The algorithm of square root Kalman filtering for the case of contaminated observations is described in the paper. This algorithm is suitable for the parallel computer implementation allowing to treat dynamic linear systems with large number of state variables in a robust recursive way.Square root Kalman filter; Robust; Parallel algorithm;

Cipra, T.

Romera, Rosario

Rubio, A.

English

Research Papers in Economics

Working Paper 92-09 
March 1992 
Divisi6n de Economia 
Universidad Carlos III de Madrid 
Calle Madrid,  126 
28903 Getafe (Madrid) 
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SQUARE ROOT KALMAN FILTER WITH CONTAl\1INATED 
OBSERVATIONS 
T.  Cipra· 
R.  Romera·· 
A.  Rubio··· 
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Abstract"-­ _ 
The  algorithm  of  square  root  Kalman  filtering  for  the  case  of  contaminated 
observations is  described  in  the paper. This algorithm is  suitable for the parallel computer 
implementation allowing to treat dynamic linear systems with large number of state variables 
in a robust recursive way. 
c 
Key  words:  Square root Kalman  filter,  robust, parallel algorithm. 
( 
( 
.. 
... 
Statistics Department, Charles University of Prague. 
Artificial Intelligence Department, Facultad de Informatica,  Universidad Politecnica 
de Madrid; and Statistics and  Econometrics Department,  Universidad Carlos  III de 
Madrid. 
Mathematics Department, Universidad de Extremadura. ie 
I 
SQUARE ROOT KALMAN FILTER WITH CONTAMINATED
 
