Statistics Reading Guides | Willams Batista

01 / 06 · Research library

Time Series and Spatial Statistics

How does dependence change when observations are indexed by time, space or both?

Time series and spatial statistics share probabilistic tools for dependence, but temporal ordering, spatial neighbourhoods and space–time indexing require different modelling assumptions. The three short routes below introduce these connections.

Recommended background: Probability, regression and introductory mathematical statistics; linear algebra for covariance and conditional models.

Shared foundations

Start with random variables and fields, stationarity, covariance, conditional distributions and inference under dependence. These ideas are shared; their existence, ordering and asymptotic assumptions are not interchangeable.

Three ways to explore

Time series

ARMA, temporal dependence, innovations, forecasting and diagnostics, including non-Gaussian extensions. Time provides a natural chronological ordering.

Start with: Peter J. Brockwell (2016)

Spatial and spatio-temporal statistics

Random fields, geostatistics, covariance and kriging, space–time models and Bayesian hierarchies. Lattice observations, point-referenced data and event locations have different sampling structures.

Start with: Noel A. C. Cressie (1993) · Peter J. Diggle (2007) · Håvard Rue (2005)

The ARMA bridge: from time to two-dimensional lattices

My methodological research connects conditional regression with two-dimensional spatial ARMA dependence. Moving from temporal recursion to spatial neighbourhoods requires explicit ordering, compatible conditional laws and model-specific inference.

Start with: Julian Besag (1974) · Dag Tjøstheim (1978)

Six starting references

Introduction to Time Series and Forecasting, 3rd ed. ↗
Peter J. Brockwell; Richard A. Davis · 2016
An accessible route into temporal dependence, ARMA and forecasting.
Statistics for Spatial Data ↗
Noel A. C. Cressie · 1993
A common foundation for geostatistics, lattice models and spatial prediction.
Spatial Interaction and the Statistical Analysis of Lattice Systems ↗
Julian Besag · 1974
An original source on lattice interactions and conditional-model compatibility.
Statistical Spatial Series Modelling ↗
Dag Tjøstheim · 1978
A specialized bridge to unilateral spatial series and multidimensional ARMA structure.
Model-based Geostatistics ↗
Peter J. Diggle; Paulo J. Ribeiro Jr. · 2007
Model-based spatial inference and prediction with geostatistical data.
Gaussian Markov Random Fields: Theory and Applications ↗
Håvard Rue; Leonhard Held · 2005
Conditional independence, sparse precision and a route to Bayesian spatial modelling.

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02 / 06 · Research library

Statistical inference and asymptotic theory

How do model assumptions, information measures and asymptotic theory justify estimation and hypothesis testing?

Classical and Bayesian inference answer related but distinct questions about evidence and uncertainty. Divergences can construct estimators and tests in several inferential frameworks; they are mathematical tools, not a separate school of inference.

Recommended background: Probability, likelihood, mathematical statistics and calculus; real analysis for advanced asymptotics.

Shared foundations

Identifiability, regularity, information and uncertainty are common concerns. Consistency, limiting laws, robustness and test calibration must be established for the specified model and sampling regime; dependent observations may require different asymptotics.

Three introductory routes

Classical statistical inference

Point estimation, likelihood and estimating equations; confidence intervals and likelihood-ratio, Wald and score tests. Study their assumptions, efficiency and finite- or large-sample validity.

Start with: E. L. Lehmann (1998) · E. L. Lehmann (2005)

Bayesian statistical inference

Prior, likelihood and posterior; estimation, credible regions, posterior predictive assessment and model comparison. Bayes factors require suitable prior specifications, including care with improper priors.

Start with: Andrew Gelman (2013) · José M. Bernardo (1994)

Statistical information and divergence-based inference

Fisher information, Kullback–Leibler, φ-divergences and density-power divergence can support estimation and hypothesis tests. Their robustness and null distributions are method- and model-specific; geodesic-distance tests form a related geometric bridge.

