# Fuzzy Regression Analysis for Fuzzy Data

## Statistical Methods to Determine Variables Based on Fuzzy Data

Regression analysis is a statistical technique to arrive at the best mathematical expression that can correctly determine the relationship between independent variables depending upon a response. The parameters of the regression are determined by minimizing the difference between the real value and observed data, which may have errors. As observations are dependent on human appraisal besides other physical conditions, these always contain the possibility of errors and the regression analysis is used to estimate the value of these errors in order to reach a correct value.

Fuzzy regression analysis for fuzzy data may be done with three categories of dependent variables. The first is where both the input and the resultant output are clear numbers. The second case may be where in spite of the input data being non-fuzzy and clear, the output obtained is of a fuzzy nature. The third and most likely case is where both input and output data are fuzzy and dependent on certain conditions for their values.

In such fuzzy regression analysis for fuzzy data, it is quite likely that the greater the value of the independent variables, the greater will be the width of the dependent variables in determining the value of the center point for the correct regression, which may be vague.

Fuzzy logic is just a different way of determining the probability of an event or value. In fuzzy logic, the value of a variable is determined by logic that governs the variable and is, therefore, determined by the IF and THEN equation that governs it. Probability theory, on the other hand, is based on determining the value based on its probable correctness in a given situation. While fuzzy logic determines values based on possibilities, the probability theory decides these values based on the application to a particular problem.

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## Regression Analysis

Regression analysis is a widely used tool to determine the relationship between variables based on certain values that are observed during set conditions. While making these observations, the researcher may find that the values are dependent on other parameters which they include in their observations and thus render the observed data as fuzzy or determinable only by certain conditions. The regression analysis then turns into fuzzy regression analysis for fuzzy data. The regression or curve fitting can then be completed based on estimations through linear programming or using the least squares method using the fuzzy data. Vaguely specified data, which is present in almost all applications where human judgments are utilized, is suitable for fuzzy regression analysis.

It has been found that using the least squares regression analysis for fuzzy data is more likely to give the correct curve than any analysis using the linear method. Software to determine the least squares is widely available today.