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FIE ' Proceedings 35th Annual Conference. In: 12th International Conference "Mathematics. Uitgeverij Eburon, Delft, Netherlands, pp. Ireland, pp. In: 14th International Conference "Mathematics. International Journal of Technology, Knowledge and Society, 3 6. Physics Education, 43 3. In: 13th International Conference "Mathematics. Computers in Human Behavior, 24 6. Academic Conferences Ltd, Reading, pp.
Computers in the Schools, 35 4. Physics Education, 54 3. Cambridge Journal of Education, 48 6. Journal of Hospitality and Tourism Technology, 7 1. Computers in Human Behavior, Early Years. Critical Publishing, St Albans, pp. Routledge Research Companions in Business Economics. Routledge, New York.
Critical Publishing, Norwich. Critical Publishing, Norwich, pp. Current debates in educational psychology. Notice that each time patch often contains several time steps e. Figure 2 shows, as an example, how to decompose a given time series into T time patches of equal length where each time patch will be processed by an STCN block. In this case, K denotes the number of time steps allocated to the time patch, whereas M defines the width of each STCN block.
The sequence is split into T time patches with even length. Each time patch is used to train an STCN block that employs information from the previous block as prior knowledge Full size image In short, the LSTCN model can be defined as a collection of STCN blocks, each processing a specific time patch and passing knowledge to the next block. The aggregation procedure creates a chained neural structure that allows for long-term predictions since the learned knowledge is used when performing reasoning in the current iteration.
In the figure, blue boxes represent STCN blocks, while orange boxes denote learning processes. The weights learned in the current block are aggregated using Eqs. Therefore, the long-term component refers to how we process the whole sequence, which is done by transferring the knowledge in the form of weights from one STCN block to another.
Once we have processed the whole sequence, the model narrows down to the last STCN in the pipeline. We ought to outline how short-term and long-term dependencies in temporal data are captured in both models to address this topic.
An RNN network in an unfolded state can be illustrated as a sequence of neural layers. The hidden layers in an RNN are responsible for window-based time series processing. In an RNN, the values computed by the network for the previous time step are used as input when processing the current time step. Due to the cyclic nature of the entire process, training an RNN is challenging.
The input signals tend to either decay or grow exponentially. Graves et al. The most significant difference between the RNN and the LSTM model is that the latter adds a forgetting mechanism at each hidden layer. The LSTM model processes the data using a windowing technique in which the number of hidden layers is equal to the length of the window.
This window is responsible for processing and recognizing short-term dependencies in time series. The forgetting mechanism in each layer acts as a symbolic switch that either retains the incoming signal or forgets it. Please note that this switch is not binary. Thus, the forgetting mechanism in LSTM adds flexibility that allows the network to accumulate long-term temporal contextual information in its internal states, but at the same time, short-term dependencies are also modeled because the processing scheme is still sequential and windows-based.
At the same time, the short-term dependencies in the time series are processed in a conventional way, in each STCN block see Fig. The added value of using a ridge regression approach is regularizing the model and preventing overfitting. In our network, overfitting is likely to happen when splitting the original time series into too many time patches covering few observations.
This generalized inverse is computed using singular value decomposition and is defined and unique for all real matrices. Remark that this learning rule assumes that the activation values in the inner layer are standardized. As far as standardization is concerned, these calculations are based on standardized activation values.
When the final weights are returned, they are adjusted back into their original scale. However, there are three main differences between these models. Secondly, while the hidden layer of ELMs is of arbitrary width, the number of neurons in an STCN is given by the number of steps ahead to be predicted and the number of features in the multivariate time series.
Finally, each neuron also referred to as neural concept represents the state of a problem feature in a given time step. While this constraint equips our model with interpretability features, it might also limit its approximation capabilities. This matrix is expected to be partially provided by domain experts or computed from a previous learning process e. The smoothed time series is obtained using the moving average method for a given window size. Finally, we generate some white noise over the computed weights to compensate for the moving average operation.
The intuition dictates that the training error will go down as more time patches are processed. Of course, such time patches should not be too small to avoid overfitting. In some cases, we might obtain an optimal performance using a single time patch containing the whole sequence such that we will have a single STCN block. In other cases, it might occur that we do not have access to the whole sequence e. Interpretability As mentioned, the architecture of our neural system allows explaining the forecasting since both neurons and weights have a precise meaning for the problem domain being modeled.
However, the interpretability cannot be confined to the absence of hidden components in the network since the structure might involve hundreds or thousands of edges. In this subsection, we introduce a measure to quantify the influence of each feature in the forecasting of multivariate time series.
