From 2702a8cd2613179a595214df7457ac0f4d7bb61a Mon Sep 17 00:00:00 2001
From: Lillian Weng <lillianweng@berkeley.edu>
Date: Tue, 23 Apr 2024 12:06:50 -0700
Subject: [PATCH] fix pca1 note

---
 docs/pca_1/images/dataset3_outlier.png     | Bin 0 -> 28258 bytes
 docs/pca_1/pca_1.html                      |  25 ++++++++++++---------
 docs/search.json                           |   4 ++--
 docs/site_libs/bootstrap/bootstrap.min.css |   2 +-
 pca_1/pca_1.qmd                            |   2 ++
 5 files changed, 19 insertions(+), 14 deletions(-)
 create mode 100644 docs/pca_1/images/dataset3_outlier.png

diff --git a/docs/pca_1/images/dataset3_outlier.png b/docs/pca_1/images/dataset3_outlier.png
new file mode 100644
index 0000000000000000000000000000000000000000..a5d8fd0455e26029dc6d1ab214d4a7a9f0e7763b
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diff --git a/docs/pca_1/pca_1.html b/docs/pca_1/pca_1.html
index 16733a17..54f1f20c 100644
--- a/docs/pca_1/pca_1.html
+++ b/docs/pca_1/pca_1.html
@@ -106,7 +106,7 @@
 require.undef("plotly");
 requirejs.config({
     paths: {
-        'plotly': ['https://cdn.plot.ly/plotly-2.25.2.min']
+        'plotly': ['https://cdn.plot.ly/plotly-2.30.0.min']
     }
 });
 require(['plotly'], function(Plotly) {
@@ -495,9 +495,9 @@ <h2 data-number="24.1" class="anchored" data-anchor-id="visualization-revisited"
 </details>
 <div class="cell-output cell-output-display">
 
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@@ -709,6 +709,9 @@ <h2 data-number="24.2" class="anchored" data-anchor-id="dimensionality"><span cl
 <img src="images/dataset3.png" width="400vw">
 </center>
 <p>What happens when there are outliers? Below, we’ve added one outlier point to the dataset above, and just that one point is enough to change the rank of the matrix from 1 to 2 dimensions. However, the data is still <em>approximately</em> 1-dimensional.</p>
+<center>
+<img src="images/dataset3_outlier.png" width="400vw">
+</center>
 <p><strong>Dimensionality reduction</strong> is generally an <strong>approximation of the original data</strong> that’s achieved by <strong>projecting</strong> the data onto a desired dimension. In the example below, our original datapoints (blue dots) are 2-dimensional. We have a few choices if we want to project them down to 1-dimension: project them onto the <span class="math inline">\(x\)</span>-axis (left), project them onto the <span class="math inline">\(y\)</span>-axis (middle), or project them to a line <span class="math inline">\(mx + b\)</span> (right). The resulting datapoints after the projection is shown in red. Which projection do you think is better? How can we calculate that?</p>
 <center>
 <img src="images/diff_reductions.png" width="800vw">
diff --git a/docs/search.json b/docs/search.json
index 3b8bc294..88c177d7 100644
--- a/docs/search.json
+++ b/docs/search.json
@@ -809,7 +809,7 @@
     "href": "pca_1/pca_1.html#dimensionality",
     "title": "24  PCA I",
     "section": "24.2 Dimensionality",
-    "text": "24.2 Dimensionality\nSuppose we have a dataset of:\n\n\\(N\\) observations (datapoints/rows)\n\\(d\\) attributes (features/columns)\n\nLet’s “rename” this in terms of linear algebra so that we can be more clear with our wording. Using linear algebra, we can view our matrix as:\n\n\\(N\\) row vectors in a \\(d\\)-Dimensional space, OR\n\\(d\\) column vectors in an \\(N\\)-Dimensions space\n\nThe intrinsic dimension of a dataset is the minimal set of dimensions needed to approximately represent the data. In linear algebra terms, it is the dimension of the column space of a matrix, or the number of linearly independent columns in a matrix; this is equivalently called the rank of a matrix.\nIn the examples below, Dataset 1 has 2 dimensions because it has 2 linearly independent columns. Similarly, Dataset 2 has 3 dimensions because it has 3 linearly independent columns.\n\n\n\n\nWhat about Dataset 4 below?\n\n\n\nIt may be tempting to say that it has 4 dimensions, but the Weight (lbs) column is actually just a linear transformation of the Weight (kg) column. Thus, no new information is captured, and the matrix of our dataset has a (column) rank of 3! Therefore, despite having 4 columns, we still say that this data is 3-dimensional.\nPlotting the weight columns together reveals the key visual intuition. While the two columns visually span a 2D space as a line, the data does not deviate at all from that singular line. This means that one of the weight columns is redundant! Even given the option to cover the whole 2D space, the data below does not. It might as well not have this dimension, which is why we still do not consider the data below to span more than 1 dimension.\n\n\n\nWhat happens when there are outliers? Below, we’ve added one outlier point to the dataset above, and just that one point is enough to change the rank of the matrix from 1 to 2 dimensions. However, the data is still approximately 1-dimensional.\nDimensionality reduction is generally an approximation of the original data that’s achieved by projecting the data onto a desired dimension. In the example below, our original datapoints (blue dots) are 2-dimensional. We have a few choices if we want to project them down to 1-dimension: project them onto the \\(x\\)-axis (left), project them onto the \\(y\\)-axis (middle), or project them to a line \\(mx + b\\) (right). The resulting datapoints after the projection is shown in red. Which projection do you think is better? How can we calculate that?\n\n\n\n\nIn general, we want the projection which is the best approximation for the original data (the graph on the right). In other words, we want the projection that captures the most variance of the original data. In the next section, we’ll see how this is calculated."