Start with: Leandro Pardo (2006) · Ayanendranath Basu (1998) · A. W. van der Vaart (1998)

Seven selected readings

Theory of Point Estimation, 2nd ed. ↗

E. L. Lehmann; George Casella · 1998
A rigorous basis for estimation, optimality and decision-theoretic criteria.
Testing Statistical Hypotheses, 3rd ed. ↗

E. L. Lehmann; Joseph P. Romano · 2005
Develops test construction, optimality and large-sample testing.
Bayesian Data Analysis, 3rd ed. ↗

Andrew Gelman; John B. Carlin; Hal S. Stern; David B. Dunson; Aki Vehtari; Donald B. Rubin · 2013
A model-based introduction to posterior inference, computation and predictive checks.
Bayesian Theory ↗

José M. Bernardo; Adrian F. M. Smith · 1994
A mathematical route through Bayesian decision theory, prior specification and information.
Statistical Inference Based on Divergence Measures ↗

Leandro Pardo · 2006
Develops divergence-based estimation and hypothesis tests, especially through φ-divergences.
Robust and efficient estimation by minimising a density power divergence ↗

Ayanendranath Basu; Ian R. Harris; Nils L. Hjort; M. C. Jones · 1998
An original route to robust estimation with a tunable efficiency–robustness trade-off.
Asymptotic Statistics ↗

A. W. van der Vaart · 1998
The mathematical bridge to M-estimation, efficiency and empirical processes.

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03 / 06 · Research library

Statistical Image Processing

How do acquisition physics, spatial dependence and uncertainty shape statistical methods across radar, optical and medical images?

Statistical image processing connects observation models with estimation, restoration, segmentation and image-derived inference. This guide offers three application routes with a shared mathematical foundation; expertise in SAR/PolSAR does not imply established work in optical or medical imaging.

Recommended background: Probability, statistical inference, regression, linear algebra and basic image formation; learn the acquisition and noise model of each modality.

Shared foundations

Start with image formation, noise and spatial dependence; then study inverse problems, statistical filtering, segmentation, classification, uncertainty and reproducible validation. Speckle in coherent radar, radiometric effects in optical imagery and modality-specific medical noise are not interchangeable. Evaluate images and predictions at the appropriate spatial, field, patient or acquisition level.

Three application routes

Established research activity

Radar imaging and SAR/PolSAR

Model coherent backscatter, speckle, polarimetric covariance and spatial dependence. Connect statistical estimation and classification to crops, vegetation and environmental monitoring.

Start with: Chris Oliver (2004) · Jong-Sen Lee (2009)

Developing research direction

Optical remote sensing and environmental imaging

Study multispectral and hyperspectral observations, radiometric correction, land cover, vegetation and environmental change. Passive optical satellite sensors differ from radar, while LiDAR uses active laser ranging; SAR–optical fusion requires explicit registration and uncertainty checks.

Start with: John A. Richards (2022) · Zhe Zhu (2017)

Emerging research interest

Medical and biomedical image analysis

Explore statistical segmentation and uncertainty for MRI, CT, pathology and lesion or tumour images. Account for modality-specific artefacts, expert labels, patient-level validation and clinical domain shift; this is a methodological research interest, not a claim of clinical expertise.

Start with: Dzung L. Pham (2000) · Geert Litjens (2017)

Seven selected readings

Computer Vision: Algorithms and Applications, 2nd ed. ↗

Richard Szeliski · 2022

A cross-domain foundation for image formation, estimation, restoration, registration and recognition.
Understanding Synthetic Aperture Radar Images ↗

Chris Oliver; Shaun Quegan · 2004

A practical and statistical introduction to SAR formation, noise and image analysis.
Polarimetric Radar Imaging: From Basics to Applications ↗

Jong-Sen Lee; Eric Pottier · 2009

Develops the polarimetric representation and relevant processing techniques.
Remote Sensing Digital Image Analysis, 6th ed. ↗

John A. Richards · 2022

Introduces optical remote-sensing image acquisition, spectral features and statistical classification.
Change detection using landsat time series: A review of frequencies, preprocessing, algorithms, and applications ↗

Zhe Zhu · 2017

Reviews radiometric preprocessing and approaches to environmental change detection with optical Landsat series.
Current methods in medical image segmentation ↗

Dzung L. Pham; Chenyang Xu; Jerry L. Prince · 2000

A foundational survey of segmentation methods and their assumptions for medical images.
A survey on deep learning in medical image analysis ↗

Geert Litjens et al. · 2017

Surveys learning-based classification, detection, segmentation and validation across medical imaging applications.

