{"title": "Recursive Estimation of Dynamic Modular RBF Networks", "book": "Advances in Neural Information Processing Systems", "page_first": 239, "page_last": 245, "abstract": null, "full_text": "Recursive Estimation of Dynamic \n\nModular  RBF  Networks \n\nVisakan Kadirkamanathan \n\nAutomatic Control &  Systems Eng.  Dept. \n\nUniversity of Sheffield,  Sheffield  Sl 4DU, UK \n\nvisakan@acse.sheffield.ac. uk \n\nMaha Kadirkamanathan \n\nDragon Systems  UK \n\nCheltenham GL52  4RW, UK \n\nmaha@dragon.co.uk \n\nAbstract \n\nIn this paper, recursive estimation algorithms for dynamic modular \nnetworks  are  developed.  The  models  are  based  on Gaussian  RBF \nnetworks  and  the  gating network  is  considered  in two  stages:  At \nfirst,  it  is  simply  a  time-varying  scalar  and  in  the  second,  it  is \nbased  on the state, as in the mixture of local experts scheme.  The \nresulting  algorithm  uses  Kalman  filter  estimation  for  the  model \nestimation and the gating probability estimation.  Both, 'hard' and \n'soft' competition based estimation schemes are developed where in \nthe former, the most probable network is  adapted and in the latter \nall networks  are  adapted by  appropriate weighting of the data. \n\n1 \n\nINTRODUCTION \n\nThe problem of learning multiple modes in  a  complex nonlinear system is  increas(cid:173)\ningly  being  studied  by  various  researchers  [2,  3,  4,  5,  6],  The  use  of a  mixture  of \nlocal experts  [5,  6],  and a  conditional mixture density network  [3]  have  been devel(cid:173)\noped  to model  various  modes  of a  system.  The  development  has  mainly been  on \nmodel estimation from  a given set  of block  data,  with  the  model likelihood  depen(cid:173)\ndent  on the input  to the networks.  A recursive  algorithm for  this static  case  is the \napproximate iterative procedure  based  on the block estimation schemes  [6]. \n\nIn this  paper,  we  consider  dynamic systems  - developing  a  recursive  algorithm is \ndifficult  since  mode  transitions  have  to  be  detected  on-line  whereas  in  the  block \nscheme,  search  procedures  allow  optimal detection.  Block  estimation schemes  for \ngeneral architectures have been described in [2,  4].  However, unlike in those schemes, \nthe  algorithm developed  here  uses  relationships  based  on  Bayes  law  and  Kalman \nfilters  and  attempts  to  describe  the  dynamic  system  explicitly,  The  modelling  is \ncarried out by radial basis function  (RBF) networks for  their property that by pre(cid:173)\nselecting the centres and widths, the problem can be reduced  to a linear estimation. \n\n\f240 \n\nV.  KADIRKAMANATHAN.  M.  KADIRKAMANATHAN \n\n2  DYNAMIC  MODULAR RBF NETWORK \n\nThe  dynamic  modular  RBF  network  consists  of a  number of models  (or  experts) \nto  represent  each  nonlinear  mode in  a  dynamical system.  The  models  are  based \non  the  RBF  networks  with  Gaussian function,  where  the  RBF  centre  and  width \nparameters  are  chosen  a  priori  and  the  unknown  parameters  are  only  the  linear \ncoefficients  w.  The functional form of the  RBF network  can be expressed  as, \n\nK \n\nf(XiP) = L wkgk(X) = w T g \n\n(1) \n\nk=l \n\nis \n\nthe  linear  weight  vector  and  g \n\n[ . . . , Wk, .. Y  E \n\nwhere  w  = \n[ ... , gk(X), .. . ]T  E lR~ are the  radial basis functions,  where, \ngk(X) = exp {-O.5r-21Ix - mk112} \n\nlRK \n\n(2) \nmk  E  lRM  are  the  RBF  centres  or  means  and  r  the  width.  The  RBF  networks \nare  used  for  their  property that having  chosen  appropriate  RBF  centre  and  width \nparameters  mk,  r,  only  the  linear  weights  w  need  to be  estimated  for  which  fast, \nefficient  and optimal algorithms exist. \n\nEach  model  has  an  associated  probability  score  of being  the  current  underlying \nmodel for  the  given  observation.  