A variaional radial basis funcion approximaion for diffusion processes. Michail D. Vreas, Dan Cornford and Yuan Shen {vreasm, d.cornford, y.shen}@ason.ac.uk Ason Universiy, Birmingham, UK hp://www.ncrg.ason.ac.uk ESANN conference, -4 April 9, Brugge, Belgium Michail D. Vreas, e al. Radial basis funcion - VGPA /
Ouline Inroducion and moivaion Approximae inference in diffusion processes Radial basis funcion approximaion Resuls on sae and parameer esimaion Summary and fuure work Michail D. Vreas, e al. Radial basis funcion - VGPA /
Inroducion and moivaion The moivaion for his work is inference of he sae and (hyper-) parameers in models of real dynamical sysems (e.g. weaher predicion models, sysems biology, ec.). Our inference framework relies on variaional Bayesian principles. Thus far, he oucome of his work is a smoohing-like algorihm (hereafer VGPA ) for inference in diffusion processes. This presenaion shows an exension of his newly proposed algorihm, employing radial basis funcions. VGPA sands for Variaional Gaussian Process Approximaion. Michail D. Vreas, e al. Radial basis funcion - VGPA 3/
Approximae inference in diffusion processes Diffusion processes are a class of coninuous-ime sochasic processes, wih coninuous sample pahs []. Their applicaion is popular when modelling real-world sochasic (or random) dynamical sysems. The ime evoluion of a diffusion process can be described by a sochasic differenial equaion, henceforh SDE, (o be inerpreed in he Iō sense): dx() = f θ (, X())d + Σ / dw(), () where f θ (, X()) R D is he (usually non-linear) drif funcion, Σ = diag{σ,..., σ D } is he sysem noise covariance marix and dw() is a D dimensional Wiener process. Problem : Exac soluions o SDEs are rarely available. Soluion : Apply numerical approximaion mehods... Michail D. Vreas, e al. Radial basis funcion - VGPA 4/
Approximae inference in diffusion processes In our work, he key idea is o approximae he rue (laen) poserior process, p(x()) by anoher one ha belongs o a family of racable ones (e.g. Gaussian processes), q(x()). We do so by minimizing he KL[q p ] divergence [4], beween he approximaion process and he rue one. The Gaussian process assumpion implies a linear SDE: dx() = ( A()X() + b())d + Σ / dw() () where A() R D D and b() R D define he linear drif. N.B. These ime varying funcions, A() and b(), need o be opimized and we will deal wih hem shorly as hey are he main subjec of his presenaion... Michail D. Vreas, e al. Radial basis funcion - VGPA 5/
Approximae inference in diffusion processes The ime evoluion of his Gaussian process can be expressed by a se of ordinary differenial equaions: ṁ() = A()m() + b() Ṡ() = A()S() S()A() + Σ To enforce hese consrains he following Lagrangian is formulaed: f { L = E() r{ψ()(ṡ() + A()S() + S()A() Σ)} } λ() (ṁ() + A()m() b()) d (3) where E() R is he energy erm, λ() R D and Ψ() R D D are ime dependen Lagrange mulipliers. The minimisaion of his cos funcion Eq.(3) will lead o he opimal poserior process. VGPA deails Furher deails of his variaional algorihm can be found in [5, 7]. For an example see Appendix. Michail D. Vreas, e al. Radial basis funcion - VGPA 6/
Radial basis funcion approximaion - (he D case) The main idea is o approximae he variaional (conrol) parameers, A() and b(), by basis funcion expansions [, 3]. In he original VGPA, hese funcions are discreized wih a small ime sep (e.g. δ =.), resuling in a se of discree ime variables. N oal = (D + ) D f δ (4) }{{} T where D is he sysem dimension, and f are he iniial and final imes. Michail D. Vreas, e al. Radial basis funcion - VGPA 7/
