The synchrosqueezing transform (SST) is a novel approach for time-frequency (T-F) representation of non-stationary signals. By synchrosqueezing and reassigning the T-F spectrum of the wavelet transform (WT) or the short time Fourier transform (STFT) of a signal, the SST can obtain a high-resolution T-F spectrum. In the light of the superiority of S-transform (ST) over the WT and the STFT, especially, in representing a high-frequency weak-amplitude signal on its T-F spectrum, we propose a synchrosqueezing S-transform (SSST) which is realized by synchrosqueezing the spectrum of the ST. The formulas for the SSST and its inverse transform are derived. Synthetic examples show that the SSST has obviously higher resolution than the ST, and is superior to the SST like the ST to the WT. We then applied the SSST to perform the spectral decomposition of a marine seismic data for natural gas hydrate exploration. The results illustrate that the SSST can be used to well detect frequency spectral anomalies correlated with the gas hydrate and free-gas accumulations. We can also conclude that the SSST is a good potential technique to assist seismic interpretation.

Synchrosqueezing S-Transform and Its Application in Seismic Spectral Decomposition

Control of blood glucose is essential for diabetes management. Current digital therapeutic approaches for subjects with type 1 diabetes mellitus such as the artificial pancreas and insulin bolus calculators leverage machine learning techniques for predicting subcutaneous glucose for improved control. Deep learning has recently been applied in healthcare and medical research to achieve state-of-the-art results in a range of tasks including disease diagnosis, and patient state prediction among others. In this paper, we present a deep learning model that is capable of forecasting glucose levels with leading accuracy for simulated patient cases (root-mean-square error (RMSE) = 9.38  $\pm$ 0.71 [mg/dL] over a 30-min horizon, RMSE = 18.87  $\pm$ 2.25 [mg/dL] over a 60-min horizon) and real patient cases (RMSE = 21.07  $\pm$ 2.35 [mg/dL] for 30 min, RMSE = 33.27  $\pm$ 4.79% for 60 min). In addition, the model provides competitive performance in providing effective prediction horizon ( $\text{PH}_{\text{eff}}$ ) with minimal time lag both in a simulated patient dataset ( $\text{PH}_{\text{eff}}$ = 29.0 $\pm$ 0.7 for 30 min and $\text{PH}_{\text{eff}}$ = 49.8  $\pm$ 2.9 for 60 min) and in a real patient dataset ( $\text{PH}_{\text{eff}}$ = 19.3  $\pm$ 3.1 for 30 min and $\text{PH}_{\text{eff}}$ = 29.3  $\pm$ 9.4 for 60 min). This approach is evaluated on a dataset of ten simulated cases generated from the UVA/Padova simulator and a clinical dataset of ten real cases each containing glucose readings, insulin bolus, and meal (carbohydrate) data. Performance of the recurrent convolutional neural network is benchmarked against four algorithms. The proposed algorithm is implemented on an Android mobile phone, with an execution time of 6 ms on a phone compared to an execution time of 780 ms on a laptop.

Convolutional Recurrent Neural Networks for Glucose Prediction

Data-driven modeling and prediction of blood glucose dynamics: Machine learning applications in type 1 diabetes

Adaptive short-time Fourier transform and synchrosqueezing transform for non-stationary signal separation

