# 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

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

then

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

where is the arithmetic mean of the random errors of the initial observations. This means that if

then

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) |

**How to Cite This Entry:**

Processing of observations.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Processing_of_observations&oldid=18464