c  OBSERVATIONS
 
T. Cipra, R. Romera, A.  Rubio. 
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. C 
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Abstract:  The algorithm of square root Kalman filtering for the case of contaminated I 
observations is  described in  the paper.  This algorithm is suitable for the parallel computer 
implementation allowing to treat dynamic linear systems with large number of state variables 
in a robust recursive way.  c 
Key  words:  Square root Kalman filter,  robust,  parallel algorithm. 
Classification:  62M20,  60G35, 93Ell. 
1. Introduction.  This paper attempts to treat simultaneously two problems connected 
with practical implementation of the Kalman  filtering: 
(1) If  the number n of state variables (the dimension of state vector) is large then the 
Kalman  filter  procedure  is  expensive  requiring  0(n3)  operations  for  each  state  update. 
Applications with enormous number of state variables appear e.g. in aerodynamics including 
aircraft testing, medicine, robotics, seismology. Some applications in the framework of state­
le  space modelling of time series with  large numbers of state variables are given e.g. in  [3]. 
I  However, the complexity of Kalman filter can be reduced by the parallel implementation on 
parallel  computers  that  are  used  in  practice  with  increasing  popularity  (see  e.g.  [1]). 
Nowadays  there  are  numerous  parallel  Kalman  filter  algorithms  suggested  for  various 
practical situations.  One of possible approaches to this problem consists in the square root 
formulation of Kalman  filter allowing to  reduce the costs to O(n)  operations for each state 
update if the algorithm is implemented on a parallel machine (see e.g. [5], [7]). 
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j (2)  In  practice  the  Kalman  filter  must  frequently  face  to  various  forms  of 
contaminated data. The occurrence of outliers and non-Gaussian distributions in the dynamic 
linear systems treated in practice has motivated a number of robust versions of Kalman filter 
(there are even suggestions concerning the robust Kalman filtering for nonlinear systems, see 
c'	  s.g. [2]). For instance, the approach to the Kalman filter robustification by [4] based on the 
M-estimation principle seems to provide good practical results. 
With respect to  the mentioned problems (1) and  (2),  this paper shows that it is not 
difficult to  rewrite the robust Kalman filtering from [4]  to the square root form.  The robust 
Kalman filter [4]  for models with contaminated observations is briefly reminded in Section  c·  2,  its  square  root  form  is  described  in  Section  3  and  the  special  case  with  scalar 
(contaminated) observations is considered in Section 4. 
2.  Robust  Kalman  filter.  Let  us  consider  a  dynamic  linear  system  with  contaminated 
observations of the form 
(2.1) 
Ye  = Ht?'e+vel  ve-iid e-contamina ted  N( 0 I  Re)  I  (2.2) 
where the residuals {wt}  and {Vt} are mutually independent. Moreover, some initial conditions 
are required  to  be fulfilled.  The  state equation  (2.1) describes  the development of an  n­
dimensional state vector X t in time while the observations equation (2.2) assigns the state Xt 
to  an  m-dimensional  observation  vector  Yt.  The  matrices  FI!  Ht,  QI!  Rt of appropriate 
dimensions are supposed to be known at time 1.  The e-contamination of the residual V t means 
that its  normal  distribution  N(O,RJ  with  acceptable  variances  is  contaminated  by  a  small c 
fraction e (e.g. e= 0.05) of a symmetric distribution with heavy tails which enables to model 
outliers in  observed data.  In  the  standard  normal case without contamination the Kalman 
filter provides recursive formulas for the minimum variance state estimator  ~tt  =  E(xtIyt) 
and its covariance matrix P\ =  E [(Xt  - ~t~  (Xt - ~tt)'  /  yt]  using all available information 
yt = {  Yo,  yl' .....  'yt} at time t. 
In the contaminated case (2.1),  (2.2), the work [4]  replaces  these formulas  by the 
approximative ones of the form 
(2.3) 
(2.4) 
where the predictive values ~\+l  =  E(xt+l /Yl) and Ptt+l  =  E[(xt+l - ~\+I)(Xt+1  - ~\+l)' /Y1 for 
time t+1 at time t are constructed as 
c 
2 
c 
------------_.----------­(2.5) 
(2.6) 
The symbol Rt 
l/2 denotes the square root matrix of Rt 
mxm diagonal matrix with 
w·  =  ]t 
and Wt  =  diag {wlt, ....  ,Wmt}  is  the 
(2.7) 
where 'lrl , .....  , 'Irm are suitable robustifying psi-functions and 
b lt  Slt 
B t  =  =  R-1/ 2H 
t  t'  St =  =  R~1/2Yt  (2.8) 
bmt  Smt 
c 
(R;112  is the inverse matrix of ~1/2 and b/s are the 1 x n rows of the matrix BJ· 
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The formulas (2.3) - (2.6) can be easily rewritten to the predictive form 
~;+l  = F0g-1+Kt(yt-Ht X~-l)  ,  (2.9) 
(2.10) 
c. 
where Ret is the innovation covariance matrix of the form 
R  = H  pt-1H  .. +R1/ 2 W-1R1/ 2 
et  t  t  t  t  t  t 
(2.11) 
and ~ is the Kalman gain matrix of the form 
(2.12) 
Specially, if m =1 so that the observations  Yt  are scalar in  the observation equation 
and, in  addition, if we use the Huber's psi-function 'lrH of the form 
3 ( 
z  ,  Izl~c, 
(2.13) 
c  sgn(z)  ,  Izl>c 
c 
(it  can  be  shown  that  this  choice  of psi-function  in  the  case  of e-contaminated  normal 
