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[HDDA21](https://statomics.github.io/HDDA21/index.html) - [About](https://statomics.github.io/HDDA21/about.html) - [Lectures](https://statomics.github.io/HDDA21/lsGeometricInterpretation.html) - [1\. Introduction](https://statomics.github.io/HDDA21/intro.html) - [2\. Singular Value Decomposition](https://statomics.github.io/HDDA21/svd.html) - [2\.3. Geometric Interpretation SVD](https://statomics.github.io/HDDA21/svdGeometricInterpretation.html) - [2\.7. Link MDS and Gram Distance Matrix](https://statomics.github.io/HDDA21/MDS_linkGramDistanceMatrix.html) - [3\. Prediction with High Dimensional Predictors](https://statomics.github.io/HDDA21/prediction.html) - [4\. Sparse Singular Value Decomposition](https://statomics.github.io/HDDA21/sparseSvd.html) - [5\. Linear Discriminant Analysis](https://statomics.github.io/HDDA21/lda.html) - [6\. Large Scale Inference](https://statomics.github.io/HDDA21/lsi.html) - [Paper 1: Model-based Clustering](https://sites.stat.washington.edu/people/raftery/Research/PDF/fraley1998.pdf) - [Paper 1: Intro Hierarchical Clustering](https://statomics.github.io/HDDA21/hclust.html) - [Paper 1: EM algorithm](https://statomics.github.io/HDDA21/em.html) - [Labs](https://statomics.github.io/HDDA21/lsGeometricInterpretation.html) - [Lab 1](https://statomics.github.io/HDDA21/Lab1-Intro-SVD.html) - [Lab 2](https://statomics.github.io/HDDA21/Lab2-PCA.html) - [Lab 3](https://statomics.github.io/HDDA21/Lab3-Penalized-Regression.html) - [Lab 4](https://statomics.github.io/HDDA21/Lab4-Sparse-PCA-LDA.html) - [Lab 5](https://statomics.github.io/HDDA21/Lab5-Large-Scale-Inference.html) - [statOmics](http://statomics.github.io/) Code - [Show All Code](https://statomics.github.io/HDDA21/lsGeometricInterpretation.html) - [Hide All Code](https://statomics.github.io/HDDA21/lsGeometricInterpretation.html) - [Download Rmd](https://statomics.github.io/HDDA21/lsGeometricInterpretation.html) # 1\. Introduction: Linear Regression - Geometric interpretation #### Lieven Clement #### statOmics, Ghent University (<https://statomics.github.io>) # 1 Intro ## 1\.1 Data Dataset with 3 observation (X,Y): ``` library(tidyverse) ``` ``` ## ── Attaching packages ─────────────────────────────────────── tidyverse 1.3.1 ── ``` ``` ## ✔ ggplot2 3.3.5 ✔ purrr 0.3.4 ## ✔ tibble 3.1.5 ✔ dplyr 1.0.7 ## ✔ tidyr 1.1.4 ✔ stringr 1.4.0 ## ✔ readr 2.0.1 ✔ forcats 0.5.1 ``` ``` ## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ── ## ✖ dplyr::filter() masks stats::filter() ## ✖ dplyr::lag() masks stats::lag() ``` ``` data <- data.frame(x=1:3,y=c(1,2,2)) data ``` ## 1\.2 Model \\\[ y\_i=\\beta\_0+\\beta\_1x + \\epsilon\_i \\\] If we write the model for each observation: \\\[ \\begin{array} {lcl} 1 &=& \\beta\_0+\\beta\_1 1 + \\epsilon\_1 \\\\ 2 &=& \\beta\_0+\\beta\_1 2 + \\epsilon\_2 \\\\ 2 &=& \\beta\_0+\\beta\_1 2 + \\epsilon\_3 \\\\ \\end{array} \\\] We can also write this in matrix form \\\[ \\mathbf{Y} = \\mathbf{X}\\boldsymbol{\\beta}+\\boldsymbol{\\epsilon} \\\] with \\\[ \\mathbf{Y}=\\left\[ \\begin{array}{c} 1\\\\ 2\\\\ 2\\\\ \\end{array}\\right\], \\quad \\mathbf{X}= \\left\[ \\begin{array}{cc} 1&1\\\\ 1&2\\\\ 1&3\\\\ \\end{array} \\right\], \\quad \\boldsymbol{\\beta} = \\left\[ \\begin{array}{c} \\beta\_0\\\\ \\beta\_1\\\\ \\end{array} \\right\] \\quad \\text{and} \\quad \\boldsymbol{\\epsilon}= \\left\[ \\begin{array}{c} \\epsilon\_1\\\\ \\epsilon\_2\\\\ \\epsilon\_3 \\end{array} \\right\] \\\] ``` lm1 <- lm(y~x,data) data$yhat <- lm1$fitted data %>% ggplot(aes(x,y)) + geom_point() + ylim(0,4) + xlim(0,4) + stat_smooth(method = "lm", color = "red", fullrange = TRUE) + geom_point(aes(x=x, y =yhat), pch = 2, size = 3, color = "red") + geom_segment(data = data, aes(x = x, xend = x, y = y, yend = yhat), lty = 2 ) ``` ``` ## `geom_smooth()` using formula 'y ~ x' ``` ``` ## Warning in max(ids, na.rm = TRUE): no non-missing arguments to max; returning ## -Inf ```  # 2 Least Squares (LS) - Minimize the residual sum of squares \\\[\\begin{eqnarray\*} RSS(\\boldsymbol{\\beta})&=&\\sum\\limits\_{i=1}^n e^2\_i\\\\ &=&\\sum\\limits\_{i=1}^n \\left(y\_i-\\beta\_0-\\sum\\limits\_{j=1}^p x\_{ij}\\beta\_j\\right)^2 \\end{eqnarray\*}\\\] - or in matrix notation \\\[\\begin{eqnarray\*} RSS(\\boldsymbol{\\beta})&=&(\\mathbf{Y}-\\mathbf{X\\beta})^T(\\mathbf{Y}-\\mathbf{X\\beta})\\\\ &=&\\Vert \\mathbf{Y}-\\mathbf{X\\beta}\\Vert^2\_2 \\end{eqnarray\*}\\\] with the \\(L\_2\\)\-norm of a \\(p\\)\-dim. vector \\(v\\) \\(\\Vert \\mathbf{v} \\Vert\_2=\\sqrt{v\_1^2+\\ldots+v\_p^2}\\) \\(\\rightarrow\\) \\(\\hat{\\boldsymbol{\\beta}}=\\text{argmin}\_\\beta \\Vert \\mathbf{Y}-\\mathbf{X\\beta}\\Vert^2\\) ## 2\.1 Minimize RSS \\\[ \\begin{array}{ccc} \\frac{\\partial RSS}{\\partial \\boldsymbol{\\beta}}&=&\\mathbf{0}\\\\\\\\ \\frac{(\\mathbf{Y}-\\mathbf{X\\beta})^T(\\mathbf{Y}-\\mathbf{X}\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}}&=&\\mathbf{0}\\\\\\\\ -2\\mathbf{X}^T(\\mathbf{Y}-\\mathbf{X}\\boldsymbol{\\beta})&=&\\mathbf{0}\\\\\\\\ \\mathbf{X}^T\\mathbf{X\\beta}&=&\\mathbf{X}^T\\mathbf{Y}\\\\\\\\ \\hat{\\boldsymbol{\\beta}}&=&(\\mathbf{X}^T\\mathbf{X})^{-1}\\mathbf{X}^T\\mathbf{Y} \\end{array} \\\] \\\[ \\hat{\\boldsymbol{\\beta}}=(\\mathbf{X}^T\\mathbf{X})^{-1}\\mathbf{X}^T\\mathbf{Y} \\\] ## 2\.2 Fitted values: \\\[ \\begin{array}{lcl} \\hat{\\mathbf{Y}} &=& \\mathbf{X}\\hat{\\boldsymbol{\\beta}}\\\\ &=& \\mathbf{X} (\\mathbf{X}^T\\mathbf{X})^{-1}\\mathbf{X}^T\\mathbf{Y}\\\\ \\end{array} \\\] # 3 Geometric interpretation of Linear regression There is also another picture to interpret linear regression\! Linear regression can also be seen as a projection\! Fitted values: \\\[ \\begin{array}{lcl} \\hat{\\mathbf{Y}} &=& \\mathbf{X}\\hat{\\boldsymbol{\\beta}}\\\\ &=& \\mathbf{X} (\\mathbf{X}^T\\mathbf{X})^{-1}\\mathbf{X}^T\\mathbf{Y}\\\\ &=& \\mathbf{HY} \\end{array} \\\] with \\(\\mathbf{H}\\) the projection matrix also referred to as the hat matrix. ``` X <- model.matrix(~x,data) X ``` ``` ## (Intercept) x ## 1 1 1 ## 2 1 2 ## 3 1 3 ## attr(,"assign") ## [1] 0 1 ``` ``` XtX <- t(X)%*%X XtX ``` ``` ## (Intercept) x ## (Intercept) 3 6 ## x 6 14 ``` ``` XtXinv <- solve(t(X)%*%X) XtXinv ``` ``` ## (Intercept) x ## (Intercept) 2.333333 -1.0 ## x -1.000000 0.5 ``` ``` H <- X %*% XtXinv %*% t(X) H ``` ``` ## 1 2 3 ## 1 0.8333333 0.3333333 -0.1666667 ## 2 0.3333333 0.3333333 0.3333333 ## 3 -0.1666667 0.3333333 0.8333333 ``` ``` Y <- data$y Yhat <- H%*%Y Yhat ``` ``` ## [,1] ## 1 1.166667 ## 2 1.666667 ## 3 2.166667 ``` ## 3\.1 What do these projections mean geometrically? The other picture to linear regression is to consider \\(X\_0\\), \\(X\_1\\) and \\(Y\\) as vectors in the space of the data \\(\\mathbb{R}^n\\), here \\(\\mathbb{R}^3\\) because we have three data points. ``` originRn <- data.frame(X1=0,X2=0,X3=0) data$x0 <- 1 dataRn <- data.frame(t(data)) library(plotly) ``` ``` ## ## Attaching package: 'plotly' ``` ``` ## The following object is masked from 'package:ggplot2': ## ## last_plot ``` ``` ## The following object is masked from 'package:stats': ## ## filter ``` ``` ## The following object is masked from 'package:graphics': ## ## layout ``` ``` p1 <- plot_ly( originRn, x = ~ X1, y = ~ X2, z= ~ X3) %>% add_markers(type="scatter3d") %>% layout( scene = list( aspectmode="cube", xaxis = list(range=c(-4,4)), yaxis = list(range=c(-4,4)), zaxis = list(range=c(-4,4)) ) ) p1 <- p1 %>% add_trace( x = c(0,1), y = c(0,0), z = c(0,0), mode = "lines", line = list(width = 5, color = "grey"), type="scatter3d") %>% add_trace( x = c(0,0), y = c(0,1), z = c(0,0), mode = "lines", line = list(width = 5, color = "grey"), type="scatter3d") %>% add_trace( x = c(0,0), y = c(0,0), z = c(0,1), mode = "lines", line = list(width = 5, color = "grey"), type="scatter3d") %>% add_trace( x = c(0,1), y = c(0,1), z = c(0,1), mode = "lines", line = list(width = 5, color = "black"), type="scatter3d") %>% add_trace( x = c(0,1), y = c(0,2), z = c(0,3), mode = "lines", line = list(width = 5, color = "black"), type="scatter3d") ``` ``` p2 <- p1 %>% add_trace( x = c(0,Y[1]), y = c(0,Y[2]), z = c(0,Y[3]), mode = "lines", line = list(width = 5, color = "red"), type="scatter3d") %>% add_trace( x = c(0,Yhat[1]), y = c(0,Yhat[2]), z = c(0,Yhat[3]), mode = "lines", line = list(width = 5, color = "red"), type="scatter3d") %>% add_trace( x = c(Y[1],Yhat[1]), y = c(Y[2],Yhat[2]), z = c(Y[3],Yhat[3]), mode = "lines", line = list(width = 5, color = "red", dash="dash"), type="scatter3d") p2 ``` ### 3\.1.1 How does this projection work? \\\[ \\begin{array}{lcl} \\hat{\\mathbf{Y}} &=& \\mathbf{X} (\\mathbf{X}^T\\mathbf{X})^{-1}\\mathbf{X}^T\\mathbf{Y}\\\\ &=& \\mathbf{X}(\\mathbf{X}^T\\mathbf{X})^{-1/2}(\\mathbf{X}^T\\mathbf{X})^{-1/2}\\mathbf{X}^T\\mathbf{Y}\\\\ &=& \\mathbf{U}\\mathbf{U}^T\\mathbf{Y} \\end{array} \\\] - \\(\\mathbf{U}\\) is a new orthonormal basis in \\(\\mathbb{R}^2\\), a subspace of \\(\\mathbb{R}^3\\) - The space spanned by U and X is the column space of X, e.g. it contains all possible linear combinantions of X. \\(\\mathbf{U}^t\\mathbf{Y}\\) is the projection of Y on this new orthonormal basis ``` eigenXtX <- eigen(XtX) XtXinvSqrt <- eigenXtX$vectors %*%diag(1/eigenXtX$values^.5)%*%t(eigenXtX$vectors) U <- X %*% XtXinvSqrt p3 <- p1 %>% add_trace( x = c(0,U[1,1]), y = c(0,U[2,1]), z = c(0,U[3,1]), mode = "lines", line = list(width = 5, color = "blue"), type="scatter3d") %>% add_trace( x = c(0,U[1,2]), y = c(0,U[2,2]), z = c(0,U[3,2]), mode = "lines", line = list(width = 5, color = "blue"), type="scatter3d") p3 ``` - \\(\\mathbf{U}^T\\mathbf{Y}\\) is the projection of \\(\\mathbf{Y}\\) in the space spanned by \\(\\mathbf{U}\\). - Indeed \\(\\mathbf{U}\_1^T\\mathbf{Y}\\) ``` p4 <- p3 %>% add_trace( x = c(0,Y[1]), y = c(0,Y[2]), z = c(0,Y[3]), mode = "lines", line = list(width = 5, color = "red"), type="scatter3d") %>% add_trace( x = c(0,U[1,1]*(U[,1]%*%Y)), y = c(0,U[2,1]*(U[,1]%*%Y)), z = c(0,U[3,1]*(U[,1]%*%Y)), mode = "lines", line = list(width = 5, color = "red",dash="dash"), type="scatter3d") %>% add_trace( x = c(Y[1],U[1,1]*(U[,1]%*%Y)), y = c(Y[2],U[2,1]*(U[,1]%*%Y)), z = c(Y[3],U[3,1]*(U[,1]%*%Y)), mode = "lines", line = list(width = 5, color = "red", dash="dash"), type="scatter3d") p4 ``` - and \\(\\mathbf{U}\_2^T\\mathbf{Y}\\) ``` p5 <- p4 %>% add_trace( x = c(0,Y[1]), y = c(0,Y[2]), z = c(0,Y[3]), mode = "lines", line = list(width = 5, color = "red"), type="scatter3d") %>% add_trace( x = c(0,U[1,2]*(U[,2]%*%Y)), y = c(0,U[2,2]*(U[,2]%*%Y)), z = c(0,U[3,2]*(U[,2]%*%Y)), mode = "lines", line = list(width = 5, color = "red",dash="dash"), type="scatter3d") %>% add_trace( x = c(Y[1],U[1,2]*(U[,2]%*%Y)), y = c(Y[2],U[2,2]*(U[,2]%*%Y)), z = c(Y[3],U[3,2]*(U[,2]%*%Y)), mode = "lines", line = list(width = 5, color = "red", dash="dash"), type="scatter3d") p5 ``` ``` p6 <- p5 %>% add_trace( x = c(0,Yhat[1]), y = c(0,Yhat[2]), z = c(0,Yhat[3]), mode = "lines", line = list(width = 5, color = "red"), type="scatter3d") %>% add_trace( x = c(Y[1],Yhat[1]), y = c(Y[2],Yhat[2]), z = c(Y[3],Yhat[3]), mode = "lines", line = list(width = 5, color = "red", dash="dash"), type="scatter3d") %>% add_trace( x = c(U[1,1]*(U[,1]%*%Y),Yhat[1]), y = c(U[2,1]*(U[,1]%*%Y),Yhat[2]), z = c(U[3,1]*(U[,1]%*%Y),Yhat[3]), mode = "lines", line = list(width = 5, color = "red", dash="dash"), type="scatter3d") %>% add_trace( x = c(U[1,2]*(U[,2]%*%Y),Yhat[1]), y = c(U[2,2]*(U[,2]%*%Y),Yhat[2]), z = c(U[3,2]*(U[,2]%*%Y),Yhat[3]), mode = "lines", line = list(width = 5, color = "red", dash="dash"), type="scatter3d") p6 ``` ## 3\.2 The Error vector Note, that it is also clear from the equation in the derivation of the least squares solution that the residual is orthogonal on the column space: \\\[ -2 \\mathbf{X}^T(\\mathbf{Y}-\\mathbf{X}\\boldsymbol{\\beta}) = 0 \\\] # 4 Curse of dimensionality? - Imagine what happens when p approaches n \\(p=n\\) or becomes much larger than p \>\> n!!