A Two-Filter Approach for State Estimation Utilizing Quantized Output Data.

Filtering and smoothing algorithms are key tools to develop decision-making strategies and parameter identification techniques in different areas of research, such as economics, financial data analysis, communications, and control systems. These algorithms are used to obtain an estimation of the system state based on the sequentially available noisy measurements of the system output. In a real-world system, the noisy measurements can suffer a significant loss of information due to (among others): (i) a reduced resolution of cost-effective sensors typically used in practice or (ii) a digitalization process for storing or transmitting the measurements through a communication channel using a minimum amount of resources. Thus, obtaining suitable state estimates in this context is essential. In this paper, Gaussian sum filtering and smoothing algorithms are developed in order to deal with noisy measurements that are also subject to quantization. In this approach, the probability mass function of the quantized output given the state is characterized by an integral equation. This integral was approximated by using a Gauss-Legendre quadrature; hence, a model with a Gaussian mixture structure was obtained. This model was used to develop filtering and smoothing algorithms. The benefits of this proposal, in terms of accuracy of the estimation and computational cost, are illustrated via numerical simulations.

An Estimation Algorithm for General Linear Single Particle Tracking Models with Time-Varying Parameters.

Single Particle Tracking (SPT) is a powerful class of methods for studying the dynamics of biomolecules inside living cells. The techniques reveal the trajectories of individual particles, with a resolution well below the diffraction limit of light, and from them the parameters defining the motion model, such as diffusion coefficients and confinement lengths. Most existing algorithms assume these parameters are constant throughout an experiment. However, it has been demonstrated that they often vary with time as the tracked particles move through different regions in the cell or as conditions inside the cell change in response to stimuli. In this work, we propose an estimation algorithm to determine time-varying parameters of systems that discretely switch between different linear models of motion with Gaussian noise statistics, covering dynamics such as diffusion, directed motion, and Ornstein-Uhlenbeck dynamics. Our algorithm consists of three stages. In the first stage, we use a sliding window approach, combined with Expectation Maximization (EM) to determine maximum likelihood estimates of the parameters as a function of time. These results are only used to roughly estimate the number of model switches that occur in the data to guide the selection of algorithm parameters in the second stage. In the second stage, we use Change Detection (CD) techniques to identify where the models switch, taking advantage of the off-line nature of the analysis of SPT data to create non-causal algorithms with better precision than a purely causal approach. Finally, we apply EM to each set of data between the change points to determine final parameter estimates. We demonstrate our approach using experimental data generated in the lab under controlled conditions.

Model Segmentation in Single Particle Tracking.

In this paper, we implement and compare two different change detection techniques applied to determining the time points in Single Particle Tracking (SPT) data where the particle changes the dynamic model of motion. The goal is to use this change detection to segment the data in order to estimate the relevant parameters of such models. We consider two well-known statistics commonly used for change detection: the likelihood ratio test (LRT) and the Kullback-Leibler divergence (KLD). We assume that our time-varying system is subject to step-like changes in the parameters that drive the process. The techniques are then applied to experimental data acquired on a microscope under controlled settings to validate our results.

A time-varying approach to single particle tracking with a nonlinear observation model.

Single Particle Tracking (SPT) is a powerful class of tools for analyzing the dynamics of individual biological macromolecules moving inside living cells. The acquired data is typically in the form of a sequence of camera images that are then post-processed to reveal details about the motion. In this work, we develop a local time-varying estimation algorithm for estimating motion model parameters from the data considering nonlinear observations. Our approach uses several well-known existing tools, namely the Expectation Maximization (EM) algorithm combined with an Unscented Kalman filter (UKF) and an Unscented Rauch-Tung-Striebel smoother (URTSS), and applies them to the time-varying case through a sliding window methodology. Due to the shot noise characteristics of the photon generation process, this model uses a Poisson distribution to capture the measurement noise inherent in imaging. In order to apply our time-varying approach to the UKF, we first need to transform the measurements into a model with additive Gaussian noise. This is carried out using a variance stabilizing transform. Results from simulations show that our approach is successful in tracing time-varying diffusion constants at a range of physically relevant signal levels. We also discuss the initialization for the EM algorithm based on the available data.

Estimation of general time-varying single particle tracking linear models using local likelihood.

In this work, we study a general approach to the estimation of single particle tracking models with time-varying parameters. The main idea is to use local Maximum Likelihood (ML), applying a sliding window over the data and estimating the model parameters in each window. We combine local ML with Expectation Maximization to iteratively find the ML estimate in each window, an approach that is amenable to generalization to nonlinear models. Results using controlled-experimental data generated in our lab show that our proposed algorithm is able to track changes in the parameters as they evolve during a trajectory under real-world experimental conditions, outperforming other algorithms of similar nature.

A 2-step algorithm for the estimation of time-varying single particle tracking models using Maximum Likelihood.

Single particle tracking (SPT) is a powerful class of methods for studying the dynamics of biomolecules inside living cells. The techniques reveal both trajectories of individual particles, with a resolution well below the diffraction limit of light, and the parameters defining the motion model, such as diffusion coefficients and confinement lengths. Existing algorithms assume these parameters are constant throughout an experiment. However, it has been demonstrated that they often vary with time as the tracked particles move through different regions in the cell or as conditions inside the cell change in response to stimuli. In this work we apply the method of local Maximum Likelihood (ML) estimation to the SPT application combined with change detection. Local ML uses a sliding window over the data, estimating the model parameters in each window. Once we have found the values for the parameters before and after the change, we apply offline change detection to know the exact time of the change. Then, we reestimate these parameters and show that there is an improvement in the estimation of key parameters found in SPT. Preliminary results using simulated data with a basic diffusion model with additive Gaussian noise show that our proposed algorithm is able to track abrupt changes in the parameters as they evolve during a trajectory.