F2E5216/TS1002 Adaptive Filtering and Change Detection. Course Organization. Lecture plan. The Books. Lecture 1

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1 Adaptive Filtering and Change Detection Bo Wahlberg (KTH and Fredrik Gustafsson (LiTH Course Organization Lectures and compendium: Theory, Algorithms, Applications, Evaluation Toolbox and manual: Algorithms, Evaluation Implementations Lecture The goal of the course is to get an understanding for the theory of model based filtering and change detection, with particular attention to applications. Projects: Application Evaluation of Algorithms Exercises: To each part available from homepage Examination, 6 credits: Three-day take-home exam. Lecture, 2005 Lecture, The Books Steven Kay: Selected parts of Fundamentals of statistical signal processing: Volume II, detection theory. Prentice Hall, 998. Only Lecture Fredrik Gustafsson: Adaptive filtering and change detection. John Wiley & Sons, 200. Lecture plan Lecture : Detection theory from Kay s book. Lecture 2: Applications. Detection of mean changes: likelihood principles; Chapters, 2, 3. Lecture 3: Detection of mean changes: The parallel filter concept. Chapters 3,4, Appendix 4. Lecture 4: Parameter estimation: recursive least squares. Residual based detection. Chapter 5 Lecture 5: Parameter estimation: parallel filters and filter bank change detection. Chapter 6-7 Lecture 6: State estimation: Kalman, EKF and particle filters. Residual based detection. Chapter 8 Lecture 7: State estimation: parallel filters and filter bank change detection. Chapters 9, 0. Given by Fredrik Lecture 8: State estimation: the parity space approach. Chapter Lecture 9: State estimation: parity space diagnosability and PCA. Lecture, Lecture,

2 Probability Theory Lecture Stochastic variable: X Basic Probability Theory Basic Detection Theory Neyman -Pearson s theorem Basic Hypothesis tests Generalized Likelihood Ratio Tests Pages in Kay Probability density function (PDF: Mean: E(X = Variance: V ar(x = xp(xdx p(x (x E(X 2 p(xdx Cummulative distribution function: Φ(x = P(X < x = Right tail probability: Q(x = P(X > x = x p(xdx x p(xdx Lecture, Lecture, p(x 2πσ 2 e (x µ2 2σ 2 E(x µ Var(x σ 2 Gaussian or Normal N(µ, σ 2 Q(x 2π(x µ/σ e (x µ2 2σ 2 E(x n (n σ 2 Matlab Q(x=0.5*erfc((x-mu/sigma/sqrt(2 Exponential Exp(µ p(x µ e µ, x 0 E(x µ Var(x µ 2 Q(x e x µ Lecture, Lecture,

3 Chi-squared with ν degrees χ 2 ν p(x 2 ν/2 Γ(ν/2 x ν/2 e x/2,x > 0 Γ(u t u e t dt, Γ(n = (n! 0 Γ(u = (u Γ(u, Γ(0.5 = π E(x ν Var(x 2ν Q(x e x/2 ν/2 (x/2 k k=0,ν 2 and even k! Comments χ 2 2 = Exp(2. If x = ν i= x2 i, x i N(0,, then x χ 2 ν approximately Gaussian for large ν Non-central chi-squared with ν degrees p(x I r (u E(x Var(x Comments 2 k=0 χ 2 ν(λ ( x λν 2 4 e (x+λ/2 I ν/2 ( λx,x > 0 ν + λ 2ν + 4λ (u/2 2k+r Bessel function k!γ(r+k+ If x = ν i= x2 i, x i N(µ i,, then x χ 2 ν( i µ i Lecture, Lecture, Detection Principles Given: Data x[n] with known distribution under the null hypothesis H 0 and the alternative hypothesis H, respectively. General test: Form a test statistic T(x. Decide H 0 Decide H Basic design parameters: if T(x < γ (threshold if T(x > γ Probability of false alarm (Type I error P FA = P(H H 0 = x:t(x>γ p(x H 0dx Probability of detection (Power P D = P(H H = x:t(x>γ p(x H dx Miss P M = P(H 0 H = P D (Type II error The only hard result in detection theory Neyman-Pearson s theorem: To maximize the power P D for a given probability of false alarm P FA = α, take T(x = p(x H p(x H 0. The test is called LRT, the Likelihood Ratio Test. The threshold is found from P FA = x:t(x>γ p(x H 0 dx = α Lecture, 2005 Lecture,

