# Tagged Questions

Questions about (linear or nonlinear) least-squares, an estimation method used in statistics, signal processing and elsewhere. See https://en.wikipedia.org/wiki/Least_squares

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### Proving OLS estimator of variance

According to Gujarati, author notes that in a simple linear equation form $Y_i=\alpha +\beta X_i + \epsilon_i$ where regression model is defined as $\hat Y_i =\hat \alpha + \hat \beta X_i$ OLS method ...
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### The OLS Estimator of $\sigma^2$

I have a question regarding the OLS Estimator of $\sigma^2$. In Gujarati's book on Econometrics author derives $E(\sum_{i=1}^n \hat u_i^2)$ (aka the expected value of residuals) to be $(n-2)\sigma$. ...
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### Solving a homogenous, weighted LSE problem

I search the non trivial solution for the system $\bf{Ax}=\bf{b} = \bf{0}$, where the equations are weighted with the matrix $\bf{W}$. I found 2 different approaches for each part, which I would ...
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### how to minimize $tr (X^T W - Y^T W_p)(X^T W -Y^TW_p)^T$ in closed form

Assume we are dealing with matrices. Then how to minimize $$E(W,W_p) = tr (X^T W - Y^T W_p)(X^T W -Y^TW_p)^T$$ w.r.t both $W, W_p$ simultaneously? I can calculate the derivatives of $W$ and $W_p$ ...
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### Proof related to the least squares method

I've seen this exercise in several statistics text, but how they get to the final formula is something that I don't quite get. How do two squared terms suddenly become a binomial term? I've been ...
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### Solve an overdetermined system of linear equations

I have doubt to solve this system of equations \begin{cases} x+y=r_1\\ x+z=c_1\\ x+w=d_1\\ y+z=d_2\\ y+w=c_2\\ z+w=r_2 \end{cases} Is it an overdetermined system because I see there are more ...
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### Explicit least-squares method for horizontal shifts of a function

I have a sequence of $N$ strictly positive real values $y_n$. They form some kind of peak; for simplicity, let's assume $f(x, \mu) = A \exp^{-(x-\mu)^2}$ is the shape, with $A$ and $\mu$ real (in the ...
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### How to calculate a projection matrix for nonnegative constrained least squares?

Suppose we have a data vector $\boldsymbol{z}$ in R^{p} and a training data matrix $\boldsymbol{X}$ in $R^{p \times N}$, where N (N>p) is the number of samples in the training data matrix. If we'd ...
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