Simultaneous Measurements of Noncommuting Observables: Positive Transformations and Instrumental Lie Groups

We formulate a general program for describing and analyzing continuous, differential weak, simultaneous measurements of noncommuting observables, which focuses on describing the measuring instrument autonomously, without states. The Kraus operators of such measuring processes are time-ordered products of fundamental differential positive transformations, which generate nonunitary transformation groups that we call instrumental Lie groups. The temporal evolution of the instrument is equivalent to the diffusion of a Kraus-operator distribution function, defined relative to the invariant measure of the instrumental Lie group. This diffusion can be analyzed using Wiener path integration, stochastic differential equations, or a Fokker-Planck-Kolmogorov equation. This way of considering instrument evolution we call the Instrument Manifold Program. We relate the Instrument Manifold Program to state-based stochastic master equations. We then explain how the Instrument Manifold Program can be used to describe instrument evolution in terms of a universal cover that we call the universal instrumental Lie group, which is independent not just of states, but also of Hilbert space. The universal instrument is generically infinite dimensional, in which case the instrument's evolution is chaotic. Special simultaneous measurements have a finite-dimensional universal instrument, in which case the instrument is considered principal, and it can be analyzed within the differential geometry of the universal instrumental Lie group. Principal instruments belong at the foundation of quantum mechanics. We consider the three most fundamental examples: measurement of a single observable, position and momentum, and the three components of angular momentum. As these measurements are performed continuously, they converge to strong simultaneous measurements. For a single observable, this results in the standard decay of coherence between inequivalent irreducible representations. For the latter two cases, it leads to a collapse within each irreducible representation onto the classical or spherical phase space, with the phase space located at the boundary of these instrumental Lie groups.

Simultaneous Momentum and Position Measurement and the Instrumental Weyl-Heisenberg Group

The canonical commutation relation, [Q,P]=iℏ, stands at the foundation of quantum theory and the original Hilbert space. The interpretation of P and Q as observables has always relied on the analogies that exist between the unitary transformations of Hilbert space and the canonical (also known as contact) transformations of classical phase space. Now that the theory of quantum measurement is essentially complete (this took a while), it is possible to revisit the canonical commutation relation in a way that sets the foundation of quantum theory not on unitary transformations but on positive transformations. This paper shows how the concept of simultaneous measurement leads to a fundamental differential geometric problem whose solution shows us the following. The simultaneous P and Q measurement (SPQM) defines a universal measuring instrument, which takes the shape of a seven-dimensional manifold, a universal covering group we call the instrumental Weyl-Heisenberg (IWH) group. The group IWH connects the identity to classical phase space in unexpected ways that are significant enough that the positive-operator-valued measure (POVM) offers a complete alternative to energy quantization. Five of the dimensions define processes that can be easily recognized and understood. The other two dimensions, the normalization and phase in the center of the IWH group, are less familiar. The normalization, in particular, requires special handling in order to describe and understand the SPQM instrument.

Stochastic Differential Equation-based Mixed Effects Model of the Fluid Volume in the Fasted Stomach in Healthy Adult Human

The rate and extent of drug dissolution and absorption from a solid oral dosage form depend largely on the fluid volume along the gastrointestinal tract. Hence, a model built upon the gastric fluid volume profiles can help to predict drug dissolution and subsequent absorption. To capture the great inter- and intra-individual variability (IAV) of the gastric fluid volume in fasted human, a stochastic differential equation (SDE)-based mixed effects model was developed and compared with the ordinary differential equation (ODE)-based model. Twelve fasted healthy adult subjects were enrolled and had their gastric fluid volume measured before and after consumption of 240 mL of water at pre-determined intervals for up to 2 hours post ingestion. The SDE- and ODE-based mixed effects models were implemented and compared using extended Kalman filter algorithm via NONMEM. The SDE approach greatly improved the goodness of fit compared with the ODE counterpart. The proportional and additive measurement error of the final SDE model decreased from 14.4 to 4.10% and from 17.6 to 4.74 mL, respectively. The SDE-based mixed effects model successfully characterized the gastric volume profiles in the fasted healthy subjects, and provided a robust approximation of the physiological parameters in the very dynamic system. The remarkable IAV could be further separated into system dynamics terms and measurement error terms in the SDE model instead of only empirically attributing IAV to measurement errors in the traditional ODE method. The system dynamics were best captured by the random fluctuations of gastric emptying coefficient Kge.