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04 / 06 · Research library

Regression Models and Estimating Equations

When should regression target a marginal mean, conditional random effects, or several parameters of a response distribution?

Regression is broader than generalized estimating equations. GEE focuses on marginal mean models for clustered or repeated outcomes; mixed and hierarchical models explicitly represent latent effects; GAM and GAMLSS let regression functions or distributional parameters vary with covariates. These approaches can be connected, but their estimands and assumptions differ.

Recommended background: Linear models, GLMs, probability, likelihood and introductory inference; matrix calculus and conditional distributions for advanced work.

Shared foundations

Define the observational unit, response support, covariates, link and estimand before choosing an estimator. Distinguish within-cluster dependence from independent clusters, marginal from conditional coefficients, and model-based from robust uncertainty. A working correlation does not repair an incorrect mean model; ordinary GEE sandwich inference may be unreliable with few independent clusters.

Three research pathways

Generalized Linear Models and GEE

Start with GLM link and variance functions, then study marginal regression for repeated or clustered data, working correlation, estimating equations and sandwich uncertainty. GEE does not normally specify a full joint law; the target and number of independent clusters matter.

Start with: Kung-Yee Liang (1986) · James W. Hardin (2013) · Scott L. Zeger (1988)

Mixed-Effects and Hierarchical Regression

GLMMs and hierarchical models specify variation through random or latent effects; estimation may be likelihood-based or Bayesian. Paula Moraga's geospatial-health work provides a Bayesian spatial example using supported INLA/SPDE constructions, not a GEE method.

Start with: Norman E. Breslow (1993) · Paula Moraga (2019)

Flexible and Distributional Regression

GAMs introduce smooth covariate effects, while GAMLSS can model location, scale and shape for non-Gaussian responses. This route connects to positive SAR measurements and environmental or biomedical outcomes; spatial dependence requires an explicit, validated extension.

Start with: Simon N. Wood (2017) · Robert A. Rigby (2005)

Eight selected readings

Generalized Linear Models, 2nd ed. ↗

P. McCullagh; J. A. Nelder · 1989

The core reference for the generalized linear model framework.
Longitudinal data analysis using generalized linear models ↗

Kung-Yee Liang; Scott L. Zeger · 1986

Introduces generalized estimating equations for correlated responses.
Generalized Estimating Equations, 2nd ed. ↗

James W. Hardin; Joseph M. Hilbe · 2013

A dedicated GEE textbook with working correlations, practical model comparisons, diagnostics and R examples.
Models for longitudinal data: a generalized estimating equation approach ↗

Scott L. Zeger; Kung-Yee Liang; Paul S. Albert · 1988

Compares population-averaged and subject-specific modelling for longitudinal responses; their coefficients need not coincide.
Approximate Inference in Generalized Linear Mixed Models ↗

Norman E. Breslow; David G. Clayton · 1993

A foundational likelihood-based treatment of GLMMs with latent random effects and approximate inference.
Geospatial Health Data: Modeling and Visualization with R-INLA and Shiny ↗

Paula Moraga · 2019

A reproducible applied route into Bayesian hierarchical spatial and spatio-temporal regression using INLA/SPDE for supported latent Gaussian models.
Generalized additive models for location, scale and shape ↗

Robert A. Rigby; D. Mikis Stasinopoulos · 2005

Introduces GAMLSS for several response-distribution parameters, including skewed and heavy-tailed families; independence is conditional on modelled effects.
Generalized Additive Models: An Introduction with R, 2nd ed. ↗

Simon N. Wood · 2017

Builds a practical and mathematical foundation for smooth regression effects and penalized-spline GAMs in R.

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05 / 06 · Research library

Information geometry

How can the geometry of probability models support statistical inference and computation?