4 Proof: Receiver Operating Characteristics Define the LRT test T = {, if, p(x H > γp(x H 0 0, if, p(x H < γp(x H 0 P D LRT chance line Let 0 T denote any test with less or equal P FA. Key inequality (Check the signs! (T T (p(x H γp(x H 0 dx 0 Re-arrange the integral: (P D P D γ(p FA P FA 0, where the last equality follows from the false alarm assumption. This shows T optimize the power! One has to be a bit more careful if p(x H = γp(x H 0 has a non-zero probability P FA The slope of the LRT ROC curve is γ! The curve is concave. The LRT maximizes any performance criterion that values low false alarm probabilities and/or high detection probabilities. Lecture, Lecture, Proof of Slope [ ] n E(T(x n p(x H H = p(x H dx = p(x H 0 [ ] n+ p(x H p(x H 0 dx = E(T(x n+ H 0 p(x H 0 which can be written t n g(t H dt = t n+ g(t H 0 dt, n Sufficient Statistics Replace x with the simpler sufficient statistics t, T(x = p(x H p(x H 0 = g(t H g(t H 0 where g(t H i is the PDF for the likelihood ratio under H i. This means that g(t H = tg(t H 0. Now and P D = g(t H dt, P F A = g(t H 0 dt γ γ dp D = dp D dp F A dγ / dp F A = g(t H dγ g(t H 0 = γ Example: x = (x 0...x N, x i N(m,σ 2, take t = N N i=0 x i Observe that γ 0 Lecture, Lecture,

5 Basic Hypothesis Test Is there a known mean A in the observed signal? H 0 :x[n] = w[n] H :x[n] = A + w[n] The more general problem of detecting a given signal H 0 :x[n] = w[n] H :x[n] = s[n] + w[n] can be rewritten to unknown mean. Example : Communication The signal s(t = 2A sin(ωt is transmitted on the carrier ω and y(t = s(t + w(t is received. Let x = T where w N(0, σ2 2T. T 0 = A + T y(t sin(ωt T 0 w(t sin(ωt = A + w, under H Lecture, Lecture, Example 2: Matched filter The signal s[n] with energy E may be contained in y[n]. Let where w N(0, Eσ2 N. x = N N n=0 y[n]s[n] = E + w under H T(x = Neyman-Pearson N e n=0 (x[n] A2 (2πσ 2 N/2 2σ 2 e (2πσ 2 N/2 N n=0 (x[n]2 2σ 2 Taking logarithm and simplifying give T(x = N N n=0 = e NA 2 N n=0 2Ax[n] 2σ 2 > γ. x[n] > σ2 NA log(γ + A 2 = γ. Lecture, Lecture,

6 ( Now T(x N ( N 0, σ2 N A, σ2 N under H 0 under H ( so P FA = P( T(x > γ H 0 = Q γ σ2 /N σ 2 γ = N Q (P FA ( ( P D = P( T(x γ A NA > γ H = Q = Q Q 2 (P FA σ2 /N σ 2 The last expression shows that P D is a function of the desired P FA and SNR, NA 2 /σ 2, only. Standard plots: ROC: Receiver Operating Characteristics plots P D versus P FA with γ being the parameter. Just guessing H with probability P FA gives P D = P FA. All clever tests give a curve above this straight line. Neyman-Pearson gives the upper bound curve for all tests. Detection performance: P D versus SNR for a given P FA. Again this performance is maximized for the Neyman-Pearson test. BER: Bit-Error Rate. In communication, both hypotheses are equal, and the design is to get P D = P FA. A BER-plot shows P D = P FA versus SNR. Lecture, Lecture, P D ROC N=5 σ= MC= LRT 0. T(x=max(x Wild guess P FA Typical Matlab code for ROC curves: gamma=[-0.:0.:4]; MC=000; N=5; A=; sigma=; x=sigma*randn(n,mc; T=mean(x; T2=max(x; for k=:length(gamma PD(k=mean(T+A>gamma(k; PD2(k=mean(T2+A>gamma(k; PFA(k=mean(T>gamma(k; PFA2(k=mean(T2>gamma(k; end Simple line search to design γ and calculate P D given P FA : PFAdesign=0.; ind=find(pfa<pfadesign; gamma=gamma(ind(, PDdesign=PD(ind(, ind=find(pfa2<pfadesign; gamma2=gamma(ind(, PD2design=PD2(ind( interp better to use. Lecture, Lecture,

