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Real-World Applications of Space Curves

Modern aerospace navigation depends on the accurate prediction of motion in three-dimensional space. In defense applications, radar systems continuously track both interceptors and moving aerial targets to find whether their flight paths will result in a collision. These motions are modeled mathematically as space curves, which represent paths that change continuously with time. Each object’s position is described by a vector function that specifies its location in terms of time-dependent...
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Level curves and contour maps provide a way to visualize functions of two variables on a two-dimensional plane. A useful example is a topographic map, where curved lines represent locations that share the same elevation. In mathematics, these curves are called level curves or contour lines. Each contour line corresponds to points in the domain where the function has a constant value. For a function of two variables written as z = f(x,y), a level curve is defined by the equation f(x,y) = k,...
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Linear Transformations and the k-Means Clustering Algorithm: Applications to Clustering Curves.

Thaddeus Tarpey1

  • 1Thaddeus Tarpey is Professor, Department of Mathematics and Statistics, Wright State University, Dayton, Ohio.

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Summary

Clustering functional data with k-means requires optimal linear transformation of regression coefficients. This method improves cluster accuracy by aligning data variability with cluster differences, enhancing functional data analysis.

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Area of Science:

  • Statistics
  • Functional Data Analysis
  • Machine Learning

Background:

  • Functional data clustering is crucial for identifying patterns in time-series or curve data.
  • K-means clustering is a common method but sensitive to data transformations.

Purpose of the Study:

  • To investigate the impact of linear transformations on k-means clustering of functional data.
  • To identify an optimal linear transformation for accurate functional data clustering.

Main Methods:

  • Functional data curves were fitted using various basis functions, resulting in different linear transformations.
  • K-means clustering was applied to raw data and transformed regression coefficients.
  • The L(2) metric in function space was used to evaluate clustering performance.

Main Results:

  • K-means clustering is not invariant to linear transformations; results vary based on curve fitting.
  • Optimal linear transformations align data variability with cluster differences, improving results.
  • Clustering raw data often yields similar results to using orthogonal design matrices.

Conclusions:

  • Effective functional data clustering via k-means necessitates a suitable linear transformation of regression coefficients.
  • This approach enhances the alignment of data variability with underlying cluster structures.
  • The method was illustrated with an antidepressant treatment example for depressed individuals.