraction and selection of suitable features have been accepted as a significant problem. Obviously, the number of features required and nature of the feature (global or local) depend on their discriminating quality. The prime characteristics required for identified features are ease of computation, invariant to noise, and transformations. For multivariate data, and more especially for hydraulic systems data, we propose the use of statistical features, which are commonly used in systems health monitoring.

Algorithm 1 shows the window-feature-based classifier model. The first step in feature extraction is to divide the original noise-free multivariate time-series data \( T_{o} \) into a set of smaller sized window segments \( W_{i} \)= \( W_{1}, W_{2},W_{3}, \ldots , W_{n} \). The entire duration is divided into 14 windows of dissimilar size. For each current window \( W_{i} \) extract eight statistical features—sum, median, mean, length, standard deviation, variance, maximum, and minimum values from five given input parameters—HEPS1, HFS, HPS1, HPS2, and HPS3. In our paper, we have explored the window label \( WF_{i} \) and it is added as feature value. Hence, for each window \( W_{i} \) nine features are extracted, and this is repeated for 14 windows, for 5 parameters.

figure a

Each parameter has 6000 samples and 2205 instances, and these 6000 samples are divided into 14 windows. For each window, we generate 2205 \(\times \) (8 statistical features + 1 window feature + 1 label) = 2205 \(\times \) 42 feature map. Since we have 14 windows and 2205 instance for each window (14 \(\times \) 2205 = 30870), the training data size is 30870 \(\times \) 42. The test data given to the model is noise-induced test data (high-level, medium-level, and low-level noise). In the traditional model, the features are extracted from the entire duration and hence we treat this to be single-window feature extraction. In this research paper, we compare our approach through the traditional approach which uses single-window feature extraction. Window-feature-based classifier is very simple, they improve classification accuracy and consistent across all three noise levels. Decision tree classification technique is used to classify the time-series data.

4 Experimental Results

This section details the process of inducing noise and its characterization. Classifier performance on an original and noise-injected multivariate time-series data will be compared under this section.

4.1 Dataset

Nikolai Helwig et al. created test data, which was experimentally acquired with hydraulic system test rig. This rig has both cooling-filter circuit and hydraulic working connected through the oil tank [2]. The hydraulic system repeats constantly with load cycles for the duration of 60 s and measures sensor values such as temperatures, pressures, and volume flows while the condition of three vital hydraulic components (valve, pump, and accumulator) differed significantly. The total number of instances are 2205. Table 3 shows the sensor attributes or samples recorded with varied sampling rates.

Table 3 Dataset parameter description
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Table 4 Noise characterization: high-level noise
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4.2 Noise Characterization

Singh et al. [20, 21] and Jim et al. [22] have reported that inducing controlled noise amounts to the original data improves the performance of the model. In our study, random noise data was produced using MATLAB. Random noise (RN) array is first produced between Max(maximum) and Min(minimum) value of the parameter, at that particular instance of time. The total number of instances are 2205, out of which 1449 instances are in stable condition and 756 instances are unstable. In this research work, we have induced 600 samples (attributes) of noise data in 100 instances for each parameter (HEPS1, HFS, HPS1, HPS2, and HPS3). The noise is categorized into three levels—high-level, medium-level, and low-level noise injection. Table 4 depicts the details of percentile of noise injected within each window for each parameter. Low-level noise represents a single parameter affected by random noise. Medium-level noise represents three parameters affected by random noise. High-level noise represents all five parameters affected by random noise.

4.3 Classifier Performance

Training data and test data are created for overall instance feature extraction and window-based feature extraction as per algorithm. Following that, decision tree is used to train and test the classifier using a tenfold cross-validation technique for both traditional approach and proposed approach. To examine the effect of noise on feature extraction and feature-level classification of multivariate data prediction. We evaluated and analyzed the working model performance on noise-free data, high-level, mid-level, and low-level noise-induced multivariate data. Three major observations are made from the experimental results.

  • The classifier accuracy is >90% for both traditional- and window-feature-based classification model. Out of eight statistical features—median, standard deviation, and variance feature values are robust and correlate with fault characteristics.

  • In case of low-level noise, the classifier accuracy drops down to 93% for the traditional model and 83.43% for the window-feature-based classification model, where a single parameter out of five affected by random noise.

  • In case of medium-level noise, the classifier accuracy drops down to 93% for the traditional model and 80.34% for the window-feature-based classification model, where three parameters out of five are affected by random noise.

  • In case of high-level noise, the classifier accuracy drops further down to 65% for the traditional model and 78.4% for the window-feature-based classification model, where all five parameters are affected by random noise (Tables 5 and 6).

Table 5 FP rate, TP rate, recall, and precision for window-feature-based classification model
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Table 6 Classification accuracy of window-feature-based classification model
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5 Conclusion

The impact of induced noise on multivariate time-series data prediction is very significant to quantify for precise prediction. This paper examines the effect of noise on feature extraction and classification model. It is observed that for noise-free data the decision tree classifier accuracy is >90% for both traditional and window-feature-based classification model. When all the five parameters are affected by random noise, the decision tree classifier accuracy decreases to 65% for the traditional model and 78.4% for the window-feature-based classification model. Though the window-based classifier is simple, it improves the classification accuracy significantly in the presence of noise. The results show that the enhancement in decision tree classification accuracy is about 13% compared to a traditional classifier.