\! - Suppose that we add a predictor \\(\\mathbf{X}\_2 = \[2,0,1\]^T\\)? \\\[ \\mathbf{Y}=\\left\[ \\begin{array}{c} 1\\\\ 2\\\\ 2\\\\ \\end{array}\\right\], \\quad \\mathbf{X}= \\left\[ \\begin{array}{ccc} 1&1&2\\\\ 1&2&0\\\\ 1&3&1\\\\ \\end{array} \\right\], \\quad \\boldsymbol{\\beta} = \\left\[ \\begin{array}{c} \\beta\_0\\\\ \\beta\_1\\\\ \\beta\_2 \\end{array} \\right\] \\quad \\text{and} \\quad \\boldsymbol{\\epsilon}= \\left\[ \\begin{array}{c} \\epsilon\_1\\\\ \\epsilon\_2\\\\ \\epsilon\_3 \\end{array} \\right\] \\\] ``` data$x2 <- c(2,0,1) fit <- lm(y~x+x2,data) # predict values on regular xy grid x1pred <- seq(-1, 4, length.out = 10) x2pred <- seq(-1, 4, length.out = 10) xy <- expand.grid(x = x1pred, x2 = x2pred) ypred <- matrix (nrow = 30, ncol = 30, data = predict(fit, newdata = data.frame(xy))) library(plot3D) ``` ``` ## Warning: no DISPLAY variable so Tk is not available ``` ``` # fitted points for droplines to surface th=20 ph=5 scatter3D(data$x, data$x2, Y, pch = 16, col="darkblue", cex = 1, theta = th, ticktype = "detailed", xlab = "x1", ylab = "x2", zlab = "y", colvar=FALSE, bty = "g", xlim=c(-1,3), ylim=c(-1,3), zlim=c(-2,4)) ```  ``` z.pred3D <- outer( x1pred, x2pred, function(x1,x2) { fit$coef[1] + fit$coef[2]*x1+fit$coef[2]*x2 }) x.pred3D <- outer( x1pred, x2pred, function(x,y) x) y.pred3D <- outer( x1pred, x2pred, function(x,y) y) scatter3D(data$x, data$x2, data$y, pch = 16, col="darkblue", cex = 1, theta = th, ticktype = "detailed", xlab = "x1", ylab = "x2", zlab = "y", colvar=FALSE, bty = "g", xlim=c(-1,4), ylim=c(-1,4), zlim=c(-2,4)) surf3D( x.pred3D, y.pred3D, z.pred3D, col="blue", facets=NA, add=TRUE) ```  Note, that the linear regression is now a plane. However, we obtain a perfect fit and all the data points are falling in the plane! 😱 This is obvious if we look at the column space of X\! ``` X <- cbind(X,c(2,0,1)) XtX <- t(X)%*%X eigenXtX <- eigen(XtX) XtXinvSqrt <- eigenXtX$vectors %*%diag(1/eigenXtX$values^.5)%*%t(eigenXtX$vectors) U <- X %*% XtXinvSqrt p7 <- p1 %>% add_trace( x = c(0,2), y = c(0,0), z = c(0,1), mode = "lines", line = list(width = 5, color = "darkgreen"), type="scatter3d") p7 ``` ``` p8 <- p7 %>% add_trace( x = c(0,U[1,1]), y = c(0,U[2,1]), z = c(0,U[3,1]), mode = "lines", line = list(width = 5, color = "blue"), type="scatter3d") %>% add_trace( x = c(0,U[1,2]), y = c(0,U[2,2]), z = c(0,U[3,2]), mode = "lines", line = list(width = 5, color = "blue"), type="scatter3d") %>% add_trace( x = c(0,U[1,3]), y = c(0,U[2,3]), z = c(0,U[3,3]), mode = "lines", line = list(width = 5, color = "blue"), type="scatter3d") p8 ``` - The column space now spans the entire \\(\\mathbb{R}^3\\)\! - With the intercept and the two predictors we can thus fit every dataset that only has 3 observations for the predictors and the response. - So the model can no longer be used to generalise the patterns seen in the data towards the population (new observations). - Problem of overfitting!!\! - If \\(p \>\> n\\) then the problem gets even worse! Then there is even no longer a unique solution to the least squares problem… ---
title: '1. Introduction: Linear Regression - Geometric interpretation'
author: "Lieven Clement"
date: "statOmics, Ghent University (https://statomics.github.io)"
always_allow_html: yes
---

