By Hung T. Nguyen
This examine monograph provides uncomplicated foundational points for a concept of records with fuzzy facts, including a suite of sensible functions. Fuzzy facts are modeled as observations from random fuzzy units. Theories of fuzzy good judgment and of random closed units are used as easy materials in development statistical techniques and systems within the context of obscure facts, together with coarse information research. The monograph additionally goals at motivating statisticians to examine fuzzy statistics to magnify the area of applicability of records quite often.
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Extra info for Fundamentals of Statistics with Fuzzy Data
Sample text
The problem is how to make an “educated guess” about T 0 ? Clearly, the above formulation is sampling surveys suggests the same structure for other statistical problems, namely a statistical model is a family of probability space {( X ,V ( X ), PT ,T 4)} . In standard statistical analysis, X is R or R d , and 4 is some finitely dimensional space (parametric statistical problems). Random variables or vectors are special cases of random closed sets. Intervals are the simplest form of imprecision about point observations (measurements).
Note that we consider non-empty random (closed) sets on Rd for practical purposes. For example, in the context of coarse data analysis, a random set S is a coarsening of some random variable X. As such, S takes non-empty subsets as values (since X is an almost selector of S). ~ On PK (Rd ), a natural metric is the following. Let H denote the Haus~ dorff metric on F(Rd) \ {I} . Then for A, B PK (Rd ), ~ H ( A, B ) sup H ( AD , BD ) 0 D d1 is a metric (see [34]), where AD is the D -level set of A.
3 Fuzzy Set-valued Random Elements 41 Let ( :, A, P) be a probability space. Let B(F) denote the Borel V field of the space of closed subsets F(Rd) of Rd. Then S : : o F(Rd) is called a random closed set if for any 3 B(F), S 1 (F) A. Similar to the situation of random vectors, it can be checked that the above measurability of S is equivalent to: for any open subset G of Rd, S 1 (FG) A, where FG = {F F : F G z I} . In view of all the above, we take the following approach (for more details, see [45]).