Threshold theorem in isolated quantum dynamics with stochastic control errors

We investigate the effect of stochastic control errors in the time-dependent Hamiltonian on isolated quantum dynamics. The control errors are formulated as time-dependent stochastic noise in the Schrödinger equation. For a class of stochastic control errors, we establish a threshold theorem that provides a sufficient condition to obtain the target state, which should be determined in noiseless isolated quantum dynamics, as a relation between the number of measurements and noise strength. The theorem guarantees that if the sum of the noise strengths is less than the inverse of computational time, the target state can be obtained through a constant-order number of measurements. If the opposite is true, the number of measurements to guarantee obtaining the target state increases exponentially with computational time. Our threshold theorem can be applied to any isolated quantum dynamics such as quantum annealing and adiabatic quantum computation. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.

Monte Carlo Simulation of Stochastic Differential Equation to Study Information Geometry

Information Geometry is a useful tool to study and compare the solutions of a Stochastic Differential Equations (SDEs) for non-equilibrium systems. As an alternative method to solving the Fokker-Planck equation, we propose a new method to calculate time-dependent probability density functions (PDFs) and to study Information Geometry using Monte Carlo (MC) simulation of SDEs. Specifically, we develop a new MC SDE method to overcome the challenges in calculating a time-dependent PDF and information geometric diagnostics and to speed up simulations by utilizing GPU computing. Using MC SDE simulations, we reproduce Information Geometric scaling relations found from the Fokker-Planck method for the case of a stochastic process with linear and cubic damping terms. We showcase the advantage of MC SDE simulation over FPE solvers by calculating unequal time joint PDFs. For the linear process with a linear damping force, joint PDF is found to be a Gaussian. In contrast, for the cubic process with a cubic damping force, joint PDF exhibits a bimodal structure, even in a stationary state. This suggests a finite memory time induced by a nonlinear force. Furthermore, several power-law scalings in the characteristics of bimodal PDFs are identified and investigated.

Inference for epidemic models with time-varying infection rates: Tracking the dynamics of oak processionary moth in the UK

Invasive pests pose a great threat to forest, woodland, and urban tree ecosystems. The oak processionary moth (OPM) is a destructive pest of oak trees, first reported in the UK in 2006. Despite great efforts to contain the outbreak within the original infested area of South‐East England, OPM continues to spread.

Here, we analyze data consisting of the numbers of OPM nests removed each year from two parks in London between 2013 and 2020. Using a state‐of‐the‐art Bayesian inference scheme, we estimate the parameters for a stochastic compartmental SIR (susceptible, infested, and removed) model with a time‐varying infestation rate to describe the spread of OPM.

We find that the infestation rate and subsequent basic reproduction number have remained constant since 2013 (with R0 between one and two). This shows further controls must be taken to reduce R0 below one and stop the advance of OPM into other areas of England.

Synthesis. Our findings demonstrate the applicability of the SIR model to describing OPM spread and show that further controls are needed to reduce the infestation rate. The proposed statistical methodology is a powerful tool to explore the nature of a time‐varying infestation rate, applicable to other partially observed time series epidemic data.

Implementation of a Commitment Machine for an Adaptive and Robust Expected Shortfall Estimation

This study proposes a metaheuristic for the selection of models among different Expected Shortfall (ES) estimation methods. The proposed approach, denominated “Commitment Machine” (CM), has a strong focus on assets cross-correlation and allows to measure adaptively the ES, dynamically evaluating which is the most performing method through the minimization of a loss function. The CM algorithm compares four different ES estimation techniques which all take into account the interaction effects among assets: a Bayesian Vector autoregressive model, Stochastic Differential Equation (SDE) numerical schemes with Exponential Weighted Moving Average (EWMA), a Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) volatility model and a hybrid method that integrates Dynamic Recurrent Neural Networks together with a Monte Carlo approach. The integration of traditional Monte Carlo approaches with Machine Learning technologies and the heterogeneity of dynamically selected methodologies lead to an improved estimation of the ES. The study describes the techniques adopted by the CM and the logic behind model selection; moreover, it provides a market application case of the proposed metaheuristic, by simulating an equally weighted multi-asset portfolio.

