Refactor notation page a little.

regression_notation.Rmd

@@ -7,9 +7,9 @@ jupyter:
       extension: .Rmd
       format_name: rmarkdown
       format_version: '1.2'
-      jupytext_version: 1.10.3
+      jupytext_version: 1.15.0
   kernelspec:
-    display_name: Python 3
+    display_name: Python 3 (ipykernel)
     language: python
     name: python3
 ---
@@ -161,61 +161,86 @@ plt.legend()
 
 ## Mathematical notation
 
+
+```{python tags=c("hide-input")}
+# Utilities to format LaTeX output.
+# You don't need to understand this code - it is to format
+# the mathematics in the notebook.
+import pandas as pd
+from IPython.display import display, Math
+
+def format_vec(x, name, break_every=5):
+    vals = [f'{v}, ' for v in x[:-1]] + [f'{x[-1]}']
+    indent = rf'\hspace{{1.5em}}'
+    for pi, pos in enumerate(range(break_every, n, break_every)):
+        vals.insert(pos + pi, r'\\' + f'\n{indent}')
+    return rf'\vec{{{name}}} = [%s]' %  ''.join(vals)
+```
+
 Our next step is to write out this model more generally and more formally in
 mathematical symbols, so we can think about *any* vector (sequence) of x values
 $\xvec$, and any sequence (vector) of matching y values $\yvec$.  But to start
 with, let's think about the actual set of values we have for $\xvec$.  We could
 write the actual values in mathematical notation as:
 
-$$
-\xvec = [ 0.389,  0.2  ,  0.241,  0.463, \\
-     4.585,  1.097,  1.642,  4.972, \\
-     7.957,  5.585,  5.527,  6.964 ]
-$$
+```{python tags=c("hide-input")}
+display(Math(format_vec(x, 'x', 6)))
+```
 
 This means that $\xvec$ is a sequence of these specific 12 values.  But we
 could write $\xvec$ in a more general way, to be *any* 12 values, like this:
 
-$$
-\xvec = [ x_1, x_2, x_3, x_4, x_5, x_6,\\
-          x_7, x_8, x_9, x_{10}, x_{11}, x_{12} ]
-$$
+```{python tags=c("hide-input")}
+indices = np.arange(1, n + 1)
+x_is = [f'x_{{{i}}}' for i in indices]
+display(Math(format_vec(x_is, 'x', 6)))
+```
 
 This means that $\xvec$ consists of 12 numbers, $x_1, x_2 ..., x_{12}$, where
 $x_1$ can be any number, $x_2$ can be any number, and so on.
 
-$x_1$ is the value for the first student, $x_2$ is the value for the second student, and so on.
+$x_1$ is the value for the first student, $x_2$ is the value for the second student, etc.
 
-In our *particular case*:
+Here's another way of looking at the relationship of the values in our
+particular case, and their notation:
 
-$$
-x_1 = 0.389 \\
-x_2 = 0.2 \\
-... \\
-x_{12} = 6.964
-$$
+```{python tags=c("hide-input")}
+df_index = pd.Index(indices, name='1-based index')
+pd.DataFrame(
+    {r'$\vec{x}$ values': x,
+     f'$x$ value notation': 
+     [f'${v}$' for v in x_is]
+    },
+    index=df_index
+)
+```
 
 We can make $\xvec$ be even more general, by writing it like this:
 
 $$
-\xvec = [ x_1, x_2, ..., x_{n} ]
+\xvec = [x_1, x_2, ..., x_{n} ]
 $$
 
 This means that $\xvec$ is a sequence of any $n$ numbers, where $n$ can be any
 whole number, such as 1, 2, 3 ...  In our specific case, $n = 12$.
 
 Similarly, for our `psychopathy` ($\yvec$) values, we can write:
 
-$$
-\yvec = [ 11.416,   4.514,  12.204,  14.835, \\
-8.416,   6.563,  17.343,  13.02, \\
-15.19 ,  11.902,  22.721,  22.324 ]
-$$
+```{python tags=c("hide-input")}
+display(Math(format_vec(y, 'y', 6)))
+```
+
+```{python tags=c("hide-input")}
+pd.DataFrame(
+    {r'$\vec{y}$ values': y,
+     f'$y$ value notation': [f'$y_{{{i}}}$' for i in indices]
+    }, index=df_index)
+```
 
 More generally we can write $\yvec$ as a vector of any $n$ numbers:
 
 $$
-\yvec = [ y_1, y_2, ..., y_{n} ]
+\yvec = [y_1, y_2, ..., y_{n} ]
 $$