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This is a practical guide to P-splines, a simple, flexible and powerful tool for smoothing. P-splines combine regression on B-splines with simple, discrete, roughness penalties. They were introduced by the authors in 1996 and have been used in many diverse applications. The regression basis makes it straightforward to handle non-normal data, like in generalized linear models. The authors demonstrate optimal smoothing, using mixed model technology and Bayesian estimation, in addition to classical tools like cross-validation and AIC, covering theory and applications with code in R. Going far beyond simple smoothing, they also show how to use P-splines for regression on signals, varying-coefficient models, quantile and expectile smoothing, and composite links for grouped data. Penalties are the crucial elements of P-splines; with proper modifications they can handle periodic and circular data as well as shape constraints. Combining penalties with tensor products of B-splines extends these attractive properties to multiple dimensions. An appendix offers a systematic comparison to other smoothers.

Smoothing (Statistics) --- Spline theory.

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Accountancy --- Accounting. --- Corporations --- Fraud. --- Income accounting. --- Smoothing (Statistics) --- Smoothing (Statistics).

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Now in its second edition, this accessible text presents a unified Bayesian treatment of state-of-the-art filtering, smoothing, and parameter estimation algorithms for non-linear state space models. The book focuses on discrete-time state space models and carefully introduces fundamental aspects related to optimal filtering and smoothing. In particular, it covers a range of efficient non-linear Gaussian filtering and smoothing algorithms, as well as Monte Carlo-based algorithms. This updated edition features new chapters on constructing state space models of practical systems, the discretization of continuous-time state space models, Gaussian filtering by enabling approximations, posterior linearization filtering, and the corresponding smoothers. Coverage of key topics is expanded, including extended Kalman filtering and smoothing, and parameter estimation. The book's practical, algorithmic approach assumes only modest mathematical prerequisites, suitable for graduate and advanced undergraduate students. Many examples are included, with Matlab and Python code available online, enabling readers to implement algorithms in their own projects.

Bayesian statistical decision theory. --- Filters (Mathematics) --- Smoothing (Statistics)

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Lissage (analyse numérique) --- Smoothing (Numerical analysis) --- Approximation et developpements --- Splines

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Title: Image Smoothing in Neuroimaging: Effect of Gaussian vs. Tissue-Specific Approaches in Statistical Analysis
Author: Jacquemin Antoine
Section: "Ingénieur civil biomédical"
Academic Year: 2024-2025
Promotor: Phillips Christophe
This study aims to improve spatial smoothing approaches in quantitative and functional magnetic resonance imaging (qMRI and fMRI) by generalizing Tissue-SPecific smOOthing compeNsated (TSPOON) and comparing its performance with Tissue-Weighted Smoothing (TWS) and traditional Gaussian Smoothing. The work utilizes a qMRI dataset from the Wellcome Trust Centre for Neuroimaging (London) and the hMRI toolbox, a new collaborative toolbox for neuroimaging research, with the potential integration of the generalized TSPOON approach into the toolbox.
To achieve these goals, the study implements TSPOON by developing optimized tissue-specific binary masks to preserve tissue specificity. TWS, by contrast, employs continuous modulated warped tissue weights. The results show that TSPOON provides consistently lower effective smoothing than TWS. It is more sensitive to pronounced signal variations near tissue boundaries, thereby enhancing specificity. In contrast, TWS is better at capturing subtle variations within homogeneous regions, offering greater sensitivity. Notably, smoothing-induced differences in both qMRI and fMRI are predominantly observed at the tissue boundaries, highlighting the effects of partial volume biases.
These findings underline the complementary nature of TWS and TSPOON. TWS is suited for exploratory studies emphasizing sensitivity, while TSPOON is optimal for analyses requiring robust tissue delineation and specificity. The choice of method should align with the study’s objectives (whether broad signal coverage or precise detection of localized effects is prioritized). Future perspectives include refining the design of tissue-specific masks, extending the evaluation of these methods to other imaging modalities such as diffusion-weighted imaging and positron emission tomography and publishing the generalized TSPOON implementation to provide a new, versatile smoothing option for neuroimaging researchers.

Spatial Smoothing --- Tissue-Specificity --- qMRI --- fMRI --- Neuroimaging --- TSPOON --- TWS --- Gaussian Smoothing --- Partial Volume Effect --- Ingénierie, informatique & technologie > Multidisciplinaire, généralités & autres