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.**

**A**ll 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

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 _{
}_{}_{}

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