multi_multiply.Rmd

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# Estimation for many voxels at the same time

We often want to fit the same design to many different voxels.

Let’s make a design with a linear trend and a constant term:

```{python}
import numpy as np
import matplotlib.pyplot as plt
# Print arrays to 4 decimal places
np.set_printoptions(precision=4)
```

```{python}
X = np.ones((12, 2))
X[:, 0] = np.linspace(-1, 1, 12)
plt.imshow(X, interpolation='nearest', cmap='gray')
```

To fit this design to any data, we take the pseudoinverse:

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

Now let’s make some data to fit to. We will draw some somples from the standar
normal distribution using numpy.random.  We use
`numpy.random.seed` to make sure the random numbers are predictable:

```{python}
np.random.seed(42)
y_0 = np.random.normal(size=12)
beta_0 = piX.dot(y_0)
beta_0
```

We can fit this same design to another set of data:

```{python}
y_1 = np.random.normal(size=12)
beta_1 = piX.dot(y_1)
beta_1
```

Now the trick. Because of the way that matrix multiplication works, we can fit
to these two sets of data with the same call to `dot`:

```{python}
Y = np.vstack((y_0, y_1)).T
betas = piX.dot(Y)
betas
```

Of course this is true for any number of columns of Y.