+    "text": "24.2 Dimensionality\nSuppose we have a dataset of:\n\n\\(N\\) observations (datapoints/rows)\n\\(d\\) attributes (features/columns)\n\nLet’s “rename” this in terms of linear algebra so that we can be more clear with our wording. Using linear algebra, we can view our matrix as:\n\n\\(N\\) row vectors in a \\(d\\)-Dimensional space, OR\n\\(d\\) column vectors in an \\(N\\)-Dimensions space\n\nThe intrinsic dimension of a dataset is the minimal set of dimensions needed to approximately represent the data. In linear algebra terms, it is the dimension of the column space of a matrix, or the number of linearly independent columns in a matrix; this is equivalently called the rank of a matrix.\nIn the examples below, Dataset 1 has 2 dimensions because it has 2 linearly independent columns. Similarly, Dataset 2 has 3 dimensions because it has 3 linearly independent columns.\n\n\n\n\nWhat about Dataset 4 below?\n\n\n\nIt may be tempting to say that it has 4 dimensions, but the Weight (lbs) column is actually just a linear transformation of the Weight (kg) column. Thus, no new information is captured, and the matrix of our dataset has a (column) rank of 3! Therefore, despite having 4 columns, we still say that this data is 3-dimensional.\nPlotting the weight columns together reveals the key visual intuition. While the two columns visually span a 2D space as a line, the data does not deviate at all from that singular line. This means that one of the weight columns is redundant! Even given the option to cover the whole 2D space, the data below does not. It might as well not have this dimension, which is why we still do not consider the data below to span more than 1 dimension.\n\n\n\nWhat happens when there are outliers? Below, we’ve added one outlier point to the dataset above, and just that one point is enough to change the rank of the matrix from 1 to 2 dimensions. However, the data is still approximately 1-dimensional.\n\n\n\nDimensionality reduction is generally an approximation of the original data that’s achieved by projecting the data onto a desired dimension. In the example below, our original datapoints (blue dots) are 2-dimensional. We have a few choices if we want to project them down to 1-dimension: project them onto the \\(x\\)-axis (left), project them onto the \\(y\\)-axis (middle), or project them to a line \\(mx + b\\) (right). The resulting datapoints after the projection is shown in red. Which projection do you think is better? How can we calculate that?\n\n\n\n\nIn general, we want the projection which is the best approximation for the original data (the graph on the right). In other words, we want the projection that captures the most variance of the original data. In the next section, we’ll see how this is calculated."
