# Modeling a single voxel#

The voxel regression page has a worked example of applying simple regression) on a single voxel.

This page runs the same calculations, but using the General Linear Model notation and matrix calculations.

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

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

```
# 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)
```

```
[<matplotlib.lines.Line2D at 0x7f557836cbe0>]
```

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

```
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')
```

```
Downloading file 'ds114_sub009_t2r1_conv.txt' from 'https://raw.githubusercontent.com/nipraxis/nipraxis-data/0.5/ds114_sub009_t2r1_conv.txt' to '/home/runner/.cache/nipraxis/0.5'.
```

```
Text(0, 0.5, '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:

```
N = len(convolved)
X = np.ones((N, 2))
X[:, 0] = convolved
plt.imshow(X, cmap='gray', aspect=0.1, interpolation='none')
```

```
<matplotlib.image.AxesImage at 0x7f55783b6830>
```

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

Our model is:

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

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

Let’s calculate the pseudoinverse for our design:

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

```
(2, 169)
```

We calculate \(\bhat\):

```
beta_hat = Xp @ voxel_time_course
beta_hat
```

```
array([ 31.185514, 2029.367689])
```

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

```
y_hat = X @ beta_hat
```

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

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

```
245.00339852605674
```

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

:

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

```
LinregressResult(slope=31.18551366491453, intercept=2029.367689291584, rvalue=0.7044637722561978, pvalue=1.1832511547748845e-26, stderr=2.4312815394118448, intercept_stderr=1.634742742490283)
```