International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(4): 338-348
Published online December 25, 2021
https://doi.org/10.5391/IJFIS.2021.21.4.338
© The Korean Institute of Intelligent Systems
Aref Safari^{1}, Rahil Hosseini^{1} , and Mahdi Mazinani^{2 }
^{1}Department of Computer Engineering, Shahr-e-Qods Branch, Islamic Azad University, Tehran, Iran
^{2}Department of Electronic Engineering, Shahr-e-Qods Branch, Islamic Azad University, Tehran, Iran
Correspondence to :
Rahil Hosseini (rahil.hosseini@qodsiau.ac.ir)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Prediction of time series is associated with nondeterministic pattern analysis for uncertain conditions. Therefore, it is necessary to develop high-quality prediction methods for real-world applications. Type-2 fuzzy systems can handle high-order uncertainties, such as sequential dependencies associated with time series. Precise and reliable prediction can help to develop reasonable strategies and assist specialists in planning the best policies for modeling events in uncertain time series. In this study, a hybrid model (dynamic type-2 fuzzy time warping [DT2FTW]) was proposed for handling high-order uncertainties in time-series prediction. A type-2 fuzzy intelligent system was developed alongside a dynamic time warping algorithm for predicting the patterns’ similarity in long-time series for time-series prediction. The results demonstrate that the proposed DT2FTW model yields more reliable predictions on global standard benchmarks such as the Mackey-Glass, Dow Jones, and NASDAQ time-series. The results also confirm that the proposed DT2FTW model has lower error rates than its counterpart algorithms in terms of the root mean square error (RMSE), mean absolute error (MAE), and mean percentage error (MPE). In addition, the results confirm the superiority of the proposed model with an average area under the ROC curve (AUC) of 94%, with the 95% confidence interval (92%-95%).
Keywords: Dynamic time warping, Interval type-2 fuzzy system, Time-series prediction
Modeling of time-series is important because many pattern-analysis problems contain the time component. These problems are typically not addressed, because the time dependence makes time-series-related problems difficult to handle. One of the most challenging issues associated with time-series is their prediction. Prediction problems are often classified into short-term, medium-term, and long-term, and are characterized by different orders of uncertainty. The uncertainty associated with time-series data is implicit, with a nonlinear pattern. On the other hand, unreliable accuracy is a major issue in time-series predictions. Different time-series models, such as fuzzy time-series, have been considered for improving the prediction accuracy.
Many intelligent models have been used for analyzing time-series patterns. Recently, the use of soft computing approaches, such as fuzzy logic, neural networks, simulated annealing, and genetic algorithms, has been reported in the literature on the time-series prediction. These approaches have been considered advantageous compared with traditional methods, because they can address nonlinearities and can approximate many types of complex dynamical systems better than linear statistical models. Fuzzy logic models have been widely adopted because of the prevalent uncertainty in the time-series data. Most related studies use the Euclidean distance metric for measuring time-series intervals. The Euclidean distance metric is widely known to be very sensitive to distortions along the time axis, but it has been very popular with many researchers. The ubiquity of the Euclidean distance metric in the face of increasing evidence of its poor accuracy for time-series prediction is almost certainly owing to its ease of implementation and its time and space efficiency. However, in our work, the problem of distortion along the time axis is addressed by dynamic time warping (DTW) alongside a type-2 fuzzy logic approach, for modeling high-order uncertainties in time-series prediction based on the footprint of uncertainty and unequal lengths of time intervals.
Analysis and prediction of time-series belong to the field of temporal pattern recognition. A time-series corresponds to a stretch of values on the same scale, indexed by a naturally occurring time, as encountered in many applications in the engineering, ecology, economy, medicine, and finance fields [1]. Five concepts are important for time-series: the starting time, pattern similarity, period range, confidence, and the endpoint time. Many techniques [2–8] have been applied for time-series prediction based on the period and data type. The DTW model has been effectively used to automatically deal with time deformations in different time-series ranges with time-dependent data, for pattern recognition and similarity analysis. The DTW approach is currently used in many areas, including online signature matching and handwriting recognition [9], gestures and sign language recognition [10], knowledge discovery, data mining and clustering [11], pattern recognition and data analysis [12], and signal processing [13,14]. In addition, in many real-world applications temporal problems are complex, uncertain, and chaotic [15]. Fuzzy logic is one of the most effective methods for handling uncertainties in dynamic and non-stationary environments [16]. Table 1 lists various hybrid fuzzy models that have been used for predicting time-series [17–22]. Fuzzy systems, especially hybrid fuzzy models, have been very promising for solving complex problems, where a model estimates and predicts the similarity between two time-series in uncertain conditions [18,23–27].
