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A common problem when analyzing models, such as mathematical modeling of a biological process, is to determine if the unknown parameters of the model can be determined from given input-output data. Identifiable models are models such that the unknown parameters can be determined to have a finite number of values given input-output data. The total number of such values over the complex numbers is called the identifiability degree of the model. Unidentifiable models are models such that the unknown parameters can have an infinite number of values given input-output data. For unidentifiable models, a set of identifiable functions of the parameters are sought so that the model can be reparametrized in terms of these functions yielding an identifiable model. In this work, we use numerical algebraic geometry to determine if a model given by polynomial or rational ordinary differential equations is identifiable or unidentifiable. For identifiable models, we present a novel approach to compute the identifiability degree. For unidentifiable models, we present a novel numerical differential algebra technique aimed at computing a set of algebraically independent identifiable functions. Several examples are used to demonstrate the new techniques.

How to quantify the uncertain information in the framework of Dempster-Shafer evidence theory is still an open issue. Quite a few uncertainty measures have been proposed in Dempster-Shafer framework, however, the existing studies mainly focus on the mass function itself, the available information represented by the scale of the frame of discernment (FOD) in the body of evidence is ignored. Without taking full advantage of the information in the body of evidence, the existing methods are somehow not that efficient. In this paper, a modified belief entropy is proposed by considering the scale of FOD and the relative scale of a focal element with respect to FOD. Inspired by Deng entropy, the new belief entropy is consistent with Shannon entropy in the sense of probability consistency. What's more, with less information loss, the new measure can overcome the shortage of some other uncertainty measures. A few numerical examples and a case study are presented to show the efficiency and superiority of the proposed method.

This paper will prove the uniqueness theorem for 3-layered complex-valued neural networks where the threshold parameters of the hidden neurons can take non-zeros. That is, if a 3-layered complex-valued neural network is irreducible, the 3-layered complex-valued neural network that approximates a given complex-valued function is uniquely determined up to a finite group on the transformations of the learnable parameters of the complex-valued neural network.

Real world processes with an ''intensity'' and ''direction'' component can be made complex by convenience of representation (vector fields, radar, sonar), and their processing directly in the field of complex numbers C is not only natural but is also becoming commonplace in modern applications. Yet, adaptive signal processing and machine learning algorithms suitable for the processing of such signals directly in C are only emerging. To this cause we introduce a second order statistical learning framework for a general class of nonlinear adaptive filters with feedback realized as recurrent neural networks (RNNs). For rigour, both the so-called proper- and improper-second order statistics of complex processes is taken into account, and the proposed augmented complex real-time recurrent learning (ACRTRL) algorithm for RNNs has been shown to be suitable for processing a wide range of both benchmark and real-world complex processes.

It should not be surprising that in solving cubic equations, sooner or later you are going to have to take the cube root of something. But it's not immediately obvious what cubic equations have to do with arc cosines. Well, in the general case of complex coefficients and roots we would need to take the cube root of a complex number

In this paper we present a hybrid control scheme, combining the advantages of task-space and joint-space control. The controller is based on a human-like adaptive design, which minimises both control effort and tracking error. Our novel hybrid adaptive controller has been tested in extensive simulations, in a scenario where a Baxter robot manipulator is affected by external disturbances in the form of interaction with the environment and tool-like end-effector perturbations. The results demonstrated improved performance in the hybrid controller over both of its component parts. In addition, we introduce a novel method for online adaptation of learning parameters, using the fuzzy control formalism to utilise expert knowledge from the experimenter. This mechanism of meta-learning induces further improvement in performance and avoids the need for tuning through trial testing.

Recently models of neural networks that can directly deal with complex numbers, complex-valued neural networks, have been proposed and several studies on their abilities of information processing have been done. Furthermore models of neural networks that can deal with quaternion numbers, which is the extension of complex numbers, have also been proposed. However they are all multilayer quaternion neural networks. This paper proposes models of fully connected recurrent quaternion neural networks, Hopfield-type quaternion neural networks. Since quaternion numbers are non-commutative on multiplication, some different models can be considered. We investigate dynamics of these proposed models from the point of view of the existence of an energy function and derive their conditions for existence.

This letter presents some results of an analysis on the decision boundaries of complex-valued neural networks whose weights, threshold values, input and output signals are all complex numbers. The main results may be summarized as follows. (1) A decision boundary of a single complex-valued neuron consists of two hypersurfaces that intersect orthogonally, and divides a decision region into four equal sections. The XOR problem and the detection of symmetry problem that cannot be solved with two-layered real-valued neural networks, can be solved by two-layered complex-valued neural networks with the orthogonal decision boundaries, which reveals a potent computational power of complex-valued neural nets. Furthermore, the fading equalization problem can be successfully solved by the two-layered complex-valued neural network with the highest generalization ability. (2) A decision boundary of a three-layered complex-valued neural network has the orthogonal property as a basic structure, and its two hypersurfaces approach orthogonality as all the net inputs to each hidden neuron grow. In particular, most of the decision boundaries in the three-layered complex-valued neural network inetersect orthogonally when the network is trained using Complex-BP algorithm. As a result, the orthogonality of the decision boundaries improves its generalization ability. (3) The average of the learning speed of the Complex-BP is several times faster than that of the Real-BP. The standard deviation of the learning speed of the Complex-BP is smaller than that of the Real-BP. It seems that the complex-valued neural network and the related algorithm are natural for learning complex-valued patterns for the above reasons.

