pca2 edits

pca_1/pca_1.qmd

@@ -284,10 +284,11 @@ $$\begin{aligned}
 
 This result implies that:
 
-* $w$ is a **unitary eigenvector** of the covariance matrix. 
+* $w$ is a **unitary eigenvector** of the covariance matrix. This means that $||w||^2 = ww^T = 1$
 * The error is minimized when $w$ is the eigenvector with the largest eigenvalue $\lambda$.
 
 This derivation can inductively be used for the next (second) principal component (not shown).
 
-The final takeaway from this derivation is that the **principal components** are the **eigenvectors** with the **largest eigenvalues** of the **covariance matrix**. These are the **directions** of the **maximum variance** of the data.
+The final takeaway from this derivation is that the **principal components** are the **eigenvectors** with the **largest eigenvalues** of the **covariance matrix**. These are the **directions** of the **maximum variance** of the data. We can construct the **latent factors (the Z matrix)** by **projecting** the centered data X onto the principal component vectors:
 
+<center><img src = "images/optimization_takeaways.png" width="450vw"></center>

pca_2/data/rectangle_data.csv

@@ -0,0 +1,101 @@
+width,height,area,perimeter
+8,6,48,28
+2,4,8,12
+1,3,3,8
+9,3,27,24
+9,8,72,34
+3,1,3,8
+4,2,8,12
+6,5,30,22
+7,1,7,16
+8,2,16,20
+5,5,25,20
+9,5,45,28
+8,4,32,24
+1,2,2,6
+2,9,18,22
+7,8,56,30
+7,5,35,24
+2,4,8,12
+2,5,10,14
+4,3,12,14
+8,1,8,18
+8,5,40,26
+9,8,72,34
+7,3,21,20
+8,3,24,22
+6,6,36,24
+6,4,24,20
+8,4,32,24
+7,2,14,18
+4,4,16,16
+4,1,4,10
+9,7,63,32
+4,2,8,12
+1,2,2,6
+8,1,8,18
+5,4,20,18
+9,5,45,28
+6,9,54,30
+3,7,21,20
+7,3,21,20
+7,7,49,28
+3,3,9,12
+5,5,25,20
+7,3,21,20
+8,2,16,20
+2,6,12,16
+7,1,7,16
+1,2,2,6
+3,5,15,16
+2,6,12,16
+5,6,30,22
+9,1,9,20
+6,3,18,18
+2,6,12,16
+6,8,48,28
+9,1,9,20
+2,8,16,20
+3,7,21,20
+6,5,30,22
+8,6,48,28
+9,2,18,22
+3,9,27,24
+2,5,10,14
+1,4,4,10
+3,3,9,12
+2,9,18,22
+6,4,24,20
+5,2,10,14
+2,6,12,16
+2,5,10,14
+1,3,3,8
+1,3,3,8
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+5,5,25,20
+3,9,27,24
+2,8,16,20
+2,2,4,8
+8,3,24,22
+1,8,8,18
+5,1,5,12
+5,7,35,24
+9,5,45,28
+3,1,3,8
+9,7,63,32
+8,5,40,26
+9,8,72,34
+4,5,20,18
+4,5,20,18
+1,5,5,12
+6,6,36,24
+8,5,40,26
+8,7,56,30
+1,4,4,10
+1,6,6,14
+2,6,12,16

probability_1/probability_1.qmd

@@ -381,7 +381,9 @@ $$ \begin{align}
 :::
 
 ### Covariance and Correlation
-We define the **covariance** of two random variables as the expected product of deviations from expectation. Put more simply, covariance is a generalization of variance to variance: $\text{Cov}(X, X) = \mathbb{E}[(X - \mathbb{E}[X])^2] = \text{Var}(X)$
+We define the **covariance** of two random variables as the expected product of deviations from expectation. Put more simply, covariance is a generalization of variance to variance: 
+
+$$\text{Cov}(X, X) = \mathbb{E}[(X - \mathbb{E}[X])^2] = \text{Var}(X)$$
 
 $$\text{Cov}(X, Y) = \mathbb{E}[(X - \mathbb{E}[X])(Y - \mathbb{E}[Y])]$$