The main objective of this study was to introduce an intelligent dynamic type-2 fuzzy time warping (DT2FTW) model for predicting long-term temporal data in realistic time-series. In this work, the proposed model aims to overcome the drawbacks of the existing methods and offer a more robust, reliable, and accurate model for predicting long time-series using type-2 fuzzy logic. The model is split into two parts: 1) high-order type-2 fuzzy logic-based time-series prediction and 2) DTW. The type-2 time-series prediction consists of several steps, and we applied the operators of Karnik-Mendel (KM) algorithm for defuzzification. Complex min-max composition operators were applied to all predictions. Then, the prediction performance was evaluated using the root mean square error (RMSE), mean absolute error (MAE), and mean percentage error (MPE) metrics, along with the statistical evaluation using the left-tailed T-test.
The remainder of this paper is organized as follows: Section 2 presents the theoretical research background and materials of the proposed model. Section 3 describes the detailed structure of the proposed model. Performance evaluation and experimental results are presented in Section 4, and the paper is concluded in Section 5.
This section presents a brief overview of DTW, followed by an overview of the concept of the interval type-2 fuzzy set (IT2FS). Finally, relevant mathematical expressions are provided.
To obtain a reliable time-series model for real-time applications, DTW and event-DTW (E-DTW) models were applied in [28], for predicting the optimal distance measure between two related time-series. Since the time is aggregated to represent a reasonable estimate, patterns may match a wide variety of actual time-series. Specifically, the pattern-detection task involves searching for a time-series,
Sequences
where
The cumulative distance for each path is as follows:
where
This section provides definitions of type-2 fuzzy sets (T2FSs) and related but essential concepts. The membership function (MF) of a T2FS of a given element is itself a type-1 fuzzy set (T1FS). A T2FS represented as
where 0 ≤
where ∫∫ represents a union over the admissible
where
The FOU for a Gaussian primary MF with an uncertain standard deviation is shown in Figure 3. The FOU is bounded by an upper bound membership function (UMF)
The block diagram of the proposed DT2FTW model is shown in Figure 1. The fuzzifier can be categorized into two types,
where
where
The mentioned sequences may well be discrete time-series or feature sequences sampled at intermediate points under chaotic and non-stationary conditions. As a predictable value of
where
where
To compute the LCM for each interval of the sequences
where
where
Moreover, at that point, the reduction was calculated using the boundary conditions of the warping window, as follows:
where
where
where
where
where ∪ is the union operation, and
where
where the switch points
In this section, the evaluation of the proposed DT2FTW model is presented. First, the metrics for the performance measurements and datasets used in this study are explained. Then, the statistical results, a comparative study, and experimental results are discussed.
This study used well-known existing time-series datasets, including the Mackey-Glass, NASDAQ, and Dow Jones time-series. The selected datasets represent a wide range of uncertainties through the time-series prediction procedure. The proposed DT2FTW model was complemented to acquire an innovative estimator of the predictions, which also permitted us to compute the uncertainties of predictions for noisy Mackey-Glass chaotic time-series. The test of the proposed model is a simulation of time-series data using the following form of the Mackey-Glass nonlinear delay differential equation
In addition, the NASDAQ is the leading United States electronic stock market. It lists around 3,300 companies. We applied 4,250 pairs of data points from the NASDAQ time-series corresponding to the window from 01/12/2018 to 01/12/2020; these data can be downloaded from Yahoo’s live daily data center. The first 3,025 pairs of the data points were used for training, while the remaining 1,225 pairs of the data points were used for validating the DT2FTW model. From the Dow Jones time-series, we used 1,250 pairs of data points, corresponding to the window from 01/03/2019 to 05/01/2020. These data can be downloaded from Yahoo’s live daily data center. The first 1,025 pairs of the data points were used for training, while the remaining 675 pairs of the data points were used for validating the DT2FTW model.
