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Sequential Joint Detection and Estimation: Optimum Tests and Applications

Yasin Yilmaz, Shang Li, Xiaodong Wang

arXiv:1411.1440v1stat.AP

TL;DR

Detection and estimation are coupled, but conventional separate approaches do not optimize their overall performance. The paper jointly optimizes stopping, detection, and estimation under a Bayesian cost constraint, and reports better performance than conventional separate methods.

  • Problem

    Detection and estimation are intrinsically coupled, but conventional approaches do not yield the overall optimum.

    Source: §I · p. 2

  • Method

    The framework jointly selects a stopping time, detector, and estimators to minimize sample size subject to a Bayesian combined-cost constraint.

    Source: §VI · p. 30

  • Results

    SJDE significantly outperforms the conventional SPRT and MMSE combination in average stopping time while achieving the same target accuracy α.

    Source: B. Numerical Results · p. 20

  • Takeaways & Limitations

    The optimum detector and estimators are strongly coupled, making separate treatment strictly suboptimal.

    Source: §VI · p. 31

Abstract

from arXiv · show

We treat the statistical inference problems in which one needs to detect and estimate simultaneously using as small number of samples as possible. Conventional methods treat the detection and estimation subproblems separately, ignoring the intrinsic coupling between them. However, a joint detection and estimation problem should be solved to maximize the overall performance. We address the sample size concern through a sequential and Bayesian setup. Specifically, we seek the optimum triplet of stopping time, detector, and estimator(s) that minimizes the number of samples subject to a constraint on the combined detection and estimation cost. A general framework for optimum sequential joint detection and estimation is developed. The resulting optimum detector and estimator(s) are strongly coupled with each other, proving that the separate treatment is strictly sub-optimum. The theoretical results derived for a quite general model are then applied to several problems with linear quadratic Gaussian (LQG) models, including dynamic spectrum access in cognitive radio, and state estimation in smart grid with topological uncertainty. Numerical results corroborate the superior overall detection and estimation performance of the proposed schemes over the conventional methods that handle the subproblems separately.

I. INTRODUCTION

Detection and estimation are coupled, yet conventional separate or GLRT approaches do not yield an overall optimum. The paper formulates a sequential joint problem that balances performance against the number of observations.

I. INTRODUCTION

  • Detection and estimation are intrinsically coupled, while separate methods and GLRT do not yield the overall optimum.Source: §I · p. 2
  • Resource constraints motivate sequential methods that minimize observations, while prior joint work was mostly fixed-sample or limited to a narrow sequential model.The cited sequential joint method treats a scalar Gaussian model with an unknown parameter only under the alternative hypothesis.Source: §I · p. 3
  • The paper addresses sequential joint detection and estimation for nonlinear vector signals, arbitrary noise, unknown parameters under both hypotheses, and general estimation costs.Source: §I · p. 3

A. Problem Formulation

  • The formulation observes measurement vectors and matrices, estimates an unknown parameter vector, and jointly accounts for binary detection and estimation costs.It assumes known hypothesis priors but specifies no distributions or system model, and allows correlated noise and observation matrices.Source: A. Problem Formulation · pp. 3–4
  • The objective minimizes expected sample count subject to a target-cost constraint, with stopping when the target accuracy is first reached.The stopping rule is optimal for stopping times determined by the observation-matrix history.Source: A. Problem Formulation · pp. 4–5
  • Stopping times depend only on the observation-matrix history, avoiding analytical complications associated with stopping based on the complete observation history.Source: A. Problem Formulation · p. 5

B. Optimum Solution

The optimum sequential solution jointly chooses a stopping time, detector, and estimators to minimize sample use subject to a combined-cost constraint. Its detector and estimators are coupled through posterior estimation costs.

B. Optimum Solution

  • The optimum MMSE estimator is a weighted average of the hypothesis-specific MMSE estimators, with weights depending on the likelihood ratio.Typically, the likelihood ratio is smaller than 1 under H0 and larger than 1 under H1, bringing the estimate closer to the corresponding hypothesis-specific conditional mean.Source: Optimum Estimators · p. 8
  • The optimum detector is coupled to the estimators through posterior estimation costs, accounting for costs across true and selected hypotheses.The decision rule selects H1 when the relevant decision expression is negative.Source: Optimum Detector · p. 10
  • The optimum stopping rule stops at the first time the optimal cost C_t is at most α, then decides and estimates under the selected hypothesis.The estimator for the selected hypothesis is given by (21).Source: Complete Solution · pp. 10–11
  • The universal procedure may require intense online computation because its statistics often lack closed-form expressions.Some statistics may be easily computed in specific cases.Source: Complete Solution · p. 11
  • The scheme is optimum for general nonlinear models, arbitrary estimation costs, noise distributions, and parameter counts, and includes the scheme in [7] as a special case.A monotonicity property for the stopping time cannot be shown for this generic model, so it is found numerically.Source: Complete Solution · p. 13

C. Separated Detection and Estimation Costs

With separated detection and estimation costs, the optimum stopping rule retains the same structure but uses a simplified optimal cost. The detector favors the hypothesis with better estimation performance.