Radial basis funcion approximaion - (he D case) Replacing he discreisaion wih RBFs leads o he following expressions: Ã() = L A i= a i φ i (), b() = b i π i () (5) L b i= where a i, b i R are he weighs, φ i (), π i () : R + R are fixed basis funcions and L A, L b N. In he absence of paricular knowledge abou he funcions we sugges he same number of Gaussian basis funcions. Hence we have L A = L b = L and φ i () = π i (), where: φ i () = e.5 ( ci λ i ) (6) c i and λ i are he i-h cenre and widh respecively and. is he Euclidean norm. Michail D. Vreas, e al. Radial basis funcion - VGPA 8/
Radial basis funcion approximaion - (he D case) Having precompued he basis funcion maps φ i () i {,,, L} and [ f ], as shown below, he opimisaion problem reduces o calculaing he weighs of he basis funcions wih L oal = L parameers. [ f ] f φ φ ( ) φ ( ) φ ( f ) φ φ ( ) φ ( ) φ ( f )....... φ L φ L ( ) φ L ( ) φ L ( f ) Table: Example of Φ() marix. Typically we expec ha L oal N oal, making he opimisaion problem smaller. Michail D. Vreas, e al. Radial basis funcion - VGPA 9/
Resuls on sae and parameer esimaion To es he sabiliy and he convergence properies of he new RBF approximaion algorihm, we consider a one dimensional double well sysem, wih drif funcion: f θ (, X ) = 4X (θ X ) θ >, (7) and consan diffusion coefficien Σ..5 8.5 U(x) 6 4 x X().5.5 3 4 5 6 7 8 9 Figure: Sample pah of a double well poenial sysem used in he experimens. The small circles indicae he noisy observaions. The inner plo (blue line) shows he poenial ha drives he sysem. Michail D. Vreas, e al. Radial basis funcion - VGPA /
Resuls on sae and parameer esimaion.5 X().5.5.5 3 4 5 6 7 8 9 Figure: HMC vs RBF variaional algorihm (on a single realisaion). 5 F(M θ, Σ, R) 9 8 7 6 5 KL 45 4 35 3 4 4 6 8 M (a) Free energy 5 4 6 8 M (b) KL divergence Figure: (a) Original VGPA vs RBF algorihm, a convergence (from realisaions). (b) Comparison of he KL(q,p), beween he rue (HMC) and original VGPA (dashed line, shaded area) and RBF (squares, dashed lines) poseriors. Boh plos are presened as funcions of RBF densiy. Michail D. Vreas, e al. Radial basis funcion - VGPA /
Resuls on sae and parameer esimaion 6 6 log(f(σ, Θ)) 5 4 3 log(f(σ, Θ)) 5 4 3.5.5.5.5.5 Θ es.5.5 Σ es.5 Θ es (a) M = Σ es (b) M = 4 Figure: Log(Energy) profiles in he parameer space (on a single realisaion, a convergence), for wo differen RBF densiies. Michail D. Vreas, e al. Radial basis funcion - VGPA /
Resuls on sae and parameer esimaion.4..95 Θ es.98.96.94.9 Σ es.9.85.8.75.9 4 6 8 M (a) θ - marginal esimaion.7 4 6 8 M (b) Σ - marginal esimaion 4 M = M = M = 3 VGPA 6 4 M = M = M = 3 VGPA F(θ Σ) 8 F(Σ θ) 8 6 6 4 4.5.6.7.8.9...3.4.5..3.4.5.6.7.8.9.. Θ es (c) θ - marginal profile Σ es (d) Σ - marginal profile Figure: (a) and (b) obained from one hundred realizaions, whereas (c) and (d) from a single (ypical) realizaion. Michail D. Vreas, e al. Radial basis funcion - VGPA 3/
Resuls on sae and parameer esimaion.6.6.4.4 Θ es..8.6.4...4.6.8..4.6 Σ es (a) M = Θ es..8.6.4...4.6.8..4.6 Σ es (b) M = 4 Figure: θ & Σ join esimaion. Michail D. Vreas, e al. Radial basis funcion - VGPA 4/
