Embedding non-linear filtering in autonomous underwater vehicle dynamics via the Kolmogorov backward equation

This paper revisits the state vector of an autonomous underwater vehicle (AUV) dynamics coupled with the underwater Markovian stochasticity in the ‘non-linear filtering’ context. The underwater stochasticity is attributed to atmospheric turbulence, planetary interactions, sea surface conditions and astronomical phenomena. In this paper, we adopt the Itô process, a homogeneous Markov process, to describe the AUV state vector evolution equation. This paper accounts for the process noise as well as observation noise correction terms by considering the underwater filtering model. The non-linear filtering of the paper is achieved using the Kolmogorov backward equation and the evolution of the conditional characteristic function. The non-linear filtering equation is the cornerstone formalism of stochastic optimal control systems. Most notably, this paper introduces the non-linear filtering theory into an underwater vehicle stochastic system by constructing a lemma and a theorem for the underwater vehicle stochastic differential equation that were not available in the literature.

Classification of Microarray Gene Expression Data Using an Infiltration Tactics Optimization (ITO) Algorithm

A number of different feature selection and classification techniques have been proposed in literature including parameter-free and parameter-based algorithms. The former are quick but may result in local maxima while the latter use dataset-specific parameter-tuning for higher accuracy. However, higher accuracy may not necessarily mean higher reliability of the model. Thus, generalized optimization is still a challenge open for further research. This paper presents a warzone inspired “infiltration tactics” based optimization algorithm (ITO)—not to be confused with the ITO algorithm based on the Itõ Process in the field of Stochastic calculus. The proposed ITO algorithm combines parameter-free and parameter-based classifiers to produce a high-accuracy-high-reliability (HAHR) binary classifier. The algorithm produces results in two phases: (i) Lightweight Infantry Group (LIG) converges quickly to find non-local maxima and produces comparable results (i.e., 70 to 88% accuracy) (ii) Followup Team (FT) uses advanced tuning to enhance the baseline performance (i.e., 75 to 99%). Every soldier of the ITO army is a base model with its own independently chosen Subset selection method, pre-processing, and validation methods and classifier. The successful soldiers are combined through heterogeneous ensembles for optimal results. The proposed approach addresses a data scarcity problem, is flexible to the choice of heterogeneous base classifiers, and is able to produce HAHR models comparable to the established MAQC-II results.

Stochastic modelling in fluid dynamics: Itô versus Stratonovich

Suppose the observations of Lagrangian trajectories for fluid flow in some physical situation can be modelled sufficiently accurately by a spatially correlated Itô stochastic process (with zero mean) obtained from data which is taken in fixed Eulerian space. Suppose we also want to apply Hamilton’s principle to derive the stochastic fluid equations for this situation. Now, the variational calculus for applying Hamilton’s principle requires the Stratonovich process, so we must transform from Itô noise in the data frame to the equivalent Stratonovich noise. However, the transformation from the Itô process in the data frame to the corresponding Stratonovich process shifts the drift velocity of the transformed Lagrangian fluid trajectory out of the data frame into a non-inertial frame obtained from the Itô correction. The issue is, ‘Will non-inertial forces arising from this transformation of reference frames make a difference in the interpretation of the solution behaviour of the resulting stochastic equations?’ This issue will be resolved by elementary considerations.

Revisiting Dynamic Potential Adopter Diffusion Models under the Influence of Irregular Fluctuations in Adoption Rate

A pragmatic innovation diffusion model is proposed in the present chapter that interpolates stochasticity in the logistic formulation of the widely-acknowledged Bass model with dynamic market size. These irregular changes are caused due to uncertainty attached to the socioeconomic and political environment in which an innovation is positioned that affects the action of potential adopters leading to their non-uniform behavior. The aim of the current study is to find the analytical solution for the two dynamic market expansion structures, namely, linear and exponential under the influence of irregular fluctuations whose closed-form solutions were not possible in the existing literature. In addition to the changeable market size, the proposed innovation diffusion also incorporates the concept of repeat purchase. The anticipated stochastic differential equation based new product diffusion model is then expounded methodically using the Itô process and Itô's integral equation. Further, the model has been used to study the growth pattern of different consumer durable products.

Probabilistic Basin of Attraction and Its Estimation Using Two Lyapunov Functions

We study stability for dynamical systems specified by autonomous stochastic differential equations of the form dX(t)=f(X(t))dt+g(X(t))dW(t), with (X(t))t≥0 an Rd-valued Itô process and (W(t))t≥0 an RQ-valued Wiener process, and the functions f:Rd→Rd and g:Rd→Rd×Q are Lipschitz and vanish at the origin, making it an equilibrium for the system. The concept of asymptotic stability in probability of the null solution is well known and implies that solutions started arbitrarily close to the origin remain close and converge to it. The concept therefore pertains exclusively to system properties local to the origin. We wish to address the matter in a more practical manner: Allowing for a (small) probability that solutions escape from the origin, how far away can they then be started? To this end we define a probabilistic version of the basin of attraction, the γ-BOA, with the property that any solution started within it stays close and converges to the origin with probability at least γ. We then develop a method using a local Lyapunov function and a nonlocal one to obtain rigid lower bounds on γ-BOA.

Heart rate from face videos under realistic conditions for advanced driver monitoring

The role of physiological signals has a large impact on driver monitoring systems, since it tells something about the human state. This work addresses the recursive probabilistic inference problem in time-varying linear dynamic systems to incorporate invariance into the task of heart rate estimation from face videos under realistic conditions. The invariance encapsulates motion as well as varying illumination conditions in order to accurately estimate vitality parameters from human faces using conventional camera technology. The solution is based on the canonical state space representation of an Itô process and a Wiener velocity model. Empirical results yield to excellent real-time and estimation performance of heart rates in presence of disturbing factors, like rigid head motion, talking, facial expressions and natural illumination conditions making the process of human state estimation from face videos applicable in a much broader sense, pushing the technology towards advanced driver monitoring systems.

Consistency issues in PDF methods

Concentrations of chemical species transported in random environments need to be statistically characterized by probability density functions (PDF). Solutions to evolution equations for the one-point one-time PDF are usually based on systems of computational particles described by Itô equations. We establish consistency conditions relating the concentration statistics to that of the Itô process and the solution of its associated Fokker-Planck equation to that of the PDF equation. In this frame, we use a recently proposed numerical method which approximates PDFs by particle densities obtained with a global random walk (GRW) algorithm. The GRW-PDF approach is illustrated for a problem of contaminant transport in groundwater.