Weighted geometric docking is a prediction algorithm that matches weighted molecular surfaces. Each molecule is represented by a grid of complex numbers, storing information about the shape of the molecule in the real part and weight information in the imaginary part. The weights are based on experimental biochemical and biophysical data or on theoretical analyses of amino acid conservation or correlation patterns in multiple-sequence alignments of homologous proteins. Only a few surface residues on either one or both molecules are weighted. In contrast to methods that use postscan filtering based on biochemical information, our method incorporates the external data in the rotation-translation search, producing a different set of docking solutions biased toward solutions in which the up-weighted residues are at the interface. Similarly, interactions involving specified residues can be impeded. The weighted geometric algorithm was applied to five systems for which regular geometric docking of the unbound molecules gave poor results. We obtained much better ranking of the nearly correct prediction and higher statistical significance when weighted geometric docking was used. The method was successful even when the weighted portion of the surface corresponded only partially and approximately to the binding site.

As is well known, the common elementary functions defined over the real numbers can be generalized to act not only over the complex number field but also over the skew (non-commuting) field of the quaternions. In this paper, we detail a number of elementary functions extended to act over the skew field of Clifford multivectors, in both two and three dimensions. Complex numbers, quaternions and Cartesian vectors can be described by the various components within a Clifford multivector and from our results we are able to demonstrate new inter-relationships between these algebraic systems. One key relationship that we discover is that a complex number raised to a vector power produces a quaternion thus combining these systems within a single equation. We also find a single formula that produces the square root, amplitude and inverse of a multivector over one, two and three dimensions. Finally, comparing the functions over different dimension we observe that $ C\ell \left (\Re^3 \right) $ provides a particularly versatile algebraic framework., 21 pages, 0 figures

Various algorithms of autoregressive (AR) recursive identification make it possible to evaluate power spectral distribution in correspondence with each sample of a time series, and time-variant spectral parameters can be calculated through the evaluation of the pole positions in the complex z-plane. In traditional analysis, the poles are obtained by zeroing the denominator of the model transfer function, expressed as a function of the AR coefficients. In this paper, two algorithms for the direct updating and tracking of movements of poles of an AR time-variant model on the basis of the innovation given to the coefficients are presented and investigated. The introduced algorithms are based upon 1) the classical linearization method and 2) a recursive method to compute the roots of a polynomial, respectively. In the present paper, applications in the field of heart rate variability (HRV) signal analysis are presented and efficient tools are proposed for quantitative extraction of spectral parameters (power and frequency of the low-frequency (LF) and high-frequency (HF) components) for the monitoring of the action of the autonomic nervous system in transient patho-physiological events. These computational methods seem to be very attractive for HRV applications, as they inherit the peculiarity of recursive time-variant identification, and provide a more immediate comprehension of the spectral process characteristics when expressed in terms of poles and AR spectral components.

This note presents an analysis of the octonionic form of the division algebraic support vector regressor (SVR) first introduced by Shilton A detailed derivation of the dual form is given, and three conditions under which it is analogous to the quaternionic case are exhibited. It is shown that, in the general case of an octonionic-valued feature map, the usual "kernel trick" breaks down. The cause of this (and its interpretation) is discussed in some detail, along with potential ways of extending kernel methods to take advantage of the distinct features present in the general case. Finally, the octonionic SVR is applied to an example gait analysis problem, and its performance is compared to that of the least squares SVR, the Clifford SVR, and the multidimensional SVR.

Most of local minima caused by the hierarchical structure can be resolved by extending the real-valued neural network to complex numbers. It was proved in 2000 that a critical point of the real-valued neural network with H-1 hidden neurons always gives many critical points of the real-valued neural network with H hidden neurons. These critical points consist of many lines in the parameter space which could be local minima or saddle points. Local minima cause plateaus which have a strong negative influence on learning. However, most of the critical points of complex-valued neural network are saddle points unlike those of the real-valued neural network. This is a prominent property of the complex-valued neural network.

Alignment of two sets containing two dimensional vectors (2D points) constitutes an important problem in medical imaging, remote sensing, and computer vision. We assume that the points in one set, called the transformed set, are constructed by translating and rotating the points in the other set, called the original set. The points in both sets are represented by complex numbers. In order to translate and then rotate a point, we add a complex constant and then multiply by a complex exponential respectively. We construct a cost function which tries to achieve the least-squares differences between a given transformed set and the set containing transformed points with respect to optimization parameters. We implement the Newton-Raphson optimization algorithm with polynomial line search in order to minimize this cost function. Simulation results with multiple datasets demonstrate that the proposed method aligns two sets efficiently and reliably.

