Finite Number Of Observations Research Articles

Aliasing with Unequally Spaced Observations

Abstract Data analysis with unequally spaced observations is approached by assuming that a finite number of observations are available at known locations in space or time. No assumptions are made about the distribution of the station locations. Two types of observational error are considered, stationary (or homogeneous) and independent. Variance transfer functions are calculated which, when multiplied by the signal spectrum or error spectrum and integrated, give the contribution to the variance of estimates. Operations such as smoothing, interpolation, estimating derivatives, gradients of two-dimensional fields, divergence of two-dimensional vector fields, and spectrum estimates are considered. Aliasing takes the form of irregular side lobes, and it is concluded that the variance transfer functions should be plotted in most practical situations involving unequally spaced observations. Simulation studies are presented where the station locations have a random distribution, but the analysis does not use thi...

On the behaviour of the Kalman-Bucy estimate if the involved Wiener-Lévy processes are approximated by physically realizable processes

It is noteworthy that the possibility of computing Kalman-Bucy estimates depends entirely on the mathematical properties of the Wiener-Lévy processes figuring in the observations, whereas no observation device ever could generate processes of that kind. However, there are physically realizable processes arbitrarily close to Wiener-Lévy processes. In this paper the consequences are investigated if the Wiener-Lévy processes in the Kalman-Bucy filter are replaced by realizable approximations. The effect is not simply that of perturbing some values in certain formulae, since the whole computation scheme of Kalman and Bucy breaks down. It is shown that estimates of Kalman-Bucy type, based on a finite number of observations, are stable with respect to the above sketched operations. A more or less controversial result is obtained if the number of observations is infinite.

On the Cost of not Knowing the Variance when Making a Fixed-Width Confidence Interval for the Mean

It is shown that the mean of a normal distribution with unknown variance $\sigma^2$ may be estimated to lie within an interval of given fixed width at a prescribed confidence level using a procedure which overcomes ignorance about $\sigma^2$ with no more than a finite number of observations. That is, the expected sample size exceeds the (fixed) sample size one would use if $\sigma^2$ were known by a finite amount, the difference depending on the confidence level $\alpha$ but not depending on the values of the mean $\mu$, the variance $\sigma^2$ and the interval width $2d$. A number of unpublished results on the moments of the sample size are presented. Some do not depend on an assumption of normality.

Filtering of Time Series with Unknown Mean Value

The regression coefficients are unknown. The sequences {pj4 and {z;} are stationary with spectral densities fp(X) and f,(X) respectively. It is desired to estimate the value of the signal at some time, which is arbitrarily chosen to be time zero, from a finite number of observations, N + 1, in such a way that the estimator is unbiased and the mean square error is miinimized. The solution to the problem of finding the estimator has been treated by several authors, [1]-[8]. Readily applied estimates of the resulting error variance are not available however. The purpose of this paper is to derive an asymptotic expression for the mean square error for a general class of regression variables. Two limiting cases are considered as N -> oc : estimation on the semi-infinite interval or endpoint estimation

Distance and decision rules

Distance between distribution functions of ten becomes a useful and convenient concept in statistics. In a class of distributions, distance can be defined in various ways. We, therefore, have to choose an adequate distance for a particular problem. Fur the r , to t r ea t the problem efficiently, we are required, or at least i t is ve ry desirable, to control all errors tha t may be committed in making decision or inference. For this purpose we have to formulate, or reformulate if necessary, the problem suitably. A proper formulation of the problem is very important for ge t t ing command over possible errors. For instance, suppose tha t we wan t to know whether or not the random variable under consideration can be considered to have mean zero, when it is given that the random variable has the Gaussian dist r ibut ion N(O, 1). In this case, if we take the problem as the one inquiring jus t whether or not the mean of the random variable is zero, we can not control possible errors. For we can find a distribution which has mean not equal to 0, bu t which is as near to the distribution N(0, 1) as desired, and this makes it impossible to control all possible errors in inference or decision making based on a finite number of observations. In terms of hypothesis testing, we can not make the first and second kinds of errors simultaneously as small as desired (i.e., while we can make the first kind of error smaller than any given (positive) value, this is not the case wi th the second kind of error). One way to avoid such inconvenience is to formulate the problem as follows. That is, introducing an adequate distance d ( . , . ) i n the space of distributions concerned, we set the problem as making decision whether the random variable under consideration has F0=N(0 , 1) or some F with d(F, Fo)>~ (>0) , where ~ is a constant which is to be predetermined from the actual situation of the problem. For the problem thus formulated we can control the errors (see Matusi ta [1], Matusita, Akaike [2]). So far, the author has t rea ted various problems with the same idea, the idea of controlling possible errors (see Matusi ta [1], [3], [5], [7], [8],

HISTORICAL EXPLANATION IN THE SOCIAL SCIENCES

THE hope which originally inspired methodology was the hope of finding a method of enquiry which would be both necessary and sufficient to guide the scientist unerringly to truth. This hope has died a natural death. Today, methodology has the more modest task of establishing certain rules and requirements which are necessary to prohibit some wrong-headed moves but insufficient to guarantee success. These rules and requirements, which circumscribe scientific enquiries without steering them in any specific direction, are of the two main kinds, formal and material. So far as I can see, the formal rules of scientific method (which comprise both logical rules and certain realistic and fruitful stipulations) are equally applicable to all the empirical sciences. You cannot, for example, deduce a universal law from a finite number of observations whether you are a physicist, a biologist, or an anthropologist. Again, a single comprehensive explanation of a whole range of phenomena is preferable to isolated explanations of each of those phenomena, whatever your field of enquiry. I shall therefore confine myself to the more disputable (I had nearly said ' more disreputable') and metaphysically impregnated part of methodology which tries to establish the appropriate material requirements which the contents of the premisses of an explanatory theory in a particular field ought to satisfy. These requirements may be called regulative principles. Fundamental differences in the subjectmatters of different sciences-differences to which formal methodolo-

Intrinsic Variability of the So-Called Fundamental Physical Constants

Several years ago I pointed out [1-3] that formulations of natural laws which imply the possibility of absolute verification through a finite number of observations are intrinsically inadequate and must in course of time give way to statistically flexible formulations of natural laws. The considerations which led to the conclusion just stated were illustrated by predictions regarding the necessarily to be expected variability of physically significant pure numbers, such as the fine structure constant α = 2πe^2/hc, the ratio of the rest masses of proton and electron r = mp/me and the ratio of the electric radius to the gravitational radius of the electron R = e^2/Γme^2, (Γ = universal gravitational constant), numbers which previously had been considered as absolute values. The programmatic value of these considerations is accentuated by recent developments in the physics of elementary particles as well as in cosmology.

On the statistical distinction between sets of two-dimensional observations

Abstract In this paper we propose to deal with the problem of how to ascertain that the number of observations available for a statistical investigation will suffice for a final conclusion in cases where each of the finite number of observations consists of or covers two numerical attributes, i. e. when the observations present themselves as vectors in a plane. The proposed method must be considered merely as a generalization of the classical method ordinarily used when each of the observations covers only a single number.