Stochastic differential equations (SDEs) are a fundamental tool for modelling dynamic processes, including gene regulatory networks (GRNs), contaminant transport, financial markets, and image generation. However, learning the underlying SDE from data is a challenging task, especially if individual trajectories are not observable. Motivated by burgeoning research in single-cell datasets, we present the first comprehensive approach for jointly identifying the drift and diffusion of an SDE from its temporal marginals. Assuming linear drift and additive diffusion, we show that non-identifiability can only arise if the initial distribution possesses generalized rotational symmetries. We further prove that even if this condition holds, the drift and diffusion can almost always be recovered from the marginals. Additionally, we show that the causal graph of any SDE with additive diffusion can be recovered from the identified SDE parameters. To complement this theory, we adapt entropy-regularized optimal transport to handle anisotropic diffusion and introduce APPEX (Alternating Projection Parameter Estimation from $X_0$), an iterative algorithm designed to estimate the drift, diffusion, and causal graph of an additive noise SDE, solely from temporal marginals, and demonstrate its effectiveness on simulated data from linear additive noise SDEs.
Parameter estimation is a fundamental challenge in machine learning, crucial for tasks such as neural network weight fitting and Bayesian inference. This paper focuses on the complexity of estimating translation $\boldsymbol{\mu} \in \mathbb{R}^l$ and shrinkage $\sigma \in \mathbb{R}_{++}$ parameters for a distribution of the form $\frac{1}{\sigma^l} f_0 \left( \frac{\boldsymbol{x} - \boldsymbol{\mu}}{\sigma} \right)$, where $f_0$ is a known density in $\mathbb{R}^l$ given $n$ samples. We highlight that while the problem is NP-hard for Maximum Likelihood Estimation (MLE), it is possible to obtain $\varepsilon$-approximations for arbitrary $\varepsilon > 0$ within $\text{poly} \left( \frac{1}{\varepsilon} \right)$ time using the Wasserstein distance.
This dissertation develops and justifies a novel method for deriving approximate formulas to estimate two parameters in stochastic volatility diffusion models with exponentially-affine characteristic functions and single- or two-factor variance. These formulas aim to improve the accuracy of option pricing and enhance the calibration process by providing reliable initial values for local minimization algorithms. The parameters relate to the volatility of the stochastic factor in instantaneous variance dynamics and the correlation between stochastic factors and asset price dynamics. The study comprises five chapters. Chapter one outlines the currency option market, pricing methods, and the general structure of stochastic volatility models. Chapter two derives the replication strategy dynamics and introduces a new two-factor volatility model: the OUOU model. Chapter three analyzes the distribution and surface dynamics of implied volatilities using principal component and common factor analysis. Chapter four discusses calibration methods for stochastic volatility models, particularly the Heston model, and presents the new Implied Central Moments method to estimate parameters in the Heston and Sch\"obel-Zhu models. Extensions to two-factor models, Bates and OUOU, are also explored. Chapter five evaluates the performance of the proposed formulas on the EURUSD options market, demonstrating the superior accuracy of the new method. The dissertation successfully meets its research objectives, expanding tools for derivative pricing and risk assessment. Key contributions include faster and more precise parameter estimation formulas and the introduction of the OUOU model - an extension of the Sch\"obel-Zhu model with a semi-analytical valuation formula for European options, previously unexamined in the literature.
Stochastic Petri Nets (SPNs) are an increasingly popular tool of choice for modeling discrete-event dynamics in areas such as epidemiology and systems biology, yet their parameter estimation remains challenging in general and in particular when transition rates depend on external covariates and explicit likelihoods are unavailable. We introduce a neural-surrogate (neural-network--based approximation of the posterior distribution) framework that predicts the coefficients of known covariate-dependent rate functions directly from noisy, partially observed token trajectories. Our model employs a lightweight 1D Convolutional Residual Network trained end-to-end on Gillespie-simulated SPN realizations, learning to invert system dynamics under realistic conditions of event dropout. During inference, Monte Carlo dropout provides calibrated uncertainty bounds together with point estimates. On synthetic SPNs with 20% missing events, our surrogate recovers rate-function coefficients with an RMSE = 0.108 and substantially runs faster than traditional Bayesian approaches. These results demonstrate that data-driven, likelihood-free surrogates can enable accurate, robust, and real-time parameter recovery in complex, partially observed discrete-event systems.
