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Description | wdiam what does it all mean Menu HOME ABOUT Posts Posted on September 2, 2013 August 6, 2020 Deriving the Linear Regression Solution In deriving the linea |
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wdiam what does it all mean Menu HOME ABOUT Posts Posted on September 2, 2013 August 6, 2020 Deriving the Linear Regression Solution In deriving the linear regression solution, we will be taking a closer look at how we “solve” the common linear regression , i.e., finding \beta in y = X\beta + \epsilon . I mention “common,” because there are actually several ways you can get an estimate for \beta based on assumptions of your data and how you can correct for various anomalies. “Common” in this case specifically refers to ordinary least squares . For this specific case, I assume you already know the punch line, that is, \beta = (X^{T}X)^{-1}X^{T}y . But, what we’re really interested in is how to get to that point. The crux is that you’re trying to find a solution \beta that minimizes the sum of the squared errors, i.e., \min\limits_{\beta} \: \epsilon^{T}\epsilon . We can find the minimum by taking the derivative and setting it to zero, i.e., \frac{d}{d\beta} \epsilon^{T}\epsilon = 0 . |
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