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matthew-brett committed Sep 14, 2023
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111 changes: 94 additions & 17 deletions _sources/glm_intro.Rmd
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Original file line number Diff line number Diff line change
Expand Up @@ -75,25 +75,33 @@ plt.xlabel('Clamminess of handshake')
plt.ylabel('Psychopathy score')
```

Let's immediately rename the values we are *predicting* to `y`, and the values we are *predicting with* to `x`:

```{python}
y = psychopathy
x = clammy
```


You have already seen a preliminary guess we made at a straight line that links
the `clammy` scores to the `psychopathy` scores. Our guess was:

```{python}
guessed_slope = 0.9
guessed_intercept = 10
b = 0.9
c = 10
```

We use this line to *predict* the `psychopathy` scores from the `clammy` scores.

```{python}
predicted_psychopathy = guessed_slope * clammy + guessed_intercept
predicted_psychopathy
predicted = b * clammy + c
predicted
```

```{python}
# Plot the data and the predictions
plt.plot(clammy, psychopathy, '+', label='Actual data')
plt.plot(clammy, predicted_psychopathy, 'ro', label='Predictions')
plt.plot(x, y, '+', label='Actual data')
plt.plot(x, predicted, 'ro', label='Predictions')
plt.xlabel('Clamminess of handshake')
plt.ylabel('Psychopathy score')
plt.title('Clammy vs psychopathy with guessed line')
Expand Down Expand Up @@ -125,7 +133,7 @@ sy_y_ind = IndexedBase("y")
vector_indices = range(1, n + 1)
sy_y_val = Matrix([sy_y_ind[i] for i in vector_indices])
Eq(sy_y, Eq(sy_y_val, Matrix(psychopathy)), evaluate=False)
Eq(sy_y, Eq(sy_y_val, Matrix(y)), evaluate=False)
```

`x` (the `clammy` score) is a *predictor*. Lets call the clammy scores $\xvec$.
Expand All @@ -137,7 +145,7 @@ student (= 0.389) and $x_i$ is the value for student $i$ where $i \in 1 .. 12$.
sy_x = symbols(r'\vec{x}')
sy_x_ind = IndexedBase("x")
sy_x_val = Matrix([sy_x_ind[i] for i in vector_indices])
Eq(sy_x, Eq(sy_x_val, Matrix(clammy)), evaluate=False)
Eq(sy_x, Eq(sy_x_val, Matrix(x)), evaluate=False)
```

In [regression notation](regression_notation.Rmd), we wrote our straight line
Expand All @@ -159,6 +167,14 @@ $\evec$ is the vector of as-yet-unknown errors $e_1, e_2, ... e_{12}$. The
values of the errors depend on the predicted values, and therefore, on our
slope $b$ and intercept $c$.

We calculate the errors as:

```{python}
e = y - predicted
```

We write the errors as an error vector:

```{python tags=c("hide-input")}
# Ignore this cell.
sy_e = symbols(r'\vec{\varepsilon}')
Expand All @@ -181,6 +197,13 @@ rhs = MatAdd(MatMul(sy_b, sy_x_val), sy_c_mat, sy_e_val)
Eq(sy_y_val, rhs, evaluate=False)
```

Confirm this is true when we calculate on our particular values:

```{python}
# Confirm that y is close as dammit to b * x + c + e
assert np.allclose(y, b * x + c + e)
```

Bear with with us for a little trick. We define $\vec{o}$ as a vector of ones,
like this:

Expand All @@ -191,6 +214,12 @@ sy_o_val = Matrix([1 for i in vector_indices])
Eq(sy_o, sy_o_val, evaluate=False)
```

In code, in our case:

```{python}
o = np.ones(n)
```

Now we can rewrite the formula as:

$$
Expand All @@ -212,13 +241,27 @@ This evaluates to the result we intend:
Eq(sy_y_val, sy_c * sy_o_val + sy_b * sy_x_val + sy_e_val, evaluate=False)
```

```{python}
# Confirm that y is close as dammit to b * x + c * o + e
assert np.allclose(y, b * x + c * o + e)
```

$\newcommand{Xmat}{\boldsymbol X} \newcommand{\bvec}{\vec{\beta}}$

We can now rephrase the calculation in terms of matrix multiplication.

Call $\Xmat$ the matrix of two columns, where the first column is $\xvec$ and
the second column is the column of ones ($\vec{o}$ above). Call $\bvec$ the
column vector:
the second column is the column of ones ($\vec{o}$ above).


