Cross Validated
2023-08-06 20:56 UTC
Score 7.0
AI-113-20230806-social-media-973e7f07
Preliminary Consider $n$ individuals each with observed data $ Z_i, i = 1, \ldots, n$ . For each individual $i$ , the longitudinal predictor $Z_i = \{Z_i(t_{i1}), \ldots, Z_i(t_{i,R_i})\}$ is measured at some finite number of observation times $t_i = (t_{i1}, \ldots, t_{i,R_i})$ . We can define a finite grid as $\bigcup_{i=1}^{n} t_i$ , covering all unique observation times for the sample, with $\tau = \max (\bigcup_{i=1}^{n} t_i)$ . The article says: Start with an arbitrary set of smooth basis functions $\psi_1(t), \ldots, \psi_K(t)$ to characterize the functional predictor, where $K$ denotes the total number of basis functions. The set of basis functions can be, for example, the empirical basis functions estimated by the conventional FPCA method , but can be flexibly extended to other basis functions. The functional data can then be rewritten in matrix notation as $(Z_1(t), \ldots, Z_n(t))^T = \bf{\lambda}\bf{\psi}(t)$ , where $\bf{\lambda} = (\lambda_1, \ldots, \lambda_n)^T$ , with $\lambda_i = (\lambda_{i,1}, \ldots, \lambda_{i,K})^T$ , and $\bf{\psi}(t) = (\psi_1(t), \ldots, \psi_K(t))^T$ , $t \in [0, \tau]$ . $\bf{M}$ is of dimension $K \times K$ , with the $(k, k')$ entry being $\langle \psi_k(t), \psi_{k'} (t) \rangle$ for $k, k' \in K$ , $t \in [0, \tau]$ . My question: As you can see $Z_{i}$ 's are irregular and sparse. So, if we were to write it in matrix notation, then we would have NAs in it. For example, #### set.seed(123) # number of observations n So, my unde…