model_one_voxel.Rmd

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# Modeling a single voxel

The [voxel regression page](regress_one_voxel.Rmd) has a worked
example of applying [simple regression](on_regression.Rmd)) on a single voxel.

This page runs the same calculations, but using the [General Linear
Model](glm_intro.Rmd) notation and matrix calculations.

Let’s get that same voxel time course back again:

```{python}
import numpy as np
import matplotlib.pyplot as plt
import nibabel as nib
# Only show 6 decimals when printing
np.set_printoptions(precision=6)
```

```{python}
# Load the function to fetch the data file we need.
import nipraxis
# Fetch the data file.
data_fname = nipraxis.fetch_file('ds114_sub009_t2r1.nii')
img = nib.load(data_fname)
data = img.get_fdata()
# Knock off the first four volumes (to avoid artefact).
data = data[..., 4:]
# Get the voxel time course of interest.
voxel_time_course = data[42, 32, 19]
plt.plot(voxel_time_course)
```

Load the convolved time course, and plot the voxel values against the convolved regressor:

```{python}
tc_fname = nipraxis.fetch_file('ds114_sub009_t2r1_conv.txt')
# Show the file name of the fetched data.
convolved = np.loadtxt(tc_fname)
# Knock off first 4 elements to match data.
convolved = convolved[4:]
# Plot.
plt.scatter(convolved, voxel_time_course)
plt.xlabel('Convolved prediction')
plt.ylabel('Voxel values')
```

As you remember, we apply the GLM by first preparing a design matrix, that has one column corresponding for each *parameter* in the *model*.

In our case we have two parameters, the *slope* and the *intercept*.

First we make our *design matrix*.  It has a column for the convolved
regressor, and a column of ones:

```{python}
N = len(convolved)
X = np.ones((N, 2))
X[:, 0] = convolved
plt.imshow(X, cmap='gray', aspect=0.1)
```

$\newcommand{\yvec}{\vec{y}}$
$\newcommand{\xvec}{\vec{x}}$
$\newcommand{\evec}{\vec{\varepsilon}}$
$\newcommand{Xmat}{\boldsymbol X} \newcommand{\bvec}{\vec{\beta}}$
$\newcommand{\bhat}{\hat{\bvec}} \newcommand{\yhat}{\hat{\yvec}}$

Our model is:

$$
\yvec = \Xmat \bvec + \evec
$$

We can get our least mean squared error (MSE) parameter *estimates* for
$\bvec$ with:

$$
\bhat = \Xmat^+y
$$

where $\Xmat^+$ is the *pseudoinverse* of $\Xmat$.  When $\Xmat$ is
invertible, the pseudoinverse is given by:

$$
\Xmat^+ = (\Xmat^T \Xmat)^{-1} \Xmat^T
$$

Let’s calculate the pseudoinverse for our design:

```{python}
import numpy.linalg as npl
Xp = npl.pinv(X)
Xp.shape
```

We calculate $\bhat$:

```{python}
beta_hat = Xp @ voxel_time_course
beta_hat
```

We can then calculate $\yhat$ (also called the *fitted data*):

```{python}
y_hat = X @ beta_hat
```

Finally, we may be interested to calculate the MSE of this model:

```{python}
# Residuals are actual minus fitted.
e_vec = voxel_time_course - y_hat
mse = np.mean(e_vec ** 2)
mse
```

Notice that the $\bhat$ parameters are the same as the slope and intercept from the Scipy calculation using `linregress`:

```{python}
import scipy.stats as sps
sps.linregress(convolved, voxel_time_course)
```