To evaluate the prediction error, MAE, RMSE, and MPE metrics were applied to the proposed DT2FTW model, as follows:
where
A T-test (left-tailed) was used for estimating the proficiency of the DT2FTW method and its robustness. The null hypothesis was
The results of learning the NASDAQ, Dow Jones, and Mackey-Glass time-series with added noise are detailed in Table 3, where the average RMSE curves and the acceptance ratios during search are shown in Figures 2 and 3, respectively. The obtained results show that the DT2FTW model achieves state-of-the-art performance for global benchmarks for uncertain time-series. These results show a significant difference between the DT2FTW model and its counterpart DTW model, clustering-DTW model [15], and the fuzzy deep artificial neural network (ANN) model [16]. The results, shown in Table 3, confirm that the proposed DT2FTW model performs better, with lower error rates in terms of the RMSE, MAE, and MPE metrics, for all scenarios.
The results of this study adhere to the uncertainty modeling theory, thus confirming that the proposed model is genuinely more reliable and accurate than its counterpart models in the literature. In addition, the results in Figure 2 show that the proposed DT2FTW model outperforms the clustering and fuzzy deep time-series models, for all scenarios, and that it also outperforms the classical DTW model. Figures 3 and 4 reveal that the proposed model better performs on realistic time series such as the Dow Jones and NASDAQ series, and performs better on Mackey-Glass time-series at different delay rates.
ROC curve analysis was conducted for obtaining a reliable estimate of the DT2FTW model’s performance. The following equations were used for assessing the performance based on the ROC curve analysis of the proposed model. In addition, standard metrics, such as precision, recall, and
Table 6 shows the comparison of ROC curve analysis results between the proposed DT2FTW model and its counterparts, revealing that the proposed DT2FTW model performs significantly better than its counterpart models. The ROC curve analysis results confirm that the proposed DT2FTW model performs 12% better than the fuzzy clustering model [15], and 4% better than the fuzzy deep ANN [16], in terms of the AUC measure.
The time-series complexity was determined from the number of computation steps required for running an algorithm as a function of the input size. Time was measured in hours, minutes, and seconds (00:00:00). The results are shown in
Tables 5 and 6 show the results for all tests on the dataset that has been divided into three different scenarios. The DT2FTW model in the first, second, and third measurements for time complexity had a greater order in several runs with different datasets and different scenarios, namely the Mackey-Glass, Dow Jones, and NASDAQ time-series.
The proposed DT2FTW model has more degrees-of-freedom than the type-1 or other related models, because of the FOU parameters in T2FSs, and owing to its potential to model non-uniform time intervals. The proficiency of T2FSs has been proven for high-order uncertainties such as non-stationary events in time-series. In addition, in the proposed model, the problem of distortion of the time axis was addressed by the DTW algorithm alongside a type-2 fuzzy logic approach for modeling high-order uncertainties in time-series prediction using the footprint of uncertainty and unequal length of the time intervals. According to the results of the ROC curve analysis, the proposed DT2FTW model was 12% better than the fuzzy clustering model [15], and 4% better than the fuzzy deep ANN model [16], in terms of the AUC metric. Similarly, the experimental results confirmed that the proposed DT2FTW model has lower error rates than the current methods. The experimental results confirmed the superiority of the DT2FTW model in terms of the RMSE, MAE, and MPE metrics.
This study presented the DT2FTW model for predicting uncertain time-series. The DT2FTW model was evaluated by applying it to three global standard datasets. It is a generalized DTW algorithm that takes advantage of both DTW and type-2 fuzzy logic, optimal alignment between time instance paths, and prediction of high-order uncertainties. The experimental results confirm that the proposed DT2FTW model has lower error rates than other state-of-the-art algorithms
No potential conflict of interest relevant to this article was reported.
Error rate comparison of the DT2FTW model with its counterparts, for global time series: (a) NASDAQ, (b) Dow Jones, and (c) Mackey-Glass.
Error rate of the DT2FTW model, for different noise levels: (a) = 1, (b) = 0.5, and (c) = 0.2, for the Mackey-Glass time series.