For people with Type 1 diabetes (T1D), forecasting of blood glucose (BG) can be used to effectively avoid hyperglycemia, hypoglycemia and associated complications. The latest continuous glucose monitoring (CGM) technology allows people to observe glucose in real-time. However, an accurate glucose forecast remains a challenge. In this work, we introduce GluNet, a framework that leverages on a personalized deep neural network to predict the probabilistic distribution of short-term (30–60 minutes) future CGM measurements for subjects with T1D based on their historical data including glucose measurements, meal information, insulin doses, and other factors. It adopts the latest deep learning techniques consisting of four components: data pre-processing, label transform/recover, multi-layers of dilated convolution neural network (CNN), and post-processing. The method is evaluated in-silico for both adult and adolescent subjects. The results show significant improvements over existing methods in the literature through a comprehensive comparison in terms of root mean square error (RMSE) ( $\text{8.88} \pm \text{0.77}$ mg/dL) with short time lag ( $\text{0.83}\pm \text{0.40}$ minutes) for prediction horizons (PH) = 30 mins (minutes), and RMSE ( $\text{19.90} \pm \text{3.17}$ mg/dL) with time lag ( $\text{16.43}\pm \text{4.07}$ mins) for PH = 60 mins for virtual adult subjects. In addition, GluNet is also tested on two clinical data sets. Results show that it achieves an RMSE ( $\text{19.28} \pm \text{2.76}$  mg/dL) with time lag ( $\text{8.03}\pm \text{4.07}$ mins) for PH = 30 mins and an RMSE ( $\text{31.83} \pm \text{3.49}$ mg/dL) with time lag ( $\text{17.78}\pm \text{8.00}$  mins) for PH = 60 mins. These are the best reported results for glucose forecasting when compared with other methods including the neural network for predicting glucose (NNPG), the support vector regression (SVR), the latent variable with exogenous input (LVX), and the auto regression with exogenous input (ARX) algorithm.

/pdf/glunet-a-deep-learning-framework-for-accurate-glucose-3oa69bpwob.pdf

GluNet: A Deep Learning Framework for Accurate Glucose Forecasting

We consider the question of 30-minute prediction of blood glucose levels measured by continuous glucose monitoring devices, using clinical data. While most studies of this nature deal with one patient at a time, we take a certain percentage of patients in the data set as training data, and test on the remainder of the patients; i.e., the machine need not re-calibrate on the new patients in the data set. We demonstrate how deep learning can outperform shallow networks in this example. One novelty is to demonstrate how a parsimonious deep representation can be constructed using domain knowledge.

/pdf/a-deep-learning-approach-to-diabetic-blood-glucose-1d4yjjqc7n.pdf

A Deep Learning Approach to Diabetic Blood Glucose Prediction

The empirical mode decomposition (EMD) algorithm, introduced by Huang et al. (Proc Roy Soc Lond Ser A Math Phys Eng Sci 454(1971):903–995, 1998), is arguably the most popular mathematical scheme for non-stationary signal decomposition and analysis. The objective of EMD is to separate a given signal into a number of components, called intrinsic mode functions (IMF’s) after which the instantaneous frequency (IF) and amplitude of each IMF are computed through Hilbert spectral analysis (HSA). On the other hand, the synchrosqueezed wavelet transform (SST), introduced by Daubechies and Maes (Wavelets in Medicine and Biology, pp. 527–546, 1996) and further developed by Daubechies et al. (Appl Comput Harmon Anal 30:243–261, 2011), is applied to estimate the IF’s of all signal components of the given signal, based on one single reference “IF function”, under the assumption that the signal components satisfy certain strict properties of a so-called adaptive harmonic model, before the signal components of the model are recovered. The objective of our paper is to develop a hybrid EMD-SST computational scheme by applying a “modified SST” to each IMF of the EMD, as an alternative approach to the original EMD-HSA method. While our modified SST assures non-negative instantaneous frequencies of the IMF’s, application of the EMD scheme eliminates the dependence on a single reference IF value as well as the guessing work of the number of signal components in the original SST approach. Our modification of the SST consists of applying vanishing moment wavelets (introduced in a recent paper by C.K. Chui, Y.-T. Lin and H.-T. Wu) with stacked knots to process signals on bounded or half-infinite time intervals, and spline curve fitting with optimal smoothing parameter selection through generalized cross-validation. In addition, we formulate a local cubic spline interpolation scheme for real-time realization of the EMD sifting process that improves over the standard global cubic spline interpolation, both in quality and computational cost, particularly when applied to bounded and half-infinite time intervals.

