bounded variogram

The variogram is a moment estimation method proposed by Motheron in 1965. It is the mathematical expectation of the square of the increment of the regionalized variable, that is, the variance of the increment of the regionalized variable. Many scholars directly refer to the semi-variogram as the semi-variogram. Variation function.

The variogram is a unique research tool in geostatistics, which can not only characterize the spatial structure of regionalized variables, but also the randomness of regionalized variables, reflecting the changes of regionalized variables in a certain direction and a certain distance range. degree.

If in the interval (a, b), the function f(x) can be expressed in the shape of Φ(x)-Ψ(x), and both Φ and Ψ are non-decreasing bounded functions, then f(x) is said to be in (a) , b) is bounded variation. It is easy to see that the sum, difference and product of two bounded variogram functions are also bounded variograms.

definition

edit

Its other definitions are as follows:

Definition one

Let the interval (a, b) be divided by the point a=x0<x1<…<xn=b, if

is often less than a constant , then the function is said to have bounded variation in (a, b). The supremum of this sum is called total variation  [1]    .

Definition 2

Let f be a function defined on the interval [a, b], and examine any set of points on [a, b]: a=x0<x1<…<xn=b, when the points change, it is called the supremum

is the total variation (or total variation) of f on [a, b]. and denoted as

   

. like

   

<+∞．Then f is called a bounded variation function (or bounded variation function) on [a, b]  [2]    ．

Definition three

Let f(x) be a function and take any partition D of [a, b]: a=x0<x1<…<xn=b,  [3]  

make

say

   

(f,D)为f(x)关于分割D的 变差，若变差

   

(f,D)都不超过某个正常数，即存在M>0，使对一切分割D，

 

(f,D)≤M，

则称f(x)为[a，b]上的有界变差函数。记

   

(f)=sup

   

(f，D)，称

   

(f)为f(x)在[a，b]上的全变差或总变差。

性质

编辑

1.单调函数是有界变差函数．

2.有限个有界变差函数的和、差、乘积仍为有界变差函数．

3.两个有界变差函数之商(分母不为零)仍为有界变差雨数．

4.(Jordan分解定理)f为[a,b]上的有界变差函数的充要条件是f可表为两个不减的非负函数之差．

5.（Lebesgue) 若f是[a,b]上的单凋函数．则f在[a,b]上几乎处处可微。

6.绝对连续函数必是有界变差函数．  [2]  

7.若f(x)是[a,b]上的有界变差函数，则∣f(x)∣在[a,b]上必为有界变差函数；

8.设f(x)是[a,b]上的有界变差函数，且a<c<b，则f(x)在[a，c]和[c，b]上均为有界变差函数，且有

   

(f)=

   

(f)+

   

(f)；

9.设f(x)，g(x)都是[a,b]上有界变差函数，α、β为两个常数，则αf(x)+βg(x)是[a,b]上的有界变差函数；

10.设f(x)，g(x)都是[a,b]上有界变差函数，则f(x)g(x)在[a,b]上亦为有界变差函数；

11.设{fn(x)}为[a,b]上的有界变差函数列，且{

   

(fn)}有界

   

=f(x)，则f(x)在[a,b]上为有界变差函数  [3]    。

推论：有界变差函数几乎处处可微  [2]    ．