Appendix A: Local Similarity

In comparing two datasets, the purpose is to estimate a smoothly varying warping function, $S_k$, required to align one dataset, $h_k$, to a reference dataset, $r_k$,

$$r_k(t) \approx h_k(S_k(t))$$ (19)

We can represent the warping function with the shifts, $g_k(t)$, as follows:

$$S_k(t) = t + g_k(t)$$ (20)

where the $t$ denotes the original independent axis and $g_k(t)$ are the shifts required to match the datasets as defined in Equation A-1. The correlation coefficient can be used to quantify the quality of the match between datasets (Hampson-Russell, 1999). The LSIM method begins with the observation that the correlation coefficient ($c$) only provides one number to describe the datasets; however, we am interested in understanding the local changes in the datasets' similarity. Therefore, the LSIM method computes local similarity $c_t$, which is a function of time, $t$. The square of $c$ can be split into a product of two factors (Fomel, 2007a):

$$c_t^2 = r_t*h_t$$ (21)

where $r_t$ and $h_t$ are the solutions to the following regularized least-squares problems, respectively

$$\begin{aligned} \min_{r_t}(\sum_{t}(a_t - r_tb_t)^2 + R[r_t]) \... ...n_{h_t}(\sum_{t}(b_t - h_ta_t)^2 + R[h_t]) \\ [3pt] \end{aligned}$$

The regularization operator, $R$, is implemented using shaping regularization (Fomel, 2007b) and designed to enforce smoothness. To estimate the solution, LSIM is calculated for a series of shifts. The results of this calculation are accumulated and displayed on a `similarity scan.'