distribution  provides  robust  estimates  that  are  optimal  in  the  min-max  sense)  then  the 
approximative formula (2.9) can be replaced by the non-approximative one of the form 
(2.14) 
where 
(2.15) 
(2.16) ( 
The covariance matrix P\+ I  is let in  the form 
(2.17) 
3.  Square root robust Kalman filter.  The square root formulation of the classical Kalman 
filter (see e.g.  [5],  [6],  [7]) takes advantage of the matrix  factorization  that can be written 
C  for a positive semidefinite matrix X as 
x = LDL",	  (3.1) 
where L is  a lower triangular matrix  with  units  on its  main  diagonal and  D is  a diagonal 
matrix.  c 
In the case of the robust Kalman filter from Section 2, its square root formulation will 
maintain the matrices Pt
l  and  ReI in  the factorized form  (3.1), i.e. 
(3.2) 
C	  where Lpll  Lel are lower triangular matrices with units on the main diagonal and D pl,  Del are 
diagonal matrices. 
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~~---~-----~  ~-----------------------c 
The input for the corresponding square root robust algorithm at time t involves the 
matrices Fu Hu Ru Qu  the observation vector Yt  and the predictive value  ~tt-l  for time t. 
c 
Let us construct matrices 
(3.3) 
c 
of the dimension (m +n) x (m +2n) (Im  is the mxm identity matrix) and 
(3.4) 
c  of the dimension (m +2n) x (m +2n). The substantial procedure of the algorithm consists in 
the following factorization 
UVU'  =  LDL',  (3.5) 
where the (m+n) x (m+2n) matrix Land (m+2n) x (m+2n) matrix D are required to be 
of the same type as  in  (3.1). It is not difficult to show that the matrices L and D will have 
the form 
(  Loo·  O.  0)  (3.6) L  -
KtLet  ,  Lp, t+l'  0 
and 
[Dot' 
a, 
;.]
D  =  0,  Dp, t+l'  (3.7) 
0,  a, 
where D· may be an arbitrary diagonal nxn  matrix. 
c 
Then the output of the algorithm at time t contains 
X;+l  =  F~g-l+ (KcLec ) L~~  (Yt-H~g-l)  ,  (3.8) 
(3.9) 
where the matrices  Let'  KtLeu  Lp,t+l'  Dp,t+l  are taken  from  (3.6) and (3.7)  (moreover,  the 
matrices Lp,t+l'  Dp,t+l  form the input for time t+ 1). 
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All procedures of this  square root robust algorithm can be performed efficiently in 
the framework of the parallel implementation. For instance,  an array of (m +n) x (m +2n) 
parallel processors using the scan-with-add operation is  suitable for  the factorization (3.5) 
(see [7]). 
4.  The  case of scalar observations.  If the observations are scalar (m = 1)  and, in addition, 
the  Huber's  psi-function  (2.13)  is  chosen  then  the  robust  Kalman  filter  from  Section  2 
reduces to the form (2.14)-(2.17). 
The corresponding square root formulation will be more simple using the matrices 
( 
1,
U= 
0, 
of the dimension (1 +n) x (1 +2n) and 
htLpt , 
FtLpt  ' 
0 J 
In 
(4.1) 
V·  (~':'  D;:,  ~J 
(4.2) 
c 
of the dimension (1 +2n)x(1 +2n). Then the matrices L,D in the factorization (3.5) will have 
the form 
( 
1,  0,  0) 
L  =  Kt'  Lp ,t+l'  0 
(4.3) 
c 
c 
and 
providing the values rell  Kt for (2.14) and Lp,t+\, Dp,t+\ for (3.9) (and for input at time t+ 1). 
The suggested square root robust Kalman filter seems to be suitable for the practical 
treatment  of  systems  with  large  numbers  of  state  variables  and  with  contaminated 
observations. Its implementation on parallel computers is  the object of continuing work. 
( 
r et  l 
D  =  0, 
0, 
0 , 
0,  ;.] 
(4.4) 
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REFERENCES 
[1]  AKL, S.  G.: The Design and Analysis of Parallel Algorithms, Prentice Hall, Englewood 
Cliffs, New Jersey,  1989. 
[2]￿  CIPRA,  T.:  Note  on  approximate  non-Gaussian  filtering  with  nonlinear  observation 
relation, Comment. Math.  Univ.  Carolinae,  31  (1990),601-605. 
[3]￿  CIPRA,  T.,  MOTYOKOVA,  I.:  Study  on  Kalman  filter  in  time  series  analysis, 
Comment. Math.  Univ.  Carolinae 28  (1987), 549-563. 
[4]￿  CIPRA,  T.,  ROMERA,  R.:  Robust  Kalman  filter  and  its  application  in  time  series 
analysis, Kybernetika 27  (1991), nO  6,  481-494. 
-. 
[5]￿  ITZKOWITZ,  H.R., BAHETI,  R.S.:  Demonstration  of square  root Kalman  filter  on 
Warp parallel computer, Proceedings of the  1989  American Control Conference. 
[6] KAILATH, T.: State-space modelling: square root algorithms, in  : Systems and Control 
CJ  .  Encyclopedia (M.G. Singh, ed.), Pergamon Press,  New York 1987, 2618-2622. 
[7] PALlS, M.A., KRECKER, D.K.: Parallel Kalman filtering on the Connection Machine, 
GE Internal Technical Report,  Advanced Technology Laboratories,  GE Aerospace, 
Moorestown, New Jersey  1989. 
C\ 
Tomas Cipra, Dept. of Statistics, Charles University of Prague, Sokolovska 83, 18600 Praha 
8,  Czechoslovakia. 
Rosario Romera, Facultad de Informatica, Universidad Politecnica de Madrid,  Campus de 
Montegancedo, Boadilla del Monte, 28660 Madrid, Spain, and Departamento de Estadistica 
y Econometria, Universidad Carlos III de Madrid, Getafe, 28903 Madrid, Spain. 
Asunci6n Rubio, Dpto. de Matematicas, Universidad de Extremadura, 10071 Caceres, Spain. 
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Square root kalman filter with contaminated observations.

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Square root kalman filter with contaminated observations.

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