Multicointegration is traditionally defined as a particular long run relationship among variables in a parametric vector autoregressive model that introduces additional cointegrating links between these variables and partial sums of the equilibrium errors. This paper departs from the parametric model, using a semiparametric formulation that reveals the explicit role that singularity of the long run conditional covariance matrix plays in determining multicointegration. The semiparametric framework has the advantage that short run dynamics do not need to be modeled and estimation by standard techniques such as fully modified least squares (FM-OLS) on the original I(1) system is straightforward. The paper derives FM-OLS limit theory in the multicointegrated setting, showing how faster rates of convergence are achieved in the direction of singularity and that the limit distribution depends on the distribution of the conditional one-sided long run covariance estimator used in FM-OLS estimation. Wald tests of restrictions on the regression coefficients have nonstandard limit theory which depends on nuisance parameters in general. The usual tests are shown to be conservative when the restrictions are isolated to the directions of singularity and, under certain conditions, are invariant to singularity otherwise. Simulations show that approximations derived in the paper work well in finite samples. The findings are illustrated empirically in an analysis of fiscal sustainability of the US government over the post-war period.
The two unobservable state variables representing the short and long term factors introduced by Schwartz and Smith in [16] for risk-neutral pricing of futures contracts are modelled as two correlated Ornstein-Uhlenbeck processes. The Kalman Filter (KF) method has been implemented to estimate the short and long term factors jointly with un- known model parameters. The parameter identification problem arising within the likelihood function in the KF has been addressed by introduc- ing an additional constraint. The obtained model parameter estimates are the conditional Maximum Likelihood Estimators (MLEs) evaluated within the KF. Consistency of the conditional MLEs is studied. The methodology has been tested on simulated data.
Atmospheric optical turbulence can be a significant source of image degradation, particularly in long range imaging applications. Many turbulence mitigation algorithms rely on an optical transfer function (OTF) model that includes the Fried parameter. We present anisoplanatic tilt statistics for spherical wave propagation. We transform these into 2D autocorrelation functions that can inform turbulence modeling and mitigation algorithms. Using these, we construct an OTF model that accounts for image registration. We also propose a spectral-ratio Fried parameter estimation algorithm that is robust to camera motion and requires no specialized scene content or sources. We employ the Fried parameter estimation and OTF model for turbulence mitigation. A numerical wave-propagation turbulence simulator is used to generate data to quantitatively validate the proposed methods. Results with real camera data are also presented.
We develop a physics-informed machine learning approach for large-scale data assimilation and parameter estimation and apply it for estimating transmissivity and hydraulic head in the two-dimensional steady-state subsurface flow model of the Hanford Site given synthetic measurements of said variables. In our approach, we extend the physics-informed conditional Karhunen-Lo\'{e}ve expansion (PICKLE) method for modeling subsurface flow with unknown flux (Neumann) and varying head (Dirichlet) boundary conditions. We demonstrate that the PICKLE method is comparable in accuracy with the standard maximum a posteriori (MAP) method, but is significantly faster than MAP for large-scale problems. Both methods use a mesh to discretize the computational domain. In MAP, the parameters and states are discretized on the mesh; therefore, the size of the MAP parameter estimation problem directly depends on the mesh size. In PICKLE, the mesh is used to evaluate the residuals of the governing equation, while the parameters and states are approximated by the truncated conditional Karhunen-Lo\'{e}ve expansions with the number of parameters controlled by the smoothness of the parameter and state fields, and not by the mesh size. For a considered example, we demonstrate that the computational cost of PICKLE increases near linearly (as $N_{FV}^{1.15}$) with the number of grid points $N_{FV}$, while that of MAP increases much faster as $N_{FV}^{3.28}$. We demonstrated that once trained for one set of Dirichlet boundary conditions (i.e., one river stage), the PICKLE method provides accurate estimates of the hydraulic head for any value of the Dirichlet boundary conditions (i.e., for any river stage).