Information geometry studies statistical models through metrics, divergences and affine connections. Three linked routes lead from the geometry of probability distributions to geodesic estimation and testing, then to geometric computation for structured models. These are study and research directions, not claims of completed original contributions.

Recommended background: Probability, likelihood, multivariable calculus and linear algebra; differential geometry for advanced reading.

Shared foundations

Begin with Fisher information as a local metric and distinguish divergences, affine connections and global geodesic distance. All three routes require a specified model and justified regularity; a useful geometric distance does not automatically provide an estimator or a calibrated hypothesis test.

Three research pathways

Statistical manifolds and dual geometry

Study Fisher–Rao metrics, exponential families, divergences, dual affine connections and geodesics. Local information geometry and global distance answer different mathematical questions.

Start with: Shun-ichi Amari (2016) · Shun-ichi Amari (2000)

Geometric estimation and hypothesis testing

Investigate estimators and tests constructed from divergences or geodesic distances. The papers of Menéndez, Morales, Pardo and Salicrú give published geodesic-test constructions; calibration, identifiability and limiting laws remain model-specific.

Start with: María Luisa Menéndez (1995) · María Luisa Menéndez (1997)

Geometric computation and structured statistical models

Learn natural-gradient optimization and numerical Fisher–Rao distances for multivariate Gaussian families. Matrix-valued models connect this route to imaging; published PolSAR classification by stochastic distances is related but is not automatically a Fisher–Rao geodesic method.

Start with: Shun-ichi Amari (1998) · Frank Nielsen (2023) · Luis Gomez (2015)

Eight selected readings

Information Geometry and Its Applications ↗

Shun-ichi Amari · 2016

A structured introduction with statistical and computational applications.
Methods of Information Geometry ↗

Shun-ichi Amari; Hiroshi Nagaoka · 2000

Advanced treatment of the geometry of statistical models and dual connections.
Geodesic submanifolds of statistical models with location parameters ↗

Jacob Burbea; Joan del Castillo · 1992

Published criteria for geodesic submanifolds and their application to tests based on information-geodesic distance.
Statistical tests based on geodesic distances ↗

María Luisa Menéndez; Domingo Morales; Leandro Pardo; Miquel Salicrú · 1995

Develops hypothesis tests based on geodesic distances and derives asymptotic test-statistic distributions under the paper's assumptions.
(h, Φ)-entropy differential metric ↗

María Luisa Menéndez; Domingo Morales; Leandro Pardo; Miquel Salicrú · 1997

Extends the geometric route to entropy-induced Riemannian metrics and illustrates geodesic testing for a Pareto family; its metric need not be Fisher–Rao.
Natural Gradient Works Efficiently in Learning ↗

Shun-ichi Amari · 1998

Introduces the natural gradient in parameter spaces with information-geometric structure, including learning and matrix models.
A Simple Approximation Method for the Fisher–Rao Distance between Multivariate Normal Distributions ↗

Frank Nielsen · 2023

Approximates Fisher–Rao distance between multivariate normal distributions by discretizing curves and using local Jeffreys divergence; the approximation is not a generic closed form.
Classification of complex Wishart matrices with a diffusion-reaction system guided by stochastic distances ↗

Luis Gomez; Luis Alvarez; Luis Mazorra; Alejandro C. Frery · 2015

An imaging bridge: PolSAR classification of complex Wishart matrices using stochastic distances and a diffusion–reaction system; no Fisher–Rao geodesic equivalence is claimed.

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06 / 06 · Research library

Causal inference

Which intervention effect is identifiable, how should it be estimated, and what changes when observations interact across space or time?

Causal inference separates the target intervention effect and its identification assumptions from the statistical procedure used to estimate it. Three routes introduce causal design, robust and machine-learning-assisted estimation, and spatial or longitudinal interventions. Prediction or an observational association alone does not identify a causal effect.

Recommended background: Probability, regression, experimental design and statistical inference; directed graphs and semiparametric theory for advanced work.

