Source-linked AI summary
Robust Inference for State-Space Models with Skewed Measurement Noise
Henri Nurminen, Tohid Ardeshiri, Robert Piché, Fredrik Gustafsson
TL;DR
The Kalman filter is optimal among filters only with normally distributed noise, while real-world outliers can cause large estimation errors and efficient methods for heavy-tailed, asymmetric time-series noise have been missing. This paper develops variational-Bayes Bayesian filters and smoothers for skew-t measurement noise. In simulations, the proposed methods outperform conventional symmetric Kalman-type algorithms when skewness is present.
Problem
Efficient estimation algorithms for time series with heavy-tailed and asymmetric noise have been missing, while outliers can cause large Kalman-filter estimation errors.
Source: §I · p. 1Method
The paper develops Bayesian filtering and smoothing approximations using variational Bayes and skew-t-distributed measurement noise.
Source: §I · p. 1Results
In simulations, the proposed methods outperform conventional symmetric Kalman-type algorithms when skewness is present; the proposed filter costs roughly 5 to 10 times as much as the Kalman filter.
Source: §VI · p. 4Takeaways & Limitations
The computational burden depends on measurement dimension and model parameters.
Source: §VI · p. 4
Abstract
from arXiv · showhide
Filtering and smoothing algorithms for linear discrete-time state-space models with skewed and heavy-tailed measurement noise are presented. The algorithms use a variational Bayes approximation of the posterior distribution of models that have normal prior and skew-t-distributed measurement noise. The proposed filter and smoother are compared with conventional low-complexity alternatives in a simulated pseudorange positioning scenario. In the simulations the proposed methods achieve better accuracy than the alternative methods, the computational complexity of the filter being roughly 5 to 10 times that of the Kalman filter.
I. INTRODUCTION
The paper addresses the need for efficient filtering and smoothing when measurement noise is both heavy-tailed and skewed, conditions that challenge the Kalman filter’s normal-noise assumptions. It proposes variational-Bayes filter and smoother approximations using skew-t measurement noise, alongside a description of the skew-t distribution and its measurement-noise assumptions.
I. INTRODUCTION
- The Kalman filter is optimal among filters only with normally distributed noise, and large errors occur more often in real-world data than the normal distribution predicts.The paper notes that this makes the filter prone to large estimation errors when outliers occur.Source: §I · p. 1
- Efficient estimation for time series with heavy-tailed and asymmetric noise was missing; t-based robust algorithms do not use skewness, while Gaussian mixtures and particle filters have computational drawbacks.The Gaussian-mixture posterior grows exponentially with measurements, and particle-filter complexity rises rapidly with state dimension.Source: §I · p. 1
- For 2,905 time-of-flight errors, BIC is 9,600 for skew t, 10,500 for Student’s t, 17,200 for normal, and 10,000 for GM2.The passage reports that the skewed distributions fit better than the symmetric distributions.Source: Figure 1 · p. 1, §I · p. 1
- The proposed Bayesian filter and smoother use variational Bayes with skew-t measurement noise to add flexibility for skewed, heavy-tailed noise while retaining Kalman-filter computational efficiency.The paper evaluates them in numerical pseudorange-positioning simulations against lightweight algorithms and a particle filter.Source: §I · p. 1
II. SKEW t-DISTRIBUTION
- The skew-t distribution has location, spread, shape, and degrees-of-freedom parameters, and approaches the normal distribution as degrees of freedom increase and shape tends to zero.The measurement-noise model is a product of independent univariate skew-t distributions, justified for statistically independent sensor data.Source: §II · pp. 1–2
III. PROBLEM FORMULATION
The problem formulation considers a linear state-space model with skew-t-distributed measurement noise. Its aim is to use variational Bayes to approximate both filtering and smoothing distributions.
III. PROBLEM FORMULATION
- The formulation considers a linear state-space model with skew-t-distributed measurement noise and a state with a prior distribution.The state transition matrix is A, and the state x_k is to be estimated.Source: §III · p. 2
- Measurements y_k are conditionally independently skew-t-distributed, with diagonal R encoding squared spread parameters, Δ encoding shape parameters, and ν encoding degrees of freedom.C is the measurement matrix; the state and measurement noise sequences are mutually independent.Source: §III · p. 2
- The aim is to derive Bayesian filtering and smoothing algorithms using variational Bayes to approximate p(x_k|y_1:k) and p(x_k|y_1:K).Source: §III · p. 2
IV. VARIATIONAL SOLUTION
Because the joint smoothing posterior is not analytically tractable, the paper approximates it with a factorized variational Bayes posterior and solves for its factors by cyclic iteration.
IV. VARIATIONAL SOLUTION
- The joint smoothing posterior is not analytically tractable, so the method seeks a factorized variational Bayes approximation.The approximation minimizes the Kullback-Leibler divergence between the true posterior and the factorized approximation.Source: §IV · p. 2
- The analytical solutions for the state, auxiliary-variable, and scale factors are obtained by cyclic iteration.The likelihood has a hierarchical representation involving a diagonal random matrix and a truncated normal variable.Source: §IV · p. 2
- The recursion converges to a local optimum, after which the auxiliary factors are discarded to obtain a normal marginal smoothing density.The smoothing algorithm outputs the mean and covariance parameters of that density.Source: §IV · pp. 2–3
V. SIMULATIONS
The simulations compare the proposed skew-t variational Bayes filter and smoother with computationally light filters and smoothers, a particle filter, and Kalman-based alternatives.