# Intro
## Data

Dataset with 3 observation (X,Y):

```{r}
library(tidyverse)
data <- data.frame(x=1:3,y=c(1,2,2))
data
```

## Model

$$
y_i=\beta_0+\beta_1x + \epsilon_i
$$

If we write the model for each observation:

$$
\begin{array} {lcl}
1 &=& \beta_0+\beta_1 1 + \epsilon_1 \\
2 &=& \beta_0+\beta_1 2 + \epsilon_2 \\
2 &=& \beta_0+\beta_1 2 + \epsilon_3 \\
\end{array}
$$

We can also write this in matrix form

$$
\mathbf{Y} = \mathbf{X}\boldsymbol{\beta}+\boldsymbol{\epsilon}
$$

with

$$
\mathbf{Y}=\left[
\begin{array}{c}
1\\
2\\
2\\
\end{array}\right],
\quad
\mathbf{X}= \left[
\begin{array}{cc}
1&1\\
1&2\\
1&3\\
\end{array}
\right],
\quad \boldsymbol{\beta} = \left[
\begin{array}{c}
\beta_0\\
\beta_1\\
\end{array}
\right]
\quad
\text{and}
\quad
\boldsymbol{\epsilon}=
\left[
\begin{array}{c}
\epsilon_1\\
\epsilon_2\\
\epsilon_3
\end{array}
\right]
$$

```{r}
lm1 <- lm(y~x,data)
data$yhat <- lm1$fitted

data %>%
  ggplot(aes(x,y)) +
  geom_point() +
  ylim(0,4) +
  xlim(0,4) +
  stat_smooth(method = "lm", color = "red", fullrange = TRUE) +
  geom_point(aes(x=x, y =yhat), pch = 2, size = 3, color = "red") +
  geom_segment(data = data, aes(x = x, xend = x, y = y, yend = yhat), lty = 2 )
```

# Least Squares (LS)

- Minimize the residual sum of squares
\begin{eqnarray*}
RSS(\boldsymbol{\beta})&=&\sum\limits_{i=1}^n e^2_i\\
&=&\sum\limits_{i=1}^n \left(y_i-\beta_0-\sum\limits_{j=1}^p x_{ij}\beta_j\right)^2
\end{eqnarray*}

