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It is far more common for collisions to occur in two dimensions; that is, the initial velocity vectors are neither parallel nor antiparallel to each other. Let's see what complications arise from this. The first idea is that momentum is a vector. Like all vectors, it can be expressed as a sum of perpendicular components (usually, though not always, an x-component and a y-component, and a z-component if necessary). Thus, when the statement of conservation of momentum is written for a...
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Elastic collision of a system demands conservation of both momentum and kinetic energy. To solve problems involving one-dimensional elastic collisions between two objects, the equations for conservation of momentum and conservation of internal kinetic energy can be used. For the two objects, the sum of momentum before the collision equals the total momentum after the collision. An elastic collision conserves internal kinetic energy, and so the sum of kinetic energies before the collision equals...
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Causal Discovery in High-dimensional, Multicollinear Datasets.

Minxue Jia1,2, Daniel Y Yuan1,2, Tyler C Lovelace1,2

  • 1Department of Computational and Systems Biology, University of Pittsburgh School of Medicine, Pittsburgh, PA, USA.

Frontiers in Epidemiology
|February 13, 2023
PubMed
Summary
This summary is machine-generated.

This study introduces a novel matrix factorization approach to improve causal discovery in large biomedical datasets. The method effectively addresses high dimensionality and multicollinearity, revealing key factors in cancer and COVID-19.

Keywords:
Causal DiscoveryCollinearityDimensionality ReductionEmpirical Bayes Matrix FactorizationLatent Factors

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Area of Science:

  • Genomics
  • Bioinformatics
  • Computational Biology

Background:

  • High-throughput genomic sequencing generates vast datasets with numerous biological features.
  • Discovering causal mechanisms of disease from complex data is challenging due to high dimensionality and multicollinearity.
  • Existing causal discovery algorithms struggle with these inherent data complexities.

Purpose of the Study:

  • To evaluate a novel non-orthogonal, general empirical Bayes matrix factorization approach for causal discovery in biomedical research.
  • To address the limitations of traditional causal discovery algorithms, specifically the curse of dimensionality and multicollinearity.
  • To identify latent factors and biological pathways associated with disease outcomes and clinical features.

Main Methods:

  • Applied a non-orthogonal, general empirical Bayes matrix factorization method to causal discovery algorithms.
  • Evaluated the strategy on simulated datasets to assess performance.
  • Tested the approach on real-world breast cancer and SARS-CoV-2 (COVID-19) datasets.

Main Results:

  • The strategy successfully inferred interpretable latent factors from observed variables.
  • In breast cancer data, identified survival-associated latent factors and enriched pathways linked to clinical features.
  • In the SARS-CoV-2 dataset, accurately predicted COVID-19 status and ICU admission, associating factors with known biological pathways.

Conclusions:

  • The empirical Bayes matrix factorization approach offers a promising solution for causal discovery in high-dimensional biomedical data.
  • This method enhances the interpretability of latent factors and their biological relevance.
  • The findings demonstrate the utility of this strategy for understanding disease mechanisms and clinical outcomes in cancer and infectious diseases.