Summary & Fuure work Summary A new variaional radial basis funcion approximaion for inference for diffusion processes has been presened. Resuls show ha he new algorihm converges o he original VGPA wih a relaively small number of basis funcions per ime uni. Reparameerisaion of he original variaional framework allows us o conrol he complexiy of he algorihm. Main benefi when esimaing (hyper-) parameers. Fuure work The exension of his algorihm o mulivariae sysems is sill open. Anoher possible approximaion is by using localised polynomials beween observaions. The laer approach can reduce he dimension of he minimisaion problem furher. Exension o he mulivariae case is sraighforward. Michail D. Vreas, e al. Radial basis funcion - VGPA 5/
References Peer E. Kloeden and Eckhard Plaen. Numerical Soluions of Sochasic Differenial Equaions, Springer-Verlag, Berlin, 99. Chrisopher M. Bishop. Neural Neworks for Paern Recogniion, Oxford Universiy Press, New York, 995. D. S. Broomhead and D. Lowe. Mulivariae funcional inerpolaion and adapive neworks. Complex Sysems, :3-355, 988. S. Kullback and R. A. Leibler, On informaion and sufficiency. Annal of Mahemaical Saisics, :79-86, 95. C. Archambeau, D. Cornford, M. Opper and J. Shawe Taylor. Gaussian process approximaion of sochasic differenial equaions. Journal of Machine Learning Research, Workshop and Conference Proceedings, :-6, 7. T. Jaakkola. Tuorial on variaional approximaion mehods. In M. Opper and D. Saad, ediors, Advanced Mean Field Mehods: Theory and Pracise, The MIT press,. C. Archambeau, M. Opper, Y. Shen, D. Cornford, J. Shawe-Taylor. Variaional Inference for Diffusion Processes. In C. Pla, D. Koller, Y. Singer and S. Roweis ediors, Neural Informaion Processing Sysems (NIPS), pages 7-4, 8. The MIT Press. Acknowledgemens This research is funded as par of he Variaional Inference in Sochasic Dynamic Environmenal Models (VISDEM) projec by EPSRC gran (EP/C5848/). The auhors also acknowledge he help of Prof. Manfred Opper whose useful commens help o beer shape he heoreical par of he RBF approximaions. Michail D. Vreas, e al. Radial basis funcion - VGPA 6/
Appendix - VGPA algorihm in a nushell Make an iniial guess for he variaional (conrol) parameers A() and b(). Propagae forward in ime he momen equaions (forward ODEs) o ge consisen mean m() and covariance S(). 3 Compue he energy of he sysem from he SDE and he observaions (E SDE (), E OBS ()). 4 Propagae backward in ime he Lagrange mulipliers Ψ() and λ() (backward ODEs), o ensure he saisfacion of he momen equaions. When you mee an observaion jump. 5 Compue he gradiens of L, w.r.. A() and b() and apply a scaled conjugae gradien (SCG) minimizaion algorihm. 6 Ierae seps -5, unil saisfacory accuracy has been achieved. Michail D. Vreas, e al. Radial basis funcion - VGPA 7/
Appendix - VGPA example in picures /4.5 8 6.5 U(x) 4 x X().5.5 3 4 5 6 7 8 Figure: Typical double well rajecory (red crosses are noisy observaions). Michail D. Vreas, e al. Radial basis funcion - VGPA 8/
Appendix - VGPA example in picures /4.5.5 A() b().5 4 6 8.5 4 6 8 4 8 m() 4 4 6 8 5 Log(F) 7 6 5..4.6.8 n Ψ() λ() 5 4 6 8 4 6 8 Figure: Iniial ieraion. Michail D. Vreas, e al. Radial basis funcion - VGPA 9/
Appendix - VGPA example in picures 3/4 4 4 3.5 A() 3 b().5 4 4 6 8 4 6 8 7 m() 4 6 8 Log(F) 6 5 4 3 3 4 n 5 5 Ψ() 5 λ() 4 6 8 5 4 6 8 Figure: Inermediae ieraion. Michail D. Vreas, e al. Radial basis funcion - VGPA /
Appendix - VGPA example in picures 4/4 A() 5 5 b() 4 6 8 4 6 8 6 m() 4 6 8 Log(F) 5 4 3 3 4 5 6 n 5 Ψ() 5 λ() 4 6 8 4 4 6 8 Figure: Final resuls (a convergence). Michail D. Vreas, e al. Radial basis funcion - VGPA /