- or in matrix notation
\begin{eqnarray*}
RSS(\boldsymbol{\beta})&=&(\mathbf{Y}-\mathbf{X\beta})^T(\mathbf{Y}-\mathbf{X\beta})\\
&=&\Vert \mathbf{Y}-\mathbf{X\beta}\Vert^2_2
\end{eqnarray*}
with the $L_2$-norm of a $p$-dim. vector $v$ $\Vert \mathbf{v} \Vert_2=\sqrt{v_1^2+\ldots+v_p^2}$
$\rightarrow$ $\hat{\boldsymbol{\beta}}=\text{argmin}_\beta \Vert \mathbf{Y}-\mathbf{X\beta}\Vert^2$


## Minimize RSS
\[
\begin{array}{ccc}
\frac{\partial RSS}{\partial \boldsymbol{\beta}}&=&\mathbf{0}\\\\
\frac{(\mathbf{Y}-\mathbf{X\beta})^T(\mathbf{Y}-\mathbf{X}\boldsymbol{\beta})}{\partial \boldsymbol{\beta}}&=&\mathbf{0}\\\\
-2\mathbf{X}^T(\mathbf{Y}-\mathbf{X}\boldsymbol{\beta})&=&\mathbf{0}\\\\
\mathbf{X}^T\mathbf{X\beta}&=&\mathbf{X}^T\mathbf{Y}\\\\
\hat{\boldsymbol{\beta}}&=&(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{Y}
\end{array}
\]

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

## Fitted values:

$$
\begin{array}{lcl}
\hat{\mathbf{Y}} &=& \mathbf{X}\hat{\boldsymbol{\beta}}\\
&=& \mathbf{X} (\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{Y}\\
\end{array}
$$

# Geometric interpretation of Linear regression

There is also another picture to interpret linear regression!

Linear regression can also be seen as a projection!

Fitted values:

$$
\begin{array}{lcl}
\hat{\mathbf{Y}} &=& \mathbf{X}\hat{\boldsymbol{\beta}}\\
&=& \mathbf{X} (\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{Y}\\
&=& \mathbf{HY}
\end{array}
$$
with $\mathbf{H}$ the projection matrix also referred to as the hat matrix.


```{r}
X <- model.matrix(~x,data)
X
```

```{r}
XtX <- t(X)%*%X
XtX
```

```{r}
XtXinv <- solve(t(X)%*%X)
XtXinv
```

```{r}
H <- X %*% XtXinv %*% t(X)
H
```


```{r}
Y <- data$y
Yhat <- H%*%Y
Yhat
```

## What do these projections mean geometrically?

The other picture to linear regression is to consider $X_0$, $X_1$ and $Y$ as vectors in the space of the data $\mathbb{R}^n$, here $\mathbb{R}^3$ because we have three data points.


```{r}
originRn <- data.frame(X1=0,X2=0,X3=0)
data$x0 <- 1
dataRn <- data.frame(t(data))

library(plotly)

p1 <- plot_ly(
    originRn,
    x = ~ X1,
    y = ~ X2,
    z= ~ X3) %>%
  add_markers(type="scatter3d") %>%
  layout(
    scene = list(
      aspectmode="cube",
      xaxis = list(range=c(-4,4)), yaxis = list(range=c(-4,4)), zaxis = list(range=c(-4,4))
      )
    )
p1 <- p1 %>%
  add_trace(
    x = c(0,1),
    y = c(0,0),
    z = c(0,0),
    mode = "lines",
    line = list(width = 5, color = "grey"),
    type="scatter3d") %>%
  add_trace(
    x = c(0,0),
    y = c(0,1),
    z = c(0,0),
    mode = "lines",
    line = list(width = 5, color = "grey"),
    type="scatter3d") %>%
  add_trace(
    x = c(0,0),
    y = c(0,0),
    z = c(0,1),
    mode = "lines",
    line = list(width = 5, color = "grey"),
    type="scatter3d") %>%
  add_trace(
    x = c(0,1),
    y = c(0,1),
    z = c(0,1),
    mode = "lines",
    line = list(width = 5, color = "black"),
    type="scatter3d") %>%
    add_trace(
    x = c(0,1),
    y = c(0,2),
    z = c(0,3),
    mode = "lines",
    line = list(width = 5, color = "black"),
    type="scatter3d")
```
```{r}
p2 <- p1 %>%
  add_trace(
    x = c(0,Y[1]),
    y = c(0,Y[2]),
    z = c(0,Y[3]),
    mode = "lines",
    line = list(width = 5, color = "red"),
    type="scatter3d") %>%
  add_trace(
    x = c(0,Yhat[1]),
    y = c(0,Yhat[2]),
    z = c(0,Yhat[3]),
    mode = "lines",
    line = list(width = 5, color = "red"),
    type="scatter3d") %>% add_trace(
    x = c(Y[1],Yhat[1]),
    y = c(Y[2],Yhat[2]),
    z = c(Y[3],Yhat[3]),
    mode = "lines",
    line = list(width = 5, color = "red", dash="dash"),
    type="scatter3d")
p2
```