In epidemic modeling, outliers can distort parameter estimation and ultimately lead to misguided public health decisions. Although there are existing robust methods that can mitigate this distortion, the ability to simultaneously detect outliers is equally vital for identifying potential disease hotspots. In this work, we introduce a robust spatiotemporal generalized additive model (RST-GAM) to address this need. We accomplish this with a mean-shift parameter to quantify and adjust for the effects of outliers and rely on adaptive Lasso regularization to model the sparsity of outlying observations. We use univariate polynomial splines and bivariate penalized splines over triangulations to estimate the functional forms and a data-thinning approach for data-adaptive weight construction. We derive a scalable proximal algorithm to estimate model parameters by minimizing a convex negative log-quasi-likelihood function. Our algorithm uses adaptive step-sizes to ensure global convergence of the resulting iterate sequence. We establish error bounds and selection consistency for the estimated parameters and demonstrate our model's effectiveness through numerical studies under various outlier scenarios. Finally, we demonstrate the practical utility of RST-GAM by analyzing county-level COVID-19 infection data in the United States, highlighting its potential to inform public health decision-making.
Parameter estimation is a foundational step in statistical modeling, enabling us to extract knowledge from data and apply it effectively. Bayesian estimation of parameters incorporates prior beliefs with observed data to infer distribution parameters probabilistically and robustly. Moreover, it provides full posterior distributions, allowing uncertainty quantification and regularization, especially useful in small or truncated samples. Utilizing the left-truncated log-logistic (LTLL) distribution is particularly well-suited for modeling time-to-event data where observations are subject to a known lower bound such as precipitation data and cancer survival times. In this paper, we propose a Bayesian approach for estimating the parameters of the LTLL distribution with a fixed truncation point \( x_L > 0 \). Given a random variable \( X \sim LL(\alpha, \beta; x_L) \), where \( \alpha > 0 \) is the scale parameter and \( \beta > 0 \) is the shape parameter, the likelihood function is derived based on a truncated sample \( X_1, X_2, \dots, X_N \) with \( X_i > x_L \). We assume independent prior distributions for the parameters, and the posterior inference is conducted via Markov Chain Monte Carlo sampling, specifically using the Metropolis-Hastings algorithm to obtain posterior estimates \( \hat{\alpha} \) and \( \hat{\beta} \). Through simulation studies and real-world applications, we demonstrate that Bayesian estimation provides more stable and reliable parameter estimates, particularly when the likelihood surface is irregular due to left truncation. The results highlight the advantages of Bayesian inference outperform the estimation of parameter uncertainty in truncated distributions for time to event data analysis.
Tuberculosis (TB) continues to pose a major public health challenge, particularly in high-burden regions such as Ethiopia, necessitating a more profound understanding of its transmission dynamics. In this study, we developed an SVEITRS compartmental model to investigate the transmission dynamics of TBs, utilizing real data from Ethiopia from 2011-2021. Model parameters were estimated via two methods: nonlinear least squares and maximum likelihood, with maximum likelihood providing more accurate and reliable results, as confirmed by a test case. The model's stability analysis indicated that there is a disease-free equilibrium in areas where the basic reproduction number ($\mathscr{R}_0$) is less than one. The results suggest that optimal conditions could lead to the elimination of TB. On the other hand, there is an endemic equilibrium in areas where $\mathscr{R}_0$ is greater than one, which means that the disease is still present. Sensitivity analysis revealed important factors affecting TB levels: higher natural death rates, vaccination rates, treatment rates, and disease-related death rates lower TB cases, whereas higher recruitment rates, contact rates, infection rates, and loss of vaccine protection increase its spread. These findings highlights to the necessity of enhancing vaccination, treatment, and recovery strategies while addressing drivers of transmission to achieve TB control in Ethiopia. This study provides useful advice for guiding TB control efforts and public health interventions in Ethiopia and similar regions.