In code, for our case:

```{python}
X = np.stack([x, o], axis=1)
X
```

Call $\bvec$ the column vector:

$$
\left[
Expand All @@ -229,6 +272,13 @@ c \\
\right]
$$

In code:

```{python}
B = np.array([b, c])
B
```

This gives us exactly the same calculation as $\yvec = b \xvec + c + \evec$
above, but in terms of matrix multiplication:

Expand All @@ -245,7 +295,6 @@ $$
\yvec = \Xmat \bvec + \evec
$$


Because of the way that matrix multiplication works, this again gives us the
intended result:

Expand All @@ -254,8 +303,15 @@ intended result:
Eq(sy_y_val, sy_xmat_val @ sy_beta_val + sy_e_val, evaluate=False)
```

We still haven’t found our best fitting line. But before we go further,
it might be obvious that we can easily add a new predictor here.
We write *matrix multiplication* in Numpy as `@`:

```{python}
# Confirm that y is close as dammit to X @ B + e
assert np.allclose(y, X @ B + e)
```

We still haven’t found our best fitting line. But before we go further, it
might be clear that we can easily add a new predictor here.


## Multiple regression
Expand All @@ -272,9 +328,22 @@ age = np.array(
```

Now rename the `clammy` predictor vector from $\xvec$ to
$\xvec_1$. Of course $\xvec_1$ has 12 values $x_{1, 1}..x_{1,
12}$. Call the `age` predictor vector $\xvec_2$. Call the slope for
`clammy` $b_1$ (slope for $\xvec_1$). Call the slope for age
$\xvec_1$.

```{python}
# clammy (our previous x) is the first column we will use to predict.
x_1 = clammy
```

Of course $\xvec_1$ has 12 values $x_{1, 1}..x_{1, 12}$. Call the `age`
predictor vector $\xvec_2$.

```{python}
# age is the second column we use to predict
x_2 = age
```

Call the slope for `clammy` $b_1$ (slope for $\xvec_1$). Call the slope for age
$b_2$ (slope for $\xvec_2$). Our model is:

$$
Expand Down Expand Up @@ -323,7 +392,15 @@ sy_Xmat = symbols(r'\boldsymbol{X}')
Eq(sy_Xmat, sy_xmat3_val, evaluate=False)
```

and therefore:
In code in our case:

```{python}
# Design X in values
X_3_cols = np.stack([x_1, x_2, o], axis=1)
X_3_cols
```

In symbols:

```{python tags=c("hide-input")}
# Ignore this cell.
Expand Down
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Expand Up @@ -605,7 +605,7 @@ <h1>Arrays as images, images as arrays<a class="headerlink" href="#arrays-as-ima
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Expand All @@ -625,7 +625,7 @@ <h1>Arrays as images, images as arrays<a class="headerlink" href="#arrays-as-ima
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Expand All @@ -641,7 +641,7 @@ <h1>Arrays as images, images as arrays<a class="headerlink" href="#arrays-as-ima
</div>
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Expand All @@ -668,7 +668,7 @@ <h1>Arrays as images, images as arrays<a class="headerlink" href="#arrays-as-ima
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Expand Down Expand Up @@ -731,7 +731,7 @@ <h1>Arrays as images, images as arrays<a class="headerlink" href="#arrays-as-ima
</div>
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Expand All @@ -748,7 +748,7 @@ <h1>Arrays as images, images as arrays<a class="headerlink" href="#arrays-as-ima
</div>
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Expand Down Expand Up @@ -785,7 +785,7 @@ <h1>Arrays as images, images as arrays<a class="headerlink" href="#arrays-as-ima
</div>
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Expand Down Expand Up @@ -932,7 +932,7 @@ <h1>Arrays as images, images as arrays<a class="headerlink" href="#arrays-as-ima
</div>
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Expand Down Expand Up @@ -1003,7 +1003,7 @@ <h1>Arrays as images, images as arrays<a class="headerlink" href="#arrays-as-ima
</div>
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Expand All @@ -1021,7 +1021,7 @@ <h1>Arrays as images, images as arrays<a class="headerlink" href="#arrays-as-ima
</div>
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Expand Up @@ -1783,9 +1783,6 @@ <h2>Slicing<a class="headerlink" href="#slicing" title="Permalink to this headin
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[9, 999, 0, 8, 20]
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Expand Down Expand Up @@ -3190,7 +3187,10 @@ <h2>Files<a class="headerlink" href="#files" title="Permalink to this heading">#
</div>
</div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Line is: MATLAB is good for matrices
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</pre></div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> MATLAB is good for matrices

Line is: Python is good for coding
</pre></div>
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Expand Up @@ -644,7 +644,7 @@ <h2>Using scipy<a class="headerlink" href="#using-scipy" title="Permalink to thi
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Expand Down Expand Up @@ -757,7 +757,7 @@ <h2>Constructing a hemodynamic response function<a class="headerlink" href="#con
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Expand Down Expand Up @@ -804,7 +804,7 @@ <h2>Constructing a hemodynamic response function<a class="headerlink" href="#con
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Expand Up @@ -730,7 +730,7 @@ <h2>Neural and hemodynamic prediction<a class="headerlink" href="#neural-and-hem
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Expand Down Expand Up @@ -824,7 +824,7 @@ <h2>Convolution is like cross-correlation with the reversed HRF<a class="headerl
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