C. Separated Detection and Estimation Costs

  • The optimum stopping rule retains the structure of Theorem 1 but uses a simplified optimal cost when wrong decisions are penalized only by detection costs.This alternative cost sets b01 = b10 = 0.Source: C. Separated Detection and Estimation… · pp. 13–14
  • When a hypothesis is selected, the estimator is optimal under that hypothesis because estimation cost is not used to penalize wrong decisions.Unlike Theorem 1, the estimator is not a mixture of the estimators under both hypotheses.Source: C. Separated Detection and Estimation… · p. 14
  • The optimum detector is biased toward the hypothesis with the lower minimum posterior estimation cost.For example, a lower minimum posterior estimation cost under H1 makes H1 easier to select.Source: C. Separated Detection and Estimation… · pp. 14–15

III. LINEAR QUADRATIC GAUSSIAN (LQG) MODEL

The section specializes the optimum sequential joint detection-and-estimation scheme to an LQG model and derives statistics used by its stopping, detection, and estimation rules. In this model, the stopping time and optimal cost depend on the Fisher information matrix scaled by σ^2.

III. LINEAR QUADRATIC GAUSSIAN (LQG) MODEL

  • The LQG specialization derives sufficient statistics for the optimum stopping time, detector, and estimator.The sufficient statistics include the conditional mean, decision likelihoods, and likelihood ratio.Source: §III · p. 15
  • The optimal cost and stopping time are functions only of U_t, the Fisher information matrix scaled by σ^2.The conditional expectation in the cost is taken over v_t given U_t.Source: §III · p. 17
  • The stopping rule can be implemented by estimating the optimal cost through Monte Carlo simulation and stopping when it is at most α.The cost can also be precomputed offline on a grid of U values.Source: §III · p. 17

A. Independent LQG Model

The independent LQG case assumes independent entries of the unknown vector and a diagonal observation matrix, then exploits the resulting independent scalar subsystems. Its reduced statistics also make an offline simulation grid more viable than under the general LQG model.

A. Independent LQG Model

  • The independent LQG case gives the sufficient statistics for the optimum scheme under its diagonal and independence assumptions.The statistics are obtained from scalar posterior and likelihood expressions for each local parameter.Source: A. Independent LQG Model · pp. 17–18
  • With independent entries of x and diagonal H_t, the linear model decomposes into N independent scalar subsystems.The passage describes this as a possible distributed sensing setup in which nodes send observations to a fusion center.Source: A. Independent LQG Model · pp. 17–18
  • The offline simulation grid is N-dimensional, smaller than the N^2+N grid under the general LQG model, making offline simulation more viable.Source: A. Independent LQG Model · p. 19

B. Numerical Results

The numerical comparison evaluates SJDE against an SPRT detector followed by an MMSE estimator in an LQG setting. At the same target accuracy, SJDE has a significantly smaller average stopping time.

B. Numerical Results

  • The conventional comparison method uses an SPRT decision followed by the corresponding MMSE estimator.The SPRT tests its log-likelihood ratio against a prescribed interval, with thresholds selected to meet the target accuracy.Source: B. Numerical Results · p. 19
  • The numerical comparison uses an LQG model with a three-dimensional unknown vector and a one-by-three observation matrix.The noise is white Gaussian with covariance I, and H_t is generated as an independent-over-time Gaussian matrix.Source: B. Numerical Results · p. 19
  • SJDE has a significantly smaller average stopping time than SPRT followed by MMSE while achieving the same target accuracy.Figure 1 plots average stopping time against target accuracy level.Source: Figure 1 · p. 20, B. Numerical Results · p. 20

A. Background

In dynamic spectrum access, secondary users detect primary-user transmissions and estimate the interference channel to manage transmit power. Because sensing time limits time available for data communication, the task is sequential joint detection and estimation.

A. Background

  • The secondary user must estimate the channel coefficient as well as detect primary-user activity to keep its transmit power within allowable interference limits.If primary-user communication is detected, underlay access remains possible subject to the interference constraint.Source: A. Background · p. 21
  • Reducing sensing time frees time for data communication and can increase secondary-user throughput.Source: A. Background · p. 21
  • The paper derives an alternative optimum scheme for the first-stage joint sensing and channel-estimation problem, omitting the later transmit-power determination stage.The cited prior procedure separates those stages.Source: A. Background · p. 21

B. Problem Formulation

The cognitive-radio formulation jointly detects PU communication and estimates interference channels when it is present, using cooperative SU observations at a fusion center. Its optimum sequential scheme couples detection with estimation and uses the resulting cost and likelihood information to determine when to stop.