This implies that our measure is a model-intrinsic feature importance measure, reflecting what the model considers important when learning the relations between the time points. Moreover, the neurons are organized temporally, which means that we have L blocks of neurons, each containing N units. Moreover, it is expected for the learning algorithm to produce sparse weights with a zero-mean normal distribution, which is an appreciated characteristic when it comes to interpretability.
The idea of computing the relevance of features from the weights in neural systems has been explored in the literature. In the first study, the feature scores indicate which features play a significant role in obtaining a given class instead of an alternative class. This type of interpretability responds to the question why not?.
Conversely, the second study measures the feature importance in obtaining the decision class. The results were contrasted with the feature scores obtained from logistic regression and both models agreed on the top features that play a role in the outcome. Both feature score measures operate on neural systems where the neurons have an explicit meaning for the problem domain. Therefore, the learned weights can be used as a proxy for interpretability.
Numerical simulations In this section, we will explore the performance forecasting error and training time of our neural system on three case studies involving univariate and multivariate time series. In the case of multivariate time series, we will also depict the feature contribution score to explain the predictions. When it comes to the pre-processing steps, we interpolate the missing values whenever applicable and normalize the series using the min-max method.
As for the performance metric, we use the mean absolute error in all simulations reported in the section. For the sake of convenience, we trimmed the training sequence by deleting the first observations such that the number of times is a multiple of L the number of steps ahead we want to forecast. In the first three models, the number of epochs was set to 20, while the batch size was obtained through hyperparameter tuning using grid search.
The candidate batch sizes were the powers of two, starting from 32 until 4, The values for the remaining parameters were retained as provided in the Keras library. In Eq. These two hyperparameters were not optimized during the hyperparameter tuning step as they were used to simulate the prior knowledge component. The values for the remaining hyperparameters were retained as provided in the library.
Finally, all experiments presented in this section were performed on a high-performance computing environment that uses two Intel Xeon Gold CPUs at 2. In this case study, the health data of one individual were extracted from the Apple Health application in the period from to In total, the time series dataset is composed of 79, instances or time steps. The Apple Health application records the step counts in small sessions during which the walking occurs.
Besides, the day of the week that each value was recorded is known. Table 1 presents descriptive statistics attached to this univariate time series before normalization. Table 1 Descriptive statistics for the steps case study Full size table The target variable number of steps follows an exponential distribution with very infrequent, extremely high step counts and very common low step counts.
Overall, the data neither follows seasonal patterns nor a trend. Table 2 shows the normalized errors attached to the models under consideration when forecasting 50 steps ahead in the Steps dataset. In addition, we portray the training and test times in seconds for the optimized models. The hyperparameter tuning reported that our neural system needed two iterations to produce the lowest forecasting errors, while the optimal batch size for RNN, GRU and LSTM was found to be Although LSTCN outperforms the remaining methods in terms of forecasting error, what is truly remarkable is its efficiency.
In this experiment, we ran all models five times with optimized hyperparameters and selected the shortest training time in each case. Hence, the time measures reported in Table 2 concern the fastest executions observed in our simulations.
In other words, we visualize the differences in the distributions of prior knowledge weights and the weights learned in the last STCN block. It is worth recalling that the prior knowledge block in that last block is what the network has learned after processing all the time patches but the last one. In contrast, the learned weights in that block adapt the prior knowledge to forecast the last time patch. This figure illustrates that the network significantly adapts the prior knowledge to the last piece of data available.
In that layer, inner and outer neurons refer to the leftmost and rightmost neurons, respectively. Observe that the learning algorithm assigns larger weights to connections between neurons processing the last steps in the input sequence and neurons processing the first steps in the output sequence. This is an expected behavior in time series forecasting that supports the rationale of the proposed feature relevance measure.
Using that knowledge, experts could estimate how many previous time steps would be needed to predict a sequence of length L without performing further simulations. Household electric power consumption The second case study concerns the energy consumption in one house in France measured each minute from December to November 47 months. Hence, records with missing values were interpolated using the nearest neighbor method.
In our experiments, we retained the following variables: global minute-averaged active power in kilowatt , global minute-averaged reactive power in kilowatt , minute-averaged voltage in volt , and global minute-averaged current intensity in ampere Table 3. On the most fine-grained scale, we observe a repeating low nighttime power consumption. We also noted a less distinct but still present pattern related to the day of the week: higher power consumption during weekend days.
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