We present a convolution-based data assimilation method tailored to neuronal electrophysiology, addressing the limitations of traditional value-based synchronization approaches. While conventional methods rely on nudging terms and pointwise deviation metrics, they often fail to account for spike timing precision, a key feature in neural signals. Our approach applies a Gaussian convolution to both measured data and model estimates, enabling a cost function that evaluates both amplitude and timing alignment via spike overlap. This formulation remains compatible with gradient-based optimization. Through twin experiments and real hippocampal neuron recordings, we demonstrate improved parameter estimation and prediction quality, particularly in capturing sharp, time-sensitive dynamics.
Sparse Bayesian learning (SBL)-aided target localization is conceived for a bistatic mmWave MIMO radar system in the presence of unknown clutter, followed by the development of an angle-Doppler (AD)-domain representation of the target-plus-clutter echo model for accurate target parameter estimation. The proposed algorithm exploits the three-dimensional (3D) sparsity arising in the AD domain of the scattering scene and employs the powerful SBL framework for the estimation of target parameters, such as the angle-of-departure (AoD), angle-of-arrival (AoA) and velocity. To handle a practical scenario where the actual target parameters typically deviate from their finite-resolution grid, a super-resolution-based improved off-grid SBL framework is developed for recursively updating the parameter grid, thereby progressively refining the estimates. We also determine the Cram\'er-Rao bound (CRB) and Bayesian CRB for target parameter estimation in order to benchmark the estimation performance. Our simulation results corroborate the superior performance of the proposed approach in comparison to the existing algorithms, and also their ability to approach the bounds derived.
Elucidating the biomechanical behavior of the myocardium is crucial for understanding cardiac physiology, but cannot be directly inferred from clinical imaging and typically requires finite element (FE) simulations. However, conventional FE methods are computationally expensive and often fail to reproduce observed cardiac motions. We propose IMC-PINN-FE, a physics-informed neural network (PINN) framework that integrates imaged motion consistency (IMC) with FE modeling for patient-specific left ventricular (LV) biomechanics. Cardiac motion is first estimated from MRI or echocardiography using either a pre-trained attention-based network or an unsupervised cyclic-regularized network, followed by extraction of motion modes. IMC-PINN-FE then rapidly estimates myocardial stiffness and active tension by fitting clinical pressure measurements, accelerating computation from hours to seconds compared to traditional inverse FE. Based on these parameters, it performs FE modeling across the cardiac cycle at 75x speedup. Through motion constraints, it matches imaged displacements more accurately, improving average Dice from 0.849 to 0.927, while preserving realistic pressure-volume behavior. IMC-PINN-FE advances previous PINN-FE models by introducing back-computation of material properties and better motion fidelity. Using motion from a single subject to reconstruct shape modes also avoids the need for large datasets and improves patient specificity. IMC-PINN-FE offers a robust and efficient approach for rapid, personalized, and image-consistent cardiac biomechanical modeling.
Summary: Errors in gradient trajectories introduce significant artifacts and distortions in magnetic resonance images, particularly in non-Cartesian imaging sequences, where imperfect gradient waveforms can greatly reduce image quality. Purpose: Our objective is to develop a general, nonlinear gradient system model that can accurately predict gradient distortions using convolutional networks. Methods: A set of training gradient waveforms were measured on a small animal imaging system, and used to train a temporal convolutional network to predict the gradient waveforms produced by the imaging system. Results: The trained network was able to accurately predict nonlinear distortions produced by the gradient system. Network prediction of gradient waveforms was incorporated into the image reconstruction pipeline and provided improvements in image quality and diffusion parameter mapping compared to both the nominal gradient waveform and the gradient impulse response function. Conclusion: Temporal convolutional networks can more accurately model gradient system behavior than existing linear methods and may be used to retrospectively correct gradient errors.
