Parameters of multivariate normal distribution for time series analysis with missing dat

Smith, W. B.

English

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SEMj%NNUAL STATUS REPORT 
June 1, 1969, t o  November 30, 1969 
TIME SERIES ANALYSIS WITH 
MISSING DATA 
National Aeronautics and Space 
Administrat ion 
Grant #NGR 44-001-097 
W. B. Smith,  Associate Professor 
I n s t i t u t e  of S t a t i s t i c s  
Texas A&M University 
College Sta.tion, Texas 77843 
S m m U  STATUS REPORT 
TIME SERIES  ANALYSIS WITH MISSING DATA 
I. Introduction. 
Hocking and Smith (1968) have presented a technique f o r  estimating 
t h e  parameters of a mul t ivar ia te  normal d i s t r i b u t i o n  from data  vectors ,  
some of which have missing elements. The purpose of t h i s  grant  i s  t o  
extend t h i s  technique to t h e  estimation of t h e  covariance. function of 
a weakly s t a t iona ry  time s e r i e s  when the  missing observations occur 
randomly. Then having estimated t h i s  covariance function one i s  able  
toes t ima te  t h e  s p e c t r a l  dens i ty  function associated with t h e  parent 
s tochas t ic  process,thus enabling an inference to be made on t h i s  parent 
process. 
sLtuation i n  which the re  i s  a random machine o r  telemetry f a i l u r e  on 
,A 
An immediate appl ica t ion  of t h i s  technique would be t o  a 
l i f e  data of an astronaut i n  space. This procedure would enable the 
s c i e n t i s t  t o  estimate t h e  missing observation, a s  wel l  a s ,  t h e  general  
process which had been occurring a s  a function of time. 
The estimation procedure of Hocking and Smith w i l l  a l so  be applied 
t o  small sample estima.tion problems to a i d  i n  determining sample s i zes ,  
means, and variances. A mathematical formulation f o r  t h e  exact amount 
of information gleaned f r o m t h e  p a r t i a l  data. vectors can be given. 
r 
The research along these l i n e s  is continuing i n  the  following fashion: 
(1) The procedure of Hocking and Smith is  being extended t o  more 
genera.1 cases and more general  problems, s o  i ts  a.pplica.tion t o  
the  problem of in t e re s t  w i l l  be more e a s i l y  a t ta ined.  
Other forms of stocha.stic processes ase being considered; mainly 
the  use of s tochas t ic  processes i n  compartmental analysis.  
(2) 
11. Research Reports. 
Technical Report #1 submitted i n  September, 1969, e n t i t l e d  "Selection 
Index Estimation from P a r t i a l  Multivariate Normal Data" represented an 
extension of the Hocking and Smith technique t o  the  estimation of the  
well-known se lec t ion  index. The se lec t ion  index is based upon multiva.riate 
time-dependent data vectors,  so  it cons t i tu tes  a. legi t imate  extension 
of t he  p a r t i a l  data technique developed by Hocking and Smith. 
procedure i s  spel led out i n  considerable de ta , i l  i n  the technica l  report  
but can be summarized as follows: 
The 
The technique i s  applied t o  the  estimation of the  select ion index, 
when incomplete mult ivar ia te  normal data. vectors  a re  avai lable .  The 
procedure u t i l i z e s  a l l  data avai lable ,  both f u l l  and p a r t i a l  vectors,  
and represents  an improvement i n  the  precis ion of t he  estimator over 
thak found by using t h e  f u l l  da ta  vectors  only. Estimates of t he  
phenotypic means and t h e  variance-covaxiance matrix a re  a l so  found 
from t he  procedure. Individuals with p a r t i a l  data vectors a re  indexed 
-3- 
by an extension of the  technique and t h i s  index i s  compared t o  t h a t  
proposed by Henderson. 
tabulated.  
The r e s u l t s  of computer simulations a re  
That is  defining 1 = b’x  where I .  i s  the  composite index value 
j 3’  J 
associated w i t h  t he  Jth member of t h e  population, b i s  an n x 1-vec tor  
of unknown coeff ic ients ,  and x 
(observations) on the  Jth member of t he  population, Smith (1936) shows 
tha t  b = P Ga. I n  t h i s  context P i s  the  n x n variance-covariance 
matrix of phenotypic va.lues, G is the  n x n matrix of genotypic values, 
i s  an n x 1 vector of phenotypic values 
