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This chapter treats the basic methods for connecting the model to empirical data: estimating and testing parameters. These are based on computer simulations of the model with provisional hypothetical values of the model’s parameters. To give some intuitive understanding, the chapter starts with a toy model. It goes on to describe how parameters are estimated in the RSiena package by the method of moments and the Robbins–Monro algorithm. Subsequently, methods are given for testing the parameters. Special attention is given to the score-type test, which is a method for testing parameters without estimating them. This can be useful, for example, for cases of complete or quasi-complete separation, where estimated parameter values would be infinitely large or nearly so. Conceptual questions about whether it is meaningful at all to use statistical models for data are discussed in a section about model-based inference. The chapter ends with a brief history of this methodology for analyzing network dynamics.
We present a search for radio emission from classical novae at 887.5 MHz using data from the Australian SKA Pathfinder Variable And Slow Transient survey. We cross-matched 43 optically discovered classical novae that erupted between 2021 September and 2025 November within the 1 200 deg2 Galactic survey footprint and found three which show significant radio emission: V6598 Sgr, V1716 Sco, and V1723 Sco. To analyse their radio light curves, we use both thermal free-free and non-thermal synchrotron emission models. We fit the data using the Markov chain Monte Carlo method to constrain parameters, including the ejected mass and ejecta velocities for the thermal models, and mass-loss rate, explosion energy, wind velocity, and filling factor for the non-thermal model. All three novae show evidence of non-thermal synchrotron emission as the dominant emission mechanism at this frequency. We use a broken power law to describe the radial density structure of a non-uniform circumbinary material, which provides a better fit than a standard wind density profile. This strong early-time synchrotron emission is strong evidence of shock-driven particle acceleration, which may be related to detections of gamma-rays from all three novae as well. In contrast to earlier studies that used multi-frequency data to distinguish between emission models, our analysis is based on single-frequency radio light curves, which can still provide useful constraints on the dominant emission mechanism when interpreted with physically motivated models.
This chapter explores Bayesian optimization and statistics, which provides a flexible framework for balancing prior knowledge about a system with new data. Many optimization techniques are either rooted in Bayesian statistics, or have a Bayesian interpretation. For example, the Tikhonov and LASSO regularized least-squares come from specific prior distributions on the unknown parameters. Bayesian methods are particularly useful when we have small data sets or complex models. They are used extensively for hyperparameter tuning of machine learning models, for experimental design, and for applications in engineering that involve complex partial differential equations. This chapter provides a high-level overview of Bayesian methods in optimization, starting with statistical estimation. Density estimation, regularization, and expectation maximization will be explored here. Estimating a statistical distribution from data is fundamentally an optimization problem, and is the basis of machine learning. The simplest approach is to assume a known distribution and estimate the parameters using optimization. If we include Bayesian prior information, this becomes maximum a posteriori estimation.
The sixth chapter provides a deeper exploration of probabilistic models, building upon concepts encountered earlier in the text. The chapter illustrates how to construct diverse models, particularly by employing the notion of conditional independence. It also outlines standard methods for estimating parameters and hidden states, as well as techniques for sampling. The chapter concludes by discussing and implementing applications such as Kalman filtering and Gibbs sampling. The chapter covers a range of topics, including parametric families of probability distributions, maximum likelihood estimation, modeling complex dependencies using conditional independence and marginalization, and applications such as linear-Gaussian models and Kalman filtering.
Chapter 2 introduces the linear regression model and discusses its assumptions, parameter estimation methods, and interpretation. Further, it highlights fundamental symmetry properties of the linear regression model (and the related Pearson correlation) which rely only on second moment information of variables. An empirical example is given to illustrate that one cannot distinguish a model (e.g., x → y) from the causally reversed alternative model (y → x), when solely focusing on second variable moments. Consequently, alternative measures are required that preserve the fundamental asymmetry in the dependence structure of variables, as proposed in subsequent chapters.
Network meta-analysis (NMA) is becoming increasingly important, especially in the field of medicine, as it allows for comparisons across multiple trials with different interventions. For time-to-event data, that is, survival data, traditional NMA based on the proportional hazards (PH) assumption simply synthesizes reported hazard ratios (HRs). Novel methods for NMA based on the non-PH assumption have been proposed and implemented using R software. However, these methods often involve complex methodologies and require advanced programming skills, creating a barrier for many researchers. Therefore, we developed an R Shiny tool, NMAsurv (https://psurvivala.shinyapps.io/NMAsurv/). NMAsurv allows users with little or zero background in R to conduct survival-data-based NMA effortlessly. The tool supports various functions such as drawing network plots, testing the PH assumption, and building NMA models. Users can input either reconstructed pseudo-individual participant data or aggregated data. NMAsurv offers a user-friendly interface for extracting parameter estimations from various NMA models, including fractional polynomial, piecewise exponential models, parametric survival models, Cox PH model, and generalized gamma model. Additionally, it enables users to effortlessly create survival and HR plots. All operations can be performed by an intuitive “point-and-click” interface. In this study, we introduce all the functionalities and features of NMAsurv and demonstrate its application using a real-world NMA example.