Estimating the governing equation parameter values is essential for integrating experimental data with scientific theory to understand, validate, and predict the dynamics of complex systems. In this work, we propose a new method for structural system identification (SI), uncertainty quantification, and validation directly from data. Inspired by generative modeling frameworks, a neural network maps random noise to physically meaningful parameters. These parameters are then used in the known equation of motion to obtain fake accelerations, which are compared to real training data via a mean square error loss. To simultaneously validate the learned parameters, we use independent validation datasets. The generated accelerations from these datasets are evaluated by a discriminator network, which determines whether the output is real or fake, and guides the parameter-generator network. Analytical and real experiments show the parameter estimation accuracy and model validation for different nonlinear structural systems.
Accurate parameter estimation in electrochemical battery models is essential for monitoring and assessing the performance of lithium-ion batteries (LiBs). This paper presents a novel approach that combines deep reinforcement learning (DRL) with an optimal experimental design (OED) framework to identify key electrochemical parameters of LiB cell models. The proposed method utilizes the twin delayed deep deterministic policy gradient (TD3) algorithm to optimize input excitation, thereby increasing the sensitivity of the system response to electrochemical parameters. The performance of this DRL-based approach is evaluated against a nonlinear model predictive control (NMPC) method and conventional tests. Results indicate that the DRL-based method provides superior information content, reflected in higher Fisher information (FI) values and lower parameter estimation errors compared to the NMPC design and conventional test practices. Additionally, the DRL approach offers a substantial reduction in experimental time and computational resources.
Mathematical models are crucial for optimizing and controlling chemical processes, yet they often face significant limitations in terms of computational time, algorithm complexity, and development costs. Hybrid models, which combine mechanistic models with data-driven models (i.e. models derived via the application of machine learning to experimental data), have emerged as a promising solution to these challenges. However, the identification of dynamic hybrid models remains difficult due to the need to integrate data-driven models within mechanistic model structures. We present an incremental identification approach for dynamic hybrid models that decouples the mechanistic and data-driven components to overcome computational and conceptual difficulties. Our methodology comprises four key steps: (1) regularized dynamic parameter estimation to determine optimal time profiles for flux variables, (2) correlation analysis to evaluate relationships between variables, (3) data-driven model identification using advanced machine learning techniques, and (4) hybrid model integration to combine the mechanistic and data-driven components. This approach facilitates early evaluation of model structure suitability, accelerates the development of hybrid models, and allows for independent identification of data-driven components. Three case studies are presented to illustrate the robustness, reliability, and efficiency of our incremental approach in handling complex systems and scenarios with limited data.
Deep neural networks have been shown to achieve exceptional performance for computer vision tasks like image recognition, segmentation, and reconstruction or denoising. Here, we evaluate the ultimate performance limits of deep convolutional neural network models for image reconstruction, by comparing them against the standard quantum limit set by shot-noise and the Heisenberg limit on precision. We train U-Net models on images of natural objects illuminated with coherent states of light, and find that the average mean-squared error of the reconstructions can surpass the standard quantum limit, and in some cases reaches the Heisenberg limit. Further, we train models on well-parameterized images for which we can calculate the quantum Cram\'er-Rao bound to determine the minimum possible measurable variance of an estimated parameter for a given probe state. We find the mean-squared error of the model predictions reaches these bounds calculated for the parameters, across a variety of parameterized images. These results suggest that deep convolutional neural networks can learn to become the optimal estimators allowed by the laws of physics, performing parameter estimation and image reconstruction at the ultimate possible limits of precision for the case of classical illumination of the object.
There is growing interest in extending average treatment effect (ATE) estimation to incorporate non-tabular data, such as images and text, which may act as sources of confounding. Neglecting these effects risks biased results and flawed scientific conclusions. However, incorporating non-tabular data necessitates sophisticated feature extractors, often in combination with ideas of transfer learning. In this work, we investigate how latent features from pre-trained neural networks can be leveraged to adjust for sources of confounding. We formalize conditions under which these latent features enable valid adjustment and statistical inference in ATE estimation, demonstrating results along the example of double machine learning. We discuss critical challenges inherent to latent feature learning and downstream parameter estimation arising from the high dimensionality and non-identifiability of representations. Common structural assumptions for obtaining fast convergence rates with additive or sparse linear models are shown to be unrealistic for latent features. We argue, however, that neural networks are largely insensitive to these issues. In particular, we show that neural networks can achieve fast convergence rates by adapting to intrinsic notions of sparsity and dimension of the learning problem.