On Deterministic and Stochastic Multiple Pathogen Epidemic Models

In this paper, we consider a stochastic epidemic model with two pathogens. In order to analyze the coexistence of two pathogens, we compute numerically the expectation time until extinction (the mean persistence time), which satisfies a stationary partial differential equation with degenerate variable coefficients, related to backward Kolmogorov equation. I use the finite element method in order to solve this equation, and we implement it in FreeFem++. The main conclusion of this paper is that the deterministic and stochastic epidemic models differ considerably in predicting coexistence of the two diseases and in the extinction outcome of one of them. Now, the main challenge would be to find an explanation for this result.

Modelling group movement with behaviour switching in continuous time

This article presents a new method for modelling collective movement in continuous time with behavioural switching, motivated by simultaneous tracking of wild or semi-domesticated animals. Each individual in the group is at times attracted to a unobserved leading point. However, the behavioural state of each individual can switch between 'following' and 'independent'. The 'following' movement is modelled through a linear stochastic differential equation, while the 'independent' movement is modelled as Brownian motion. The movement of the leading point is modelled either as an Ornstein-Uhlenbeck (OU) process or as Brownian motion (BM), which makes the whole system a higher-dimensional Ornstein-Uhlenbeck process, possibly an intrinsic non-stationary version. An inhomogeneous Kalman filter Markov chain Monte Carlo algorithm is developed to estimate the diffusion and switching parameters and the behaviour states of each individual at a given time point. The method successfully recovers the true behavioural states in simulated data sets , and is also applied to model a group of simultaneously tracked reindeer (Rangifer tarandus).

Extinction debt in local habitats: quantifying the roles of random drift, immigration and emigration

We developed a time-dependent stochastic neutral model for predicting diverse temporal trajectories of biodiversity change in response to ecological disturbance (i.e. habitat destruction) and dispersal dynamic (i.e. emigration and immigration). The model is general and predicts how transition behaviours of extinction may accumulate according to a different combination of random drift, immigration rate, emigration rate and the degree of habitat destruction. We show that immigration, emigration, the areal size of the destroyed habitat and initial species abundance distribution (SAD) can impact the total biodiversity loss in an intact local area. Among these, the SAD plays the most deterministic role, as it directly determines the initial species richness in the local target area. By contrast, immigration was found to slow down total biodiversity loss and can drive the emergence of species credits (i.e. a gain of species) over time. However, the emigration process would increase the extinction risk of species and accelerate biodiversity loss. Finally but notably, we found that a shift in the emigration rate after a habitat destruction event may be a new mechanism to generate species credits.

Stochastic SIS Modelling: Coinfection of Two Pathogens in Two-Host Communities

A pathogen can infect multiple hosts. For example, zoonotic diseases like rabies often colonize both humans and animals. Meanwhile, a single host can sometimes be infected with many pathogens, such as malaria and meningitis. Therefore, we studied two susceptible classes S 1 ( t ) and S 2 ( t ) , each of which can be infected when interacting with two different infectious groups I 1 ( t ) and I 2 ( t ) . The stochastic models were formulated through the continuous time Markov chain (CTMC) along with their deterministic analogues. The statistics for the developed model were studied using the multi-type branching process. Since each epidemic class was assumed to transmit only its own type of pathogen, two reproduction numbers were obtained, in addition to the probability-generating functions of offspring. Thus, these, together with the mean number of infections, were used to estimate the probability of extinction. The initial population of infectious classes can influence their probability of extinction. Understanding the disease extinctions and outbreaks could result in rapid intervention by the management for effective control measures.

