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Spin systems where the difference in chemical shifts of the coupled nuclei is greater than ten times J are called first-order spin systems. These nuclei are weakly coupled, and their chemical shifts and coupling constant can generally be estimated from the well-separated signals in the spectrum.
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Functional mixed effects spectral analysis.

Robert T Krafty1, Martica Hall2, Wensheng Guo3

  • 1Department of Statistics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, U.S.A. krafty@pitt.edu.

Biometrika
|February 9, 2016
PubMed
Summary
This summary is machine-generated.

This study introduces a mixed effects Cramér spectral representation to model time series data. It accounts for correlations within units, enabling analysis of covariates on power spectra for improved scientific insights.

Keywords:
Cramér representationMixed effects modelReplicated time seriesSmoothing splineSpectral analysis

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Area of Science:

  • Statistics
  • Time Series Analysis
  • Spectral Analysis

Background:

  • Multivariate time series data are common in experiments.
  • Correlations within units and across segments complicate analysis.
  • Existing methods may not fully capture complex dependencies.

Purpose of the Study:

  • To introduce a mixed effects Cramér spectral representation for analyzing time series data.
  • To model the impact of covariates on the second-order power spectrum.
  • To account for correlations among time series segments from the same unit.

Main Methods:

  • Developed a mixed effects Cramér spectral representation.
  • Modeled transfer functions with deterministic (population-average) and random (unit-specific) components.
  • Utilized functional mixed effects models for log-spectra.

Main Results:

  • Demonstrated that log-periodograms converge to a functional mixed effects model for smooth spectra.
  • Proposed an iterative estimation procedure for smoothing splines and functional covariance.
  • Enabled prediction of unit-specific random effects using best linear unbiased prediction.

Conclusions:

  • The proposed method effectively models covariate effects on power spectra in correlated time series data.
  • Functional mixed effects models provide a robust framework for spectral analysis.
  • The estimation procedure offers a data-driven approach for complex time series analysis.