Bimodality in gene expression without feedback: From Gaussian white noise to log-normal coloured noise

Extrinsic noise-induced transitions to bimodal dynamics have been largely investigated in a variety of chemical, physical, and biological systems. In the standard approach in physical and chemical systems, the key properties that make these systems mathematically tractable are that the noise appears linearly in the dynamical equations, and it is assumed Gaussian and white. In biology, the Gaussian approximation has been successful in specific systems, but the relevant noise being usually non-Gaussian, non-white, and nonlinear poses serious limitations to its general applicability. Here we revisit the fundamental features of linear Gaussian noise, pinpoint its limitations, and review recent new approaches based on nonlinear bounded noises, which highlight novel mechanisms to account for transitions to bimodal behaviour. We do this by considering a simple but fundamental gene expression model, the repressed gene, which is characterized by linear and nonlinear dependencies on external parameters. We then review a general methodology introduced recently, so-called nonlinear noise filtering, which allows the investigation of linear, nonlinear, Gaussian and non-Gaussian noises. We also present a derivation of it, which highlights its dynamical origin. Testing the methodology on the repressed gene confirms that the emergence of noise-induced transitions appears to be strongly dependent on the type of noise adopted, and on the degree of nonlinearity present in the system.Comment: Review paper, 17 pages, 8 figure

Behavioral models of nonlinear filters based on discrete time cellular neural networks

The nonlinear dynamic system modeling based on the input/output relationship results from solving the approximation problem. One can distinguish two large classes: polynomials and neural networks. The different types of neural networks draw attention. The discrete time feedforward cellular neural network is suggested for filtering non-Gaussian noise, as well as the example of nonlinear filters modeling to cancel the impulse noise is represented

Use of Bridging Strategy between the Ensemble Kalman Filter and Particle Filter for the Measurements with Various Quasi-Gaussian Noise

Filtering and estimation are two important tools of engineering. Whenever the state of the system needs to be estimated from the noisy sensor measurements, some kind of state estimator is used. If the dynamics of the system and observation model are linear under Gaussian conditions, the root mean squared error can be computed using the Kalman Filter. But practically, noise frequently enters the system as not strictly Gaussian. Therefore, the Kalman Filter does not necessarily provide the better estimate. Hence the estimation of the nonlinear system under non-Gaussian or quasi-Gaussian noise is of an acute interest. There are many versions of the Kalman filter such as the Extended Kalman filter, the Unscented Kalman filter, the Ensemble Kalman filter, the Particle filter, etc., each having their own disadvantages. In this thesis work I used a bridging strategy between the Ensemble Kalman filter and Particle filter called an Ensemble Kalman Particle filter. This filter works well in nonlinear system and non-Gaussian measurements as well. I analyzed this filter using MATLAB simulation and also applied Gaussian Noise, non-zero mean Gaussian Noise, quasi-Gaussian noise (with drift), random walk and Laplacian Noise. I applied these noises and compared the performances of the Particle filter and the Ensemble Kalman Particle filter in the presence of linear and nonlinear observations which leads to the conclusion that the Ensemble Kalman Particle filter yields the minimum error estimate. I also found the optimum value for the tuning parameter which is used to bridge the two filters using Monte Carlo Simulation

Linear theory for filtering nonlinear multiscale systems with model error

We study filtering of multiscale dynamical systems with model error arising from unresolved smaller scale processes. The analysis assumes continuous-time noisy observations of all components of the slow variables alone. For a linear model with Gaussian noise, we prove existence of a unique choice of parameters in a linear reduced model for the slow variables. The linear theory extends to to a non-Gaussian, nonlinear test problem, where we assume we know the optimal stochastic parameterization and the correct observation model. We show that when the parameterization is inappropriate, parameters chosen for good filter performance may give poor equilibrium statistical estimates and vice versa. Given the correct parameterization, it is imperative to estimate the parameters simultaneously and to account for the nonlinear feedback of the stochastic parameters into the reduced filter estimates. In numerical experiments on the two-layer Lorenz-96 model, we find that parameters estimated online, as part of a filtering procedure, produce accurate filtering and equilibrium statistical prediction. In contrast, a linear regression based offline method, which fits the parameters to a given training data set independently from the filter, yields filter estimates which are worse than the observations or even divergent when the slow variables are not fully observed

Nonlinear processing of non-Gaussian stochastic and chaotic deterministic time series

It is often assumed that interference or noise signals are Gaussian stochastic processes. Gaussian noise models are appealing as they usually result in noise suppression algorithms that are simple: i.e. linear and closed form. However, such linear techniques may be sub-optimal when the noise process is either a non-Gaussian stochastic process or a chaotic deterministic process. In the event of encountering such noise processes, improvements in noise suppression, relative to the performance of linear methods, may be achievable using nonlinear signal processing techniques. The application of interest for this thesis is maritime surveillance radar, where the main source of interference, termed sea clutter, is widely accepted to be a non-Gaussian stochastic process at high resolutions and/or at low grazing angles. However, evidence has been presented during the last decade which suggests that sea clutter may be better modelled as a chaotic deterministic process. While the debate over which model is more suitable continues, this thesis investigates whether nonlinear processing techniques can be used to improve the performance of maritime surveillance radar, relative to the performance achievable using linear techniques. Linear and nonlinear prediction of chaotic signals, sea clutter data sets, and stochastic surrogate clutter data sets is carried out. Volterra series filter networks and radial basis function networks are used to implement nonlinear predictors. A novel structure for a forward-backward nonlinear predictor, using a radial basis function network, is presented. Prediction results provide evidence to support the view that sea clutter is better modelled as a stochastic process, rather than as a chaotic process. The clutter data sets are shown to have linear predictor functions. Linear and nonlinear predictors are used as the basis of target detection algorithms. The performance of these predictor-detectors, against backgrounds of sea clutter data and against a background of chaotic noise data is evaluated. The detection results show that linear predictor-detectors perform as well as, or better than, nonlinear predictor-detectors against the non-Gaussian clutter backgrounds considered in this thesis, whilst the reverse is true for a background of chaotic noise. An existing, nonlinear inverse, noise cancellation technique, referred to as Broomhead’s filtering technique in this thesis, is re-investigated using a sine wave corrupted by broadband chaotic noise. It is demonstrated that significant improvements can be obtained using this nonlinear inverse technique, relative to results obtained using linear alternatives, despite recent work which suggested otherwise. A novel bandstop filtering approach is applied to Broomhead’s filtering method, which allows the technique to be applied to the cancellation of signals with a band of interest greater than that of a sine wave. This modified Broomhead filtering technique is shown to cancel broadband chaotic noise from a narrowband Gaussian signal better than alternative linear methods. The modified Broomhead filtering technique is shown to only perform as well as, o

