This paper aims at comparing different forecasting strategies combined with the STL decomposition method. STL is a versatile and robust time series decomposition method. The forecasting strategies we consider are as follows: three statistical methods (ARIMA, ETS, and Theta), five machine learning methods (KNN, SVR, CART, RF, and GP), and two versions of RNNs (CNN-LSTM and ConvLSTM). We conduct the forecasting test on six horizons (1, 6, 12, 18, and 24 months). Our results show that, when applied to monthly industrial M3 Competition data as a preprocessing step, STL decomposition can benefit forecasting using statistical methods but harms the machine learning ones. Moreover, the STL-Theta combination method displays the best forecasting results on four over the five forecasting horizons.

Permutation Entropy (PE) and Multiscale Permutation Entropy (MPE) have been extensively used in the analysis of time series searching for regularities. Although PE has been explored and characterized, there is still a lack of theoretical background regarding MPE. Therefore, we expand the available MPE theory by developing an explicit expression for the estimator’s variance as a function of time scale and ordinal pattern distribution. We derived the MPE Cramér–Rao Lower Bound (CRLB) to test the efficiency of our theoretical result. We also tested our formulation against MPE variance measurements from simulated surrogate signals. We found the MPE variance symmetric around the point of equally probable patterns, showing clear maxima and minima. This implies that the MPE variance is directly linked to the MPE measurement itself, and there is a region where the variance is maximum. This effect arises directly from the pattern distribution, and it is unrelated to the time scale or the signal length. The MPE variance also increases linearly with time scale, except when the MPE measurement is close to its maximum, where the variance presents quadratic growth. The expression approaches the CRLB asymptotically, with fast convergence. The theoretical variance is close to the results from simulations, and appears consistently below the actual measurements. By knowing the MPE variance, it is possible to have a clear precision criterion for statistical comparison in real-life applications.

Permutation Entropy (PE) is a powerful tool for measuring the amount of information contained within a time series. However, this technique is rarely applied directly on raw signals. Instead, a preprocessing step, such as linear filtering, is applied in order to remove noise or to isolate specific frequency bands. In the current work, we aimed at outlining the effect of linear filter preprocessing in the final PE values. By means of the Wiener–Khinchin theorem, we theoretically characterize the linear filter’s intrinsic PE and separated its contribution from the signal’s ordinal information. We tested these results by means of simulated signals, subject to a variety of linear filters such as the moving average, Butterworth, and Chebyshev type I. The PE results from simulations closely resembled our predicted results for all tested filters, which validated our theoretical propositions. More importantly, when we applied linear filters to signals with inner correlations, we were able to theoretically decouple the signal-specific contribution from that induced by the linear filter. Therefore, by providing a proper framework of PE linear filter characterization, we improved the PE interpretation by identifying possible artifact information introduced by the preprocessing steps.

Diabetic foot (DF) complications are associated with temperature variations. The occurrence of DF ulceration could be reduced by using a contactless thermal camera. The aim of our study is to provide a decision support tool for the prevention of DF ulcers. Thus, the segmentation of the plantar foot in thermal images is a challenging step for a non-constraining acquisition protocol. This paper presents a new segmentation method for plantar foot thermal images. This method is designed to include five pieces of prior information regarding the aforementioned images. First, a new energy term is added to the snake of Kass et al. in order to force its curvature to match that of the prior shape, which has a known form. Second, we defined the initial contour as the downsized prior-shape contour, which is placed inside the plantar foot surface in a vertical orientation. This choice makes the snake avoid strong false boundaries present outside the plantar region when evolving. As a result, the snake produces a smooth contour that rapidly converges to the true boundaries of the foot. The proposed method is compared to two classical prior-shape snake methods, that of Ahmed et al. and that of Chen et al. A database of 50 plantar foot thermal images was processed. The results show that the proposed method outperforms the previous two methods with a root-mean-square error of 5.12 pixels and a dice similarity coefficient of 94%. The segmentation of the plantar foot regions in the thermal images helped us to assess the point-to-point temperature differences between the two feet in order to detect hyperthermia regions. The presence of such regions is the pre-sign of ulcers in the diabetic foot. Furthermore, our method was applied to hyperthermia detection to illustrate the promising potential of thermography in the case of the diabetic foot. Associated with a friendly acquisition protocol, the proposed segmentation method is the first step for a future mobile smartphone-based plantar foot thermal analysis for diabetic foot patients.

Surface electromyography (sEMG) is a valuable technique that helps provide functional and structural information about the electric activity of muscles. As sEMG measures output of complex living systems characterized by multiscale and nonlinear behaviors, Multiscale Permutation Entropy (MPE) is a suitable tool for capturing useful information from the ordinal patterns of sEMG time series. In a previous work, a theoretical comparison in terms of bias and variance of two MPE variants—namely, the refined composite MPE (rcMPE) and the refined composite downsampling (rcDPE), was addressed. In the current paper, we assess the superiority of rcDPE over MPE and rcMPE, when applied to real sEMG signals. Moreover, we demonstrate the capacity of rcDPE in quantifying fatigue levels by using sEMG data recorded during a fatiguing exercise. The processing of four consecutive temporal segments, during biceps brachii exercise maintained at 70% of maximal voluntary contraction until exhaustion, shows that the 10th-scale of rcDPE was capable of better differentiation of the fatigue segments. This scale actually brings the raw sEMG data, initially sampled at 10 kHz, to the specific 0–500 Hz sEMG spectral band of interest, which finally reveals the inner complexity of the data. This study promotes good practices in the use of MPE complexity measures on real data.