Prediction fit results of the DT2FTW model, for different datasets: (a) NASDAQ, (b) Dow Jones, and (c)Mackey-Glass.
Table 1. Related fuzzy models for time-series prediction.
Method | Limitation | Accuracy (%) |
---|---|---|
FCM time-series [17] | No optimal parameter | 77.86 |
Fuzzy logic [18] | No data reduction | 77.18 |
Fuzzy Markov [19] | Complexity of model | 69.80 |
Fuzzy time-series [22] | High-order uncertainty is not modeled | 77.00 |
Fuzzy neural [24] | Weights are fixed | 79.24 |
Fuzzy-PSO [25] | Limited to mid-range series | 80.12 |
Gustafson-Kessel fuzzy clustering [26] | Insufficient results for long time-series | 82.93 |
PSO-fuzzy time-series [27] | Not reliable for long time-series | 82.45 |
Fuzzy-NN time-series clustering [18] | Time complexity is high | 84.73 |
Table 2. T-test results for the DT2FTW and DTW.
Fold# | DT2FTW | DTW |
---|---|---|
1 | 0.9089 | 0.6414 |
2 | 0.9071 | 0.6149 |
3 | 0.9079 | 0.6212 |
4 | 0.9109 | 0.6311 |
5 | 0.9154 | 0.7063 |
6 | 0.9237 | 0.7195 |
7 | 0.9341 | 0.7431 |
8 | 0.9472 | 0.7621 |
9 | 0.9429 | 0.7901 |
10 | 0.9503 | 0.7811 |
Mean |
Table 3. Comparison results for the DT2FTW model on the NASDAQ, Dow Jones, and Mackey-Glass data.
Method | NASDAQ | Dow Jones | Mackey-Glass | ||||||
---|---|---|---|---|---|---|---|---|---|
MAE | RMSE | MPE | MAE | RMSE | MPE | MAE | RMSE | MPE | |
Fuzzy clustering time-series [15] | 0.029 | 0.035 | 1.59 | 0.320 | 0.037 | 1.71 | 0.027 | 0.026 | 1.41 |
Fuzzy deep ANN time-series [16] | 0.019 | 0.019 | 1.52 | 0.024 | 0.028 | 1.64 | 0.021 | 0.017 | 1.39 |
DT2FTW | 0.015 | 0.013 | 1.39 | 0.019 | 0.017 | 1.41 | 0.011 | 0.009 | 1.19 |
Table 4. Comparison of average performance of the DT2FTW model with its counterpart models (unit: %).
Method | AUC | CI | Recall | Precision | F-measure |
---|---|---|---|---|---|
Fuzzy-clustering | 82 | 80–83 | 82 | 80 | 81 |
Fuzzy deep ANN | 90 | 88–91 | 92 | 91 | 90 |
DT2FTW | 94 | 92–95 | 94 | 95 | 93 |
Table 5. Average complexity of the proposed DT2FTW model.
Pseudo-code | Average complexity |
---|---|
Class type | (6 * |
Training time | (8 * |
Calculating the output | |
Calculating the error | |
The output |
Table 6. Time consumption of the DT2FTW model, for the three datasets.
Samples | Mackey-Glass | Dow Jones | NASDAQ |
---|---|---|---|
36 | 0:00:01 | 0:00:08 | 0:00:14 |
48 | 0:00:01 | 0:00:09 | 0:00:23 |
60 | 0:00:02 | 0:00:12 | 0:00:29 |
112 | 0:00:04 | 0:00:21 | 0:00:37 |
224 | 0:00:08 | 0:00:32 | 0:01:05 |
448 | 0:00:16 | 0:00:55 | 0:02:23 |
1,200 | 0:00:29 | 0:01:22 | 0:03:37 |
E-mail: safari.aref@gmail.com
E-mail: rahilhosseini@gmail.com
E-mail: mahdi mazinani@yahoo.com
International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(4): 338-348
Published online December 25, 2021 https://doi.org/10.5391/IJFIS.2021.21.4.338
Copyright © The Korean Institute of Intelligent Systems.
