Tailored to the Problem Specificity Mathematical Transforms (TPS MTs) were defined as extension of linear transforms [1], when orthogonal bases of the classical Fourier or wavelets transforms are replaced by “physiologically plausible,” non-orthogonal, and parametrized bases, more suited to the physiology or physics of the phenomena being modeled, like in a transform called Dynalet, based on the potential Hamiltonian decomposition of Liénard systems [2, 3]. This article is an effort to improve TPS MT formalism. In this sense, it is claimed that because Extreme Learning Machines (ELMs) [4] are implemented in Fast Artificial Neural Network (FANN), with an extremely fast and analytically well-formalized training cycle, we use here FANN in place of backpropagation ANN (BANN) previously used to define TPS MT.

Tailored to the Problem Specificity Mathematical Transform will be calculated in this article by using a trained Fast Artificial Neural Network (FANN), which presents a low algorithm complexity as compared, for example, with numerical methods [4]. In Figure 1, (ε₁, ε₂, …, ε_n) corresponds to the i-th input vector of the FANN, whose weight matrices [Ω^h] and [Ω⁰] contain each n columns equal, respectively, to vectors (ωi,1h,…,ωi,mh) and (ωi,10,…, ωi,p0).

The vectors (ωi,1h,…,ωi,mh) and (ωi,10,…, ωi,p0) correspond, respectively, to the synaptic coefficients from input to hidden layer and from hidden to output layer matrices; the activation vectors (A1h,…,Amh) and (A10,…,Ap0) are functions of hidden and output neurons and vectors (σmh,…,σmh) and (σp0,…,σp0) are the neuron outputs of hidden and output layers. In reference to Figure 1, using ELM within a single-layer FANN and under assumptions, which are discussed in the middle of Section 4 of this article, its training cycle can be reduced to solve the equation:

where [Ω^o] is the synaptic weight matrix from hidden to output layer, evidences learning matrix [Σ^h]⁻¹ is the inverse of the Moore–Penrose matrix of the input to hidden layer neurons for each input vector, and [T] is the target matrix, for every input vector. Finally, it is postulated that characterize the above matrices can be dealt with as if they were mathematical transforms between two vector spaces.

Biosignals have been described as being composed by different sequential components that underlie a variety of successive recorded physiological processes: mathematical models of auditory brainstem responses (ABRs), blood pressure waveforms (BPWs), hemodynamic response function (HRF), and homeostatic system dynamics (HSD) will be indeed explored in this article to illustrate TPS, for showing the generic character of our general approach.

Auditory brainstem response in the cat is assumed to be composed by waves originated in the sequential activation of the synaptic relays within the auditory pathway, from cochlear nucleus to cortical projections from the thalamic medial geniculate nucleus to the primary auditory cortical area [5]. ABR has been modeled by 5 sequential components since 1986 [6] up to 2010 [3]. Each of the ABR components is named as a Jewett component (JC). Each JC represents the effect of a global postsynaptic potential in the corresponding relay within the auditory pathway.

Blood pressure waveform in human cardiovascular system is assumed to be composed by up to three or four components [7]. Main component is due to cardiac pumping and it has been confirmed [8] the existence of two major reflection sites in the central arteries. The first occurs in the juncture between thoracic and abdominal aorta and the second occurs in the juncture between abdominal aorta and common iliac arteries. These sites produce the second and third BPW components. Each of the BPW components is named here a Latham component (LC).

Hemodynamic response function is recorded in functional magnetic resonance imaging. When a cortical area is recruited while performing a synchronized motor task, the local oxygen consumption increases, inducing a chain of events, including variations in regional blood volume, blood flow, and relative deoxyhemoglobin/oxyhemoglobin concentration that leads to an increase in the local magnetic resonance signal. It is temporally described by the hemodynamic response function (HRF). The HRF is assumed to be composed by components of a balloon model [9]. The first component corresponds to the inflation phase of the “balloon” representing the local increase in blood volume and the second component corresponds to the deflation phase. Taking into account the elastic properties of the venous compartment and corresponding transient mismatch between variations in local parameters, the balloon model of the HRF is operationally described by two mathematical equations [10, 11], both of them with two components. It is said here that the HRF is composed by Glover component (GC) or Friston component (FC).