7 Maximum Likelihood Estimation Probability density function for the observation x parameterized by the unknown parameter θ: p(x,θ ˆθ ML = arg maxp(x observed,θ θ Example: If x = (x 0...x N, x i N(θ,σ 2, then ˆθ ML = N N i=0 x i Some Properties Asymptotic distribution: N(ˆθML θ 0 N(0, [I(θ 0 ] where the Fisher information matrix equals I(θ = E [ ] [ d d log p(x,θ dθ dθ Asymptotically the best possible estimator (reach the Cramér-Rao bound ] T log p(x,θ Lecture, Lecture, The GLRT for unknown A Some more common special case: H 0 :x p 0 (x θ 0 H :x p (x θ General principle: Plug in the maximum likelihood estimate of A under H, and let L(x = max θ p (x θ max θ0 p 0 (x θ 0 = p (x ˆθ ML p 0 (x ˆθ ML Generalized Likelihood Ratio Test (GLRT Decide H 0 Decide H if L(x < γ if L(x > γ 0 H 0 : θ = θ 0 H : θ θ 0 GLRT becomes L(x = max θ p(x θ p(x θ 0 General result for asymptotic distribution: 2 log L(x { χ 2 n under H 0 χ 2 n(λ under H. = λ = (θ θ 0 T I(θ 0 (θ θ 0 where I(θ is Fisher s information matrix p(x ˆθ ML p(x θ 0 Lecture, Lecture,

8 Consider Example: H 0 : x[n] = w[n] H : x[n] = A + w[n] with unknown A and Gaussian noise with known variance σ 2. Then 2 log L(x = NÂ2 σ 2 Â = N N n=0 ( N x[n] ( N { χ 2 2 log L(x under H 0 χ 2 ( NA2 under H σ 2. 0, σ2 N A, σ2 N under H 0 under H The MLRT for unknown A H 0 : x p 0 (x θ 0 H : x p (x θ General principle: Use a prior on θ and integrate out its influence in the LRT. This is the Bayesian approach. L(x = θ p (x θ p(θ dθ θ 0 p 0 (x θ 0 p(θ 0 dθ 0 This is the Marginalized Likelihood Ratio Test (MLRT. Lecture, Lecture, Test: Mixed M/G LRT approach Nuisance parameters: classical approaches H 0 :θ = θ 0,η H :θ θ 0,η Here η is an unknown nuisance parameter. General principle using the Bayesian approach: marginalize nuisance parameters, estimate the other parameters. The LRT becomes L(x = max θ p(x θ,ηdη η p(x θ = p(x ˆθ ML,ηdη η η 0,ηdη p(x θ η 0,ηdη Non-Bayesian approaches for the test H 0 : θ = θ 0,η H : θ θ 0,η GLRT ML estimate of nuisance parameters: L(x = p(x ˆθ ML, ˆη ML p(x θ 0, ˆη ML Lecture, Lecture,