Aref Safari^{1}, Rahil Hosseini^{1} , and Mahdi Mazinani^{2 }
^{1}Department of Computer Engineering, Shahr-e-Qods Branch, Islamic Azad University, Tehran, Iran
^{2}Department of Electronic Engineering, Shahr-e-Qods Branch, Islamic Azad University, Tehran, Iran
Correspondence to:Rahil Hosseini (rahil.hosseini@qodsiau.ac.ir)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Prediction of time series is associated with nondeterministic pattern analysis for uncertain conditions. Therefore, it is necessary to develop high-quality prediction methods for real-world applications. Type-2 fuzzy systems can handle high-order uncertainties, such as sequential dependencies associated with time series. Precise and reliable prediction can help to develop reasonable strategies and assist specialists in planning the best policies for modeling events in uncertain time series. In this study, a hybrid model (dynamic type-2 fuzzy time warping [DT2FTW]) was proposed for handling high-order uncertainties in time-series prediction. A type-2 fuzzy intelligent system was developed alongside a dynamic time warping algorithm for predicting the patterns’ similarity in long-time series for time-series prediction. The results demonstrate that the proposed DT2FTW model yields more reliable predictions on global standard benchmarks such as the Mackey-Glass, Dow Jones, and NASDAQ time-series. The results also confirm that the proposed DT2FTW model has lower error rates than its counterpart algorithms in terms of the root mean square error (RMSE), mean absolute error (MAE), and mean percentage error (MPE). In addition, the results confirm the superiority of the proposed model with an average area under the ROC curve (AUC) of 94%, with the 95% confidence interval (92%-95%).
Keywords: Dynamic time warping, Interval type-2 fuzzy system, Time-series prediction
Modeling of time-series is important because many pattern-analysis problems contain the time component. These problems are typically not addressed, because the time dependence makes time-series-related problems difficult to handle. One of the most challenging issues associated with time-series is their prediction. Prediction problems are often classified into short-term, medium-term, and long-term, and are characterized by different orders of uncertainty. The uncertainty associated with time-series data is implicit, with a nonlinear pattern. On the other hand, unreliable accuracy is a major issue in time-series predictions. Different time-series models, such as fuzzy time-series, have been considered for improving the prediction accuracy.
Many intelligent models have been used for analyzing time-series patterns. Recently, the use of soft computing approaches, such as fuzzy logic, neural networks, simulated annealing, and genetic algorithms, has been reported in the literature on the time-series prediction. These approaches have been considered advantageous compared with traditional methods, because they can address nonlinearities and can approximate many types of complex dynamical systems better than linear statistical models. Fuzzy logic models have been widely adopted because of the prevalent uncertainty in the time-series data. Most related studies use the Euclidean distance metric for measuring time-series intervals. The Euclidean distance metric is widely known to be very sensitive to distortions along the time axis, but it has been very popular with many researchers. The ubiquity of the Euclidean distance metric in the face of increasing evidence of its poor accuracy for time-series prediction is almost certainly owing to its ease of implementation and its time and space efficiency. However, in our work, the problem of distortion along the time axis is addressed by dynamic time warping (DTW) alongside a type-2 fuzzy logic approach, for modeling high-order uncertainties in time-series prediction based on the footprint of uncertainty and unequal lengths of time intervals.
Analysis and prediction of time-series belong to the field of temporal pattern recognition. A time-series corresponds to a stretch of values on the same scale, indexed by a naturally occurring time, as encountered in many applications in the engineering, ecology, economy, medicine, and finance fields [1]. Five concepts are important for time-series: the starting time, pattern similarity, period range, confidence, and the endpoint time. Many techniques [2–8] have been applied for time-series prediction based on the period and data type. The DTW model has been effectively used to automatically deal with time deformations in different time-series ranges with time-dependent data, for pattern recognition and similarity analysis. The DTW approach is currently used in many areas, including online signature matching and handwriting recognition [9], gestures and sign language recognition [10], knowledge discovery, data mining and clustering [11], pattern recognition and data analysis [12], and signal processing [13,14]. In addition, in many real-world applications temporal problems are complex, uncertain, and chaotic [15]. Fuzzy logic is one of the most effective methods for handling uncertainties in dynamic and non-stationary environments [16]. Table 1 lists various hybrid fuzzy models that have been used for predicting time-series [17–22]. Fuzzy systems, especially hybrid fuzzy models, have been very promising for solving complex problems, where a model estimates and predicts the similarity between two time-series in uncertain conditions [18,23–27].