Application of Sigma Point Particle Filter Method for Passive State Estimation in Underwater

Bearings-only tracking (BOT) plays a vital role in underwater surveillance. In BOT, measurement is tangentially related to state of the system. This measurement is also corrupted with noise due to turbulent underwater environment. Hence state estimation process using BOT becomes nonlinear. This necessitates the use of nonlinear filtering algorithms in place of traditional linear filters like Kalman filter. In general, these nonlinear filters utilize the assumption of measurements being corrupted with Gaussian noise for state estimation. The measurements cannot be always corrupted with Gaussian noise because of the highly unstable sea environment. These problems indicate the necessity for development of nonlinear non-Gaussian filters like particle filter (PF) for underwater tracking. However, PF suffers from severe problems like sample degeneracy and impoverishment and also it is tedious to select an appropriate technique for resampling. To overcome these difficulties in PF implementation, the strategy of combining PF with another filter like unscented Kalman filter is proposed for target’s state estimation. The detailed analysis of the same is presented in comparison with other particle filter combinations using the simulation results obtained in Matlab

Novel Computational Methods for State Space Filtering

The state-space formulation for time-dependent models has been long used invarious applications in science and engineering. While the classical Kalman filter(KF) provides optimal posterior estimation under linear Gaussian models, filteringin nonlinear and non-Gaussian environments remains challenging.Based on the Monte Carlo approximation, the classical particle filter (PF) can providemore precise estimation under nonlinear non-Gaussian models. However, it suffers fromparticle degeneracy. Drawing from optimal transport theory, the stochastic map filter(SMF) accommodates a solution to this problem, but its performance is influenced bythe limited flexibility of nonlinear map parameterisation. To account for these issues,a hybrid particle-stochastic map filter (PSMF) is first proposed in this thesis, wherethe two parts of the split likelihood are assimilated by the PF and SMF, respectively.Systematic resampling and smoothing are employed to alleviate the particle degeneracycaused by the PF. Furthermore, two PSMF variants based on the linear and nonlinearmaps (PSMF-L and PSMF-NL) are proposed, and their filtering performance is comparedwith various benchmark filters under different nonlinear non-Gaussian models.Although achieving accurate filtering results, the particle-based filters require expensive computations because of the large number of samples involved. Instead, robustKalman filters (RKFs) provide efficient solutions for the linear models with heavy-tailednoise, by adopting the recursive estimation framework of the KF. To exploit the stochasticcharacteristics of the noise, the use of heavy-tailed distributions which can fit variouspractical noises constitutes a viable solution. Hence, this thesis also introduces a novelRKF framework, RKF-SGαS, where the signal noise is assumed to be Gaussian and theheavy-tailed measurement noise is modelled by the sub-Gaussian α-stable (SGαS) distribution. The corresponding joint posterior distribution of the state vector and auxiliaryrandom variables is estimated by the variational Bayesian (VB) approach. Four differentminimum mean square error (MMSE) estimators of the scale function are presented.Besides, the RKF-SGαS is compared with the state-of-the-art RKFs under three kinds ofheavy-tailed measurement noises, and the simulation results demonstrate its estimationaccuracy and efficiency.One notable limitation of the proposed RKF-SGαS is its reliance on precise modelparameters, and substantial model errors can potentially impede its filtering performance. Therefore, this thesis also introduces a data-driven RKF method, referred to asRKFnet, which combines the conventional RKF framework with a deep learning technique. An unsupervised scheduled sampling technique (USS) is proposed to improve theistability of the training process. Furthermore, the advantages of the proposed RKFnetare quantified with respect to various traditional RKFs

Unscented Orientation Estimation Based on the Bingham Distribution

Orientation estimation for 3D objects is a common problem that is usually tackled with traditional nonlinear filtering techniques such as the extended Kalman filter (EKF) or the unscented Kalman filter (UKF). Most of these techniques assume Gaussian distributions to account for system noise and uncertain measurements. This distributional assumption does not consider the periodic nature of pose and orientation uncertainty. We propose a filter that considers the periodicity of the orientation estimation problem in its distributional assumption. This is achieved by making use of the Bingham distribution, which is defined on the hypersphere and thus inherently more suitable to periodic problems. Furthermore, handling of non-trivial system functions is done using deterministic sampling in an efficient way. A deterministic sampling scheme reminiscent of the UKF is proposed for the nonlinear manifold of orientations. It is the first deterministic sampling scheme that truly reflects the nonlinear manifold of the orientation