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Related Concept Videos

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Learning to draw Fischer projections of molecules and understanding their relevance plays a crucial role in the visual depiction of organic molecules. A Fischer projection is a two-dimensional projection on a planar surface to simplify the three-dimensional wedge–dash representation of molecules. This is especially helpful in the case of molecules with multiple chiral centers that can be difficult to draw. Here, all the bonds of interest are represented as horizontal or vertical lines. While...
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Different notations are used to represent the three-dimensional structure of molecules on two-dimensional surfaces. One of the most commonly used representations is the dash-wedge formula. The dashed wedges, solid wedges, and the plane lines indicate the groups situated behind the plane, coming out of the plane, and in the plane, respectively.
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Relative velocity is the velocity of an object as observed from a particular reference frame, or the velocity of one reference frame with respect to another reference frame. The concept of relative velocity can be used to describe motion in two dimensions. Consider a particle P and two reference frames S and S′. The position of the origin of S′ as measured in S is , the position of P as measured in S′ is , and the position of P as measured in S is , which can be evaluated by utilizing...
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Assessing Cerebral Autoregulation via Oscillatory Lower Body Negative Pressure and Projection Pursuit Regression
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Projection pursuit in high dimensions.

Peter J Bickel1, Gil Kur2, Boaz Nadler3

  • 1Department of Statistics, University of California, Berkeley, CA 94720; boaz.nadler@weizmann.ac.il bickel@stat.berkeley.edu.

Proceedings of the National Academy of Sciences of the United States of America
|August 29, 2018
PubMed
Summary
This summary is machine-generated.

Projection pursuit in high-dimensional data can find spurious structures when sample size is small. Enforcing sparsity is crucial for reliable analysis, especially with independent component analysis.

Keywords:
dimensionality reductionindependent component analysisprojection pursuitrandom matrix theorysparsity

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Area of Science:

  • Statistics
  • Data Mining
  • Machine Learning

Background:

  • Projection pursuit is a classical exploratory data analysis technique.
  • Its application in high-dimensional settings is increasingly relevant.
  • Understanding its behavior in these settings is critical for reliable data interpretation.

Purpose of the Study:

  • To analyze the asymptotic properties of projection pursuit in high-dimensional data.
  • To investigate the impact of dimension (p) and sample size (n) on projection pursuit.
  • To determine the conditions under which projection pursuit can detect non-Gaussian structures.

Main Methods:

  • Asymptotic analysis of projection pursuit on structureless multivariate Gaussian data.
  • Examination of data with identity covariance as dimension p and sample size n tend to infinity.
  • Analysis of sparse projections involving a subset of variables.

Main Results:

  • If n << p, projections can approximate any distribution, potentially leading to spurious findings.
  • If n >> p, not all limiting distributions are possible, but non-Gaussian distributions can still be approximated.
  • Sparse projections asymptotically yield Gaussian distributions, while non-sparse projections do not.

Conclusions:

  • In high-dimensional settings (small n, large p), projection pursuit may identify statistically insignificant structures without enforced sparsity.
  • Fundamental limitations exist in detecting non-Gaussian signals in high-dimensional data using methods like independent component analysis.
  • The study highlights the importance of considering dimensionality and sparsity when applying projection pursuit.