The main objective of this study was to introduce an intelligent dynamic type-2 fuzzy time warping (DT2FTW) model for predicting long-term temporal data in realistic time-series. In this work, the proposed model aims to overcome the drawbacks of the existing methods and offer a more robust, reliable, and accurate model for predicting long time-series using type-2 fuzzy logic. The model is split into two parts: 1) high-order type-2 fuzzy logic-based time-series prediction and 2) DTW. The type-2 time-series prediction consists of several steps, and we applied the operators of Karnik-Mendel (KM) algorithm for defuzzification. Complex min-max composition operators were applied to all predictions. Then, the prediction performance was evaluated using the root mean square error (RMSE), mean absolute error (MAE), and mean percentage error (MPE) metrics, along with the statistical evaluation using the left-tailed T-test.
The remainder of this paper is organized as follows: Section 2 presents the theoretical research background and materials of the proposed model. Section 3 describes the detailed structure of the proposed model. Performance evaluation and experimental results are presented in Section 4, and the paper is concluded in Section 5.
This section presents a brief overview of DTW, followed by an overview of the concept of the interval type-2 fuzzy set (IT2FS). Finally, relevant mathematical expressions are provided.
To obtain a reliable time-series model for real-time applications, DTW and event-DTW (E-DTW) models were applied in [28], for predicting the optimal distance measure between two related time-series. Since the time is aggregated to represent a reasonable estimate, patterns may match a wide variety of actual time-series. Specifically, the pattern-detection task involves searching for a time-series,
Sequences
where
The cumulative distance for each path is as follows:
where
This section provides definitions of type-2 fuzzy sets (T2FSs) and related but essential concepts. The membership function (MF) of a T2FS of a given element is itself a type-1 fuzzy set (T1FS). A T2FS represented as
where 0 ≤
where ∫∫ represents a union over the admissible
where
The FOU for a Gaussian primary MF with an uncertain standard deviation is shown in Figure 3. The FOU is bounded by an upper bound membership function (UMF)
The block diagram of the proposed DT2FTW model is shown in Figure 1. The fuzzifier can be categorized into two types,
where
where
The mentioned sequences may well be discrete time-series or feature sequences sampled at intermediate points under chaotic and non-stationary conditions. As a predictable value of
where
where
To compute the LCM for each interval of the sequences
where
where
Moreover, at that point, the reduction was calculated using the boundary conditions of the warping window, as follows:
where
where
where
where
where ∪ is the union operation, and
where
where the switch points
In this section, the evaluation of the proposed DT2FTW model is presented. First, the metrics for the performance measurements and datasets used in this study are explained. Then, the statistical results, a comparative study, and experimental results are discussed.
This study used well-known existing time-series datasets, including the Mackey-Glass, NASDAQ, and Dow Jones time-series. The selected datasets represent a wide range of uncertainties through the time-series prediction procedure. The proposed DT2FTW model was complemented to acquire an innovative estimator of the predictions, which also permitted us to compute the uncertainties of predictions for noisy Mackey-Glass chaotic time-series. The test of the proposed model is a simulation of time-series data using the following form of the Mackey-Glass nonlinear delay differential equation
In addition, the NASDAQ is the leading United States electronic stock market. It lists around 3,300 companies. We applied 4,250 pairs of data points from the NASDAQ time-series corresponding to the window from 01/12/2018 to 01/12/2020; these data can be downloaded from Yahoo’s live daily data center. The first 3,025 pairs of the data points were used for training, while the remaining 1,225 pairs of the data points were used for validating the DT2FTW model. From the Dow Jones time-series, we used 1,250 pairs of data points, corresponding to the window from 01/03/2019 to 05/01/2020. These data can be downloaded from Yahoo’s live daily data center. The first 1,025 pairs of the data points were used for training, while the remaining 675 pairs of the data points were used for validating the DT2FTW model.