Shared foundations

State the intervention, unit, target population and estimand first; then assess consistency, exchangeability, positivity, treatment assignment and possible interference. Identification comes before estimation. A correct covariance model, Bayesian prior, GEE fit or accurate image-derived prediction cannot replace untestable causal assumptions.

Three research pathways

Causal identification and research design

Use potential outcomes and causal DAGs to define interventions and select adjustment sets. Compare experiments, observational assumptions, propensity-score designs and instrumental variables; check whether the desired estimand is identified before fitting a model.

Start with: Miguel A. Hernán (2020) · Guido W. Imbens (2015) · Judea Pearl (2009)

Semiparametric estimation and causal machine learning

After identification, study propensity scores, weighting, doubly robust estimators, influence functions and orthogonal-score machine learning. Robustness has specified model and rate conditions; cross-fitting does not automatically justify inference under spatial or clustered dependence.

Start with: Paul R. Rosenbaum (1983) · Heejung Bang (2005) · Victor Chernozhukov (2018)

Spatial and longitudinal causal inference

Study treatments changing over time, modern difference-in-differences, spatial confounding and spillovers between locations. Environmental and SAR/optical imagery can measure outcomes and exposures, but valid causal identification must address interference, design and spatially dependent uncertainty.

Start with: James M. Robins (2000) · Brantly Callaway (2021) · Brian J. Reich (2021)

Nine selected readings

Causal Inference: What If ↗

Miguel A. Hernán; James M. Robins · 2020

Connects target trials, observational identification and longitudinal methods.
Causal Inference for Statistics, Social, and Biomedical Sciences ↗

Guido W. Imbens; Donald B. Rubin · 2015

Develops design- and potential-outcomes-based inference.
Causality: Models, Reasoning, and Inference, 2nd ed. ↗

Judea Pearl · 2009

Develops structural causal models, intervention calculus and identification using graphs.
The central role of the propensity score in observational studies for causal effects ↗

Paul R. Rosenbaum; Donald B. Rubin · 1983

Establishes balancing-score arguments under strong ignorability.
Doubly Robust Estimation in Missing Data and Causal Inference Models ↗

Heejung Bang; James M. Robins · 2005

Establishes doubly robust estimation under stated treatment and outcome-model assumptions, a methodological bridge to estimating equations.
Double/debiased machine learning for treatment and structural parameters ↗

Victor Chernozhukov et al. · 2018

Develops orthogonal-score estimation with flexible nuisance learners.
Marginal structural models and causal inference in epidemiology ↗

James M. Robins; Miguel A. Hernán; Babette Brumback · 2000

Addresses time-varying confounding through marginal structural models.
Difference-in-Differences with Multiple Time Periods ↗

Brantly Callaway; Pedro H. C. Sant’Anna · 2021

Provides identification and estimation with staggered timing and heterogeneous effects.
A Review of Spatial Causal Inference Methods for Environmental and Epidemiological Applications ↗

Brian J. Reich et al. · 2021

Reviews spatial confounding, interference and space–time extensions in environmental and epidemiological causal studies.

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A starting point for each research area

Time Series and Spatial Statistics

Shared foundations

Three ways to explore

Time series

Spatial and spatio-temporal statistics

The ARMA bridge: from time to two-dimensional lattices

Six starting references

Statistical inference and asymptotic theory

Shared foundations

Three introductory routes

Classical statistical inference

Bayesian statistical inference

Statistical information and divergence-based inference

Seven selected readings

Statistical Image Processing

Shared foundations

Three application routes

Radar imaging and SAR/PolSAR

Optical remote sensing and environmental imaging

Medical and biomedical image analysis

Seven selected readings

Regression Models and Estimating Equations

Shared foundations

Three research pathways

Generalized Linear Models and GEE

Mixed-Effects and Hierarchical Regression

Flexible and Distributional Regression

Eight selected readings

Information geometry

Shared foundations

Three research pathways

Statistical manifolds and dual geometry

Geometric estimation and hypothesis testing

Geometric computation and structured statistical models

Eight selected readings

Causal inference

Shared foundations

Three research pathways

Causal identification and research design

Semiparametric estimation and causal machine learning

Spatial and longitudinal causal inference

Nine selected readings