Chapter 15 extensively examines the resource theory of asymmetry, focusing on the significance of asymmetry as a quantum resource, particularly in situations lacking a shared reference frame. The chapter begins by identifying the foundational elements, such as free states and operations within this theory, emphasizing their role in alignment of quantum reference frames. A significant part of the discussion revolves around the quantification of asymmetry, utilizing measures like the Fisher information and Wigner–Yanase–Dyson skew information to assess the degree of asymmetry in quantum states. The concept of G-twirling is introduced as a method to achieve symmetric states, serving as a key technique in analyzing and understanding asymmetry. Moreover, the chapter explores how asymmetry can enhance tasks like parameter estimation, leveraging the maximum likelihood method to improve precision.
Wireless channel propagation parameter estimation forms the foundation of channel sounding, estimation, modeling, and sensing. This paper introduces a deep learning approach for joint delay and Doppler estimation from frequency and time samples of a radio channel transfer function.
Our work estimates the 2D path parameters from a channel impulse response containing an unknown number of paths. Compared to existing deep learning-based methods, the parameters are not estimated via classification but in a quasi-grid-free manner. We employ a deterministic preprocessing scheme that incorporates a multichannel windowing to increase the estimator’s robustness and enables the use of a convolutional neural network (CNN) architecture. The proposed architecture then jointly estimates the number of paths along with the respective delay and Doppler shift parameters of the paths. Hence, it jointly solves the model order selection and parameter estimation task. We also integrate the CNN into an existing maximum-likelihood estimator framework for efficient initialization of a gradient-based iteration, to provide more accurate estimates.
In the analysis, we compare our approach to other methods in terms of estimate accuracy and model order error on synthetic data. Finally, we demonstrate its applicability to real-world measurement data from a anechoic bistatic RADAR emulation measurement.
This study extends the loss function-based parameter estimation method for diagnostic classification models proposed by Ma, de la Torre, et al. (2023, Psychometrika) to consider prior knowledge and uncertainty of sampling. To this end, we integrate the loss function-based estimation method with the generalized Bayesian method. We establish the consistency of attribute mastery patterns of the proposed generalized Bayesian method. The proposed generalized Bayesian method is compared in a simulation study and found to be superior to the previous nonparametric diagnostic classification method—a special case of the loss function-based method. Moreover, the proposed method is applied to real data and compared with previous parametric and nonparametric estimation methods. Finally, practical guidelines for the proposed method and future research directions are discussed.
For classroom teaching and learning, classifying students’ skills into more than two categories (e.g., no, basic, and advanced masteries) is more instructionally relevant. Such classifications require polytomous attributes, for which most existing cognitive diagnosis models (CDMs) are inapplicable. This paper proposes the saturated polytomous cognitive diagnosis model (sp-CDM), a general model that subsumes existing CDMs for polytomous attributes as special cases. The generalization is shown by mathematically illustrating the relationships between the proposed and existing CDMs. Moreover, algorithms to estimate the proposed model is proposed. A simulation study is conducted to evaluate the parameter recovery of the sp-CDM using the proposed estimation algorithms, as well as to illustrate the consequences of improperly fitting constrained or unnecessarily complex polytomous-attribute CDMs. A real-data example involving polytomous attributes is presented to demonstrate the practical utility of the proposed model.
A paired composition is a response (upon a dependent variable) to the ordered pair <j, k> of stimuli, treatments, etc. The present paper develops an alternative analysis for the paired compositions layout previously treated by Bechtel's [1967] scaling model. The alternative model relaxes the previous one by including row and column scales that provide an expression of bias for each pair of objects. The parameter estimation and hypothesis testing procedures for this model are illustrated by means of a small group analysis, which represents a new approach to pairwise sociometrics and personality assessment.
The well-known Rasch model is generalized to a multicomponent model, so that observations of component events are not needed to apply the model. It is shown that the generalized model has retained the property of the specific objectivity of the Rasch model. For a restricted variant of the model, maximum likelihood estimates of its parameters and a statistical test of the model are given. The results of an application to a mathematics test involving six components are described.
