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The closed form of w in Linear regression can be written as

$\hat{w}=(X^TX)^{-1}X^Ty$

How can we intuitively explain the role of $(X^TX)^{-1}$ in this equation?

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    $\begingroup$ Could you elaborate on what you mean by "intuitively"? For instance, there is a wonderfully intuitive explanation in terms of inner-product spaces presented in Christensen's Plane Answers to Complex Questions, but not everybody will appreciate that approach. As another example, there's a geometric explanation in my answer at stats.stackexchange.com/a/62147/919, but not everybody views geometrical relations as "intuitive." $\endgroup$ – whuber Aug 29 '18 at 17:02
  • $\begingroup$ Intuitively is like what does $(X^TX)^{-1} mean? Is it some kind of distance calculation or something, I don't understand it. $\endgroup$ – Darshak Aug 29 '18 at 17:09
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    $\begingroup$ That's fully explained in the answer I linked to. $\endgroup$ – whuber Aug 29 '18 at 17:27
  • $\begingroup$ This question already exists here although possibly not with a satisfying answer math.stackexchange.com/questions/2624986/… $\endgroup$ – Martijn Weterings Aug 29 '18 at 17:30
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I found these posts particularly helpful:

How to derive the least square estimator for multiple linear regression?

Relationship between SVD and PCA. How to use SVD to perform PCA?

http://www.math.miami.edu/~armstrong/210sp13/HW7notes.pdf

If $X$ is an $n \times p$ matrix then the matrix $X(X^TX)^{-1}X^T$ defines a projection onto the column space of $X$. Intuitively, you have an overdetermined system of equations, but still want to use it to define a linear map $\mathbb{R}^p \rightarrow \mathbb{R}$ that will map rows $x_i$ of $X$ to something close to values $y_i$, $i\in \{1,\dots,n\}$. So we settle for sending $X$ to the closest thing to $y$ that can be expressed as a linear combination of your features (the columns of $X$).

As far as an interpretation of $(X^TX)^{-1}$, I don't have an amazing answer yet. I know you can think of $(X^TX)$ as basically being the covariance matrix of the dataset.

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  • $\begingroup$ $(X^T X)$ is sometimes referred to as a "scatter matrix" and is just a scaled up version of the covariance matrix $\endgroup$ – JacKeown Mar 19 at 2:43
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Assuming you're familiar with the simple linear regression: $$y_i=\alpha+\beta x_i+\varepsilon_i$$ and its solution: $$\beta=\frac{\mathrm{cov}[x_i,y_i]}{\mathrm{var}[x_i]}$$

It's easy to see how $X'y$ corresponds to numerator above and $X'X$ maps to denominator. Since we're dealing with matrices the order matters. $X'X$ is KxK matrix, and $X'y$ is Kx1 vector. Hence, the order is: $(X'X)^{-1}X'y$

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  • $\begingroup$ But that analogy itself doesn't tell you if pre- or postmultiply with the inverse. $\endgroup$ – kjetil b halvorsen Aug 29 '18 at 18:17
  • $\begingroup$ @kjetilbhalvorsen, I put the order of operations $\endgroup$ – Aksakal Aug 29 '18 at 18:21
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Geometric viewpoint

A geometric viewpoint can be like the n-dimensional vectors $y$ and $X\beta$ being points in n-dimensional-space $V$. Where $X\hat\beta$ is also in the subspace $W$ spanned by the vectors $x_1, x_2, \cdots, x_m$.

projection

Two types of coordinates

For this subspace $W$ we can imagine two different types of coordinates:

  • The $\boldsymbol{\beta}$ are like coordinates for a regular coordinate space. The vector $z$ in the space $W$ are the linear combination of the vectors $\mathbf{x_i}$ $$z = \boldsymbol{\beta_1} \mathbf{x_1} + \boldsymbol{\beta_2} \mathbf{x_1} + .... \boldsymbol{\beta_m} \mathbf{x_m} $$
  • The $\boldsymbol{\alpha}$ are not coordinates in the regular sense, but they do define a point in the subspace $W$. Each $\alpha_i$ relates to the perpendicular projections onto the vectors $x_i$. If we use unit vectors $x_i$ (for simplicity) then the "coordinates" $\alpha_i$ for a vector $z$ can be expressed as:

    $$\alpha_i = \mathbf{x_i^T} \mathbf{z}$$

    and the set of all coordinates as:

$$\boldsymbol{\alpha} = \mathbf{X^T} \mathbf{z}$$


Mapping between coordinates $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$

for $\mathbf{z} = \mathbf{X}\boldsymbol{\beta}$ the expression of "coordinates" $\alpha$ becomes a conversion from coordinates $\beta$ to "coordinates" $\alpha$

$$\boldsymbol{\alpha} = \mathbf{X^T} \mathbf{X}\boldsymbol{\beta}$$

You could see $(\mathbf{X^T} \mathbf{X})_{ij}$ as expressing how much each $x_i$ projects onto the other $x_j$

Then the geometric interpretation of $(\mathbf{X^T} \mathbf{X})^{-1}$ can be seen as the map from vector projection "coordinates" $\boldsymbol{\alpha}$ to linear coordinates $\boldsymbol{\beta}$.

$$\boldsymbol{\beta} = (\mathbf{X^T} \mathbf{X})^{-1}\boldsymbol{\alpha}$$

The expression $\mathbf{X^Ty}$ gives the projection "coordinates" of $\mathbf{y}$ and $(\mathbf{X^T} \mathbf{X})^{-1}$ turns them into $\boldsymbol{\beta}$.


Note: the projection "coordinates" of $\mathbf{y}$ are the same as projection "coordinates" of $\mathbf{\hat{y}}$ since $(\mathbf{y-\hat{y}}) \perp \mathbf{X}$.

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  • $\begingroup$ A very similar account of the topic stats.stackexchange.com/a/124892/3277. $\endgroup$ – ttnphns Aug 30 '18 at 12:49
  • $\begingroup$ Indeed very similar. To me this view is very new and I had to take a night to think about it. I did always view least squares regression in terms of a projection but in this viewpoint I have never tried to realize an intuitive meaning to the part $(X^TX)^{-1}$ or I always saw it in the more indirect expression $X^T y = X^TX\beta$. $\endgroup$ – Martijn Weterings Aug 30 '18 at 12:54

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