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@@ -823,7 +823,7 @@
     "href": "pca_1/pca_1.html#principal-component-analysis-pca",
     "title": "24  PCA I",
     "section": "24.4 Principal Component Analysis (PCA)",
-    "text": "24.4 Principal Component Analysis (PCA)\nIn PCA, our goal is to transform observations from high-dimensional data down to low dimensions (often 2, as most visualizations are 2D) through linear transformations. In other words, we want to find a linear transformation that creates a low-dimension representation that captures as much of the original data’s total variance as possible.\n\n\n\nWe often perform PCA during the Exploratory Data Analysis (EDA) stage of our data science lifecycle when we don’t know what model to use. It helps us with:\n\nVisually identifying clusters of similar observations in high dimensions.\nRemoving irrelevant dimensions if we suspect that the dataset is inherently low rank. For example, if the columns are collinear, there are many attributes, but only a few mostly determine the rest through linear associations.\nCreating a transformed dataset of decorrelated features.\n\n\n\n\nThere are two equivalent ways of framing PCA:\n\nFinding directions of maximum variability in the data.\nFinding the low dimensional (rank) matrix factorization that best approximates the data.\n\nTo execute the first approach of variance maximization framing (more common), we can find the variances of each attribute with np.var and then keep the \\(k\\) attributes with the highest variance. However, this approach limits us to work with attributes individually; it cannot resolve collinearity, and we cannot combine features.\nThe second approach uses PCA to construct principal components with the most variance in the data (even higher than the first approach) using linear combinations of features. We’ll describe the procedure in the next section.\n\n24.4.1 PCA Procedure (Overview)\nTo perform PCA on a matrix:\n\nCenter the data matrix by subtracting the mean of each attribute column.\nTo find the \\(i\\)-th principal component, \\(v_i\\):\n\n\\(v\\) is a unit vector that linearly combines the attributes.\n\\(v\\) gives a one-dimensional projection of the data.\n\\(v\\) is chosen to maximize the variance along the projection onto \\(v\\). This is equivalent to minimizing the sum of squared distances between each point and its projection onto \\(v\\).\nChoose \\(v\\) such that it is orthogonal to all previous principal components.\n\n\nThe \\(k\\) principal components capture the most variance of any \\(k\\)-dimensional reduction of the data matrix.\nIn practice, however, we don’t carry out the procedures in step 2 because they take too long to compute. Instead, we use singular value decomposition (SVD) to find all principal components efficiently.\n\n\n24.4.2 Deriving PCA as Error Minimization\nIn this section, we will derive PCA keeping the following goal in mind: minimize the reconstruction loss for our matrix factorization model. You are not expected to be able to be able to redo this derivation, but understanding the derivation may help with future assignments.\nGiven a matrix \\(X\\) with \\(n\\) rows and \\(d\\) columns, our goal is to find its best decomposition such that \\[X \\approx Z W\\] Z has \\(n\\) rows and \\(k\\) columns; W has \\(k\\) rows and \\(d\\) columns.\n\n\n\nTo measure the accuracy of our reconstruction, we define the reconstruction loss below, where \\(X_i\\) is the row vector of \\(X\\), and \\(Z_i\\) is the row vector of \\(Z\\):\n\n\n\n\nThere are many solutions to the above, so let’s constrain our model such that \\(W\\) is a row-orthonormal matrix (i.e. \\(WW^T=I\\)) where the rows of \\(W\\) are our principal components.\nIn our derivation, let’s first work with the case where \\(k=1\\). Here Z will be an \\(n \\times 1\\) vector and W will be a \\(1 \\times d\\) vector.\n\\[\\begin{aligned}\nL(z,w) &= \\frac{1}{n}\\sum_{i=1}^{n}(X_i - z_{i}w)(X_i - z_{i}w)^T \\\\\n&= \\frac{1}{n}\\sum_{i=1}^{n}(X_{i}X_{i}^T - 2z_{i}X_{i}w^T + z_{i}^{2}ww^T) & \\text{(expand the loss)} \\\\\n= \\frac{1}{n}\\sum_{i=1}^{n}(-2z_{i}X_{i}w^T + z_{i}^{2}) & \\text{(First term is constant and }ww^T=1\\text{ by orthonormality)}\n\\end{aligned}\\]\n\\end{aligned}$$\nNow, we can take the derivative with respect to \\(Z_i\\). \\[\\begin{aligned}\n\\frac{\\partial{L(Z,W)}}{\\partial{z_i}} &= \\frac{1}{n}(-2X_{i}w^T + 2z_{i}) \\\\\nz_i &= X_iw^T & \\text{(Setting derivative equal to 0 and solving for }z_i\\text{)}\n\\end{aligned}\\]\nWe can now substitute our solution for \\(z_i\\) in our loss function:\n\\[\\begin{aligned}\nL(z,w) &= \\frac{1}{n}\\sum_{i=1}^{n}(-2z_{i}X_{i}w^T + z_{i}^{2}) \\\\\nL(z=X_iw^T,w) &= \\frac{1}{n}\\sum_{i=1}^{n}(-2X_iw^TX_{i}w^T + (X_iw^T)^{2}) \\\\\n&= \\frac{1}{n}\\sum_{i=1}^{n}(-X_iw^TX_{i}w^T) \\\\\n&= \\frac{1}{n}\\sum_{i=1}^{n}(-wX_{i}^TX_{i}w^T) \\\\\n&= -w\\frac{1}{n}\\sum_{i=1}^{n}(X_i^TX_{i})w^T \\\\\n&= -w\\Sigma w^T\n\\end{aligned}\\]\nNow, we need to minimize our loss with respect to \\(w\\). Since we have a negative sign, one way we can do this is by making \\(w\\) really big. However, we also have the orthonormality constraint \\(ww^T=1\\). To incorporate this constraint into the equation, we can add a Lagrange multiplier, \\(\\lambda\\). Note that lagrangian multipliers are out of scope for Data 100.\n\\[\nL(w,\\lambda) = -w\\Sigma w^T + \\lambda(ww^T-1)\n\\]\nTaking the derivative with respect to \\(w\\), \\[\\begin{aligned}\n\\frac{\\partial{L(w,\\lambda)}}{w} &= -2\\Sigma w^T + 2\\lambda w^T \\\\\n2\\Sigma w^T - 2\\lambda w^T &= 0 & \\text{(Setting derivative equal to 0)} \\\\\n\\Sigma w^T &= \\lambda w^T \\\\\n\\end{aligned}\\]\nThis result implies that:\n\n\\(w\\) is a unitary eigenvector of the covariance matrix.\nThe error is minimized when \\(w\\) is the eigenvector with the largest eigenvalue \\(\\lambda\\).\n\nThis derivation can inductively be used for the next (second) principal component (not shown).\nThe 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."