Spatial Lymphocyte Dynamics in Lymph Nodes Predicts the Cytotoxic T Cell Frequency Needed for HIV Infection Control

The surveillance of host body tissues by immune cells is central for mediating their defense function. In vivo imaging technologies have been used to quantitatively characterize target cell scanning and migration of lymphocytes within lymph nodes (LNs). The translation of these quantitative insights into a predictive understanding of immune system functioning in response to various perturbations critically depends on computational tools linking the individual immune cell properties with the emergent behavior of the immune system. By choosing the Newtonian second law for the governing equations, we developed a broadly applicable mathematical model linking individual and coordinated T-cell behaviors. The spatial cell dynamics is described by a superposition of autonomous locomotion, intercellular interaction, and viscous damping processes. The model is calibrated using in vivo data on T-cell motility metrics in LNs such as the translational speeds, turning angle speeds, and meandering indices. The model is applied to predict the impact of T-cell motility on protection against HIV infection, i.e., to estimate the threshold frequency of HIV-specific cytotoxic T cells (CTLs) that is required to detect productively infected cells before the release of viral particles starts. With this, it provides guidance for HIV vaccine studies allowing for the migration of cells in fibrotic LNs.

Nonlinear Stochastic Equation within an Itô Prescription for Modelling of Financial Market

The stochastic nonlinear model based on Itô diffusion is proposed as a mathematical model for price dynamics of financial markets. We study this model with relation to concrete stylised facts about financial markets. We investigate the behavior of the long tail distribution of the volatilities and verify the inverse power law behavior which is obeyed for some financial markets. Furthermore, we obtain the behavior of the long range memory and obtain that it follows to a distinct behavior of other stochastic models that are used as models for the finances. Furthermore, we have made an analysis by using Fokker–Planck equation independent on time with the aim of obtaining the cumulative probability distribution of volatilities P(g), however, the probability density found does not exhibit the cubic inverse law.

Assessment of the Tumbling-Snake Model against Linear and Nonlinear Rheological Data of Bidisperse Polymer Blends

We have recently solved the tumbling-snake model for concentrated polymer solutions and entangled melts in the academic case of a monodisperse sample. Here, we extend these studies and provide the stationary solutions of the tumbling-snake model both analytically, for small shear rates, and via Brownian dynamics simulations, for a bidisperse sample over a wide range of shear rates and model parameters. We further show that the tumbling-snake model bears the necessary capacity to compare well with available linear and non-linear rheological data for bidisperse systems. This capacity is added to the already documented ability of the model to accurately predict the shear rheology of monodisperse systems.

The effect of delay in viral production in within-host models during early infection

Delay in viral production may have a significant impact on the early stages of infection. During the eclipse phase, the time from viral entry until active production of viral particles, no viruses are produced. This delay affects the probability that a viral infection becomes established and timing of the peak viral load. Deterministic and stochastic models are formulated with either multiple latent stages or a fixed delay for the eclipse phase. The deterministic model with multiple latent stages approaches in the limit the model with a fixed delay as the number of stages approaches infinity. The deterministic model framework is used to formulate continuous-time Markov chain and stochastic differential equation models. The probability of a minor infection with rapid viral clearance as opposed to a major full-blown infection with a high viral load is estimated from a branching process approximation of the Markov chain model and the results are confirmed through numerical simulations. In addition, parameter values for influenza A are used to numerically estimate the time to peak viral infection and peak viral load for the deterministic and stochastic models. Although the average length of the eclipse phase is the same in each of the models, as the number of latent stages increases, the numerical results show that the time to viral peak and the peak viral load increase.

Tumbling-Snake Model for Polymeric Liquids Subjected to Biaxial Elongational Flows with a Focus on Planar Elongation

We have recently solved the tumbling-snake model for concentrated polymer solutions and entangled melts in the presence of both steady-state and transient shear and uniaxial elongational flows, supplemented by a variable link tension coefficient. Here, we provide the transient and stationary solutions of the tumbling-snake model under biaxial elongation both analytically, for small and large elongation rates, and via Brownian dynamics simulations, for the case of planar elongational flow over a wide range of rates, times, and the model parameters. We show that both the steady-state and transient first planar viscosity predictions are similar to their uniaxial counterparts, in accord with recent experimental data. The second planar viscosity seems to behave in all aspects similarly to the shear viscosity, if shear rate is replaced by elongation rate.

