1. A Similarity Analysis of Curves: A Comparison of the Distribution of Gangliosides in Brains of Old and Young Rats.
Yolanda Munoz Maldonado
Department of Statistics
Texas A&M University
Dr. Joan Staniswalis
Department of Mathematical Sciences
University of Texas at El Paso
This project was partially supported by RCMI grant 5G12-RR08124 from the National Institute of Health.

2. Overview Introduction of the Biological Problem Methodology Simulation
Data Analysis
Summary

3. Thin Silica Gel Plate

4. Ganglioside Standards

5. Standard Curves

6. Functional Object
The intensity of the gangliosides is considered as a function of distance, so the first step in the analysis is to reconstruct the entire profile on a closed interval so that it can be evaluated at any point (Ramsay and Silverman, 1998). Regression splines are used for this purpose (Eubank, 1988).

7. Regression Splines
The sampled curve Y(t), t in G, is interpolated by fitting a
linear combination of B-splines
This involves the minimization of over

8. Splines
A spline of order K with knots , is any function of the form:

9. B-splines
The i th normalized B-spline of order K for the knot sequence is denoted by
and satisfy the properties:

10. Cubic B-Splines

11. Ganglioside Profiles

12. Warping Functions The registration of the curves requires:
a monotone transformation w for each curve Y(t) such that the registered curves have more or less identical argument values for any of the characteristic features.

13. Individual Curves

14. Properties

15. Warping function
The warping functions were estimated using the Penalized Least-Squares Error Criterion by minimizing
The minimizer of this is expression is a natural cubic spline
(Shoenberg 1946). Since we want to preserve the area under the curve, the registered curve is given by

16. Warping Functions

17. Aligned Curves

18. Similarity
Similarity is based upon comparison of the functions evaluated on a common grid G . The index of similarity between two curves uses the Pearson’s sample correlation coefficient.

19. Test Statistics Three test statistics were considered:
The pooled mean similarity within groups:
The pooled variance similarity within groups: .
3. The ratio of the pooled-mean to the square-root of the pooled variance:

20. Permutation Distribution
The permutation distribution of each test statistic under the null hypothesis is obtained by permuting the 10 curves, and then dividing them into two groups “old” and “young”.
The p-value for the pooled-mean and the ratio is given by the number of permutations which yield a value of the test statistic greater than the observed value.
The p-value for the pooled-variance is obtained by the number of permutations which yield a value of the test statistic that is less than the observed value.

21. Simulation The noisy data were simulated according to
is the normal pdf.
is generated following a
is the vector of the center of the peaks of the original data.
is the variance-covariance matrix of these points.

22. Simulation
The follow a chi-square distribution with the following degrees of freedom: 20, 45, 30, 20, 20.
The are normally distributed with mean 0 and covariance
The are independent, uniformly distributed coefficients on the intervals:
min = ( 0.175, 0.25, .0.2, .0.08, 0.1)
max = (0.5 ,0.7, 0.5, 0.417, 0.4)

23. Simulated Curves under
OLD YOUNG

24. Size of the Test
0.032 *
0.008
0.024 *
0.01
0.068
0.064
0.05
0.108
0.104
0.080
0.1
Ratio Pooled Mean/Variance
Pooled
Variance
Mean
Proportion of Rejections
Significance
Level

25. Simulation under

26. Power function at a=0.05

27. Data Analysis Three data sets are studied: Medulla Locus Coeruleus
Hippocampus
The last data set was expected to show no differences between old and young rats.

28. Registered, Cut and Normalized Profiles

29. Analysis Result 0.333 235.4329 Ratio of Means/Variances 0.706 0.0059
Hippocampus
0.805
0.8347
Mean
0.014
0.028
Locus Coeruleus
0.009
0.027
0.0024
Medulla
0.046
P-value
Test Statistic
Brain Region

30. Conclusions
The result confirms the biologists expectations of differences in ganglioside concentration in the Medulla region and no difference for Hippocampus.
The result for Locus Coeruleus region provides new evidence for a significant development shift in ganglioside pattern.

31. References
Irwin, L.N. (1984). Ontogeny and Phylogeny of vertebrate brain gangliosides. In Ganglioside Structure, Function and Biomedical Potential. New York: Plenum. Edited by Leeden, R.W., Yu, R.K., Rapport, M.M. and Suzuki, K, pp
Eubank, Randall (1988). Spline smoothing and nonparametric regression. New York: Marcel Dekker, Inc.
Heckman, N. (1997). The Theory and Application of Penalized Least Squares Methods or Reproducing Kernel Hilbert Spaces Made Easy.
Kimeldorf, G. and Wahba, G. (1971). Some Results on Tchebycheffian Spline Functions.
Journal of Mathematical Analysis and Applications, vol. 33, pp
Kneip, A. and Gasser, T. (1992). Statistical Tools to Analyze Data Representing a Sample of Curves. The Annals of Statistics, Vol. 20, No. 3, pp
Ramsay, J.O. and Silverman B.W. (1997). Functional Data Analysis.
New York: Springer Series in Statistics.
Ramsay, J.O. and Silverman B.W. (1998). S-Plus Functions for FDA.