Many promising approaches to symbolic regression have been presented in recent years, yet progress in the field continues to suffer from a lack of uniform, robust, and transparent benchmarking standards. In this paper, we address this shortcoming by introducing an open-source, reproducible benchmarking platform for symbolic regression. We assess 14 symbolic regression methods and 7 machine learning methods on a set of 252 diverse regression problems. Our assessment includes both real-world datasets with no known model form as well as ground-truth benchmark problems, including physics equations and systems of ordinary differential equations. For the real-world datasets, we benchmark the ability of each method to learn models with low error and low complexity relative to state-of-the-art machine learning methods. For the synthetic problems, we assess each method's ability to find exact solutions in the presence of varying levels of noise. Under these controlled experiments, we conclude that the best performing methods for real-world regression combine genetic algorithms with parameter estimation and/or semantic search drivers. When tasked with recovering exact equations in the presence of noise, we find that deep learning and genetic algorithm-based approaches perform similarly. We provide a detailed guide to reproducing this experiment and contributing new methods, and encourage other researchers to collaborate with us on a common and living symbolic regression benchmark.
We propose to use deep learning to estimate parameters in statistical models when standard likelihood estimation methods are computationally infeasible. We show how to estimate parameters from max-stable processes, where inference is exceptionally challenging even with small datasets but simulation is straightforward. We use data from model simulations as input and train deep neural networks to learn statistical parameters. Our neural-network-based method provides a competitive alternative to current approaches, as demonstrated by considerable accuracy and computational time improvements. It serves as a proof of concept for deep learning in statistical parameter estimation and can be extended to other estimation problems.
Smoothing and sharpening are two fundamental image processing operations. The latter is usually related to the former through the unsharp masking algorithm. In this paper, we develop a new type of filter which performs smoothing or sharpening via a tuning parameter. The development of the new filter is based on (1) a new Laplacian-based filter formulation which unifies the smoothing and sharpening operations, (2) a patch interpolation model similar to that used in the guided filter which provides edge-awareness capability, and (3) the generalized Gamma distribution which is used as the prior for parameter estimation. We have conducted detailed studies on the properties of two versions of the proposed filter (self-guidance and external guidance). We have also conducted experiments to demonstrate applications of the proposed filter. In the self-guidance case, we have developed adaptive smoothing and sharpening algorithms based on texture, depth and blurriness information extracted from an image. Applications include enhancing human face images, producing shallow depth of field effects, focus-based image enhancement, and seam carving. In the external guidance case, we have developed new algorithms for combining flash and no-flash images and for enhancing multi-spectral images using a panchromatic image.
Knowing the geometrical and acoustical parameters of a room may benefit applications such as audio augmented reality, speech dereverberation or audio forensics. In this paper, we study the problem of jointly estimating the total surface area, the volume, as well as the frequency-dependent reverberation time and mean surface absorption of a room in a blind fashion, based on two-channel noisy speech recordings from multiple, unknown source-receiver positions. A novel convolutional neural network architecture leveraging both single- and inter-channel cues is proposed and trained on a large, realistic simulated dataset. Results on both simulated and real data show that using multiple observations in one room significantly reduces estimation errors and variances on all target quantities, and that using two channels helps the estimation of surface and volume. The proposed model outperforms a recently proposed blind volume estimation method on the considered datasets.
Accurate camera calibration is a precondition for many computer vision applications. Calibration errors, such as wrong model assumptions or imprecise parameter estimation, can deteriorate a system's overall performance, making the reliable detection and quantification of these errors critical. In this work, we introduce an evaluation scheme to capture the fundamental error sources in camera calibration: systematic errors (biases) and uncertainty (variance). The proposed bias detection method uncovers smallest systematic errors and thereby reveals imperfections of the calibration setup and provides the basis for camera model selection. A novel resampling-based uncertainty estimator enables uncertainty estimation under non-ideal conditions and thereby extends the classical covariance estimator. Furthermore, we derive a simple uncertainty metric that is independent of the camera model. In combination, the proposed methods can be used to assess the accuracy of individual calibrations, but also to benchmark new calibration algorithms, camera models, or calibration setups. We evaluate the proposed methods with simulations and real cameras.