Sensing emerges as a critical challenge in 6G networks, which require simultaneous communication and target sensing capabilities. State-of-the-art super-resolution techniques for the direction of arrival (DoA) estimation encounter significant performance limitations when the number of targets exceeds antenna array dimensions. This paper introduces a novel sensing parameter estimation algorithm for orthogonal frequency-division multiplexing (OFDM) multiple-input multiple-output (MIMO) radar systems. The proposed approach implements a strategic two-stage methodology: first, discriminating targets through delay and Doppler domain filtering to reduce the number of effective targets for super-resolution DoA estimation, and second, introducing a fusion technique to mitigate sidelobe interferences. The algorithm enables robust DoA estimation, particularly in high-density target environments with limited-size antenna arrays. Numerical simulations validate the superior performance of the proposed method compared to conventional DoA estimation approaches.
Opinion dynamics models such as the bounded confidence models (BCMs) describe how a population can reach consensus, fragmentation, or polarization, depending on a few parameters. Connecting such models to real-world data could help understanding such phenomena, testing model assumptions. To this end, estimation of model parameters is a key aspect, and maximum likelihood estimation provides a principled way to tackle it. Here, our goal is to outline the properties of statistical estimators of the two key BCM parameters: the confidence bound and the convergence rate. We find that their maximum likelihood estimators present different characteristics: the one for the confidence bound presents a small-sample bias but is consistent, while the estimator of the convergence rate shows a persistent bias. Moreover, the joint parameter estimation is affected by identifiability issues for specific regions of the parameter space, as several local maxima are present in the likelihood function. Our results show how the analysis of the likelihood function is a fruitful approach for better understanding the pitfalls and possibilities of estimating the parameters of opinion dynamics models, and more in general, agent-based models, and for offering formal guarantees for their calibration.
Solving inverse problems in cardiovascular modeling is particularly challenging due to the high computational cost of running high-fidelity simulations. In this work, we focus on Bayesian parameter estimation and explore different methods to reduce the computational cost of sampling from the posterior distribution by leveraging low-fidelity approximations. A common approach is to construct a surrogate model for the high-fidelity simulation itself. Another is to build a surrogate for the discrepancy between high- and low-fidelity models. This discrepancy, which is often easier to approximate, is modeled with either a fully connected neural network or a nonlinear dimensionality reduction technique that enables surrogate construction in a lower-dimensional space. A third possible approach is to treat the discrepancy between the high-fidelity and surrogate models as random noise and estimate its distribution using normalizing flows. This allows us to incorporate the approximation error into the Bayesian inverse problem by modifying the likelihood function. We validate five different methods which are variations of the above on analytical test cases by comparing them to posterior distributions derived solely from high-fidelity models, assessing both accuracy and computational cost. Finally, we demonstrate our approaches on two cardiovascular examples of increasing complexity: a lumped-parameter Windkessel model and a patient-specific three-dimensional anatomy.
We introduce an approach to topic modelling with document-level covariates that remains tractable in the face of large text corpora. This is achieved by de-emphasizing the role of parameter estimation in an underlying probabilistic model, assuming instead that the data come from a fixed but unknown distribution whose statistical functionals are of interest. We propose combining a convex formulation of non-negative matrix factorization with standard regression techniques as a fast-to-compute and useful estimate of such a functional. Uncertainty quantification can then be achieved by reposing non-parametric resampling methods on top of this scheme. This is in contrast to popular topic modelling paradigms, which posit a complex and often hard-to-fit generative model of the data. We argue that the simple, non-parametric approach advocated here is faster, more interpretable, and enjoys better inferential justification than said generative models. Finally, our methods are demonstrated with an application analysing covariate effects on discourse of flavours attributed to Canadian beers.