Pattern Recognition and Machine Learning

13.3. Linear Dynamical Systems 641 Figure 13.22 An illustration of a linear dy- namical system being used to track a moving obje ...

642 13. SEQUENTIAL DATA 13.3.2 Learning in LDS So far, we have considered the inference problem for linear dynamical systems, as ...

13.3. Linear Dynamical Systems 643 μnew 0 = E[z 1 ] (13.110) Vnew 0 = E[z 1 zT 1 ]−E[z 1 ]E[zT 1 ]. (13.111) Similarly, to optim ...

644 13. SEQUENTIAL DATA We have approached parameter learning in the linear dynamical system using maximum likelihood. Inclusion ...

13.3. Linear Dynamical Systems 645 13.3.4 Particle filters For dynamical systems which do not have a linear-Gaussian, for exampl ...

646 13. SEQUENTIAL DATA straightforward since, again using Bayes’ theorem p(zn+1|Xn)= ∫ p(zn+1|zn,Xn)p(zn|Xn)dzn = ∫ p(zn+1|zn)p ...

Exercises 647 p(zn|Xn) p(zn+1|Xn) p(xn+1|zn+1) p(zn+1|Xn+1) z Figure 13.23 Schematic illustration of the operation of the partic ...

648 13. SEQUENTIAL DATA 13.5 ( ) Verify the M-step equations (13.18) and (13.19) for the initial state probabili- ties and trans ...

Exercises 649 using modified forms of (13.18 ) and (13.19) given by πk = ∑R r=1 γ(z( 1 rk)) ∑R r=1 ∑K j=1 γ(z (r) 1 j) (13.124) ...

650 13. SEQUENTIAL DATA (13.68) where the quantitiesω(zn)are defined by (13.70). Show that the initial condition for this recurs ...

Exercises 651 a linear dynamical system governed by (13.75) and (13.76), with latent variables {z 1 ,...,zN}in whichCbecomes the ...

652 13. SEQUENTIAL DATA 13.34 ( ) Verify the results (13.115) and (13.116) for the M-step equations forCandΣ in the linear dynam ...

14 Combining Models In earlier chapters, we have explored a range of different models for solving classifi- cation and regressio ...

654 14. COMBINING MODELS model combination is to select one of the models to make the prediction, in which the choice of model i ...

14.2. Committees 655 In the case of our Gaussian mixture example, this leads to a distribution of the form p(x)= ∑K k=1 πkN(x|μk ...

656 14. COMBINING MODELS that when we trained multiple polynomials using the sinusoidal data, and then aver- aged the resulting ...

14.3. Boosting 657 Exercise 14.2 then we obtain ECOM= 1 M EAV. (14.14) This apparently dramatic result suggests that the average ...

658 14. COMBINING MODELS Figure 14.1 Schematic illustration of the boosting framework. Each base classifierym(x)is trained on a ...

14.3. Boosting 659 Make predictions using the final model, which is given by YM(x) = sign (M ∑ m=1 αmym(x) ) . (14.19) We see ...

660 14. COMBINING MODELS m=1 −1 0 1 2 −2 0 (^2) m=2 −1 0 1 2 −2 0 (^2) m=3 −1 0 1 2 −2 0 2 m=6 −1 0 1 2 −2 0 (^2) m=10 −1 0 1 2 ...