Cells sense their surrounding by employing intracellular signaling pathways that transmit hormonal signals from the cell membrane to the nucleus. TGF-$\beta$/SMAD signaling encodes various cell fates, controls tissue homeostasis and is deregulated in diseases such as cancer. The pathway shows strong heterogeneity at the single-cell level, but quantitative insights into mechanisms underlying fluctuations at various time scales are still missing, partly due to inefficiency in the calibration of stochastic models that mechanistically describe signaling processes. In this work we analyze single-cell TGF-$\beta$/SMAD signaling and show that it exhibits temporal stochastic bursts which are dose-dependent and whose number and magnitude correlate with cell migration. We propose a stochastic modeling approach to mechanistically describe these pathway fluctuations with high computational efficiency. Employing high-order numerical integration and fitting to burst statistics we enable efficient quantitative parameter estimation and discriminate models that assume noise in different reactions at the receptor level. This modeling approach suggests that stochasticity in the internalization of TGF-$\beta$ receptors into endosomes plays a key role in the observed temporal bursting. Further, the model predicts the single-cell dynamics of TGF-$\beta$/SMAD signaling in untested conditions, e.g., successfully reflects memory effects of signaling noise and cellular sensitivity towards repeated stimulation. Taken together, our computational framework based on burst analysis, noise modeling and path computation scheme is a suitable tool for the data-based modeling of complex signaling pathways, capable of identifying the source of temporal noise.
Statistical signal processing applications usually require the estimation of some parameters of interest given a set of observed data. These estimates are typically obtained either by solving a multi-variate optimization problem, as in the maximum likelihood (ML) or maximum a posteriori (MAP) estimators, or by performing a multi-dimensional integration, as in the minimum mean squared error (MMSE) estimators. Unfortunately, analytical expressions for these estimators cannot be found in most real-world applications, and the Monte Carlo (MC) methodology is one feasible approach. MC methods proceed by drawing random samples, either from the desired distribution or from a simpler one, and using them to compute consistent estimators. The most important families of MC algorithms are Markov chain MC (MCMC) and importance sampling (IS). On the one hand, MCMC methods draw samples from a proposal density, building then an ergodic Markov chain whose stationary distribution is the desired distribution by accepting or rejecting those candidate samples as the new state of the chain. On the other hand, IS techniques draw samples from a simple proposal density, and then assign them suitable weights that measure their quality in some appropriate way. In this paper, we perform a thorough review of MC methods for the estimation of static parameters in signal processing applications. A historical note on the development of MC schemes is also provided, followed by the basic MC method and a brief description of the rejection sampling (RS) algorithm, as well as three sections describing many of the most relevant MCMC and IS algorithms, and their combined use.
In a recent methodological paper, we showed how to learn chaotic dynamics along with the state trajectory from sequentially acquired observations, using local ensemble Kalman filters. Here, we more systematically investigate the possibility to use a local ensemble Kalman filter with either covariance localisation or local domains, in order to retrieve the state and a mix of key global and local parameters. Global parameters are meant to represent the surrogate dynamical core, for instance through a neural network, which is reminiscent of data-driven machine learning of dynamics, while the local parameters typically stand for the forcings of the model. Aiming at joint state and parameter estimation, a family of algorithms for covariance and local domain localisation is proposed. In particular, we show how to rigorously update global parameters using a local domain ensemble Kalman filter (EnKF) such as the local ensemble transform Kalman filter (LETKF), an inherently local method. The approach is tested with success on the 40-variable Lorenz model using several of the local EnKF flavors. A two-dimensional illustration based on a multi-layer Lorenz model is finally provided. It uses radiance-like non-local observations. It features both local domains and covariance localisation in order to learn the chaotic dynamics and the local forcings. This paper more generally addresses the key question of online estimation of both global and local model parameters.