Dynalet transform (DT) was defined to represent the HSD. DT produces another example of mathematical transform-based model [3, 12, 13]. It is represented by the general Liénard systems, which obey a non-linear damping equation, generalizing the van der Pol oscillator [14, 15]. Its solution has two components: a Potential (P) and a Hamiltonian (H). DT is said here to be composed by Demongeot components (DCs).

Vector spaces are commonly used to represent mathematical transforms (MTs) in vector spaces (VSs) [1]. In the linear case, however, these structures are restrained to conditions to be accomplished by a subspace to be a basis of MT.

In opposition with linear analysis, where orthogonal basis is used because of mathematical convenience, TPS MT emphasizes the “physiological plausibility” of the modeling approach. In order to evaluate the impact of levering constraints for vectors considered as physiologically plausible, as being part of basis B, the general linear analysis is briefly visited and breakpoints are established providing TPS formalization. Whether V is ABR, BPW, HFR, or HDS, linear combination of physiologically plausible elements conforming a basis B, could span the corresponding vector space.

The principles of mathematical transforms recall Jean Baptiste Joseph Fourier's approach [29, 30], while studying heat propagation. Fourier findings were generalized in linear analysis [1], where linear combination of elements of its orthogonal basis, B, spans the corresponding vector space V. In his study (done in Grenoble between 1804 and 1822), Fourier realized that “functions of a variable, whether continuous or discontinuous, can be expanded in a series of sines of multiples of the variable.” Later on, Dirichlet [31] restricted this knowledge to functions defined on [–π, π], linear combinations of the elements of the following orthonormal basis B:

Linear combinations of elements of B span the set of partial continuous functions on [−π, π], denoted C([–π, π]) [32]. Findings of Fourier gave rise to the series and transforms named honoring him. As it is shown in the next section, coefficients of Fourier series are analytically established because, on one hand, basis is orthogonal and, on the other hand, Fourier coefficients weight the linear combinations of the basis vectors, spanning C([–π, π]).

At this stage, basis orthogonality is the necessary condition for analytical solution of mathematical transforms (MTs).

Depending on the orthogonal vector set selected as basis B of V, different orthogonal MTs are defined: Fourier, Laplace, Legendre, Hermite, and Laguerre polynomial transforms and orthonormal wavelet transform. These MTs satisfy the condition of orthogonality and belong to Linear transforms [1].

Dynalet transform can be considered as a transition from Linear transforms to TPS MT. Despite having a non-orthogonal basis, it approximately solves the anharmonic relaxation pendulum problem, with the desired precision level, depending on the number of harmonics considered in the solution. Fourier solves exactly the harmonic pendulum problem and Wavelet solves its damped analog.

In previous studies [3, 16, 17, 28], backpropagation learning rule was implicitly or explicitly proposed and/or used to train Single Layer Feedforward Neural Network (SLFNN) to solve system equations like (29) and to define a TPS. In this study, it is claimed that ELM [4], implemented as an ANN with SLFNN architecture, provides better tools to formalize, understand, and rapidly simulate a TPS MT. Equation 1 is at the center of this claim. According to Figure 1, let:

where each column of the matrix [Ω^o] is ωqo⃗, with 1 ≤ q ≤ p.

Finally, because the proposed ELM uses single hidden layer FANN architecture, it is a universal approximator [32] for every input matrix [Σ^o], it can be as close as required to the corresponding target matrix [T]. In such a case, [Σ^o] can be reasonably replaced by [T] and:

Multiplying both the sides of Equation (31) to the left by [Σ^h]⁻¹, which denotes the generalized Moore–Penrose inverse of [Σ^h], generates the Equation (1):

In summary, TPS transform may be not orthogonal and its basis, which can be parametrized, spans ABR space combining non-linearly its elements. Using this kind of transforms maintains the unified vision provided by the linear analysis in the sense that they correspond to a direct transformation from vector space V onto another space, containing the corresponding coefficients and parameters.