### How does this projection work?

$$
\begin{array}{lcl}
\hat{\mathbf{Y}} &=& \mathbf{X} (\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{Y}\\
&=& \mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1/2}(\mathbf{X}^T\mathbf{X})^{-1/2}\mathbf{X}^T\mathbf{Y}\\
&=& \mathbf{U}\mathbf{U}^T\mathbf{Y}
\end{array}
$$


- $\mathbf{U}$ is a new orthonormal basis in $\mathbb{R}^2$, a subspace of $\mathbb{R}^3$

- The space spanned by U and X is the column space of X, e.g. it contains all possible linear combinantions of X.
$\mathbf{U}^t\mathbf{Y}$ is the projection of Y on this new orthonormal basis

```{r}
eigenXtX <- eigen(XtX)
XtXinvSqrt <- eigenXtX$vectors %*%diag(1/eigenXtX$values^.5)%*%t(eigenXtX$vectors)
U <- X %*% XtXinvSqrt

p3 <- p1 %>%
  add_trace(
    x = c(0,U[1,1]),
    y = c(0,U[2,1]),
    z = c(0,U[3,1]),
    mode = "lines",
    line = list(width = 5, color = "blue"),
    type="scatter3d") %>%
  add_trace(
    x = c(0,U[1,2]),
    y = c(0,U[2,2]),
    z = c(0,U[3,2]),
    mode = "lines",
    line = list(width = 5, color = "blue"),
    type="scatter3d")

p3
```


- $\mathbf{U}^T\mathbf{Y}$ is the projection of $\mathbf{Y}$ in the space spanned by $\mathbf{U}$.
- Indeed $\mathbf{U}_1^T\mathbf{Y}$

```{r}
p4 <- p3 %>%
  add_trace(
    x = c(0,Y[1]),
    y = c(0,Y[2]),
    z = c(0,Y[3]),
    mode = "lines",
    line = list(width = 5, color = "red"),
    type="scatter3d") %>%
  add_trace(
    x = c(0,U[1,1]*(U[,1]%*%Y)),
    y = c(0,U[2,1]*(U[,1]%*%Y)),
    z = c(0,U[3,1]*(U[,1]%*%Y)),
    mode = "lines",
    line = list(width = 5, color = "red",dash="dash"),
    type="scatter3d") %>% add_trace(
    x = c(Y[1],U[1,1]*(U[,1]%*%Y)),
    y = c(Y[2],U[2,1]*(U[,1]%*%Y)),
    z = c(Y[3],U[3,1]*(U[,1]%*%Y)),
    mode = "lines",
    line = list(width = 5, color = "red", dash="dash"),
    type="scatter3d")
p4
```

- and $\mathbf{U}_2^T\mathbf{Y}$
```{r}
p5 <- p4 %>%
  add_trace(
    x = c(0,Y[1]),
    y = c(0,Y[2]),
    z = c(0,Y[3]),
    mode = "lines",
    line = list(width = 5, color = "red"),
    type="scatter3d") %>%
  add_trace(
    x = c(0,U[1,2]*(U[,2]%*%Y)),
    y = c(0,U[2,2]*(U[,2]%*%Y)),
    z = c(0,U[3,2]*(U[,2]%*%Y)),
    mode = "lines",
    line = list(width = 5, color = "red",dash="dash"),
    type="scatter3d") %>% add_trace(
    x = c(Y[1],U[1,2]*(U[,2]%*%Y)),
    y = c(Y[2],U[2,2]*(U[,2]%*%Y)),
    z = c(Y[3],U[3,2]*(U[,2]%*%Y)),
    mode = "lines",
    line = list(width = 5, color = "red", dash="dash"),
    type="scatter3d")
p5
```

```{r}
p6 <- p5 %>%
  add_trace(
    x = c(0,Yhat[1]),
    y = c(0,Yhat[2]),
    z = c(0,Yhat[3]),
    mode = "lines",
    line = list(width = 5, color = "red"),
    type="scatter3d") %>%
  add_trace(
    x = c(Y[1],Yhat[1]),
    y = c(Y[2],Yhat[2]),
    z = c(Y[3],Yhat[3]),
    mode = "lines",
    line = list(width = 5, color = "red", dash="dash"),
    type="scatter3d") %>%
  add_trace(
    x = c(U[1,1]*(U[,1]%*%Y),Yhat[1]),
    y = c(U[2,1]*(U[,1]%*%Y),Yhat[2]),
    z = c(U[3,1]*(U[,1]%*%Y),Yhat[3]),
    mode = "lines",
    line = list(width = 5, color = "red", dash="dash"),
    type="scatter3d")  %>%
  add_trace(
    x = c(U[1,2]*(U[,2]%*%Y),Yhat[1]),
    y = c(U[2,2]*(U[,2]%*%Y),Yhat[2]),
    z = c(U[3,2]*(U[,2]%*%Y),Yhat[3]),
    mode = "lines",
    line = list(width = 5, color = "red", dash="dash"),
    type="scatter3d")
p6
```

## The Error vector

Note, that it is also clear from the equation in the derivation of the least squares solution that the residual is orthogonal on the column space:

\[
 -2 \mathbf{X}^T(\mathbf{Y}-\mathbf{X}\boldsymbol{\beta}) = 0
\]


# Curse of dimensionality?

- Imagine what happens when p approaches n $p=n$ or becomes much larger than p >> n!!!

