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Added vector diagrams
authorNeil Smith <neil.git@njae.me.uk>
Wed, 12 Mar 2014 12:33:21 +0000 (12:33 +0000)
committerNeil Smith <neil.git@njae.me.uk>
Wed, 12 Mar 2014 12:33:21 +0000 (12:33 +0000)
slides/caesar-break.html patch | blob | history
slides/vector-dot-product.svg [new file with mode: 0644] patch | blob
slides/vector-subtraction.svg [new file with mode: 0644] patch | blob

diff --git a/slides/caesar-break.html b/slides/caesar-break.html
index d552209419fe6f6e7074499ea7998f4c55a7465c..c6d402393db42b83f61d3684fa218ad2090a1353 100644 (file)
--- a/slides/caesar-break.html
+++ b/slides/caesar-break.html
@@ -37,6 +37,9 @@
         text-shadow: 0 0 20px #333;
         padding: 2px 5px;
       }
         text-shadow: 0 0 20px #333;
         padding: 2px 5px;
       }
+       .float-right {
+        float: right;
+      }
     </style>
   </head>
   <body>
     </style>
   </head>
   <body>
@@ -44,7 +47,7 @@
 
 # Breaking caesar ciphers
 
 
 # Breaking caesar ciphers
 
-![centre-aligned Caesar wheel](caesarwheel1.gif)
+![center-aligned Caesar wheel](caesarwheel1.gif)
 
 ---
 
 
 ---
 
@@ -67,6 +70,7 @@ What steps do we know how to do?
 # How close is it to English?
 
 What does English look like?
 # How close is it to English?
 
 What does English look like?
+
 * We need a model of English.
 
 How do we define "closeness"?
 * We need a model of English.
 
 How do we define "closeness"?
@@ -97,26 +101,56 @@ The distance between the vectors is how far from English the text is.
 
 ---
 
 
 ---
 
-# Vector distances
+# Frequencies of English
+
+But before then how do we count the letters?
+
+* Read a file into a string
+```python
+open()
+read()
+```
+* Count them
+```python
+import collections
+```
+
+---
+
+# Canonical forms
+
+Counting letters in _War and Peace_ gives all manner of junk.
+
+* Convert the text in canonical form (lower case, accents removed, non-letters stripped) before counting
 
 
+---
+
+# Vector distances
 
 
-## FIXME! Diagram of vector subtraction here
+.float-right[![right-aligned Vector subtraction](vector-subtraction.svg)]
 
 Several different distance measures (__metrics__, also called __norms__):
 
 
 Several different distance measures (__metrics__, also called __norms__):
 
-* L<sub>2</sub> norm (Euclidean distance):  `\(|\mathbf{x} - \mathbf{y}| = \sqrt{\sum_i (\mathbf{x}_i - \mathbf{y}_i)^2} \)`
+* L<sub>2</sub> norm (Euclidean distance): 
+`\(|\mathbf{a} - \mathbf{b}| = \sqrt{\sum_i (\mathbf{a}_i - \mathbf{b}_i)^2} \)`
 
 
-* L<sub>1</sub> norm (Manhattan distance, taxicab distance):  `\(|\mathbf{x} - \mathbf{y}| = \sum_i |\mathbf{x}_i - \mathbf{y}_i| \)`
+* L<sub>1</sub> norm (Manhattan distance, taxicab distance): 
+`\(|\mathbf{a} - \mathbf{b}| = \sum_i |\mathbf{a}_i - \mathbf{b}_i| \)`
 
 
-* L<sub>3</sub> norm:  `\(|\mathbf{x} - \mathbf{y}| = \sqrt[3]{\sum_i |\mathbf{x}_i - \mathbf{y}_i|^3} \)`
+* L<sub>3</sub> norm: 
+`\(|\mathbf{a} - \mathbf{b}| = \sqrt[3]{\sum_i |\mathbf{a}_i - \mathbf{b}_i|^3} \)`
 
 The higher the power used, the more weight is given to the largest differences in components.
 
 (Extends out to:
 
 
 The higher the power used, the more weight is given to the largest differences in components.
 
 (Extends out to:
 
-* L<sub>0</sub> norm (Hamming distance):  `\(|\mathbf{x} - \mathbf{y}| = \sum_i \left\{\begin{matrix} 1 &amp;\mbox{if}\ \mathbf{x}_i \neq \mathbf{y}_i , \\ 0 &amp;\mbox{if}\ \mathbf{x}_i = \mathbf{y}_i \end{matrix} \right| \)`
+* L<sub>0</sub> norm (Hamming distance): 
+`$$|\mathbf{a} - \mathbf{b}| = \sum_i \left\{
+\begin{matrix} 1 &amp;\mbox{if}\ \mathbf{a}_i \neq \mathbf{b}_i , \\
+ 0 &amp;\mbox{if}\ \mathbf{a}_i = \mathbf{b}_i \end{matrix} \right. $$`
 
 
-* L<sub>&infin;</sub> norm:  `\(|\mathbf{x} - \mathbf{y}| = \max_i{(\mathbf{x}_i - \mathbf{y}_i)} \)`
+* L<sub>&infin;</sub> norm: 
+`\(|\mathbf{a} - \mathbf{b}| = \max_i{(\mathbf{a}_i - \mathbf{b}_i)} \)`
 
 neither of which will be that useful.)
 ---
 
 neither of which will be that useful.)
 ---
@@ -125,9 +159,9 @@ neither of which will be that useful.)
 