In  the first  stage  of the  development,  this  prob(cid:173)\nability  is  not  determined  from  parametrised  gating  network  as  in  the  mixture  of \nlocal experts  [5]  and the mixture density network [3],  but is determined on-line as it \nvaries with time.  In dynamic systems, time information must be taken into account \nwhereas  the  mixture  of local  experts  use  only  the  state  information which  is  not \nsufficient  in  general,  unless  the  states  contain  the  necessary  information.  In  the \nsecond stage,  the probability is extended to represent  both the time and state infor(cid:173)\nmation explicitly using the expressions from the mixture of local experts.  Recently, \ntime and state information have  been  combined in developing  models for  dynamic \nsystems  such  as  the mixture of controllers  [4]  and  the  Input  - Output  HMM  [2]. \nHowever,  the  scheme  developed  here  is  more  explicit  and  is  not  as  general  as  the \nabove schemes  and is  recursive  as opposed to block estimation. \n\n3  RECURSIVE ESTIMATION \n\nThe problem of recursive  estimation with  RBF  networks  have  been studied  previ(cid:173)\nously [7,  8]  and the algorithms developed here is a continuation of that process.  Let \nthe set of input - output observations from which the model is  to be estimated be, \n(3) \n\n2 N  = {zn  1 n  = 1, ... , N} \n\nwhere,  2N  includes  all observations upto the  Nth data and Zn  is the nth data, \n\n( 4 ) \nThe underlying system generating the observations  are  assumed to be multi-modal \n(with known  H  modes), with each observation satisfying the  nonlinear relation, \n\nZn  = {( X n , Yn)  1 Xn  E lRM , Yn  E lR} \n\nY =  fh(X) + 1] \n\n(5) \nwhere 1]  is the noise with unknown distribution and fh (.)  : lRM  1-+  lR  is the unknown \nunderlying  nonlinear function  for  the  hth mode  which  generated  the  observation. \nUnder  assumptions of zero  mean Gaussian noise  and that the  model  can  approxi(cid:173)\nmate the underlying function  arbitrarily closely,  the probability distribution, \nh)12} \n(  I  h \nP Zn  W  ,M  = Mh, 2 n - 1  =  271\") \n\n_1  -t  {1  -11 \n\n-\"2Ro  Yn  -!h Xn; W \n\n2  Ro  exp \n\n( \n\n(6) \n\nn \n\n) \n\n( \n\n\fRecursive Estimation of Dynamic Modular RBF Networks \n\n241 \n\nis  Gaussian.  This  is  the  likelihood of the observation Zn  for  the model Mh,  which \nin  our  case  is  the  GRBF  network,  given  model  parameters  wand that  the  nth \nobservation  was  generated  by  Mh.  Ro  is  the  variance  of the  noise  TJ .  In  general \nhowever,  the model generating  the  nth observation is  unknown  and  the  likelihood \nof the nth observation is  expanded  to include I~ the  indicator variable,  as  in [6], \n\np(zn\"nIW,M,Zn-l) = IT [p(znlw\\Mn =  Mh,Zn_dp(Mn =  Mhlxn,zn-l)r\" \n\nk \n\nH \n\nh=l \n\n(7) \n\nBayes law can be  applied to the on-line or recursive  parameter estimation, \n\np(WIZn,M) = p(znIW,M, Zn-dp(WIZn-l,M) \n\nP(ZnIZn-l,M) \n\n(8) \nand  the  above  equation  is  applied  recursively  for  n  =  1, ... , N .  The  term \np(zn IZn-l, M)  is  the  evidence.  