The G-DINA (generalized deterministic inputs, noisy “and” gate) model is a generalization of the DINA model with more relaxed assumptions. In its saturated form, the G-DINA model is equivalent to other general models for cognitive diagnosis based on alternative link functions. When appropriate constraints are applied, several commonly used cognitive diagnosis models (CDMs) can be shown to be special cases of the general models. In addition to model formulation, the G-DINA model as a general CDM framework includes a component for item-by-item model estimation based on design and weight matrices, and a component for item-by-item model comparison based on the Wald test. The paper illustrates the estimation and application of the G-DINA model as a framework using real and simulated data. It concludes by discussing several potential implications of and relevant issues concerning the proposed framework.
A general model is presented for homogeneous, dichotomous items when the answer key is not known a priori. The model is structurally related to the two-class latent structure model with the roles of respondents and items interchanged. For very small sets of respondents, iterative maximum likelihood estimates of the parameters can be obtained by existing methods. For other situations, new estimation methods are developed and assessed with Monte Carlo data. The answer key can be accurately reconstructed with relatively small sets of respondents. The model is useful when a researcher wants to study objectively the knowledge possessed by members of a culturally coherent group that the researcher is not a member of.
A method is presented to provide estimates of parameters of specified nonlinear equations from ordinal data generated from a crossed design. The analytic method, NOPE, is an iterative method in which monotone regression and the Gauss-Newton method of least squares are applied alternatively until a measure of stress is minimized. Examples of solutions from artificial data are presented together with examples of applications of the method to experimental results.
In this paper it is demonstrated how statistical inference from multistage test designs can be made based on the conditional likelihood. Special attention is given to parameter estimation, as well as the evaluation of model fit. Two reasons are provided why the fit of simple measurement models is expected to be better in adaptive designs, compared to linear designs: more parameters are available for the same number of observations; and undesirable response behavior, like slipping and guessing, might be avoided owing to a better match between item difficulty and examinee proficiency. The results are illustrated with simulated data, as well as with real data.
Taking a step-by-step approach to modelling neurons and neural circuitry, this textbook teaches students how to use computational techniques to understand the nervous system at all levels, using case studies throughout to illustrate fundamental principles. Starting with a simple model of a neuron, the authors gradually introduce neuronal morphology, synapses, ion channels and intracellular signalling. This fully updated new edition contains additional examples and case studies on specific modelling techniques, suggestions on different ways to use this book, and new chapters covering plasticity, modelling extracellular influences on brain circuits, modelling experimental measurement processes, and choosing appropriate model structures and their parameters. The online resources offer exercises and simulation code that recreate many of the book's figures, allowing students to practice as they learn. Requiring an elementary background in neuroscience and high-school mathematics, this is an ideal resource for a course on computational neuroscience.
Modelling a neural system involves the selection of the mathematical form of the model’s components, such as neurons, synapses and ion channels, plus assigning values to the model’s parameters. This may involve matching to the known biology, fitting a suitable function to data or computational simplicity. Only a few parameter values may be available through existing experimental measurements or computational models. It will then be necessary to estimate parameters from experimental data or through optimisation of model output. Here we outline the many mathematical techniques available. We discuss how to specify suitable criteria against which a model can be optimised. For many models, ranges of parameter values may provide equally good outcomes against performance criteria. Exploring the parameter space can lead to valuable insights into how particular model components contribute to particular patterns of neuronal activity. It is important to establish the sensitivity of the model to particular parameter values.
Chapter 2 discusses methods of estimation for the parameters of the GLM, with a strong emphasis on ordinary least squares estimation (OLS). OLS estimation minimizes the squared difference between observed and estimated values of the dependent variable, in units of this variable. A total of nine optimization criteria is discussed. The OLS solution is explained in detail
In this work, a new adaptive digital predistorter (DPD) is proposed to linearize radio frequency power amplifiers (PA). The DPD structure is composed of two sub-models. A Feedback–Wiener sub-model, describing the main inverse nonlinearities of the PA, combined with a second sub-model based on a memory polynomial (MP) model. The interest of this structure is that only the MP model is identified in real time to compensate deviations from the initial behavior and thus further improve the linearization. The identification architecture combines offline measurement and online parameter estimation with small number of coefficients in the MP sub-model to track the changes in the PA characteristics. The proposed structure is used to linearize a class AB 75 W PA, designed by Telerad society for aeronautical communications in Ultra High Frequency (UHF) / Very High Frequency (VHF) bands. The obtained results, in terms of identification of optimal DPD and the performances of the digital processing, show a good trade-off between linearization performances and computational complexity.