An Application of Mindsadbesd Regression
Table of Contents
In this work, an application of the modified minsadbed (minimizing sum of absolute differences between deviations) approach for a fuzzy environment is given. This type of regression was used for a statistical model with two real parameters and experimental observations which implies real numbers (see Arthanary and Dodge). We develop minsadbed to minsadbesd (minimizing sum of absolute differences between squared deviations) which is more suitable for our model on vague sets. The models on fuzzy sets are described by Ming, Friedman and Kandel; these authors estimate the parameters pre-eminently using least squares. We make an attempt for another method, as in the following writing.
The minsadbesd approach
Consider the model composed by observations which are put in the forms , where , , , are real functions defined on closed interval (see Goetschel&Voxman, Ming, Friedman and Kandel). The model is approximately described by a regression line given by the equation , of form , where const., const.. Thus we have the initial relation . For the inputs the distance between an observed value and the corresponding theoretical value is:
Case 1: .
In this case we solve the problem under the assumption that .
The minsadbesd algorithm lead us to solve the problem
For all , , we make the substitutions:
Thus (1.2) is equivalent to
Let . For function , we have
The sign of the discriminant is unknown. We have four cases which depends on signs of ; consequently, the graph of has one of the forms shown in Fig. 1-4.
The "easy" case appears when all the discriminants are negative, namely. In this situation the functions have the forms shown in Fig. 3 or Fig. 4.
The problem (1.4) is equivalent to
where ( this writing means that some of the terms are positively and the others are negatively, depending on the concrete signs of
The unique minimizing point for the function (see also Fig. 5) is
have random signs.
The graph of the continuous function is composed from small pieces which are parts from the functions given by the equations are real numbers with general form (see Fig. 6). We consider the following sets:
Thus the feasible set is which is relatively easy to settle, as in Section 2.
Case 2: .
The problem (1.7) is equivalent with
We denote . For function , we have
First, we consider the case .
Thus the graph of has one of the two forms shown in Fig. 3 and Fig. 4.
Then the problem (1.9) is equivalent with
The unique minimizing point for function is
The approach is the same as in first case but with other coefficients.
In both situation, is obtained from the condition that the line pass through the initial fixed point.
All the comments stored in case 1.2 keeps their validity.
For we have and (1.4) is equivalent with
Figure 1. The graph of (green) if and
Figure 2. The graph of (green) if and
Figure 3. The graph of if and
Figure 4. The graph of if and
Figure 5. The graphs of the functions, when ; the surface bounded by the graph of is colored in gray
Figure 6. The graph of the function when have random signs
We test the method for fixed point and the fuzzy data:
Case 1: We search the minimizing points for the function
Accordingly to the facts proved in the preceding chapters, namely Section 1, case 1.2, the set of feasible points is ( notice: for this example we obtain
Case 2: We search the minimizing point for and the minimum is attained in on we conclude that the estimators for are the real numbers where and depends by the desired threshold of error. If then is a better estimator.
At last, we have and for all and the final solution for this problem is
From the preceding theoretical facts and numerical example we obtain the following conclusions:
1) For , , we evaluate and .
If thus the solution is .
If thus the solution is .
2) For , or , it is necessary to make small supplementary calculations which implies the special properties of the functions
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