On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions

In a recent article (Jentzen et al. 2016 Commun. Math. Sci.14, 1477-1500 (doi:10.4310/CMS.2016.v14.n6.a1)), it has been established that, for every arbitrarily slow convergence speed and every natural number d∈{4,5,…}, there exist d-dimensional stochastic differential equations with infinitely often differentiable and globally bounded coefficients such that no approximation method based on finitely many observations of the driving Brownian motion can converge in absolute mean to the solution faster than the given speed of convergence. In this paper, we strengthen the above result by proving that this slow convergence phenomenon also arises in two (d=2) and three (d=3) space dimensions.

Nonstationary Stochastic Dynamics Underlie Spontaneous Transitions between Active and Inactive Behavioral States

The neural basis of spontaneous movement generation is a fascinating open question. Long-term monitoring of fish, swimming freely in a constant sensory environment, has revealed a sequence of behavioral states that alternate randomly and spontaneously between periods of activity and inactivity. We show that key dynamical features of this sequence are captured by a 1-D diffusion process evolving in a nonlinear double well energy landscape, in which a slow variable modulates the relative depth of the wells. This combination of stochasticity, nonlinearity, and nonstationary forcing correctly captures the vastly different timescales of fluctuations observed in the data (∼1 to ∼1000 s), and yields long-tailed residence time distributions (RTDs) also consistent with the data. In fact, our model provides a simple mechanism for the emergence of long-tailed distributions in spontaneous animal behavior. We interpret the stochastic variable of this dynamical model as a decision-like variable that, upon reaching a threshold, triggers the transition between states. Our main finding is thus the identification of a threshold crossing process as the mechanism governing spontaneous movement initiation and termination, and to infer the presence of underlying nonstationary agents. Another important outcome of our work is a dimensionality reduction scheme that allows similar segments of data to be grouped together. This is done by first extracting geometrical features in the dataset and then applying principal component analysis over the feature space. Our study is novel in its ability to model nonstationary behavioral data over a wide range of timescales.

The ISI distribution of the stochastic Hodgkin-Huxley neuron

The simulation of ion-channel noise has an important role in computational neuroscience. In recent years several approximate methods of carrying out this simulation have been published, based on stochastic differential equations, and all giving slightly different results. The obvious, and essential, question is: which method is the most accurate and which is most computationally efficient? Here we make a contribution to the answer. We compare interspike interval histograms from simulated data using four different approximate stochastic differential equation (SDE) models of the stochastic Hodgkin-Huxley neuron, as well as the exact Markov chain model simulated by the Gillespie algorithm. One of the recent SDE models is the same as the Kurtz approximation first published in 1978. All the models considered give similar ISI histograms over a wide range of deterministic and stochastic input. Three features of these histograms are an initial peak, followed by one or more bumps, and then an exponential tail. We explore how these features depend on deterministic input and on level of channel noise, and explain the results using the stochastic dynamics of the model. We conclude with a rough ranking of the four SDE models with respect to the similarity of their ISI histograms to the histogram of the exact Markov chain model.

Density-dependent state-space model for population-abundance data with unequal time intervals

The Gompertz state-space (GSS) model is a stochastic model for analyzing time-series observations of population abundances. The GSS model combines density dependence, environmental process noise, and observation error toward estimating quantities of interest in biological monitoring and population viability analysis. However, existing methods for estimating the model parameters apply only to population data with equal time intervals between observations. In the present paper, we extend the GSS model to data with unequal time intervals, by embedding it within a state-space version of the Ornstein-Uhlenbeck process, a continuous-time model of an equilibrating stochastic system. Maximum likelihood and restricted maximum likelihood calculations for the Ornstein-Uhlenbeck state-space model involve only numerical maximization of an explicit multivariate normal likelihood, and so the extension allows for easy bootstrapping, yielding confidence intervals for model parameters, statistical hypothesis testing of density dependence, and selection among sub-models using information criteria. Ecologists and managers previously drawn to models lacking density dependence or observation error because such models accommodated unequal time intervals (for example, due to missing data) now have an alternative analysis framework incorporating density dependence, process noise, and observation error.

