Random functions

On purely kinetic grounds, however, the term random must be used carefully in describing a MaxweUian gas. The probabUity of a MaxweUian gas entering a duct is not a random function. This probabUity is proportional to the cosine of the angle between the molecular trajectory and the normal to the entrance plane of the duct. The latter assumption is consistent with the second law of thermodynamics, whereas assuming a random distribution entry is not. 

If the laminae orientation is a random function of z as in Figure 7-53c, define V as the spatial average of the individual Vj(/ B,o) will be treated alike) ... 

Consider an order W system and a random function 4> which maps each of the H = 2 possible binary states Si to unique successor states Sj = cyclic structure of the corresponding state transition graph... 

The starting point is the (pseudo-) randomization function supplied with most computers it generates a rectangular distribution of events, that is, if called many times, every value between 0 and 1 has an equal probability of being hit. For our purposes, many a mathematician s restraint regarding randomization algorithms (the sequence of numbers is not perfectly random because of serial correlation, and repeats itself after a very large number of... 

The randomization function is initialized with a different seed every time the program is run. 

Probabilistic techniques of estimation provide some Insights Into the potential error of estimation. In the case of krlglng, the variable pCic) spread over the site A is first elevated to the status of a random function PC c). An estimator P (2c) is then built to minimize the estimation variance E [P(2c)-P (2c) ], defined as the expected squared error ( ). The krlglng process not only provides the estimated values pCiyc) from which a kriged map can be produced, but also the corresponding minimum estimation variances 0 (39 ) ... 

The first approach consists of assuming some multivariate distribution model for the random function P(x), xeA A convenient... 

Here, I, I, and I are angular momentum operators, Q is the quadrupole moment of the nucleus, the z component, and r the asymmetry parameter of the electric field gradient (efg) tensor. We wish to construct the Hamiltonian for a nucleus if the efg jumps at random between HS and LS states. For this purpose, a random function of time / (f) is introduced which can assume only the two possible values +1. For convenience of presentation we assume equal... 

In random functions, each realization of the function can be conceived as the sum of a structured component and an erratic or stochastic component. The structured component is the one assuring that the observed values have a systematic variation or, in other words, that if we are, for instance, in an area where different measurements above a normal value have been obtained, there exist a high probability for additional measurements to be high as well. On the other hand, the random component is the one making difficult the exact prediction of these hypothetical measurements, since unpredictable fluctuations exist. 

V. S. Pugachev, Teoriya Sluchainikh Funktzii (Theory of Random Functions), Gos. Izd. Fiziko-Matem. Liter., 2ndEd.., Moscow, 1960, Section 46, pp. 191-202. 3E. Parzen, Modern Probability Theory and Its Applications, Wiley and Sons, New York, 1960, Chapter 3, Section 3, pp. 136-147. 

A. A. Sveshnikov, Problems in Probability Theory, Mathematical Statistics and Theory of Random Functions, Dover, 1968, Chapter VIII, pp. 231-274. 

Equation (2.2) can be considered as the fundamental governing equation for the concentration of an inert constituent in a turbulent flow. Because the flow in the atmosphere is turbulent, the velocity vector u is a random function of location and time. Consequently, the concentration c is also a random fimction of location and time. Thus, the dispersion of a pollutant (or tracer) in the atmosphere essentiaUy involves the propagation of the species molecules through a random medium. Even if the strength and spatial distribution of the source 5 are assumed to be known precisely, the concentration of tracer resulting from that source is a random quantity. The instantaneous, random concentration, c(x, y, z, t), of an inert tracer in a turbulent fluid with random velocity field u( c, y, z, t) resulting from a source distribution S x, y, z, t) is described by Eq. (2.2). 

Random Functions and Regionalized Variables. In univariate statistics, an observation y. is defined as a realization of a random... 

An example of a random function is the distribution of lead in the top 5 cm of soil within a mile radius of a lead smelter. An example of a ReV would be the set of measurements obtained from taking say 100 core samples around the lead smelter. The important thing to remember is that these 100 measurements constitute only a single realization of the random function. 

Second Order Stationarity. With only a single realization of the random function it would be impossible to make any meaningful inferences about the random function if we did not make some assumptions about its stationarity. A random function is said to be strictly stationary if the joint probability density function for k arbitrary points is invariant under simultaneous translation of all... 

In practice, only the first two moments of the random function are of interest. The first order moment is the expectation (mean) of the random function at an arbitrary location x, which is defined to be... 

The second order moment can be expressed either in terms of the covariance or the variogram. The covariance of the random function at points and is defined to be... 

The random function is said to be weakly or second order stationary when its first two moments are invariant under simultaneous translation by h. That is, for every x and h ... 

That is, the random function has a constant mean and the covariance... 

Thus, if the assumption of second order stationarity holds, then statistical inferences about the first two moments become possible since each pair of observations that are separated by a distance h can be considered a different realization of the random function. 

Intrinsic Hypothesis. The assumption of second order stationarity assumes that the variance exists (i.e., it is not equal to infinity). This assumption is still stronger than necessary. A random function is said to be intrinsic (i.e., satisfies the intrinsic hypothesis) when for every ... 

To use what is termed simple kriging, only the assumption that the random function is intrinsic needs to be made. The problem with this assumption is that the expected value of the phenomena of interest is rarely a constant. For example, the expected concentration of lead in the soil around a smelter would decrease as the distance from the smelter increased. If this decrease (or trend) is gradual enough, it is often assumed that within a limited neighborhood the random function has a "local stationarity" and then simple kriging is used, since generally only the observations within the limited neighborhood are used in the estimation process. 

Intrinsic Random Function of Order K. When the expected value of the random function cannot be assumed to be a constant, even within a limited neighborhood, then the random function is assumed to be the sum of two terms. That is,... 

The second order increment is an intrinsic random function of order two (IRF-2). 

To use what is termed universal kriging, it is assumed that Z(2 ) is an intrinsic random function of order k. But the problem of identifying the drift and the semi-variogram when they are both unknown is still present. However, Matheron (11) defined a family of functions called the generalized covariance, K(h). and the variance of the generalized increment of order k can be defined in terms of K(h ). That is. 

Simple kriging is actually a subset of universal kriging since the assumption that Z(2 ) is an intrinsic random function of order 0 is the same as the assumption that ZCjc) is intrinsic. Additionally, when l x) is intrinsic, the generalized covariance and the semi-variogram are related as follows ... 

In the previous section, an overview of the kriging assumptions was given. When these assumptions are accepted, a kriging system of linear equations can be developed. Whether the random function,... 

If the distance between two points is less than the range, then the value at one point is correlated with the value at the other point. If the distance between two points is greater than the range, then the points are independent. The sill is the bound on the semi-variogram and provides an estimate of the overall variability. When a semi-variogram is bounded then the random function is second order stationary and... 

Generalized Covariance Models. When l x) is an intrinsic random function of order k, an alternative to the semi-variogram is the generalized covariance (GC) function of order k. Like the semi-variogram model, the GC model must be a conditionally positive definite function so that the variance of the linear functional of ZU) is greater than or equal to zero. The family of polynomial GC functions satisfy this requirement. The polynomial GC of order k is... 

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