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

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 AM University
• E-mail ymunoz_at_stat.tamu.edu
• Dr. Joan Staniswalis
• Department of Mathematical Sciences
• University of Texas at El Paso
• E-mail joan_at_math.utep.edu
• This project was partially supported by RCMI
  grant 5G12-RR08124 from the National Institute of
  Health.

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Overview

• Introduction of the Biological Problem
• Methodology
• Simulation
• Data Analysis
• Summary

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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).
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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
.
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Splines
A spline of order K with knots
, is any function of the form
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B-splines
The i th normalized B-spline of order K for the
knot sequence is
denoted by
and satisfy the properties
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Cubic B-Splines
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Ganglioside Profiles
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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.

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Individual Curves
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Properties
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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
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Warping Functions
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Aligned Curves
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Similarity
Similarity is based upon comparison of the
functions evaluated on a common grid G . The
index of similarity between two curves
uses the Pearsons sample
correlation coefficient.
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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

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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.

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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.

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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)

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Simulated Curves under
OLD YOUNG
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Size of the Test
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Simulation under
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Power function at a0.05
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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.
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Registered, Cut and Normalized Profiles
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Analysis Result
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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.

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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
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• Ramsay, J.O. and Silverman B.W. (1997).
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• New York Springer Series in Statistics.
• Ramsay, J.O. and Silverman B.W. (1998). S-Plus
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