If  the  underlying  system  is  unimodal,  this  will \nresult  in the  optimal Kalman estimator and if we  assign  the  prior probability dis(cid:173)\ntribution for  the model parameters  p(wh IMhk to  be  Gaussian  with mean  Wo  and \ncovariance  matrix  (positive  definite)  Po  E  1R  xK,  which  combines  the  likelihood \nand the prior to give  the  posterior probability distribution which at time n  is given \nby p(whlZn, Mh)  which  is  also Gaussian, \n\np(whIZn,Mh) = (27r)-4Ip~l-t exp { _~(wh - W~fp~-l (wh - w~)}  (9) \n\n1 \n\n1  {1 \n\nIn  the  multimodal case  also,  the  estimation for  the  individual  model  parameters \ndecouple naturally with the only modification being that the likelihood used for the \nparameter estimation is  now  based on weighted  data and given by, \n\n' \n\nh\n\nh- 1 \n\n1  h I \n\np(znlw  ,Mh,Zn-l)=(27r)-~(Roln  )-~exp  -'2 Ro  In  Yn-ih(Xn;W) \n\nh 12} \n(10) \nThe  Bayes  law  relation  (8)  applies  to  each  model.  Hence,  the  only  modification \nin  the  Kalman filter  algorithm is  that  the  noise  variance  for  each  model  is  set  to \nRoh~ and  the  resulting  equations  can  be  found  in  [7].  It increases  the  apparent \nuncertainty in the measurement output  according  to how  likely the model is  to be \nthe true underlying mode, by increasing the noise variance term of the Kalman filter \nalgorithm.  Note that the term p(Mn = Mhlxn, zn-l) is  a time-varying scalar and \ndoes  not influence  the parameter estimation process. \n\nThe evidence  term can  also  be  determined directly from the Kalman filter, \n\nwhere  the  e~ is  the  prediction error and  R~ is  the innovation variance with, \n\neh \nn \nRh n \n\nhT \n\nYn  - wn-1gn \nROln  + gnP n-lgn \nT \n\nh- 1 \n\nh \n\n(11) \n\n(12) \n\n(13) \n\nThis  is  also the  likelihood of the  nth observation given  the  model M  and  the past \nobservations  Zn-l.  The  above  equation  shows  that  the  evidence  term  used  in \nBayesian model selection  [9]  is  computed recursively,  but for  the specific  priors  Ro, \nPo.  On-line Bayesian model selection can be  carried out by choosing many different \npriors,  effectively  sampling the prior space,  to  determine  the  best  model to fit  the \ngiven data, as  discussed  in  [7]. \n\n\f242 \n\nV.  KADIRKAMANATHAN.  M.  KADIRKAMANATHAN \n\n4  RECURSIVE  MODEL SELECTION \n\nBayes  law  can  be  invoked  to  perform recursive  or on-line model selection  and  this \nhas  been  used  in  the  derivation of the  multiple  model  algorithm  [1] .  The  multiple \nmodel algorithm has been used for  the recursive  identification of dynamical nonlin(cid:173)\near systems  [7].  Applying Bayes  law gives  the following relation: \n\n(14) \n\nwhich  can  be computed recursively  for  n  = 1, ... , N.  p(ZnIMh, Zn-1)  is  the likeli(cid:173)\nhood given in (11) and p(MhIZn) is the posterior probability of model Mh being the \nunderlying model for the nth data given the observations Zn\u00b7  The term p(Zn IZn-1) \nis  the normalising term given by, \n\nH \n\nP(ZnI Zn-1) =  Lp(znIMh,Zn-1)p(MhI Zn-1) \n\nh=l \n\n(15) \n\nThe  initial  prior  probabilities  for  models  are  assigned  to  be  equal  to  1/ H.  The \nequations (11),  (14)  combined with the Kalman filter estimation equations is known \nas  the multiple model algorithm [1] . \n\nAmongst  all  the  networks  that  are  attempting  to identify  the  underlying  system, \nthe identified model is  the one with the highest  posterior probability p(MhIZn)  at \neach time n,  ie., \n\n(16) \n\nand hence  can vary from time to time.  This is preferred over the averaging of all the \nH  models  as  the  likelihood is  multimodal and  hence  modal estimates  are sought. \nPredictions  are based on this most probable model. \n\nSince the system is  dynamical, if the underlying model for  the dynamics is  known, \nit can be used to predict the estimates at the next time instant based on the current \nestimates, prior to observing the next  data.  Here,  a first  order Markov assumption \nis  made for  the  mode  transitions.  