+    "text": "24.4 Principal Component Analysis (PCA)\nIn PCA, our goal is to transform observations from high-dimensional data down to low dimensions (often 2, as most visualizations are 2D) through linear transformations. In other words, we want to find a linear transformation that creates a low-dimension representation that captures as much of the original data’s total variance as possible.\n\n\n\nWe often perform PCA during the Exploratory Data Analysis (EDA) stage of our data science lifecycle when we don’t know what model to use. It helps us with:\n\nVisually identifying clusters of similar observations in high dimensions.\nRemoving irrelevant dimensions if we suspect that the dataset is inherently low rank. For example, if the columns are collinear, there are many attributes, but only a few mostly determine the rest through linear associations.\nCreating a transformed dataset of decorrelated features.\n\n\n\n\nThere are two equivalent ways of framing PCA:\n\nFinding directions of maximum variability in the data.\nFinding the low dimensional (rank) matrix factorization that best approximates the data.\n\nTo execute the first approach of variance maximization framing (more common), we can find the variances of each attribute with np.var and then keep the \\(k\\) attributes with the highest variance. However, this approach limits us to work with attributes individually; it cannot resolve collinearity, and we cannot combine features.\nThe second approach uses PCA to construct principal components with the most variance in the data (even higher than the first approach) using linear combinations of features. We’ll describe the procedure in the next section.\n\n24.4.1 PCA Procedure (Overview)\nTo perform PCA on a matrix:\n\nCenter the data matrix by subtracting the mean of each attribute column.\nTo find the \\(i\\)-th principal component, \\(v_i\\):\n\n\\(v\\) is a unit vector that linearly combines the attributes.\n\\(v\\) gives a one-dimensional projection of the data.\n\\(v\\) is chosen to maximize the variance along the projection onto \\(v\\). This is equivalent to minimizing the sum of squared distances between each point and its projection onto \\(v\\).\nChoose \\(v\\) such that it is orthogonal to all previous principal components.\n\n\nThe \\(k\\) principal components capture the most variance of any \\(k\\)-dimensional reduction of the data matrix.\nIn practice, however, we don’t carry out the procedures in step 2 because they take too long to compute. Instead, we use singular value decomposition (SVD) to find all principal components efficiently.\n\n\n24.4.2 Deriving PCA as Error Minimization\nIn this section, we will derive PCA keeping the following goal in mind: minimize the reconstruction loss for our matrix factorization model. You are not expected to be able to be able to redo this derivation, but understanding the derivation may help with future assignments.\nGiven a matrix \\(X\\) with \\(n\\) rows and \\(d\\) columns, our goal is to find its best decomposition such that \\[X \\approx Z W\\] Z has \\(n\\) rows and \\(k\\) columns; W has \\(k\\) rows and \\(d\\) columns.\n\n\n\nTo measure the accuracy of our reconstruction, we define the reconstruction loss below, where \\(X_i\\) is the row vector of \\(X\\), and \\(Z_i\\) is the row vector of \\(Z\\):\n\n\n\n\nThere are many solutions to the above, so let’s constrain our model such that \\(W\\) is a row-orthonormal matrix (i.e. \\(WW^T=I\\)) where the rows of \\(W\\) are our principal components.\nIn our derivation, let’s first work with the case where \\(k=1\\). Here Z will be an \\(n \\times 1\\) vector and W will be a \\(1 \\times d\\) vector.