Last updated: 2019-10-30

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Introduction

In a previous post, we mentioned that trend filtering is a special case of generalized lasso with \(X=I\) and due to the special band structure of \(D\), it can be refomularized into a lasso problem.

Here, I’ll show the equivalence of 0-order trend filtering and random walk dynamic linear model.

To recap, \(k\)th order Trend filter can be written as

\[\begin{equation} \hat{\alpha} = argmin_\alpha \frac{1}{2}||y-H\alpha||_2^2 +\lambda\sum_{j=k+2}^n|\alpha_j|, \end{equation}\] and \(\hat{\beta} = H\hat{\alpha}\). When \(k=0\), \[\begin{equation} H=\left[\begin{array}{cccc}{1} & {0} & {\ldots} & {0} \\ {1} & {1} & {\ldots} & {0} \\ {\vdots} & {} & {} & {} \\ {1} & {1} & {\ldots} & {1}\end{array}\right] \end{equation}\]

Random walk dynamic linear model is \[\begin{equation} \begin{split} y_t = \beta_t+\epsilon_t, \epsilon_t|\sigma_t\sim N(0,\sigma^2) \\ \beta_t-\beta_{t-1} = \alpha_t, \alpha_t\sim g(\cdot), t=2,3,...,n \\ \beta_1\sim \pi(\cdot), \end{split} \end{equation}\] where \(g(\cdot)\) is sparsity-inducing prior and \(\pi(\cdot)\) is a noninformative prior.

Hence, we can rewrite \(y_t = \beta_1+\sum_{i=2}^t\alpha_i+\epsilon_t\). Posterior of \(\beta_1\) and \(\alpha:=\{\alpha_2,...,\alpha_n\}\) is \[p(\beta_1,\alpha|y,g,\pi)\propto p(y|\beta_1,\alpha)\pi(\beta_1)g(\alpha)\]

The MAP estimator is then \[\hat{\beta}_1,\hat{\alpha} = argmax \sum_{t=1}^n (y_t-\beta_1-\sum_{i=2}^t\alpha_i)^2+\log\pi(\beta_1)+\sum_{t=2}^n \log g(\alpha_t)\]

The equivalence holds if we choose Laplace prior on \(\alpha\). Of course, we can choose any sparsity-inducing prior.

Similarly, we can show the equivalence of order \(k\geq 1\).