 Frequency distributions drawn from different sources will have different lengths. For a fair comparison we need to scale them. 
 
 
 Frequency distributions drawn from different sources will have different lengths. For a fair comparison we need to scale them. 
 
-* Eucliean scaling (vector with unit length): `\( \hat{\mathbf{x}} = \frac{\mathbf{x}}{\| \mathbf{x} \|} = \frac{\mathbf{x}}{ \sqrt{\mathbf{x}_1^2 + \mathbf{x}_2^2 + \mathbf{x}_3^2 + \dots } }\)`
+* Eucliean scaling (vector with unit length): `$$ \hat{\mathbf{x}} = \frac{\mathbf{x}}{\| \mathbf{x} \|} = \frac{\mathbf{x}}{ \sqrt{\mathbf{x}_1^2 + \mathbf{x}_2^2 + \mathbf{x}_3^2 + \dots } }$$`
 
 
-* Normalisation (components of vector sum to 1): `\( \hat{\mathbf{x}} = \frac{\mathbf{x}}{\| \mathbf{x} \|} = \frac{\mathbf{x}}{ \mathbf{x}_1 + \mathbf{x}_2 + \mathbf{x}_3 + \dots }\)`
+* Normalisation (components of vector sum to 1): `$$ \hat{\mathbf{x}} = \frac{\mathbf{x}}{\| \mathbf{x} \|} = \frac{\mathbf{x}}{ \mathbf{x}_1 + \mathbf{x}_2 + \mathbf{x}_3 + \dots }$$`
 
 ---
 
 
 ---
 
@@ -135,6 +169,8 @@ Frequency distributions drawn from different sources will have different lengths
 
 Rather than looking at the distance between the vectors, look at the angle between them.
 
 
 Rather than looking at the distance between the vectors, look at the angle between them.
 
+.float-right[![right-aligned Vector dot product](vector-dot-product.svg)]
+
 Vector dot product shows how much of one vector lies in the direction of another: 
 `\( \mathbf{A} \bullet \mathbf{B} = 
 \| \mathbf{A} \| \cdot \| \mathbf{B} \| \cos{\theta} \)`
 Vector dot product shows how much of one vector lies in the direction of another: 
 `\( \mathbf{A} \bullet \mathbf{B} = 
 \| \mathbf{A} \| \cdot \| \mathbf{B} \| \cos{\theta} \)`
@@ -144,17 +180,47 @@ But,
 and `\( \| \mathbf{A} \| = \sum_i \mathbf{A}_i^2 \)`
 
 A bit of rearranging give the cosine simiarity:
 and `\( \| \mathbf{A} \| = \sum_i \mathbf{A}_i^2 \)`
 
 A bit of rearranging give the cosine simiarity:
-`\( \cos{\theta} = \frac{ \mathbf{A} \bullet \mathbf{B} }{ \| \mathbf{A} \| \cdot \| \mathbf{B} \| } = 
-\frac{\sum_i \mathbf{A}_i \cdot \mathbf{B}_i}{\sum_i \mathbf{A}_i^2 \times \sum_i \mathbf{B}_i^2} \)`
+`$$ \cos{\theta} = \frac{ \mathbf{A} \bullet \mathbf{B} }{ \| \mathbf{A} \| \cdot \| \mathbf{B} \| } = 
+\frac{\sum_i \mathbf{A}_i \cdot \mathbf{B}_i}{\sum_i \mathbf{A}_i^2 \times \sum_i \mathbf{B}_i^2} $$`
 
 This is independent of vector lengths!
 
 
 This is independent of vector lengths!
 
-Cosine similarity is 1 if in same direction, 0 if at 90⁰, -1 if antiparallel.
+Cosine similarity is 1 if in parallel, 0 if perpendicular, -1 if antiparallel.
+
+---
+
+# An infinite number of monkeys
+
+What is the probability that this string of letters is a sample of English?
+
+Given 'th', 'e' is about six times more likely than 'a' or 'i'.
+
+## Naive Bayes, or the bag of letters
+
+Ignore letter order, just treat each letter individually.
+
+Probability of a text is `\( \prod_i p_i \)`
+
+(Implmentation issue: this can often underflow, so get in the habit of rephrasing it as `\( \sum_i \log p_i \)`)
+
+---
+
+# Which is best?
+
+   | Euclidean | Normalised
+---|-----------|------------  
+L1 |     x     |      x
+L2 |     x     |      x
+L3 |     x     |      x
+Cosine |     x     |      x
+
+And the probability measure!
+
+* Nine different ways of measuring fitness.
 
 
-## FIXME! Cosine distance bug: frequencies2 length not squared.
+## Computing is an empircal science
 
 
 
 
-## FIXME! Diagram of vector dot product
 
     </textarea>
     <script src="http://gnab.github.io/remark/downloads/remark-0.6.0.min.js" type="text/javascript">
 
     </textarea>
     <script src="http://gnab.github.io/remark/downloads/remark-0.6.0.min.js" type="text/javascript">
diff --git a/slides/vector-dot-product.svg b/slides/vector-dot-product.svg
new file mode 100644 (file)
index 0000000..05d69c1
--- /dev/null
+++ b/slides/vector-dot-product.svg
@@ -0,0 +1,324 @@
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Teacher training using ciphers