\n\\[\\begin{aligned}\nL(z,w) &= \\frac{1}{n}\\sum_{i=1}^{n}(X_i - z_{i}w)(X_i - z_{i}w)^T \\\\\n&= \\frac{1}{n}\\sum_{i=1}^{n}(X_{i}X_{i}^T - 2z_{i}X_{i}w^T + z_{i}^{2}ww^T) & \\text{(expand the loss)} \\\\\n= \\frac{1}{n}\\sum_{i=1}^{n}(-2z_{i}X_{i}w^T + z_{i}^{2}) & \\text{(First term is constant and }ww^T=1\\text{ by orthonormality)} \\\\\n\\end{aligned}\\]\nNow, we can take the derivative with respect to \\(Z_i\\). \\[\\begin{aligned}\n\\frac{\\partial{L(Z,W)}}{\\partial{z_i}} &= \\frac{1}{n}(-2X_{i}w^T + 2z_{i}) \\\\\nz_i &= X_iw^T & \\text{(Setting derivative equal to 0 and solving for }z_i\\text{)}\\end{aligned}\\]\nWe can now substitute our solution for \\(z_i\\) in our loss function:\n\\[\\begin{aligned}\nL(z,w) &= \\frac{1}{n}\\sum_{i=1}^{n}(-2z_{i}X_{i}w^T + z_{i}^{2}) \\\\\nL(z=X_iw^T,w) &= \\frac{1}{n}\\sum_{i=1}^{n}(-2X_iw^TX_{i}w^T + (X_iw^T)^{2}) \\\\\n&= \\frac{1}{n}\\sum_{i=1}^{n}(-X_iw^TX_{i}w^T) \\\\\n&= \\frac{1}{n}\\sum_{i=1}^{n}(-wX_{i}^TX_{i}w^T) \\\\\n&= -w\\frac{1}{n}\\sum_{i=1}^{n}(X_i^TX_{i})w^T \\\\\n&= -w\\Sigma w^T\n\\end{aligned}\\]\nNow, we need to minimize our loss with respect to \\(w\\). Since we have a negative sign, one way we can do this is by making \\(w\\) really big. However, we also have the orthonormality constraint \\(ww^T=1\\). To incorporate this constraint into the equation, we can add a Lagrange multiplier, \\(\\lambda\\). Note that lagrangian multipliers are out of scope for Data 100.\n\\[\nL(w,\\lambda) = -w\\Sigma w^T + \\lambda(ww^T-1)\n\\]\nTaking the derivative with respect to \\(w\\), \\[\\begin{aligned}\n\\frac{\\partial{L(w,\\lambda)}}{w} &= -2\\Sigma w^T + 2\\lambda w^T \\\\\n2\\Sigma w^T - 2\\lambda w^T &= 0 & \\text{(Setting derivative equal to 0)} \\\\\n\\Sigma w^T &= \\lambda w^T \\\\\n\\end{aligned}\\]\nThis result implies that:\n\n\\(w\\) is a unitary eigenvector of the covariance matrix. This means that \\(||w||^2 = ww^T = 1\\)\nThe error is minimized when \\(w\\) is the eigenvector with the largest eigenvalue \\(\\lambda\\).\n\nThis derivation can inductively be used for the next (second) principal component (not shown).\nThe 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:"
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diff --git a/docs/site_libs/bootstrap/bootstrap.min.css b/docs/site_libs/bootstrap/bootstrap.min.css
index d3c0f959..020a55e1 100644
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+++ b/docs/site_libs/bootstrap/bootstrap.min.css
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diff --git a/pca_1/pca_1.qmd b/pca_1/pca_1.qmd
index e622888a..bd79ab5f 100644
--- a/pca_1/pca_1.qmd
+++ b/pca_1/pca_1.qmd
@@ -143,6 +143,8 @@ Plotting the weight columns together reveals the key visual intuition. While the
 
 What happens when there are outliers? Below, we've added one outlier point to the dataset above, and just that one point is enough to change the rank of the matrix from 1 to 2 dimensions. However, the data is still *approximately* 1-dimensional. 
 
+<center><img src = "images/dataset3_outlier.png" width="400vw"></center>
+
 **Dimensionality reduction** is generally an **approximation of the original data** that's achieved by **projecting** the data onto a desired dimension. In the example below, our original datapoints (blue dots) are 2-dimensional. We have a few choices if we want to project them down to 1-dimension: project them onto the $x$-axis (left), project them onto the $y$-axis (middle), or project them to a line $mx + b$ (right). The resulting datapoints after the projection is shown in red. Which projection do you think is better? How can we calculate that? 
 
 <center><img src = "images/diff_reductions.png" width="800vw"></center>