B. Problem Formulation

  • The formulation assumes K cooperating SUs observe simultaneous PU pilot transmissions and report observations centrally to a fusion center.The fusion center may be one of the SUs; under energy and bandwidth constraints, SUs can report necessary and sufficient statistics using level-triggered sampling [1].Source: B. Problem Formulation · pp. 21–22
  • The task first detects PU activity; when it is absent, an SU transmits at Pmax, and when present, estimated PU–SU channels determine transmission powers.The SU with the highest permitted transmission power then begins data communication.Source: B. Problem Formulation · p. 22
  • The model represents the interference channels between two PUs and K SUs, with PU pilot signals in a diagonal observation matrix.Channel coefficients have specified means and variances; the observations and Gaussian noise vectors define the binary hypothesis test.Source: B. Problem Formulation · p. 22
  • The channel-estimation cost is assigned only under H1, since estimating interference channels is meaningful only when PU communication occurs.Under H0, the estimate is treated as zero, and the resulting objective combines the two detection-error costs.Source: B. Problem Formulation · p. 23

C. Optimum Solution

  • Corollary 2 gives the optimum sequential scheme for joint spectrum sensing and channel estimation, including its stopping time, detector, and estimation rule.The likelihood ratio is formed from independent PU–SU channel likelihoods, while the optimal cost can be estimated by Monte Carlo or offline over a two-dimensional grid.Source: C. Optimum Solution · pp. 23–24
  • The optimum detector uses its own channel estimates as side information: estimates farther from zero favor H1, while estimates nearer zero favor H0.When H1 is selected, the optimum estimator is the MMSE estimator because channel estimation is meaningful only during PU communication.Source: C. Optimum Solution · p. 24
  • The paper presents an alternative derivation to earlier work on a similar complex-channel problem and explains how the dynamic-spectrum-access method fits its general sequential theory.The passage also notes that optimal stopping can be computed efficiently through Monte Carlo simulations without the monotonicity property for optimal cost.Source: C. Optimum Solution · pp. 24–25

V. STATE ESTIMATION IN SMART GRID WITH TOPOLOGICAL UNCERTAINTY

The smart-grid application jointly identifies a possible topology error or outage and estimates the grid state. It uses a multi-hypothesis LQG formulation and an optimum detector that accounts for estimation performance.

A. Background and Problem Formulation

  • The candidate topologies represent possible errors in periodically estimated switch-status data and power outages that alter both topology and the state prior.Normal operation is the no-error, no-outage hypothesis; the formulation uses multiple hypotheses and penalizes wrong decisions with detection costs.Source: A. Background and Problem Formulation · pp. 25–26

B. Optimum Solution

C. Numerical Results

In the IEEE-4 bus evaluation, the proposed SJDE scheme is compared with ML detection and MMSE estimation under the same stopping-time framework. SJDE reports better combined detection-and-estimation performance and requires fewer average samples at a given target accuracy.

C. Numerical Results

  • The IEEE-4 bus test uses five candidate topologies: the default network and four cases with one specified link outage.The state has three phase-angle components, and the eight measurements yield an 8-by-3 topology matrix.Source: C. Numerical Results · p. 28
  • The evaluation assumes unit link impedances and assigns a Gaussian state distribution under each candidate topology, with white Gaussian measurement noise.The stated task is to select among the five topologies while estimating the state vector.Source: C. Numerical Results · pp. 28–29
  • SJDE significantly outperforms the ML-detector/MMSE-estimator combination on combined detection-and-estimation cost, and uses fewer average samples at the same target accuracy.The comparison baseline uses the optimum stopping time, so it is not a completely conventional scheme.Source: C. Numerical Results · pp. 29–30, Figure 3 · p. 30

VI. CONCLUSION

The paper develops a Bayesian framework for sequential joint detection and estimation that minimizes sample size subject to overall performance. Its theoretical results apply to LQG problems, including cognitive-radio spectrum access and smart-grid state estimation.

VI. CONCLUSION

  • The framework jointly selects a stopping time, detector, and estimators to minimize sample size while guaranteeing the best overall detection-and-estimation performance under a Bayesian setup.The conventional separate treatment is strictly suboptimal because the optimum detector and estimators are strongly coupled.Source: §VI · pp. 30–31
  • The general theory is applied to LQG models for dynamic spectrum access in cognitive radio and state estimation in smart grids, with numerical results supporting the theoretical findings.Source: §VI · p. 31
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