This paper constructs a full dimensional (6D) radio map to predict the channel gain between any transmitter location and any receiver location based on received signal strength (RSS) measurements between low-altitude aerial nodes and ground nodes. The main challenge is how to describe the signal strength attenuation due to the blockage from the environment. Conventional interpolation-type approaches fail to exploit the close relation between the radio map and the geometry of the environment. This paper proposes to construct radio maps by first estimating and constructing a multi-class 3D virtual obstacle map that describes the geometry of the environment with radio semantics. Mathematically, a least-squares problem is formulated to jointly estimate the virtual obstacle map and the propagation parameters. This problem is found to have a partial quasiconvexity that leads to the development of an efficient parameter estimation and radio map construction algorithm. Numerical experiments confirm that the proposed method substantially reduces the amount of measurement required for achieving the same radio map accuracy. It is also demonstrated that in a unmanned aerial vehicle (UAV)-aided relay communication scenario, a radio map assisted approach for UAV placement can achieve more than 50% capacity gain.
Deep neural networks (DNN) have an impressive ability to invert very complex models, i.e. to learn the generative parameters from a model's output. Once trained, the forward pass of a DNN is often much faster than traditional, optimization-based methods used to solve inverse problems. This is however done at the cost of lower interpretability, a fundamental limitation in most medical applications. We propose an approach for solving general inverse problems which combines the efficiency of DNN and the interpretability of traditional analytical methods. The measurements are first projected onto a dense dictionary of model-based responses. The resulting sparse representation is then fed to a DNN with an architecture driven by the problem's physics for fast parameter learning. Our method can handle generative forward models that are costly to evaluate and exhibits similar performance in accuracy and computation time as a fully-learned DNN, while maintaining high interpretability and being easier to train. Concrete results are shown on an example of model-based brain parameter estimation from magnetic resonance imaging (MRI).
Portfolio optimization has been a central problem in finance, often approached with two steps: calibrating the parameters and then solving an optimization problem. Yet, the two-step procedure sometimes encounter the "error maximization" problem where inaccuracy in parameter estimation translates to unwise allocation decisions. In this paper, we combine the prediction and optimization tasks in a single feed-forward neural network and implement an end-to-end approach, where we learn the portfolio allocation directly from the input features. Two end-to-end portfolio constructions are included: a model-free network and a model-based network. The model-free approach is seen as a black-box, whereas in the model-based approach, we learn the optimal risk contribution on the assets and solve the allocation with an implicit optimization layer embedded in the neural network. The model-based end-to-end framework provides robust performance in the out-of-sample (2017-2021) tests when maximizing Sharpe ratio is used as the training objective function, achieving a Sharpe ratio of 1.16 when nominal risk parity yields 0.79 and equal-weight fix-mix yields 0.83. Noticing that risk-based portfolios can be sensitive to the underlying asset universe, we develop an asset selection mechanism embedded in the neural network with stochastic gates, in order to prevent the portfolio being hurt by the low-volatility assets with low returns. The gated end-to-end with filter outperforms the nominal risk-parity benchmarks with naive filtering mechanism, boosting the Sharpe ratio of the out-of-sample period (2017-2021) to 1.24 in the market data.
Power grid parameter estimation involves the estimation of unknown parameters, such as inertia and damping coefficients, using observed dynamics. In this work, we present a comparison of data-driven algorithms for the power grid parameter estimation problem. First, we propose a new algorithm to solve the parameter estimation problem based on the Sparse Identification of Nonlinear Dynamics (SINDy) approach, which uses linear regression to infer the parameters that best describe the observed data. We then compare its performance against two benchmark algorithms, namely, the unscented Kalman filter (UKF) approach and the physics-informed neural networks (PINN) approach. We perform extensive simulations on IEEE bus systems to examine the performance of the aforementioned algorithms. Our results show that the SINDy algorithm outperforms the PINN and UKF algorithms in being able to accurately estimate the power grid parameters over a wide range of system parameters (including high and low inertia systems). Moreover, it is extremely efficient computationally and so takes significantly less time than the PINN algorithm, thus making it suitable for real-time parameter estimation.
