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Examoogle • User • info@examoogle.com • EE-2024-8821
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As per least square estimation, the estimated value of X (Xest) is represented by (Where Z is measurement vector and H is the linear relation between Z and X)
Xest=(HTH)−1HTHZ
Xest=(HTH)−1HTZ
Xest=(HTH)−1HZ
None of above
Xest=(HTH)−1HTZ
Quick Summary: In least square estimation for a linear system defined by $Z = HX + \epsilon$, the goal is to minimize the sum of squared residuals $J = \epsilon^T \epsilon = (Z - HX)^T (Z - HX)$. Setting the derivative of $J$ with respect to $X$ to zero yields the normal equations, which result in the estimator $X_{est} = (H^T H)^{-1} H^T Z$.
In least square estimation for a linear system defined by Z=HX+ϵ, the goal is to minimize the sum of squared residuals J=ϵTϵ=(Z−HX)T(Z−HX). Setting the derivative of J with respect to X to zero yields the normal equations, which result in the estimator Xest=(HTH)−1HTZ.
J=(Z−HX)T(Z−HX) — Objective function to minimize
Xest=(HTH)−1HTZ — Least square estimator solution
The principle relies on projecting the measurement vector Z onto the column space of the matrix H. By minimizing the Euclidean norm of the residual vector, we obtain the Moore-Penrose pseudoinverse (HTH)−1HT. This method assumes that HTH is non-singular and invertible, which corresponds to the system being observable.
The estimator is based on the minimization of the sum of the squares of the errors (residuals).
It assumes the errors are independent and identically distributed with zero mean.
If the matrix H is not of full column rank, HTH is singular and the unique inverse does not exist.
This method is widely used in power system state estimation for processing SCADA data.
Provides a mathematically optimal solution when errors are Gaussian.
Computational efficiency for linear systems.
Sensitive to outliers (bad data) in the measurement vector Z.
Requires high computational overhead for extremely large sparse matrices.
Power system state estimation (monitoring bus voltages and angles).
Signal processing and sensor fusion.
The term (HTH)−1HT is known as the left Moore-Penrose pseudoinverse of H.
Option C is dimensionally inconsistent as (HTH)−1 is n×n and H is m×n, making their product undefined unless m=n.
B is correct — The least square estimator for a linear model Z=HX is derived from the normal equations, resulting in Xest=(HTH)−1HTZ.
Always ensure dimensions match in matrix algebra; in (HTH)−1HTZ, if H is m×n, then (HTH) is n×n, and the total expression simplifies to an n×1 vector.