Time deconvolution

Let us consider a linear model: \begin{equation} \label{eq:fir} \mathbf{y} = \mathbf{X} \mathbf{b} + \mathbf{e} \end{equation} where \(\mathbf{y} \in \Re^{T\,\times\,1}\) is a response time series over \(T\) timepoints, \(\mathbf{X} \in \Re^{T\,\times\,P}\) is a design matrix of \(P\) predictors such that \(\|\mathbf{X}'\mathbf{X}\| > 0\), \(\mathbf{b} \in \Re^{P\,\times\,1}\) is an unknown weight vector, \(\mathbf{e} \in \Re^{T\,\times\,1}\) is a Gaussian noise vector \(\mathbf{e} \sim \mathcal{N} (\mathbf{0}, \sigma^2\mathbf{I})\).

When the \(\mathbf{X}\) is a finite impulse respnose (FIR) design matrix (e.g., Toeplitz matrix), the linear model describes a convolution between one column in \(\mathbf{X}\) (e.g., \(\mathbf{x}^{(1)}\) where \(\)) and the vector \(\mathbf{b}\). Then, solving this model to find \(\mathbf{b}\) is equivalent to finding a deconvoultion from \(\mathbf{y}\) with \(\mathbf{X}\) to recover \(\mathbf{b}\).

For that an FIR model is prone to overfitting to noise due to its flexibility, the stability of solution can be earned by introducing a small bias such as \(l^2\) penalty: \begin{equation} \mathbf{ {\hat b} } = (\mathbf{X}’ \mathbf{X} + \lambda \mathbf{I})^{-1} \mathbf{y} \end{equation} which is a well-known problem.

In fMRI terminologies, \(\mathbf{b}\) is the HRF kernel for a given design vector \(\mathbf{x}^{(1)}\), which is assumed to be identical to underlying neural activity. Thus, without knowing the true HRF, estimating the true neural activity (\(\mathbf{x}^{(1)}\)) from \(\mathbf{y}\) is met with underdeterminedness.