- Suppose that we add a predictor $\mathbf{X}_2 = [2,0,1]^T$?

$$
\mathbf{Y}=\left[
\begin{array}{c}
1\\
2\\
2\\
\end{array}\right],
\quad
\mathbf{X}= \left[
\begin{array}{ccc}
1&1&2\\
1&2&0\\
1&3&1\\
\end{array}
\right],
\quad \boldsymbol{\beta} = \left[
\begin{array}{c}
\beta_0\\
\beta_1\\
\beta_2
\end{array}
\right]
\quad
\text{and}
\quad
\boldsymbol{\epsilon}=
\left[
\begin{array}{c}
\epsilon_1\\
\epsilon_2\\
\epsilon_3
\end{array}
\right]
$$


```{r}
data$x2 <- c(2,0,1)
fit <- lm(y~x+x2,data)
# predict values on regular xy grid
x1pred <- seq(-1, 4, length.out = 10)
x2pred <- seq(-1, 4, length.out = 10)
xy <- expand.grid(x = x1pred,
x2 = x2pred)
ypred <- matrix (nrow = 30, ncol = 30,
data = predict(fit, newdata = data.frame(xy)))

library(plot3D)


# fitted points for droplines to surface
th=20
ph=5
scatter3D(data$x,
  data$x2,
  Y,
  pch = 16,
  col="darkblue",
  cex = 1,
  theta = th,
  ticktype = "detailed",
  xlab = "x1",
  ylab = "x2",
  zlab = "y",
  colvar=FALSE,
  bty = "g",
  xlim=c(-1,3),
  ylim=c(-1,3),
  zlim=c(-2,4))


z.pred3D <- outer(
  x1pred,
  x2pred,
  function(x1,x2)
  {
    fit$coef[1] + fit$coef[2]*x1+fit$coef[2]*x2
  })

x.pred3D <- outer(
  x1pred,
  x2pred,
  function(x,y) x)

y.pred3D <- outer(
  x1pred,
  x2pred,
  function(x,y) y)

scatter3D(data$x,
  data$x2,
  data$y,
  pch = 16,
  col="darkblue",
  cex = 1,
  theta = th,
  ticktype = "detailed",
  xlab = "x1",
  ylab = "x2",
  zlab = "y",
  colvar=FALSE,
  bty = "g",
  xlim=c(-1,4),
  ylim=c(-1,4),
  zlim=c(-2,4))

surf3D(
  x.pred3D,
  y.pred3D,
  z.pred3D,
  col="blue",
  facets=NA,
  add=TRUE)
```

Note, that the linear regression is now a plane.

However, we obtain a perfect fit and all the data points are falling in the plane! `r set.seed(4);emo::ji("fear")`

This is obvious if we look at the column space of X!

```{r}
X <- cbind(X,c(2,0,1))
XtX <- t(X)%*%X
eigenXtX <- eigen(XtX)
XtXinvSqrt <- eigenXtX$vectors %*%diag(1/eigenXtX$values^.5)%*%t(eigenXtX$vectors)
U <- X %*% XtXinvSqrt

p7 <- p1 %>%
  add_trace(
    x = c(0,2),
    y = c(0,0),
    z = c(0,1),
    mode = "lines",
    line = list(width = 5, color = "darkgreen"),
    type="scatter3d")
p7
```

```{r}
p8 <- p7 %>%
  add_trace(
    x = c(0,U[1,1]),
    y = c(0,U[2,1]),
    z = c(0,U[3,1]),
    mode = "lines",
    line = list(width = 5, color = "blue"),
    type="scatter3d") %>%
  add_trace(
    x = c(0,U[1,2]),
    y = c(0,U[2,2]),
    z = c(0,U[3,2]),
    mode = "lines",
    line = list(width = 5, color = "blue"),
    type="scatter3d") %>%
  add_trace(
    x = c(0,U[1,3]),
    y = c(0,U[2,3]),
    z = c(0,U[3,3]),
    mode = "lines",
    line = list(width = 5, color = "blue"),
    type="scatter3d")

p8
```

- The column space now spans the entire  $\mathbb{R}^3$!
- With the intercept and the two predictors we can thus fit every dataset that only has 3 observations for the predictors and the response.
- So the model can no longer be used to generalise the patterns seen in the data towards the population (new observations).
- Problem of overfitting!!!

- If $p >> n$ then the problem gets even worse! Then there is even no longer a unique solution to the least squares problem...
 *** This work is licensed under the [CC BY-NC-SA 4.0](https://creativecommons.org/licenses/by-nc-sa/4.0) licence.