Merge pull request #90 from kk-Syuer/patch-2

update of some typos found

AA2324/course/04_pca_svd_high_dim/04_pca_svd_high_dim.ipynb

@@ -298,7 +298,7 @@
    "source": [
     "# Geometry of SVD \n",
     "\n",
-    "Source: wikipedia\n",
+    "Source: Wikipedia\n",
     "<img width='50%' src=\"https://upload.wikimedia.org/wikipedia/commons/b/bb/Singular-Value-Decomposition.svg\" />"
    ]
   },
@@ -354,7 +354,7 @@
     }
    },
    "source": [
-    "### Find projection that maximizes the spread of the data\n",
+    "### Find a projection that maximizes the spread of the data\n",
     "\n",
     "- Find $\\mathbf{u}\\in \\mathbb{R}^k$ $\\left\\|\\mathbf{u}\\right\\|_2=1$ for which you can project the data $\\mathbf{x}^{\\prime}=\\mathbb{P}_{\\mathbf{u}}\\mathbf{x}$\n",
     "\n",
@@ -480,12 +480,12 @@
     }
    },
    "source": [
-    "# What happens in high dimension?\n",
+    "# What happens in high dimensions?\n",
     "\n",
     "- Are we supposed to think about the world in a 3 dimensional space?\n",
     "- Our brain can think in a 3 dimensional world.\n",
     "- Maybe with relativity theory we arrive into a 4-D space if we consider time also.\n",
-    "- So why we need higher dimensions?"
+    "- So why do we need higher dimensions?"
    ]
   },
   {
@@ -499,7 +499,7 @@
     "# High Dimensional Space\n",
     "\n",
     "- Visualizing one hundred dimensional space is incredibly difficult for humans.\n",
-    "- Most of time the representation for an input datum is a vector in **high dimensional space.**"
+    "- Most of the time the representation for an input datum is a vector in **high dimensional space.**"
    ]
   },
   {
@@ -631,10 +631,10 @@
     "## Best Practice: Always look at the data before start working!\n",
     "\n",
     "- Do not do simple summary statistics on the dataset\n",
-    "- Always look at a large random samples of the dataset\n",
+    "- Always look at a large random sample of the dataset\n",
     "- The data may **contain noise** that you want to be aware of!\n",
     "- Do not give for granted that the data **and** the labels are noise-free\n",
-    "- You could try to plot the data to lower dimension also (i.e. with PCA)"
+    "- You could try to plot the data to a lower dimension also (i.e. with PCA)"
    ]
   },
   {
@@ -824,7 +824,7 @@
    "source": [
     "# Sampling\n",
     "samples = np.random.uniform(0, 255, (100, 62, 47)).astype(np.uint8)\n",
-    "# sampling unifirmly 100 points in 62x47 space.\n",
+    "# sampling uniformly 100 points in 62x47 space.\n",
     "\n",
     "# Plot the faces\n",
     "N_ax, N_img = 10, 10  # 10 rows with 10 images per row\n",
@@ -862,7 +862,7 @@
    "source": [
     "# Sampling\n",
     "samples = np.random.uniform(0, 255, (100, 62, 47)).astype(np.uint8)\n",
-    "# sampling unifirmly 100 points in 62x47 space.\n",
+    "# sampling uniformly 100 points in 62x47 space.\n",
     "\n",
     "# Plot the faces\n",
     "N_ax, N_img = 10, 10  # 10 rows with 10 images per row\n",
@@ -884,7 +884,7 @@
    },
    "source": [
     "### The probability of hitting a face is very small!\n",
-    "The phenomenon is also know as..."
+    "The phenomenon is also known as..."
    ]
   },
   {
@@ -1002,9 +1002,9 @@
    "source": [
     "### Distances in high dimensional space\n",
     "\n",
-    "Distances in high dimension follow a pattern:\n",
+    "Distances in high dimensions follow a pattern:\n",
     "\n",
-    "- Distance of points sampled on the unit sphere goes to zero as D increases.\n",
+    "- The distance of points sampled on the unit sphere goes to zero as D increases.\n",
     "- Euclidean distance of points goes to $4\\sqrt{D}$.\n",
     "- The variance of distances between two random points keeps shrinking as D increases.\n",
     "\n",
@@ -1331,7 +1331,7 @@
     }
    ],
    "source": [
-    "############### Fittin with 3 component ######\n",
+    "############### Fittin with 3 components ######\n",
     "pca = PCA(n_components=3) # retain 3 components\n",
     "pca.fit(faces.data)\n",
     "#############################################\n",
@@ -1474,7 +1474,7 @@
    "source": [
     "# Why compression\n",
     "\n",
-    "This example is just used for illustrative purpose since **Machine Learning** is much related to **Information Theory**.\n",
+    "This example is just used for illustrative purposes since **Machine Learning** is much related to **Information Theory**.\n",
     "\n",
     "<img width=\"50%\" src=\"figs/compression_net.png\" />"
    ]
@@ -1609,7 +1609,7 @@
     }
    },
    "source": [
-    "# Any question of previous lectures before moving on?\n",
+    "# Any questions of previous lectures before moving on?\n",
     "\n",
     "- We will review a few concept of PCA at the end of matrix calculus"
    ]
@@ -1624,7 +1624,7 @@
    "source": [
     "# Matrix Calculus\n",
     "\n",
-    "Part of this lectures are taken from:\n",
+    "Part of these lectures are taken from:\n",
     "\n",
     "- [CS229 LinAlg](http://cs229.stanford.edu/summer2019/cs229-linalg.pdf)\n",
     "- [CS229 Calculus Recap](https://www.youtube.com/watch?v=b0HvwszmqcQ)\n",
@@ -1698,7 +1698,7 @@
     "\n",
     "\n",
     "- Let's take a complex function $$f(x) = \\sin(x^x)$$ over the $[0, 3]$. \n",
-    "- Its behaviour is not simple to understand."
+    "- Its behavior is not simple to understand."
    ]
   },
   {
@@ -2059,7 +2059,7 @@
     "$$\\mbf{x}^T \\mbf{A}\\mbf{x}$$\n",
     "\n",
     "- It is vector to scalar function\n",
-    "- It used for characterizing **Definiteness** of matrices"
+    "- It is used for characterizing **Definiteness** of matrices"
    ]
   },
   {