j 
-1 
and a i s  an n x l v e c t o r  of economic weights associated w i t h  the  n 
t r a i t s .  
#1 by estimating P the  matrix of phenotypic values. 
both full and incomplete data  vectors  represents an extension d i r e c t l y  
The problem of estimating b i s  attacked i n  Technical Report 
This estimation from 
of the  Hocking and Smith technique. 
and mathematical formulation of it see Technica.1 Report #l. 
For fu r the r  d e t a i l s  on t h e  procedure 
Another area of research cur ren t ly  being ca.rried out i s  the  contents 
of Technical Report #2, “Stochastic Compartmental Analysis: Model 
and Least Squares Estimakion from Time Ser ies  Data”. This repor t ,  by 
J. H. Matis and H. 0. Hartley, submitted November, 1969, can be summarized 
as  follows: 
E f f i c i en t  estimation of t h e  t r a n s i t i o n  r a t e  parameters of a s tochast ic  
m compartmental system requires  the associated d is t r ibu t ion .  T h i s  report  
advances the  d i s t r ibu t ion  theory  considerably by providing f o r  t he  first 
-4- 
time a compact ana ly t ic  solut ion.  Often i n  p rac t i ce  individual 
compartments a re  inaccessible  f o r  observation and instead of t i m e  
s e r i e s ,  data i s  avai lable  only on the  passage of mater ia l  t o  the  system 
exter ior .  The covariance kernel  of such observations is  derived 
i n  t h i s  paper and u t i l i z e d  f o r  e f f i c i e n t  parameter estimation. 
t h e  estimation procedure i s  demonstrated from simulated data .  
This research is  i n t o  an a rea  which represents a new branch of 
Final ly ,  
biomathematical modeling. This branch, ca.lled compartmental analysis ,  
assumes a system may be divided i n t o  homogeneous components, or 
compartments. Various cha rac t e r i s t i c s  of the  system are  determined 
by observing the movement of t r a c e r  elements through these compartments. 
Mathematical models f o r  such movements have been under the  impl ic i t  
assumption of determinis t ic  flow pat terns .  I n  prac t ice  such pa t te rns  
are  e n t i r e l y  un rea l i s t i c .  This report  represents  a f i rs t  attempt 
a t  using s tochas t ic  flow pa t te rns .  
111. Future Research. 
The l i n e s  of fu ture  research under t h i s  grant a r e  three-fold: 
F i r s t ,  t he  extension of t he  Hocking-Smith technique t o  time s e r i e s  data 
i n  which there  a re  a. combination of types of data., t h a t  i s ,  d i sc re t e  
and continuous random variables .  An extension of the  procedure t o  
t h i s  combinaction w i l l  be s ign i f icant  and i s  being car r ied  on a.t the 
present time. Secondly, fur ther  research on compartmental analysis  and 
r e s u l t s  of ac tua l  experiments on the  same w i l l  be reported OK. Fina.lly, 
-5-, 
t he  basic  problem of estimating the  spec t r a l  densi ty  function from 
the covariance kernel  of a s tochas t ic  process i n  the  presence of 
p a r t i a l  da ta  w i l l  be attacked using the  advanced methods developed 
by e a r l i e r  papers and by e a z l i e r  work i n  t h i s  contract .  
IV. Technical Reports. 
Technical Report #1, "Selection Index Estimation from P a r t i a l  Mu1 
Multivariate Normal Data", by W. B. Smith and R. C .  Pfaffenberger, 
September, 1969. (Submitted t o  Biometrics). 
Technical Report #2, "Stochastic Compartmental Analysis: Model 
and Least Squares Estimation from Time Ser ies  Data", by J. H. Matis 
and H. 0. Hartley, November, 1969. (To be submitted f o r  publ icat ion) .  
V. References 
1. Hocking, R .  R.,and W. B. Smith, (1968). "Estimation of Parameters 
i n  the  Multiva,riate Normal Distr ibut ion with Missing Observations", 
Journal of t h e  American S t a t i s t i c a l  Association, - 63, 159-173. 
2. Henderson, C. R. ,  (1963). "Selection Index and the  Expected Genetic 
Advance", NAS-NRC Publication 982, 141-163. 
3. Smith, H. F., (1936). "A Discriminant Function f o r  Plant Selection",  
Ann. Eugen. Lon$. , - 7, 240-250. 


Time series analysis with missing data Semiannual status report, 1 Jun. - 30 Nov. 1969

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