There can also exist an inverse transform from parameter space onto V, the set of the b_i (t), which are possibly non-orthogonal and can represent independent components spanning V. Nevertheless, TPS basis is chosen because of its physiological plausibility, not because of mathematical convenience. In TPS, the conditions for an orthogonal and non-parametrized basis are removed while, on the contrary, from the perspective of modeling, physiological plausibility of basis generators is strengthened.

Left side of Figure 7 illustrates the mapping of a temporal function onto the corresponding amplitude A and phase φ spectra in a frequency space characterizing direct Fourier transform F. In the opposite direction, inverse Fourier transform F⁻¹ is performed from the Fourier vector space to the signal vector space. Right side of the Figure 7 illustrates the direct TPS transform J mapping from the original temporal signal space onto the corresponding parameter space. In the opposite direction, inverse TPS transform J⁻¹ performs mapping from the parameter space onto temporal signal vector space.

In a previous study, an eight steps methodology was defined to solve TPS MT [3]. Steps considered are:

As previously pointed out, if TPS is possible, basis could express causality better than linear transforms and “fundamental and technical analysis” would be congruent between each other. Yet it remains the following steps:

This article can be considered as a trial to use ELM instead of backpropagation learning rule in solving TPS, such as Jewett transform, as well as an informal venture into the definition of Glover, Friston, and Latham transforms and a first attempt to link Dynalet transform with TPS transforms.

A recent improvement of TPS MT approach is the functional principal component analysis (FPCA), a tool of dimension reduction with high correlations in functional data analysis [33, 34], which is used in epidemiologic problems as those related to coronavirus disease 2019 (COVID-19) outbreak dynamics, as given in Figure 8 [35–45].

Let {x_i(t)}_{i = 1,m} be a given set of functions and let denote α = (α₁, …, α_m) a weight vector, FPCA is calculated such as it finds, in the set of the C₁ functions on the time interval of observation, the first principal component weight function α₁(t), for which the first principal component score is given by:

while maximizing ∑i=1,mfi12 is subjected to:

Next, the weight function α₂(t) is calculated and the second principal component score maximizes ∑i=1,mfi22, and is subjected to the constraints ‖α2‖2=1 and:

Then, the process is repeated for many iterations. In our analysis, we used a tool called pca.fd for the principal component analysis. As given in Figure 8, the 4 PCs values plots throughout the days considered for all the m (m = 101) French departments providing functional data being before vaccination started and during vaccination. Table 1 gives the weight of each PC for the curves of hospitalization before and after vaccination has started.

This last example shows that the TPS approach is still here relevant and can be considered as a valid concrete alternative to realistically approach the modeling of temporal data in epidemiological [46–51] or physiological [24, 52] examples.

To conclude, even with the recent techniques of the functional principal component analysis (FCPA), the TPS MT approach is relevant, in the sense that it makes it possible to consider the experimental signal as a linear combination of functions that are the weighted sum (not necessarily orthogonal) of the main empirical variables of the system source of the recorded signal, in a decreasing order of importance, which allows the interpretation of the respective role of the empirical variables in the generation of this signal.

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Ethical review and approval was not required for the study on human participants in accordance with the local legislation and institutional requirements. The patients/participants provided their written informed consent to participate in this study.

JD and AG: conceptualization, methodology, validation, formal analysis, resources, writing—original draft preparation, supervision, and project administration. KO and CT: software. AG, JD, RS, SC, PR, AA, GT, MS, HZ, CT, and KO: investigation. AA, GT, MS, HZ, CT, and KO: data curation. JD and KO: writing—review and editing. KO, CT, and AG: visualization. All authors have read and agreed to the final version of the manuscript.

The authors acknowledge to support Chilean Grants DIUV 44/11 from U. de Valparaiso, Fondecyt MEC 80110027, and FONDEF IT13I20060 from Conicyt.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.