Processing of observations
The application of mathematical methods to results of observations, in order to form conclusions about the true values of unknown quantities. Any observational result arising in some way from measurements involves errors of various origins. Errors are divided into three groups: gross errors, systematic errors and random errors (concerning gross errors see Errors, theory of; in the rest of this article it will be assumed that the observations involve no gross errors). The result of a measurement of some quantity
is usually assumed to be a random variable; the measurement error
is then also a random variable. Let
be its mathematical expectation. Then
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The quantity is called the systematic error, and
the random error; the expectation of
equals zero. The systematic error
is frequently known in advance and is then easily eliminated. For example, in astronomy, when the angle between the direction of a star and the plane of the horizon is being measured, the systematic error is the sum of two errors: a systematic error originating in the instrument at the reading of the specific angle (the instrumental error) and a systematic error due to the refraction of light rays in the atmosphere. The instrumental error may be determined by consulting a correction table or graph for the instrument; the error due to refraction (for zenith distances less than
) may be calculated theoretically with adequate accuracy.
The effect of random errors is estimated by methods of the theory of errors. If are the results of
independent measurements of
, carried out under identical conditions and by identical means, one usually puts
![]() | (1) |
where is the systematic error.
If it is required to compute the value of a function at a point
, where
is estimated on the basis of
independent observations
, one approximates the required value by
![]() | (2) |
Let be the expectation of
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then
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Therefore is the systematic error and
the random error of the approximation (2). If the random errors in the independent observations
obey the same distribution and if the function
is "nearly" linear in a neighbourhood of
, then
and
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where is the arithmetic mean of the random errors of the initial observations. This means that if
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then
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as .
In the case of several unknown parameters, the observations are often processed using the method of least squares (cf. Least squares, method of).
If one is studying the dependence between two random variables and
on the basis of a sequence of
independent observations, each of which is a vector
,
, subject to the (unknown) joint distribution of
and
, one uses the theory of correlation to process the observations.
Whenever processing observations, one must make certain assumptions about the nature of the functional dependence, the distribution of the random errors, etc. It is therefore necessary to check the agreement between such assumptions and the results of the observations (both those actually used and others). See Statistical hypotheses, verification of.
References
[1] | E. Whittaker, G. Robinson, "The calculus of observations" , Blackie (1944) |
[2] | Yu.V. Linnik, "Methode der kleinste Quadraten in moderner Darstellung" , Deutsch. Verlag Wissenschaft. (1961) (Translated from Russian) |
Processing of observations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Processing_of_observations&oldid=18464