In least square estimation method, The weight-age of error in measurement vector is

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Examoogle · Accepted Answer

Equal

Explanation: In Ordinary Least Squares (OLS) estimation, the objective is to minimize the sum of the squares of the differences between observed and fitted values. This method implicitly assumes that every measurement in the measurement vector is assigned an equal weight, meaning no individual measurement is prioritized over others.

Working Principle: The method functions by minimizing the cost function $J = \sum_{i=1}^{n} (z_i - h_i(x))^2$. Since each residual is squared and added without a specific scaling coefficient, the contribution of every measurement to the final estimation is treated with equal importance. If measurements possess different levels of uncertainty, a Weighted Least Squares (WLS) approach is employed instead, using a diagonal weight matrix $W$ to assign importance.

Key Formulas: $J(x) = \sum_{i=1}^{m} [z_i - h_i(x)]^2$ — Standard OLS cost function objective; $W = I$ — Identity matrix used as weight matrix in OLS

Key Points: Ordinary Least Squares assumes homoscedasticity (constant variance) across all measurement samples.; The method minimizes the Euclidean norm of the residual vector.; If errors are Gaussian and have equal variance, OLS provides the Best Linear Unbiased Estimator (BLUE).; To handle measurements with varying reliability, one must shift from OLS to Weighted Least Squares (WLS).

Advantages: Computationally simple and efficient; No prior knowledge of noise statistics required

Disadvantages: Sensitive to outliers as errors are squared; Does not account for differences in measurement precision

Applications: Static State Estimation in Power Systems; Linear Regression modeling

Additional Info: In power system state estimation, Weighted Least Squares (WLS) is preferred over OLS because real-world sensors (PMUs, SCADA) have varying accuracy.. Option B refers to Weighted Least Squares (WLS), which is a generalization, not the default assumption of basic Least Squares.

Answer: A is correct — The ordinary least square estimation method mathematically treats the weight of each error in the measurement vector as equal.

Exam Tip: Always distinguish between Ordinary Least Squares (OLS) and Weighted Least Squares (WLS); OLS is a specific case of WLS where the weight matrix is the identity matrix.

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