To evaluate the prediction error, MAE, RMSE, and MPE metrics were applied to the proposed DT2FTW model, as follows:
where
A T-test (left-tailed) was used for estimating the proficiency of the DT2FTW method and its robustness. The null hypothesis was
The results of learning the NASDAQ, Dow Jones, and Mackey-Glass time-series with added noise are detailed in Table 3, where the average RMSE curves and the acceptance ratios during search are shown in Figures 2 and 3, respectively. The obtained results show that the DT2FTW model achieves state-of-the-art performance for global benchmarks for uncertain time-series. These results show a significant difference between the DT2FTW model and its counterpart DTW model, clustering-DTW model [15], and the fuzzy deep artificial neural network (ANN) model [16]. The results, shown in Table 3, confirm that the proposed DT2FTW model performs better, with lower error rates in terms of the RMSE, MAE, and MPE metrics, for all scenarios.
The results of this study adhere to the uncertainty modeling theory, thus confirming that the proposed model is genuinely more reliable and accurate than its counterpart models in the literature. In addition, the results in Figure 2 show that the proposed DT2FTW model outperforms the clustering and fuzzy deep time-series models, for all scenarios, and that it also outperforms the classical DTW model. Figures 3 and 4 reveal that the proposed model better performs on realistic time series such as the Dow Jones and NASDAQ series, and performs better on Mackey-Glass time-series at different delay rates.
ROC curve analysis was conducted for obtaining a reliable estimate of the DT2FTW model’s performance. The following equations were used for assessing the performance based on the ROC curve analysis of the proposed model. In addition, standard metrics, such as precision, recall, and
Table 6 shows the comparison of ROC curve analysis results between the proposed DT2FTW model and its counterparts, revealing that the proposed DT2FTW model performs significantly better than its counterpart models. The ROC curve analysis results confirm that the proposed DT2FTW model performs 12% better than the fuzzy clustering model [15], and 4% better than the fuzzy deep ANN [16], in terms of the AUC measure.
The time-series complexity was determined from the number of computation steps required for running an algorithm as a function of the input size. Time was measured in hours, minutes, and seconds (00:00:00). The results are shown in
Tables 5 and 6 show the results for all tests on the dataset that has been divided into three different scenarios. The DT2FTW model in the first, second, and third measurements for time complexity had a greater order in several runs with different datasets and different scenarios, namely the Mackey-Glass, Dow Jones, and NASDAQ time-series.
The proposed DT2FTW model has more degrees-of-freedom than the type-1 or other related models, because of the FOU parameters in T2FSs, and owing to its potential to model non-uniform time intervals. The proficiency of T2FSs has been proven for high-order uncertainties such as non-stationary events in time-series. In addition, in the proposed model, the problem of distortion of the time axis was addressed by the DTW algorithm alongside a type-2 fuzzy logic approach for modeling high-order uncertainties in time-series prediction using the footprint of uncertainty and unequal length of the time intervals. According to the results of the ROC curve analysis, the proposed DT2FTW model was 12% better than the fuzzy clustering model [15], and 4% better than the fuzzy deep ANN model [16], in terms of the AUC metric. Similarly, the experimental results confirmed that the proposed DT2FTW model has lower error rates than the current methods. The experimental results confirmed the superiority of the DT2FTW model in terms of the RMSE, MAE, and MPE metrics.
This study presented the DT2FTW model for predicting uncertain time-series. The DT2FTW model was evaluated by applying it to three global standard datasets. It is a generalized DTW algorithm that takes advantage of both DTW and type-2 fuzzy logic, optimal alignment between time instance paths, and prediction of high-order uncertainties. The experimental results confirm that the proposed DT2FTW model has lower error rates than other state-of-the-art algorithms
Block diagram of the proposed DT2FTW model.
Error rate comparison of the DT2FTW model with its counterparts, for global time series: (a) NASDAQ, (b) Dow Jones, and (c) Mackey-Glass.
Error rate of the DT2FTW model, for different noise levels: (a) = 1, (b) = 0.5, and (c) = 0.2, for the Mackey-Glass time series.
Prediction fit results of the DT2FTW model, for different datasets: (a) NASDAQ, (b) Dow Jones, and (c)Mackey-Glass.
