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Exact linear-Gaussian Kalman filtering — Examples
1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Areas of Elementary Plane Figures
- Binary Operations, Monoids, Groups and Subgroups
- Central Limit Theorems
- Characteristic Functions Inversion and Continuity
- Compactness
- Compactness in Metric Spaces
- Complete Metrizability, Čech-Completeness, and Baire Category
- Completeness, Completion, and Uniform Continuity
- Complex Lp Spaces and Test-Function Conventions
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Conditional Distributions and Regular Conditional Probability
- Conditional Expectation
- Congruences, the Integers Modulo n and the Chinese Remainder Theorem
- Conjugate families and Gaussian conditioning
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Countability Axioms and Cardinal Functions
- Cyclic Groups and Direct Products
- Density Separability and Convolution in Lᵖ
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Direct Matrix Factorisations: LU, Cholesky and QR
- Discrete Time Martingales
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Divisibility, Greatest Common Divisors and Bézout's Identity
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Estimators and Randomized Procedures
- Exact linear-Gaussian Kalman filtering
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Finite Probability and the Probabilistic Method
- Foundations of the Real Numbers for Analysis
- Fourier Transform Convolution and Approximate Identities
- Fubini and Change of Variables
- Fundamental Trigonometric Identities
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Hausdorff via the Diagonal
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Improper and Parameter-Dependent Multiple Integrals
- Improper Integrals
- Independence Borel Cantelli and Zero One Laws
- Infinite Product Measures and Kolmogorov Extension
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Lebesgue Measure on Euclidean Space
- Lebesgue-Stieltjes Measures and Distribution Functions
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Markov Kernels and Markov Chains
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Metric Spaces
- Mixed effects and longitudinal models
- Mixed Partials, Taylor Formulae, and Extrema
- Modes of Convergence Egorov and Lusin
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Normal Subgroups and Quotient Groups
- Order, Zorn's Lemma, and the Axiom of Choice
- Outer Measure and the Caratheodory Extension Theorem
- Partitions of Unity and Paracompactness
- Pointwise Extended-Real Function Products
- Polynomial Rings, the Division Algorithm and Roots
- Power Series and Real-Analytic Functions
- Primes, Euclid's Lemma and the Fundamental Theorem of Arithmetic
- Probability Spaces Random Variables and Expectation
- Product Measures and the Fubini Tonelli Theorems
- Properties of the Integral and the Working FTC
- Radon Measures and the Riesz Markov Kakutani Theorem
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Scientific model contracts and applicability
- Separation Axioms: the Hierarchy
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sigma Algebras and Borel Sets
- Signed and Complex Measures Hahn and Jordan
- Simple Field Extensions and the Construction of the Complex Numbers
- Sine, Cosine, and the Definition of Pi
- Splitting Fields
- Standard-Borel Real Codings and Determining Classes
- State-space model laws
- State-space models and exact Bayesian filtering
- Stationary means and sample autocovariance
- Statistical Experiments, Models and Identifiability
- Subspaces, Products, and Quotients
- Suprema and Infima
- Sylow's Theorems, p-Groups and Nilpotent Groups
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Exponential Function
- The Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
- The Inverse and Implicit Function Theorems
- The Lebesgue and Riemann Integrals Compared
- The Lebesgue Integral and the Convergence Theorems
- The Logarithm and General Powers
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Maximal Function and Lebesgue Differentiation
- The Radon Nikodym Theorem and Lebesgue Decomposition
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Spectral Theorem, Positive Operators and Singular Value Decomposition
- The Topology of Euclidean Space
- The Total Derivative in ℝᵐ → ℝⁿ
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Urysohn's Lemma and the Tietze Extension Theorem
- Vector Spaces, Linear Subspaces, Span and Direct Sums
- Weak Convergence Tightness and Representation
2 · Summary
Objective
These examples compute a scalar update and a two-dimensional update with singular state covariance. They verify that only the innovation covariance needs to be positive definite, while the exact filtered law may remain degenerate.
3 · Logical flowchart
4 · Definitions, theorems and proofs
None yet.
5 · Examples, counterexamples and false statements
A one-step scalar Kalman update
Example
Take a one-step scalar model with , transition for , and observation for , with the three Gaussian variables independent. At the observation , the predicted law is , the innovation variance is , the gain is , and the selected filtering-kernel value is .
Facts & Assumptions
Given: AC and the stated independent-noise linear-Gaussian model.
AC is assumed through the Gaussian model and conditional-law theorem. The Axiom of Choice.
Under AC, the Kalman recursion with positive innovation covariance defines an everywhere filtering-kernel version of the exact conditional state law. The finite-horizon linear-Gaussian Kalman filter is exact.
The model permits singular positive-semidefinite initial, process, and measurement covariances with mutually independent centered Gaussian noises. Linear-Gaussian state-space models with independent noises.
Every positive-semidefinite covariance defines a multivariate-normal law under AC. Multivariate normal law, including singular covariance.
Verification
Set , , , and . The covariances are positive semidefinite and the noises satisfy [F2]. The prediction is and . Thus , and for the innovation is .
Solving the scalar innovation system gives . Therefore and . By [F1], the selected conditional-law version at is exactly .
Singular prior covariance with positive innovation variance
Example
Let and take , so the initial covariance is which is singular; below denotes the identity matrix. Use , , , and , with measurement noise independent of . At , the innovation variance is , and the selected filtering-kernel value is The filtered covariance remains singular.
Facts & Assumptions
Given: AC and the stated one-step linear-Gaussian model.
AC is assumed through the degenerate Gaussian model and conditional-law theorem. The Axiom of Choice.
The exact Kalman filtering theorem permits singular state and filtered covariances whenever the innovation covariance is positive definite. The finite-horizon linear-Gaussian Kalman filter is exact.
Positive-semidefinite covariance matrices, including zero and singular ones, define multivariate-normal laws under AC. Multivariate normal law, including singular covariance.
A positive-semidefinite operator has a positive-semidefinite square root, so the rank-one covariance is an admissible Gaussian covariance. A non-negative operator has a unique non-negative square root.
Initial, process, and observation covariances in the linear-Gaussian model may be positive semidefinite, and the noises are mutually independent. Linear-Gaussian state-space models with independent noises.
Verification
The matrix is positive semidefinite because ; it is singular because . Since and , the Gaussian realization has covariance . With and , prediction gives and .
Since and , . Hence and . The covariance update is , which is still singular and positive semidefinite. By [F1] this is the exact filtering-kernel covariance at , despite the singular prior covariance.