9 General result for asymptotic distribution: { χ 2 2 log L(x dim(θ under H 0 χ 2 dim(θ (λ under H. λ = (θ θ 0 ( T I θ,θ I θ,η Iη,ηI η,θ (θ θ 0 }{{} [I ] θ,θ Use Wald test L(x = (ˆθ θ 0 T [I ( θ,η H ] θ,θ (ˆθ θ 0 where I(θ is Fisher s information matrix partitioned as I = ( I θ,θ I η,θ I θ,η I η,η where θ,η H is the ML estimate under H. Same performance as GLRT. evaluated at the true θ 0,η 0 (also θ denotes the true value under H. Note that the underbraced expression [I ] θ,θ is the Schur complement to compute the upper left corner of the matrix inverse. Lecture, Lecture, Locally Most Powerful (LMP test Use L(x = d log(p(x θ,η dθ Rao test where ˆη H 0 is the ML estimate under H 0. Same performance as GLRT. T [I d log(p(x θ,η (θ 0, ˆη H 0 ] θ,θ θ 0,ˆη H 0 dθ Note that θ does not have to be estimated here, as it has in the Wald test. θ0,ˆη H 0 The Locally Most Powerful (LMP test is a special case of the Rao test for the case of scalar parameter, no nuisance parameter and one-sided test: Take the test statistic L(x = dlog(p(x θ dθ I(θ0 θ0 H 0 :θ = θ 0 H :θ > θ 0 { N(0, under H0 N( I(θ 0 (θ θ 0, under H for small changes θ θ 0 > 0. The derivation is based on local analysis by Taylor expansion. Lecture, Lecture,

10 Bayesian extensions Priors: Assume priors p(h 0 and p(h, respectively. Bayesian decision rule, compare the a posteriori distribution of each hypothesis, p(h x H 0 H p(h 0 x p(x H p(h p(x p(x H p(x H 0 H 0 p(x H 0 p(h 0 H p(x H 0 p(h H p(h 0 = γ Multiple tests The use of prior enables multiple hypotheses tests: ˆk = arg max k p(h k x = arg maxp(x H k p(h k k which shows that Bayes rule provides a way to design the threshold in the LRT test. Lecture, Lecture, Define a cost to each decision Bayes Risk C ij = P(H i H j that is, C 0 is the false alarm cost and C 0, is the missed detection cost. Then p(x H p(x H 0 H 0 (C 0 C 00 p(h H (C 0 C p(h 0 = γ minimizes the Bayes risk or expected cost R = E(C = i=0 C ij P(H i H j p(h j j=0 Model: Y = Φ T θ + E where Cov(E = C and p(y θ = Linear regressions (2π N/2 det(c e 2 (Y ΦT θ T C (Y Φ T θ. Lecture, Lecture,

11 Test: Known parameter { H0 : θ = 0 H : θ = θ Neyman-Pearson gives LRT: ( p(y θ log P(Y θ = 0 = 0.5(Y Φ T θ T C (Y Φ T θ + 0.5Y T C Y = Y T C Φ T θ 0.5θ T ΦC Φ T θ The last term in independent of data, so take Constant in noise: L(Y = Y T C Φ T θ N(θ T ΦC Φ T θ,θ T ΦC Φ T θ, ( P D = Q Q (P FA θ T ΦC Φ T θ Correlation detector or Matched Filter Step Pre-whiten the data: Ȳ = C /2 Y, Φ = C /2 Φ Ȳ = Φ T θ + Ē Cov(Ē = I Step 2: L = Ȳ T M, where M = Φ T θ Correlation detector: L = N i= ȳ[i]m[i] Matched filter: Convolution l(t = t i=0 h[i]ȳ[t i] Impulse response h[i] = m[n i], (matched to the signal Test statistics: L = l(n Lecture, Lecture, Test ML estimate Unknown parameter H 0 : θ = 0 H : θ 0 ˆθ = (ΦΦ T ΦY GLRT ( p(y ˆθ L(x = 2 log P(Y θ = 0 = (Y Φ T ˆθ T C (Y Φ T ˆθ + Y T C Y = 2Y T C Φ T ˆθ ˆθT ΦC Φ T ˆθ =... = Y T C Φ T (ΦC Φ T ΦC Y χ 2 dim(θ See Chapter 6. in Gustafsson for more calculations on the linear model. Lecture, Lecture,

12 Exercises Check the lecture notes and report errors to me Groups of at most five persons I want one TEX version of problems + solution per group Kay:.,.2,.4, 2.2, 3., 3.8, 3.3 Lecture,