Which of the following method is generally adapted for curve fitting

Question

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Option A: Weighted least square

Explanation & Concepts:
The Method of Least Squares is a standard mathematical approach in regression analysis to approximate the solution of overdetermined systems by minimizing the sum of the squares of the residuals (the differences between the observed values and the values provided by the model). It is the most commonly adapted technique for curve fitting because it provides a closed-form solution for linear regression models.

Working Principle: Given a set of data points $(x_i, y_i)$, the method aims to find a function $f(x)$ that minimizes the objective function $S = \sum_{i=1}^{n} (y_i - f(x_i))^2$. By taking partial derivatives of $S$ with respect to the parameters of the model (like slope $m$ and intercept $c$) and setting them to zero, we obtain a set of 'normal equations' that can be solved to find the best-fit parameters.

Key Formulas: $S = \sum_{i=1}^{n} (y_i - (mx_i + c))^2$ — Objective function for simple linear regression; $\frac{\partial S}{\partial m} = 0, \frac{\partial S}{\partial c} = 0$ — Normal equations for parameter estimation

Key Points: Minimizes the sum of the squares of vertical deviations.; Efficient for linear and polynomial models.; Highly sensitive to outliers in the data set.; Provides a unique 'best fit' solution under Gaussian error distribution.

Advantages: Mathematically simple and computationally efficient.; Provides statistically optimal estimates under normal error assumptions.

Disadvantages: High sensitivity to outliers.; Assumes error is normally distributed (Gaussian).

Applications: Load forecasting in Power Systems.; Trend analysis in financial modeling.; Sensor data calibration.

Additional Info: Weighted least squares is a variation used when errors have non-constant variance (heteroscedasticity).. Option B is technically incorrect as 'Extrapolation least square' is not a standard terminology for curve fitting.

Answer: C is correct — The method of least squares is the most widely adopted mathematical technique for fitting a curve to a given set of data points by minimizing the error squared.

Exam Tip: In Power System load forecasting, remember that the least squares method is frequently used to identify the trend component of long-term load growth patterns.

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Option B: Extrapolation least square

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Option C: Least square

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Option D: None of above

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