Dynamical systems with binary-valued observations are widely used in information industry, technology of biological pharmacy and other fields. Though there have been much efforts devoted to the identification of such systems, most of the previous investigations are based on first-order gradient algorithm which usually has much slower convergence rate than the Quasi-Newton algorithm. Moreover, persistence of excitation(PE) conditions are usually required to guarantee consistent parameter estimates in the existing literature, which are hard to be verified or guaranteed for feedback control systems. In this paper, we propose an online projected Quasi-Newton type algorithm for parameter estimation of stochastic regression models with binary-valued observations and varying thresholds. By using both the stochastic Lyapunov function and martingale estimation methods, we establish the strong consistency of the estimation algorithm and provide the convergence rate, under a signal condition which is considerably weaker than the traditional PE condition and coincides with the weakest possible excitation known for the classical least square algorithm of stochastic regression models. Convergence of adaptive predictors and their applications in adaptive control are also discussed.
We introduce a unified framework, formulated as general latent space models, to study complex higher-order network interactions among multiple entities. Our framework covers several popular models in recent network analysis literature, including mixture multi-layer latent space model and hypergraph latent space model. We formulate the relationship between the latent positions and the observed data via a generalized multilinear kernel as the link function. While our model enjoys decent generality, its maximum likelihood parameter estimation is also convenient via a generalized tensor decomposition procedure.We propose a novel algorithm using projected gradient descent on Grassmannians. We also develop original theoretical guarantees for our algorithm. First, we show its linear convergence under mild conditions. Second, we establish finite-sample statistical error rates of latent position estimation, determined by the signal strength, degrees of freedom and the smoothness of link function, for both general and specific latent space models. We demonstrate the effectiveness of our method on synthetic data. We also showcase the merit of our method on two real-world datasets that are conventionally described by different specific models in producing meaningful and interpretable parameter estimations and accurate link prediction. We demonstrate the effectiveness of our method on synthetic data. We also showcase the merit of our method on two real-world datasets that are conventionally described by different specific models in producing meaningful and interpretable parameter estimations and accurate link prediction.
The purpose of this article is to introduce a new L\'evy process, termed Variance Gamma++ process, to model the dynamic of assets in illiquid markets. Such a process has the mathematical tractability of the Variance Gamma process and is obtained applying the self-decomposability of the gamma law. Compared to the Variance Gamma model, it has an additional parameter representing the measure of the trading activity. We give a full characterization of the Variance Gamma++ process in terms of its characteristic triplet, characteristic function and transition density. In addition, we provide efficient path simulation algorithms, both forward and backward in time. We also obtain an efficient "integral-free" explicit pricing formula for European options. These results are instrumental to apply Fourier-based option pricing and maximum likelihood techniques for the parameter estimation. Finally, we apply our model to illiquid markets, namely to the calibration of European power future market data. We accordingly evaluate exotic derivatives using the Monte Carlo method and compare these values to those obtained using the Variance Gamma process and give an economic interpretation of the obtained results. Finally, we illustrate an extension to the multivariate framework.
Performance of parameter estimation is one of the most important issues in array signal processing. The root mean square error, probability of success, resolution probabilities, and computational complexity are frequently used indexes for evaluating the performance of the parameter estimation methods. However, a common characteristic of these indexes is that they are unsupervised indexes, and are passively used for evaluating the estimation results. In other words, these indexes cannot participate in the design of estimation methods. It seems that exploiting a validity supervised index for the parameter estimation that can guide the design of the methods will be an interesting and meaningful work. In this study, we exploit an index to build a supervised learning model of the parameter estimation. With the developed model we refine the signal subspace so as to enhance the performance of the parameter estimation method. The main characteristic of the proposed model is a circularly applied feedback of the estimated parameter for refining the estimated subspace. It is a closed loop and supervised method not reported before. This research opens a specific way for improving the performance of the parameter estimation by a supervised index. However, the proposed method is still unsatisfying in some scopes of signal-to-noise ratio (SNR). We believe that exploiting a validity index for the parameter estimation in array signal processing is still a general and interesting problem.