[FDA] 4. Curve alignment

Curve alignment

In FDA, we usually use mean function to capture overall trend of functions. However, it is not fitted properly when each curve aligns with different time points. Thus we need to align these each curves and it is called 'Curve alignment'

 

There are two components in curves. These are

• Amplitude variation
• phase variation

Amplitude variation means height variation and phase variation menas domain variation. We want to focus on amplitude variation by eliminating phase variation in many cases. (want to treat curves at the same time point)

 

Warping functions

So, we model phase variation by warping function as $X_n(t) = X_n^{*} (h_n(t))$ where $X_n$ is unaligned curve, $X_n^*$ is the unobserved aligned curve, and $h_n$ is a subject specific warping function unobserved. i.e. our observed unaligned curve can be represented as function of unobserved aligned curve.

Difference between $X_n(t)$ and $X_n^*(t)$ describes phase variation. And variation between $X_n^*(t)$ describes amplitude variation.

 

warping functions $h_n(t)$ have some property.

1. $h(0)=0$
2. $h(T)=T$
3. it is monotone increasing function 

Then, aligned curve can be represented as follows:

$$ X_n^*(t)=X_n(h_n^{-1}(t))$$

 

Then, how can we align the curves? First approach is landmark registration. It is the process that choose landmark from each function to align such as minimum or maximum point.

Let $t_{n1}, \dots,t_{nJ}$ be the selected time point and $t_1,\dots,t_J$ be the selected points of mean function. Then our goal is to find $h_n$ such that

$$h_n(t_j)=t_{nj}$$

 

Another approach is continuous registration. It is much easier to automate. This method need to choose a specific curve (mean function in general) and align "all" time points to that curve.

Basically, choose $h_n$ that minimizes

$$\int (X(h_n^{-1}(t))-\bar{X}(t))^2 \ dt$$

 

Practice

In practice, I aligned following curves using continuous registration.

50 curves with 10 time points

The unaligned curves seems to have two main trends (left peak and right peak).

And aligned curves are as follows:

aligned curve

Result is quite good in perspective of alignment. But it forces to have same trend with centered peak.

Thus, I think that curve alignment has some disadvantage since it obscures the different trend in curves.