Signal analysis via instantaneous frequency estimation of signal components

Decomposition of functions in terms of their primary building blocks is one of the most fundamental problems in mathematical analysis and its applications. Indeed, atomic decomposition of functions in the Hardy space H^p for 0<p≤1 as infinite series of “atoms” that have the property of vanishing moments with order at least up to 1 / p has significant impacts, not only to the advances of harmonic and functional analyses, but also to the birth of wavelet analysis, which in turn allows the construction of sufficiently large dictionaries of wavelet-like basis functions for the success of atomic decomposition of more general functions or signals, by such mathematical tools as “basis pursuit” and “nonlinear basis pursuit”. However, such dictionaries are necessarily huge for atomic decomposition of real-world signals. The spirit of the present paper is to construct the atoms directly from the data, without relying on a large dictionary. Following Gabor, the starting point of this line of thought is to observe that “any” signal a can be written as a(t)=A(t)cosϕ(t) via complex extension using the Hilbert transform. Hence, if a given signal f has been decomposed by whatever available methods or schemes, as the sum of sub-signals f_k, then each sub-signal can be written as f_k(t)=Ak(t)cosϕ_k(t). Whether or not f_k is an atom of the given signal f depends on whether any of the sub-signals f_k can be further decomposed in a meaningful way. In this regard, the most popular decomposition scheme in the current literature is the sifting process of the empirical mode decomposition (EMD), where the sub-signals f_k are called intrinsic mode functions (IMF’s). The main contribution of our present paper is firstly to demonstrate that IMF’s may not be atoms, and secondly to give a computational scheme for decomposing such IMF’s into finer and meaningful signal building blocks. Our innovation is to apply the signal separation operator (SSO), introduced by the first two authors, with a clever choice of parameters, first to extract the instantaneous frequencies (IF’s) of each IMF obtained from the sifting process, and then (by using the same parameters for the SSO, with the IF’s as input) to construct finer signal building blocks of the IMF. In other words, we replace the Hilbert transform of the EMD scheme by the SSO in this present paper, first for frequency extraction, and then for constructing finer signal building blocks. As an example, we consider the problem in super-resolution of separating two Dirac delta functions that are arbitrarily close to each other. This problem is equivalent to finding the two cosine building blocks of a two-tone signal with frequencies that are arbitrarily close. While the sifting process can only yield one IMF when the frequencies are too close together, the SSO applied to this IMF extracts the two frequencies and recover the two cosine building blocks (or atoms). For this reason, we coin our scheme of sifting + SSO as “superEMD”, where “super” is used as an abbreviation of super-resolution.

Data-driven atomic decomposition via frequency extraction of intrinsic mode functions

Empirical mode decomposition (EMD) is a popular, novel, user-friendly algorithm to decompose a given signal into its constituting components, utilizing spline interpolation. In this paper, we equip EMD with a shape-preserving interpolation scheme based on quadratic B-splines. Using numerical experiments, we show that our scheme, which we coin Geometric EMD, or GEMD, outperforms the original EMD, both qualitatively and quantitatively.

Empirical mode decomposition with shape-preserving spline interpolation

We develop a local polynomial spline interpolation scheme for arbitrary spline order on bounded intervals. Our method’s local formulation, eective boundary considerations and optimal interpolation error rate make it particularly useful for real-time implementation in real-world applications. Mathematics Subject Classification: 41A05, 41A15

/pdf/real-time-local-spline-interpolation-schemes-on-bounded-1piddiqr8w.pdf

Maria D. van der Walt

Papers

A Deep Learning Approach to Diabetic Blood Glucose Prediction

Signal analysis via instantaneous frequency estimation of signal components

Data-driven atomic decomposition via frequency extraction of intrinsic mode functions

Empirical mode decomposition with shape-preserving spline interpolation

Real-Time, Local Spline Interpolation Schemes on Bounded Intervals