Given  that  at  the  time instant  n  - 1  the  given \nmode is  j, it is  predicted  that the  probability of the mode  at time instant n  being \nh  is  the  transition  probability  Phj .  With  H  modes,  2: Phj  =  1.  The  predicted \nprobability of the mode being  h at time n  therefore  is  given by, \n\nH \n\nPnln-l(MhI Zn-1) =  L  Phjp(Mj IZn-1) \n\nj=l \n\n(17) \n\nThis  can  be  viewed  as  the  prediction stage of the model selection  algorithm.  The \npredicted  output of the system is  obtained from  the output of the model that has \nthe highest  predicted  probability. \nGiven  the  observation  Zn,  the  correction  is  achieved  through  the  multiple  model \nalgorithm of (14)  with the following modification: \n\np(MhIZn) = p(znIMh, Zn-1)Pnln-1(MhI Zn-1) \n\np(znIZn-d \n\n(18) \n\nwhere  modification  to  the  prior  has  been  made.  Note  that  this  probability  is  a \ntime-varying scalar value and does  not depend on the states. \n\n\fRecursive Estimation of Dynamic Modular RBF Networks \n\n243 \n\n5  HARD  AND  SOFT  COMPETITION \n\nThe  development  of the  estimation  and  model  selection  algorithms  have  thus  far \nassumed  that  the  indicator  variable  'Y~  is  known.  The  'Y~  is  unknown  and  an \nexpected  value must be used  in the  algorithm, which  is  given by, \n\n(3h  _  p(znlMn = Mh, Zn-I)Pnln_1(Mn = MhIZn-I) \n\nn  -\n\nP(ZnI Zn-1) \n\n(19) \n\nTwo possible methodologies can be  used for  choosing the values for  'Y~.  In the first \nscheme, \n\n'Y~ =  1  if,B~ > ,B~ for  all  j  1=  h, \n\n(20) \nThis results in  'hard'  competition where,  only the model with the highest  predicted \nprobability undergoes  adaptation using  the Kalman filter  algorithm while  all other \nmodels are prevented from adapting.  Alternatively, the expected  value can be used \nin the  algorithm, \n\nand  0  otherwise \n\n(21) \nwhich results in  'soft' competition and all models are allowed to undergo adaptation \nwith  appropriate  data weighting  as  outlined  in  section  3.  This  scheme  is  slightly \ndifferent  from  that presented  in [7].  Since  the  posterior  probabilities of each mode \neffectively  indicate  which  mode is  dominant  at each  time n,  changes  can  then  be \nused  as  means of detecting mode transitions. \n\n6  EXPERIMENTAL RESULTS \n\nThe problem chosen for the experiment is learning the inverse robot kinematics used \nin  [3].  This  is  a  two  link  rigid  arm manipulator for  which,  given joint  arm angles \n(0 1 , O2 ),  the end effector  position in cartesian  co-ordinates is  given by, \n\nL1  COS(Ol)  - L2 COS(Ol  + O2) \n:l:1 \n:l:2  =  L1 sin(Ol) - L2 sin(Ol + O2) \n\n(22) \n\nL1  =  0.8,  L2  =  0.2  being the arm lengths.  The inverse kinematics learning problem \nrequires  the  identification  of  the  underlying  mapping from  (:l:1' :l:2) \n(0 1 , O2 ), \nwhich is bi-modal.  Since the algorithm is developed for the identification of dynam(cid:173)\nical systems, the data are generated with the joint angles being excited  sinusoidally \nwith differing frequencies  within the intervals  [0.3,1.2] x  ['71\"/2,371\"/2].  The first  1000 \nobservations are used for  training and the next 1000 observations are used for  test(cid:173)\ning  with  the  adaptation  turned  off.  The  models  use  28  RBFs  chosen  with  fixed \nparameters,  the  centres  being uniformly placed on  a  7 x  4 grid. \n\n-\n\nc>  g \n\n,  .-'~',\"--'-\"-':\"\",  (,  :',-1 \n:  r .::: \n-\n~: I \nl : :  ~: \n0 __  ::  \".. \n0 _\"7  I ! :  \n' \"  \n\n. '   'I  ~~  .,  ! ' ..  ~\\  :.\"  .,  :\" \n: .... ::  I \nt : \n\n~,  . . .  : \n\n: : : \" \n\n::,  .. : \n\n' :  \n\nI. \n\nII \n\nI \n\n\u2022  ;:\"';'d:',~r\":rbv:~b\u00b7::\":~~:-:-:-~W-,~:~,g ::-~~.  \" :'  \" \" '\\'1  \"  ':  '\\  . \n\n\" \n\n, I   :' \n\n:l ,'  .\" \n:,  I :,1  '~,  I .   i ~  I \n'1: ::': \n,. I \n! \n\n'I \nI \n\n~ \n\nI \n\nt \n\n,.'  ~ . ' \n\nI \n\nI \n\n' I   \" \n\n: .   I ,   I~ \n' :   : :   i  I  : :   : :   : : : :   I ,  \nI  : \nI :: \n~;~ I \n~: \n, \n~ \n\n. :,,: I \n::!! \ny. ~ \n\n,'I \n\n'0 \n\n\" \n\n\" \n\nI \n\nI \n\nI \n: \n\n'\n\nI \n\" \n:  ,\",:  : \n\n.::: I \n:: : \n\"  ~ \n\n\" \n\n, \n\n. ,   \" \n\n:  ::~:: \n. :: :: I \n~ :: \ni \n~  ~ \n\n0\n\nI \n. \n\n_ e \n\n1 0 . &\n0__ \n::=~I  I  I : ~ \n\nt.U\\JJW~ :,'  :'llJr: , .l/v  :. \n\n, ,1.111.;,  l,!111. ::,  tU \\\",:  ,i \n\\ _ 'J \n\n300  \n\n_00  \n\n:zoo \n\n..,  co \n\n\u00b0 0  \n\n\u2022 \n\n, \n\n: .~ \n\n::'  L ~ \n\n: : : : '  \n\n: : ;  \n\n:,:,  ,:, \n\nI  \u2022 \u2022  t \n\n. .  00  \n\"TI......,_ \n\n- \\  \n\n! \n\" \n: \n~: \n,,,, \n\" \n' \n\nI.  I '  . \" \n\n:  VI f;'  :: \n,,' \n\nI '   .\n.,  I' \n\nI... \n: \n. : .  \nI.  I. \n\n' \n\nFigure  1:  Learning inverse  kinematics  (,hard'  competition):  Model probabilities. \n\nFigure  1  shows  the  model  probabilities  during  training  and  shows  the  switching \ntaking place between  the  two modes. \n\n\f244 \n\n0.9 \n\n0.8 \n\n0.7 \n\n0.6 \n\n\\l 0.5 \n\n0.4 \n\n0.3 \n\n02 \n\n0. ' \n\nV.  KADIRKAMANATHAN.  M.  KADIRKAMANATHAN \n\nModtl1  Teat Ca.  errora in the EncI.n.ctcr Poei6oo S~ \n\nModtl 2 Teat Data enoN In ttw End .n.ckH' PoeItion Space \n\n0.9 \n\n0.8 \n\n0.7 \n\n0.6 \n\n\\l OS \n\n0.4 \n\n0.3 \n\n02 \n\n0.' \n\n~~~0.'--~02~0~.3~0~.' ~0.~5~0.6~~OJ~0~.8~0~.9~ \n\n~~~0.~'~02~0~.3~0~.4~0.~5~0.6~~0.7~0~.8~0~.9 ~ \n\nFigure  2:  End effector  position errors  (test  data)  ('hard' competition) :  (a)  Model \n1 prediction (b)  Model  2 prediction. \n\n\" \n\n\" \n\nFigure 2 show the end effector position errors on the test data by both models 1 and \n2 separately  under  the  'hard' competition scheme.  The figure  indicates  the  errors \nachieved  by  the best  model used  in the  prediction - both models predicting in the \ncentre  of the  input  space  where  the  function  is  multi-modal.  This  demonstrates \nthe successful  operation of the algorithm in the two RBF networks  capturing some \nelements of the two underlying modes of the  relationship.  The best  results  on this \nlearning task are: The RMSE on test  data for  this problem by the Mixture Density \n\nTable 1:  Learning Inverse  Kinematics:  Results \n\nHard Competition  Soft  Competition \n\nRMSE  (Train) \nRMSE  (Test) \n\n0.0213 \n0.0084 \n\n0.0442 \n0.0212 \n\nNetwork  is  0.0053  and  by  a  single  network  is  0.0578  [3].  Note  however  that  the \nalgorithm here  did not  use state information and used  only  the time dependency. \n\n7  PARAMETRISED  GATING  NETWORKS \n\nThe model parameters were  determined explicitly based on the  time information in \nthe dynamical system . If the gating model probabilities are expressed  as  a function \nof the states,  similar to [6], \n\np(Mhlxn, Zn-l) = exp{ahT g} /  L exp{ahT g} = a~ \n\nH \n\n(23) \n\nh=l \n\nwhere  a h  are the gating network parameters.  Note  that the gating network  shares \nthe same basis functions  as  the expert  models. \nThis extension to the gating networks  does  not  affect  the model parameter estima(cid:173)\ntion procedure .  The likelihood in  (7)  decomposes  into a  part for  model parameter \nestimation involving output  prediction error  and  a  part for  gating parameter esti(cid:173)\nmation involving the  indicator variable Tn .  The second  part can  be  approximated \nto a  Gaussian of the form, \n\nP(Tnlxn,a  ,Zn-d ~ (21r)-~RgO ~  exp  -\"2 Rgo \n\nh \n\n1  h-l.  {1  h- 1  h \n\nh 2} \nI'Yn  - ani \n\n(24) \n\n\fRecursive Estimation of Dynamic Modular RBF Networks \n\n245 \n\nThis  approximation  allows  the  extended  Kalman  filter  algorithm  to  be  used  for \ngating network  parameter estimation.  The  model  selection  equations  of section  4 \ncan  be  applied  without  any  modification with  the  new  gating  probabilities.  The \nchoice  of the  indicator variable 'Y~  can  be  based as  before,  resulting  in either hard \nor  soft  competition.  The  necessary  expressions  in  (21)  are  obtained  through  the \nKalman filter  estimates  and  the  evidence  values,  for  both  the  model  and  gating \nparameters.  Note  that  this is  different  from  the estimates used  in  [6]  in the sense \nthat, marginalisation over  the model and gating parameters have  been  done  here. \n\n8  CONCLUSIONS \n\nRecursive estimation algorithms for  dynamic modular RBF networks  have been  de(cid:173)\nveloped .  The models are based on Gaussian RBF networks and the gating is simply \na  time-varying  scalar.  The  resulting  algorithm uses  Kalman filter  estimation for \nthe model parameters and the multiple model algorithm for  the gating probability. \nBoth,  (hard' and  (soft' competition based  estimation schemes  are  developed  where \nin the former,  the most probable network is  adapted and in the latter all networks \nare  adapted  by  appropriate weighting of the  data.  Experimental results  are  given \nthat demonstrate the capture of the switching in the dynamical system by the mod(cid:173)\nular RBF networks.  Extending the method to include the gating probability to be \na  function  of the  state  are  then  outlined  briefly.  Work  is  currently  in  progress  to \nexperimentally demonstrate the  operation of this extension. \n\nReferences \n\n[1]  Bar-Shalom, Y. and Fortmann, T. E.  Tracking  and  data  association,  Academic \n\nPress,  New  York,  1988. \n\n[2]  Bengio,  Y.  and  Frasconi,  P. \n\n\"An  input  output  HMM  architecture\", \n\nIn \n\nG .  Tesauro,  D.  S.  Touretzky  and  T .  K.  Leen  (eds.)  Advances  in  Neural  In(cid:173)\nformation  Processing  Systems  7,  Morgan Kaufmann, CA:  San Mateo,  1995. \n\n[3]  Bishop,  C.  M.  \"Mixture  density  networks\",  Report  NCRG/4288,  Computer \n\nScience  Dept., Aston  University,  UK,  1994. \n\n[4]  Cacciatore,  C.  W.  and Nowlan,  S.  J.  \"Mixtures of controllers  for jump linear \n\nand  nonlinear  plants\",  In J.  Cowan,  G.  Tesauro,  and  J.  Alspector  (eds.)  Ad(cid:173)\nvances  in  Neural  Information  Processing  Systems  6,  Morgan  Kaufmann, CA: \nSan Mateo,  1994. \n\n[5]  Jacobs,  R.  A.,  Jordan,  M.  I.,  Nowlan,  S.  J .  and  Hinton,  G.  E.  \"Adaptive \n\nmixtures of local experts\",  Neural  Computation,  9:  79-87,  1991. \n\n[6]  Jordan,  M. I.  and Jacobs,  R.  A.  \"Hierarchical mixtures of experts and the EM \n\nalgorithm\" ,  Neural  Computation,  6:  181-214,  1994. \n\n[7]  Kadirkamanathan, V.  \"Recursive nonlinear identification using multiple model \nalgorithm\",  In  Proceedings  of the  IEEE  Workshop  on  Neural  Networks  for \nSignal  Processing  V,  171-180,  1995. \n\n[8]  Kadirkamanathan, V. \n\n((A  statistical inference  based  growth  criterion for  the \nRBF network\",  In  Proceedings  of the IEEE  Workshop  on  Neural Networks for \nSignal  Processing  IV,  12-21,  1994. \n\n[9]  MacKay,  D.  J.  C.  \"Bayesian interpolation\",  Neural  Computation,  4:  415-447, \n\n1992. \n\n\f", "award": [], "sourceid": 1122, "authors": [{"given_name": "Visakan", "family_name": "Kadirkamanathan", "institution": null}, {"given_name": "Maha", "family_name": "Kadirkamanathan", "institution": null}]}