NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

HO3

- 2 I -5119 6 April 1953 2339

Progress Report for Jan 0 - March 1953

on

Applications of the Theory of Stochastic

Processes to the Study of Trajectories

(NBS Project 1103-21-5119)

The publication, r

unless permission

25, D, C. Such pt

cally prepared If

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

r In part, Is prohibited

Itandards, Washington

eport has been speclfireport for Its own use,

PROGRESS REPORT FOR JAN. - MARCH 19^3

ON

APPLICATION OF THE THEORY OF STOCHASTIC PROCESSES

TO THE STUDY OF TRAJECTORIES

(NBS Project 1103-21-^119)

I . Summary

This report contains a summary of the work done during

the quarter. Results of this work are briefly stated. Technical

reports written In connection with this project are mentioned

but are transmitted separately.

II . Discussion of work done during the quarter . It was found desirable to construct an artificial

trajectory whibh would permit to demonstrate how the mean

value function of a fundamental random process could be estimated

A polynomial of degree four' was selected to represent the ’’true”'

path and equidistant points on this curve were determined.

Suitably chosen random numbers were attached as ’’observational

errors" to the coordinates of this curve. In this manner

’’’simulated observations” were obtained which will be used in

studying the proposed estimation procedures.

A stochastic process y(t) is studied and it is assumed that

y ( t ) = x ( t) + f(t) where

f(t) = (t) + K ()(t) + ... + K ()(t)

and where x(t) is a fundamental random process (Wiener process).

The first problem is the estimation of the parameters

K

g

on the basis of observations taken on y(t)

over' a time interval [0,T]. In the proposed example the

’’simulated observations” will represent the y(t) process.

It is known that estimates of the parameters K„Is. . .K

can be obtained in the following manner

Let j |sJr

i

j

be the inverse of the matrix [ | ^

j j j+r-1 j,r=l,...s

and denote by <j)^ r = -

jrT

A

the estimate K of K

J j

i s then found to be

s

)

This method of estimating the mean value curve f(t) , becomes quite convenient as soon as tfcable;s of the matrix | |S^ r

j ] are available. It became therefore necessary to compute

tables of this matrix. This was done by I . R. Savage and E .Lu-kacs

in N.B.S„S report No. 2279 entitled ^Tables useful in estimating

the mean value function of a fundamental random process”

-Since the tables and their method of computation are of some

independent interest a second paper entitled ^Tables of ^inverses,

of finite segments of the Hilbert Matrix” was prepared by the

same authors. This is a condensed version of NBS report 2279

and it is intended to submit it to a technical journal 0 It might be worthwhile to remark that it seems that the

results obtained during this quarter will also be applicable

to work under progress in the Bureau's Ordnance Development

Division. A paper by Eugene Lukacs ”0n strongly continuous

stochastic processes** was submitted to a technical journal.

This paper is the result of the author's work performed in the

previous quarter.

This is a special progress report to the U.S. Naval

Ordnance Test Station which sponsors NBS Project 1103=21=5119 p Application of the theory of stochastic processes to the

study of tra jec tories o

F „ L. Alt

Acting Chiefs National Applied

Mathematics Laboratories