A fast and accurate algorithm is developed for the numerical computation of the family of complex linear canonical transforms (CLCTs), which represent the input-output relationship of complex quadratic-phase systems. Allowing the linear canonical transform parameters to be complex numbers makes it possible to represent paraxial optical systems that involve complex parameters. These include lossy systems such as Gaussian apertures, Gaussian ducts, or complex graded-index media, as well as lossless thin lenses and sections of free space and any arbitrary combinations of them. Complex-ordered fractional Fourier transforms (CFRTs) are a special case of CLCTs, and therefore a fast and accurate algorithm to compute CFRTs is included as a special case of the presented algorithm. The algorithm is based on decomposition of an arbitrary CLCT matrix into real and complex chirp multiplications and Fourier transforms. The samples of the output are obtained from the samples of the input in approximately N log N time, where N is the number of input samples. A space-bandwidth product tracking formalism is developed to ensure that the number of samples is information-theoretically sufficient to reconstruct the continuous transform, but not unnecessarily redundant.

Jim Blinn discusses a good practical closed-form solution process for cubic equations and makes some observations about the relation between iterative and closed-form solutions.

We have presented a complex graph matrix representation to characterize proteomics maps obtained from 2D-gel electrophoresis. In this method, each bubble in a 2D-gel proteomics map is represented by a complex number with components which are charge and mass. Then, a graph with complex weights is constructed by connecting the vertices in the relative order of abundance. This yields adjacency matrices and distance matrices of the proteomics graph with complex weights. We have computed the spectra, eigenvectors, and other properties of complex graphs and the Euclidian/graph distance obtained from the complex graphs. The leading eigenvalues and eigenvectors and, likewise, the smallest eigenvalues and eigenvectors, and the entire graph spectral patterns of the complex matrices derived from them yield novel weighted biodescriptors that characterize proteomics maps with information of charge and masses of proteins. We have also applied these eigenvector and eigenvalue maps to contrast the normal cells and cells exposed to four peroxisome proliferators, namely, clofibrate, diethylhexyl phthalate (DEHP), perfluorodecanoic acid (PFDA), and perfluoroctanoic acid (PFOA). Our complex eigenspectra show that the proteomic response induced by DEHP differs from the corresponding responses of other three chemicals consistent with their chemical structures and properties.

Complex numbers appear naturally in biology whenever a system can be analyzed in the frequency domain, such as physiological data from magnetoencephalography (MEG). For example, the MEG steady state response to a modulated auditory stimulus generates a complex magnetic field for each MEG channel, equal to the Fourier transform at the stimulus modulation frequency. The complex nature of these data sets, often not taken advantage of, is fully exploited here with new methods. Whole-head, complex magnetic data can be used to estimate complex neural current sources, and standard methods of source estimation naturally generalize for complex sources. We show that a general complex neural vector source is described by its location, magnitude, and direction, but also by a phase and by an additional perpendicular component. We give natural interpretations of all the parameters for the complex equivalent-current dipole by linking them to the underlying neurophysiology. We demonstrate complex magnetic fields, and their equivalent fully complex current sources, with both simulations and experimental data., 23 pages, 1 table, 5 figures; to appear in Journal of Neuroscience Methods

The origin of the concept of a molecular complex between oxygen and hemoglobin can be traced to Stokes, a century and a half ago. Subsequently, physicochemical concepts of equilibria provided a path to quantitative formulations of these ligand-receptor interactions. Then, it took a quarter of a century before a proper format was prepared in terms of four stoichiometric equilibria and their associated binding constants. Since then, experimental measurements of these stoichiometric binding constants have consistently disclosed that successive values of K(1) to K(4) are accentuated above those expected if every subunit of hemoglobin maintained the same, intrinsic, unchanging affinity for oxygen. An alternative analysis of the observed cooperative interactions has been obtained by extracting the roots of the polynomial of the stoichiometric binding equation and deriving an alternative binding equation containing constants that for O(2)-Hb are complex numbers. These constants have the dimensions and properties of equilibrium constants. They provide some novel phenomenological insights into ligand-receptor equilibria.

Two patients with severe amnesia following rupture of anterior communicating artery aneurysms were able to learn a complex algorithm for mentally squaring two-digit numbers. Although both patients learned the algorithm at a similar rate, one patient's improvement was accounted for by savings in the steps of the algorithm. The other patient, however, showed little improvement in the steps while performance of the whole algorithm improved dramatically. Neither patient showed savings in the Hebb Digit Span procedure. The results suggested a dissociation between amnesics in their capacity to learn the constituent “productions” or steps of a skill and their capacity to organize those productions into a single coherent act.