V. SIMULATIONS
- The STVBF is compared with the TVBF, bootstrap particle filter, KF, and KF with measurement-validation gating.The gating alternative discards individual measurement components whose normalized squared innovation exceeds the chi-square distribution’s 99% quantile.Source: §V · p. 3
- The STVBS is compared with the TVBS and a Rauch-Tung-Striebel smoother with gating.KF and RTSS use the true measurement-noise mean and covariance, while TVBF and TVBS use the true mean and (ν −2)/ν times the true covariance as the shape matrix.Source: §V · p. 3
A. One-dimensional positioning
In one-dimensional positioning, the STVBF has the lowest reported RMSE, while the compared methods show distinct error behavior; its accuracy improves with VB iterations, at added computational cost.
A. One-dimensional positioning
- The VB iterations stop when the estimate changes by less than 0.01; the simulation uses 100-step random walks.The simulation description also specifies A = 1, Q = 1, and C = 13×1.Source: A. One-dimensional positioning · p. 3
- The STVBF has the lowest RMSE; TVBF and KF-G have negative bias, while KF has the highest error standard deviation and positive skew.Table III reports estimation-error statistics for one-dimensional positioning.Source: A. One-dimensional positioning · p. 3, Table III · p. 3
- Figure 3 illustrates TVBF’s negative bias and KF’s sensitivity to outliers, with the measurement error of 300 at time 49 omitted.TVBF and KF-G react relatively slowly to positive errors, while KF is excessively sensitive to large positive measurement errors.Source: Figure 3 · p. 3, A. One-dimensional positioning · p. 3
- In Figure 4, for q = 10, ten STVBF iterations are enough to outperform TVBF, while one additional VB iteration gives the same accuracy gain as 100 additional PF particles.Source: Figure 4 · p. 4
B. Pseudorange positioning
The pseudorange experiment uses a GNSS-type measurement model with skew-t noise and evaluates three-dimensional position RMSE across Monte Carlo trajectories.
B. Pseudorange positioning
- The simulated GNSS-type pseudorange measurements use satellite position, receiver position, a bias state, and measurement noise.The model is linearized, with negligible linearization error because the satellites are far from the receiver.Source: B. Pseudorange positioning · p. 4
- Results use 1,000 Monte Carlo replications of 100-step trajectories, averaging 7.6 measured satellites and computing RMSE for the three position components.The state is a three-dimensional random walk with process-noise covariance Q = diag(q^2, q^2, 0.5^2), and the constant bias has prior N(0, 0.75^2).Source: B. Pseudorange positioning · p. 4
1) Evaluation of the filter:
The STVBF outperforms comparison methods in most replications when q ≥ 1, but its performance depends on signal-to-noise ratio and model conditions.
1) Evaluation of the filter:
- With q ≥ 1, the STVBF outperforms comparison methods in a significant majority of replications, but works well only when SNR is not very low.For q = 0.1, measuring only sums of x_k and u_k makes them posterior-correlated, causing the VB approximation to underestimate posterior variance.Source: 1) Evaluation of the filter · p. 4
- After 30 VB iterations, RMSE reduction is negligible, while 10 iterations suffice to outperform TVBF, which requires 5.In this example, one extra VB iteration gives the same accuracy gain as 100 additional PF particles.Source: 1) Evaluation of the filter · p. 4
- Convergence depends on model parameters: larger δ slows it, and large q or many sensors can increase the required iterations.Source: 1) Evaluation of the filter · p. 4
2) Real-world noise:
With noise drawn from a real time-of-flight error histogram, the STVBF has lower RMSE than TVBF in 61% of replications; comparisons with skewed measurements favor STVBF when SNR is high enough.
2) Real-world noise:
- 61% of 1000 replications had lower RMSE for STVBF than TVBF with noise generated from the time-of-flight error histogram.The result indicates robustness to small deviations from the model that appear in real data.Source: 2) Real-world noise · p. 4, Figure 6 · p. 4
- With skewed measurements, STVBF outperforms the comparison methods when SNR is high enough.Source: Figure 5 · p. 4
3) Evaluation of the smoother:
The STVBS outperforms TVBS at low SNR, while its advantage over TVBS at high SNR is smaller than the corresponding filter difference.
3) Evaluation of the smoother:
- At low SNR, STVBS outperforms TVBS, but the high-SNR difference is smaller than between the corresponding filters.The evaluation used 30 VB iterations for STVBS and 10 for TVBS, observed to provide convergence.Source: 3) Evaluation of the smoother · p. 4, Figure 7 · p. 4
VI. CONCLUSIONS
The paper proposes a variational-Bayes filter and smoother that account for skewness and heavy-tailed measurement noise; in the presented simulations they outperform conventional symmetric Kalman-type algorithms when skewness is present.
VI. CONCLUSIONS
- In the presented simulations, the proposed methods outperform conventional symmetric Kalman-type algorithms when skewness is present.Source: §VI · p. 4
- The proposed filter’s computational cost is roughly 5 to 10 times the Kalman filter’s in the presented simulations.Computational burden depends on measurement dimension and model parameters.Source: §VI · p. 4
- The proposed filter and smoother account for skewness and heavy-tailedness of measurement noise using a variational Bayes approximation.Source: §VI · p. 4