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An Application of Mindsadbesd Regression
Ciprian Costin Popescu
Sudradjat Supian
Keywords
mindsadbesd regression, application, fuzzy models
Table of Contents
The minsadbesd approach
Example
Conclusions
References
Abstract
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:
if
and
if
.
Case 1:
.
In this case we solve the problem under the assumption that
.
The minsadbesd algorithm lead us to solve the problem
(1.1)
or
(1.2)
For all
,
,
we make the substitutions:
,
,
,
.
Thus (1.2) is equivalent to
(1.3)
or
(1.4)
where
,
,
.
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.
Case 1.1.
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
(1.5)
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
Case 1.2.
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:
.
(1.6)
gives
(1.7)
The problem (1.7) is equivalent with
(1.8)
which becomes
(1.9)
if
.
We denote
.
For function
,
we have
Case 2.1.
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
(1.10)
where
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.
Case 2.2.
All the comments stored in case 1.2 keeps their validity.
For
we have
and (1.4) is equivalent with
(1.11)
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
Example
We test the method for fixed point
and the fuzzy data:
Then
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2 |
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3 |
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1 |
9.66 |
32.66 |
16.50 |
15.5 |
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2 |
0.58 |
108.66 |
21.00 |
20.00 |
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3 |
43.33 |
55.33 |
46.66 |
36.00 |
and
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
Conclusions
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
References
1. Arthanary T. S., Yadolah Dodge, Mathematical Programming in
Statistics, John Wiley and Sons,
2. Wu Cong-Xin and Ma Ming: Embedding problem of fuzzy number space: Part I, Fuzzy Sets and Systems 44, 1991, p. 33-38
3. Wu Cong-Xin and Ma Ming: Embedding problem of fuzzy number space: Part III, Fuzzy Sets and Systems 46, 1992, p. 281-286
4. P. Diamond and P. Kloeden, Metric spaces of fuzzy sets, Fuzzy Sets and Systems 35, 1990, p. 241-249
5. P. Diamond, Fuzzy least squares, Inform. Sci. 46, 1988, p. 141-157
6. P. Diamond and P. Kloeden, Metric spaces of fuzzy sets, Corrigendum, Fuzzy Sets and Systems 45, 1992, p. 123
7. R. Goetschel, W. Voxman, Elementary Calculus, Fuzzy Sets and Systems 18, 1986, p. 31-43
8. I.M. Hammerbacher and R. R. Yager, Predicting television revenues using fuzzy subsets, TIMS Stud. Management Sci. 20, 1984, p. 469-477
9. A. Katsaras and D. B. Liu, Fuzzy vector spaces and fuzzy topological vector spaces, J. Math. Anal. Appl. 58, 1977, p. 135-146
10. Ma Ming, M. Friedman, A. Kandel, General fuzzy least squares, Fuzzy Sets and Systems 88, 1997, p. 107-118
11. C. V. Negoita and D. A. Ralescu, Applications of Fuzzy Sets to Systems
Analysis, Wiley,
12. H. Prade, Operations research with fuzzy data, in: P. P. Wang and
S. K. Chang, Eds., "Fuzzy sets: Theory and Application to Policy Analysis and Information
Systems", plenum,
13. M. L. Puri and D. A. Ralescu, Differentials for fuzzy functions, J. Math. Anal. Appl. 91, 1983, p. 552-558
14. H. Tanaka, H. Isibuchi and
15. H. Tanaka,
16. H. J. Zimmermann, Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and Systems 1, 1978, p. 45-55
17. R. R. Yager, Fuzzy prediction based upon regression models, Inform. Sci. 26, 1982, p. 45-63