1. Curve Registration The rigid metric of physical time may not be directly relevant to the internal dynamics of many real-life systems. Rather, there can be a sort of biological or meteorological time scale that can be nonlinearly related to physical time…

2. Amplitude Variation vs Phase Variation Shift Registration Landmark Registration More General Registration Techniques

3. Shift Registration The goal is to estimate a δ for each curve such that the criteria (REGSSE) is minimized The Procrustes Method involves estimating a delta for each curve. Then, one reestimates with the newly registered curves the mean curve and repeats the process.

4. Numerical method for a simple shift How do we estimate each delta? Newton-Raphson 1. Choose an initial delta, perhaps using visual landmarks. 2. Modify delta by 3. Update the mean and continue Procrustes Method. In the text, R & S use this method to improve visual alignment of the pinch force data.

5. Landmark Registration Mark important structural events (characteristic points). Sometimes this is done visually. Other times, events in the derivative (minima and maxima) may be useful. How do we use these events to align the curve? Time Warping Function—a strictly increasing transformation of time. h(t) where h(0)=0 and h(T)=T for our time in [0,T] Must use a reference function (often the mean) and h(t 0f )=t if for each feature f. Gasser et al use average times to do this. So, there is no target function. Our registered sample functions will be x(h(t))=x*(t). These together will yield the structural average.

6. Interpolation In R&S, the example uses linear extrapolation to fit h(t) through each of the adjusted values for the handwriting data. This is data from Ramsay’s “ A guide to curve registration.” Here a smooth interpolation is used. Another example is Gasser et all using piecewise cubic polynomials to piece together human growth curve data. One problem with curve registration is the regularity of landmark features across the sample. Sometimes there is ambiguity in maxima and minima and visual or intuitive methods must be used.

7. General Method The problem is obtaining a strictly increasing function that aligns h(0)=0 and h(T)=T. We restrict h to be in this family and to be twice differentiable. Thus, Which has the solution,

8. Modeling warping using SDE’s Ramsey and Wang mention that an alternative model would be Where everything is as before except z is a stochastic process If we take B to be Brownian motion (w.o. drift) there is a solution This may be “… envisaged as a clock that is running fast or slow from instant to instant, constantly undergoing a percentage change in rate in a memoryless chaotic manner.”

9. Finding w(t) Now, we look for a w(t) that will minimize Lambda is a penalty on the second derivative of h(t), which ensures a smooth h. Ramsay and Li recommend an order 1 B-spline to approximate w. This allows for a closed form solution of h(t)—which, while not absolutely necessary, is desirable.

10. Height Acceleration This was done with lambda= 0.01 and breakvalues 4,7,10,12,14,16,18.

11. An Extension Alternatively, you could minimize This allows for differing weights over the intervals, across derivatives, and over vectors if the function is vector-valued.

12. An alternate minimizing criteria What if the sample functions are multiples of the target function (the mean function)? Minimize the log of the smallest eigenvalue of If these really are multiples, the smallest eigenvalue should be zero.

13. Other Sources Silverman includes the warping function as part of his principal component analysis. He uses a parametric model for the warping function. Sakoe and Chiba did a much earlier version by minimizing a weighted distance between curves (with appropriate restrictions on h) using dynamic programming Kneip and Gasser (1992) went through a detailed analysis of the statistical properties of using warping functions.