A Similarity Analysis of Curves: A Comparison of the Distribution of Gangliosides in Brains of Old and Young Rats.

Yolanda Munoz MaldonadoDepartment of Statistics

Texas A&M University

E-mail: ymunoz@stat.tamu.edu

Dr. Joan StaniswalisDepartment of Mathematical Sciences

University of Texas at El Paso

E-mail: joan@math.utep.edu

This project was partially supported by RCMI grant 5G12-RR08124 from the National Institute of Health.

Overview

Introduction of the Biological Problem

Methodology Simulation Data Analysis Summary

Thin Silica Gel Plate

Ganglioside Standards

Standard Curves

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).

Regression Splines

d

dd t )(

DB2 Y

The sampled curve Y(t), t in G, is interpolated by fitting a

linear combination of B-splines .

This involves the minimization of over .

Splines

A spline of order with knots , is any function of the form:

nttt ... 2 1

1

0 1

1)(K

k

n

i

Kii

kk ttttS

B-splines

The i th normalized B-spline of order for the knot sequence is denoted by

1, )](,...,)[()(

KiKiiKiKi ttttttS

and satisfy the properties:

support itson positive is )(S .3

1)()( .2

],[ if ,0)( .1

Ki,

1

1

1,,

,

t

ttttStS

ttttS

ijj

j

KjiKiKi

KiiKi

nttt ... 2 1

Cubic B-Splines

Ganglioside Profiles

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.

)(twY

Individual Curves

Properties

jj

n

o

w

Ttw

tw

)( 3.

)( .2

0)( .1

Warping function

The warping functions were estimated using the Penalized Least-Squares Error Criterion by minimizing

T

jjj dww

0

2)2(7

1

2 )( )(

w

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

)]([ )1( twYtw

Warping Functions

Aligned Curves

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.

GttYtY ji , )( , )( **

ijr** , ji YY

Test Statistics

Three test statistics were considered:

2. The pooled mean similarity within groups:

3. The pooled variance similarity within groups:

.

3. The ratio of the pooled-mean to the square-root of the pooled variance:

2yo mm

2

22yo ss

22yo

yo

ss

mm

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.

SimulationThe 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.

)(0075.0)ˆ,ˆ;()( 25

1

ttatY iii

i

t ˆ,ˆ,ˆ,ˆ,ˆ 54321 ooN ,5

o

o

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)

2ˆ i

t ts

ka , max,min kk

Simulated Curves under

OLD YOUNG

oH

Size of the Test

0.032 *0.0080.024 *0.01

0.0680.0680.0640.05

0.1080.1040.0800.1

Ratio Pooled Mean/Variance

Pooled

Variance

Pooled

Mean

Proportion of RejectionsSignificance

Level

Simulation under aH

Power function at

Data Analysis

Three data sets are studied:

1. Medulla2. Locus Coeruleus3. Hippocampus

The last data set was expected to show no differences between old and young rats.

Registered, Cut and Normalized Profiles

Analysis Result

0.333235.4329Ratio of Means/Variances

0.7060.0059VarianceHippocampus

0.8050.8347Mean

0.014109.815Ratio of Means/Variances

0.0280.00987VarianceLocus Coeruleus

0.0090.736441Mean

0.027588.8824Ratio of Means/Variances

0.0270.0024VarianceMedulla

0.0460.84114Mean

P-valueTest StatisticBrain Region

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.

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. 319-329.

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.

http://www.stat.ubc.ca/people/nancy.

Kimeldorf, G. and Wahba, G. (1971). Some Results on Tchebycheffian Spline Functions.

Journal of Mathematical Analysis and Applications, vol. 33, pp.82-95.

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. 1266-1305.

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. http://www.psych.mcgill.ca/faculty/ramsay.