Discovering rare behaviours in stochastic differential equations using decision procedures: applications to a minimal cell cycle model

Stochastic Differential Equation (SDE) models are used to describe the dynamics of complex systems with inherent randomness. The primary purpose of these models is to study rare but interesting or important behaviours, such as the formation of a tumour. Stochastic simulations are the most common means for estimating (or bounding) the probability of rare behaviours, but the cost of simulations increases with the rarity of events. To address this problem, we introduce a new algorithm specifically designed to quantify the likelihood of rare behaviours in SDE models. Our approach relies on temporal logics for specifying rare behaviours of interest, and on the ability of bit-vector decision procedures to reason exhaustively about fixed-precision arithmetic. We apply our algorithm to a minimal parameterised model of the cell cycle, and take Brownian noise into account while investigating the likelihood of irregularities in cell size and time between cell divisions.

Stochastic modeling of systems mapping in pharmacogenomics

As a basis of personalized medicine, pharmacogenetics and pharmacogenomics that aim to study the genetic architecture of drug response critically rely on dynamic modeling of how a drug is absorbed and transported to target tissues where the drug interacts with body molecules to produce drug effects. Systems mapping provides a general framework for integrating systems pharmacology and pharmacogenomics through robust ordinary differential equations. In this chapter, we extend systems mapping to more complex and more heterogeneous structure of drug response by implementing stochastic differential equations (SDE). We argue that SDE-implemented systems mapping provides a computational tool for pharmacogenetic or pharmacogenomic research towards personalized medicine.

A Multiresolution Method for Parameter Estimation of Diffusion Processes

Diffusion process models are widely used in science, engineering and finance. Most diffusion processes are described by stochastic differential equations in continuous time. In practice, however, data is typically only observed at discrete time points. Except for a few very special cases, no analytic form exists for the likelihood of such discretely observed data. For this reason, parametric inference is often achieved by using discrete-time approximations, with accuracy controlled through the introduction of missing data. We present a new multiresolution Bayesian framework to address the inference difficulty. The methodology relies on the use of multiple approximations and extrapolation, and is significantly faster and more accurate than known strategies based on Gibbs sampling. We apply the multiresolution approach to three data-driven inference problems - one in biophysics and two in finance - one of which features a multivariate diffusion model with an entirely unobserved component.

Bayesian parameter inference for stochastic biochemical network models using particle Markov chain Monte Carlo

Computational systems biology is concerned with the development of detailed mechanistic models of biological processes. Such models are often stochastic and analytically intractable, containing uncertain parameters that must be estimated from time course data. In this article, we consider the task of inferring the parameters of a stochastic kinetic model defined as a Markov (jump) process. Inference for the parameters of complex nonlinear multivariate stochastic process models is a challenging problem, but we find here that algorithms based on particle Markov chain Monte Carlo turn out to be a very effective computationally intensive approach to the problem. Approximations to the inferential model based on stochastic differential equations (SDEs) are considered, as well as improvements to the inference scheme that exploit the SDE structure. We apply the methodology to a Lotka-Volterra system and a prokaryotic auto-regulatory network.

Stochastic functional data analysis: a diffusion model-based approach

This article presents a new modeling strategy in functional data analysis. We consider the problem of estimating an unknown smooth function given functional data with noise. The unknown function is treated as the realization of a stochastic process, which is incorporated into a diffusion model. The method of smoothing spline estimation is connected to a special case of this approach. The resulting models offer great flexibility to capture the dynamic features of functional data, and allow straightforward and meaningful interpretation. The likelihood of the models is derived with Euler approximation and data augmentation. A unified Bayesian inference method is carried out via a Markov chain Monte Carlo algorithm including a simulation smoother. The proposed models and methods are illustrated on some prostate-specific antigen data, where we also show how the models can be used for forecasting.

Recurrent events and the exploding Cox model

Counting process models have played an important role in survival and event history analysis for more than 30 years. Nevertheless, almost all models that are being used have a very simple structure. Analyzing recurrent events invites the application of more complex models with dynamic covariates. We discuss how to define valid models in such a setting. One has to check carefully that a suggested model is well defined as a stochastic process. We give conditions for this to hold. Some detailed discussion is presented in relation to a Cox type model, where the exponential structure combined with feedback lead to an exploding model. In general, counting process models with dynamic covariates can be formulated to avoid explosions. In particular, models with a linear feedback structure do not explode, making them useful tools in general modeling of recurrent events.