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@@ -85,7 +85,7 @@ The answer of course boils down to linear algebra, and we will begin by translat
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| Prediction Error / Residuals | A residual vector $e \in \mathbb{R}^{n \times 1}$ | Difference between actual targets and predictions. |
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| Training / "best fit" | Optimization: minimizing the norm of the residual vector | To find the "best" model by finding a model which makes the norm of the residual vector as small as possible. |
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So our matrix $X$ will represent our data set, our vector $y$ is the target, and $\beta$ is our vector of parameters. We will often be interested in understanding data with "intercepts", i.e., when there is a base value given in our data. So we will augment a column of 1's (denoted by $\mathbb 1$) to $X$ and append a parameter $\beta_0$ to the top of $\beta$, yielding
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So our matrix $X$ will represent our data set, our vector $y$ is the target, and $\beta$ is our vector of parameters. We will often be interested in understanding data with "intercepts", i.e., when there is a base value given in our data. So we will augment a column of 1's (denoted by $\mathbb{1}$) to $X$ and append a parameter $\beta_0$ to the top of $\beta$, yielding
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$$ \tilde{X} = \begin{bmatrix} \mathbb{1} & X \end{bmatrix} \text{ and } \tilde{\beta} = \begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \vdots \\ \beta_p \end{bmatrix}. $$
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@@ -131,7 +131,7 @@ So a least-squares solution to the equation $Ax = b$ is trying to find a vector
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$$ \text{Col}(A) = \{Ax \mid x \in \mathbb{R}^n\} $$
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of $A$. We know this to be the projection of the vector $b$ onto the column space.
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)
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> **Theorem**: The set of least-squares solutions of $Ax = b$ coincides with solutions of the **normal equations** $A^TAx = A^Tb$. Moreover, the normal equations always have a solution.
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@@ -357,7 +357,7 @@ df.plot(
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plt.show()
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```
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)
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Of course, these figures are somewhat meaningless due to how unpopulated our data is.
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README.new.md
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README.new.md
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