Table 1 . Related fuzzy models for time-series prediction.
Method | Limitation | Accuracy (%) |
---|---|---|
FCM time-series [17] | No optimal parameter | 77.86 |
Fuzzy logic [18] | No data reduction | 77.18 |
Fuzzy Markov [19] | Complexity of model | 69.80 |
Fuzzy time-series [22] | High-order uncertainty is not modeled | 77.00 |
Fuzzy neural [24] | Weights are fixed | 79.24 |
Fuzzy-PSO [25] | Limited to mid-range series | 80.12 |
Gustafson-Kessel fuzzy clustering [26] | Insufficient results for long time-series | 82.93 |
PSO-fuzzy time-series [27] | Not reliable for long time-series | 82.45 |
Fuzzy-NN time-series clustering [18] | Time complexity is high | 84.73 |
Table 2 . T-test results for the DT2FTW and DTW.
Fold# | DT2FTW | DTW |
---|---|---|
1 | 0.9089 | 0.6414 |
2 | 0.9071 | 0.6149 |
3 | 0.9079 | 0.6212 |
4 | 0.9109 | 0.6311 |
5 | 0.9154 | 0.7063 |
6 | 0.9237 | 0.7195 |
7 | 0.9341 | 0.7431 |
8 | 0.9472 | 0.7621 |
9 | 0.9429 | 0.7901 |
10 | 0.9503 | 0.7811 |
Mean |
Table 3 . Comparison results for the DT2FTW model on the NASDAQ, Dow Jones, and Mackey-Glass data.
Method | NASDAQ | Dow Jones | Mackey-Glass | ||||||
---|---|---|---|---|---|---|---|---|---|
MAE | RMSE | MPE | MAE | RMSE | MPE | MAE | RMSE | MPE | |
Fuzzy clustering time-series [15] | 0.029 | 0.035 | 1.59 | 0.320 | 0.037 | 1.71 | 0.027 | 0.026 | 1.41 |
Fuzzy deep ANN time-series [16] | 0.019 | 0.019 | 1.52 | 0.024 | 0.028 | 1.64 | 0.021 | 0.017 | 1.39 |
DT2FTW | 0.015 | 0.013 | 1.39 | 0.019 | 0.017 | 1.41 | 0.011 | 0.009 | 1.19 |
Table 4 . Comparison of average performance of the DT2FTW model with its counterpart models (unit: %).
Method | AUC | CI | Recall | Precision | F-measure |
---|---|---|---|---|---|
Fuzzy-clustering | 82 | 80–83 | 82 | 80 | 81 |
Fuzzy deep ANN | 90 | 88–91 | 92 | 91 | 90 |
DT2FTW | 94 | 92–95 | 94 | 95 | 93 |
Table 5 . Average complexity of the proposed DT2FTW model.
Pseudo-code | Average complexity |
---|---|
Class type | (6 * |
Training time | (8 * |
Calculating the output | |
Calculating the error | |
The output |
Table 6 . Time consumption of the DT2FTW model, for the three datasets.
Samples | Mackey-Glass | Dow Jones | NASDAQ |
---|---|---|---|
36 | 0:00:01 | 0:00:08 | 0:00:14 |
48 | 0:00:01 | 0:00:09 | 0:00:23 |
60 | 0:00:02 | 0:00:12 | 0:00:29 |
112 | 0:00:04 | 0:00:21 | 0:00:37 |
224 | 0:00:08 | 0:00:32 | 0:01:05 |
448 | 0:00:16 | 0:00:55 | 0:02:23 |
1,200 | 0:00:29 | 0:01:22 | 0:03:37 |
Block diagram of the proposed DT2FTW model.
|@|~(^,^)~|@|Error rate comparison of the DT2FTW model with its counterparts, for global time series: (a) NASDAQ, (b) Dow Jones, and (c) Mackey-Glass.
|@|~(^,^)~|@|Error rate of the DT2FTW model, for different noise levels: (a) = 1, (b) = 0.5, and (c) = 0.2, for the Mackey-Glass time series.
|@|~(^,^)~|@|Prediction fit results of the DT2FTW model, for different datasets: (a) NASDAQ, (b) Dow Jones, and (c)Mackey-Glass.