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Linear Approximation in Frequency Domain01:26

Linear Approximation in Frequency Domain

254
Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....
254
Aliasing01:18

Aliasing

405
Accurate signal sampling and reconstruction are crucial in various signal-processing applications. A time-domain signal's spectrum can be revealed using its Fourier transform. When this signal is sampled at a specific frequency, it results in multiple scaled replicas of the original spectrum in the frequency domain. The spacing of these replicas is determined by the sampling frequency.
If the sampling frequency is below the Nyquist rate, these replicas overlap, preventing the original...
405

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Related Experiment Video

Updated: Nov 27, 2025

Combined Invasive Subcortical and Non-invasive Surface Neurophysiological Recordings for the Assessment of Cognitive and Emotional Functions in Humans
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De-Biased Graphical Lasso for High-Frequency Data.

Yuta Koike1

  • 1Mathematics and Informatics Center and Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan.

Entropy (Basel, Switzerland)
|December 8, 2020
PubMed
Summary
This summary is machine-generated.

This study introduces novel statistical inference methods for high-dimensional precision matrices from high-frequency data. It enables accurate estimation, interval construction, and hypothesis testing for precision matrix entries.

Keywords:
Malliavin calculusasymptotic mixed normalityfactor modelhigh-dimensionsprecision matrixsparsity

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Area of Science:

  • Statistics
  • Econometrics
  • Data Science

Background:

  • High-frequency financial data presents unique challenges for statistical analysis due to its high dimensionality.
  • Understanding the precision matrix is crucial for modeling complex financial systems and risk management.

Purpose of the Study:

  • To develop a robust statistical inference theory for the precision matrix in high-dimensional, high-frequency data settings.
  • To extend existing methods to include interval estimation and hypothesis testing beyond point estimation.
  • To accommodate scenarios with known factor structures within the data.

Main Methods:

  • Development of an abstract asymptotic theory for the weighted graphical Lasso and its de-biased variants.
  • Extension of the theory to incorporate known factor structures in the data.
  • Application of the theory using realized covariance matrices as initial estimators.

Main Results:

  • Established a feasible asymptotic distribution theory for precision matrix entries.
  • Enabled the construction of simultaneous confidence intervals for precision matrix elements.
  • Developed multiple testing procedures for hypotheses concerning precision matrix entries.

Conclusions:

  • The new theory provides a comprehensive framework for statistical inference on precision matrices with high-frequency data.
  • The methods are applicable in practical settings, offering tools for robust financial data analysis.
  • This work advances the understanding and application of graphical models in high-dimensional econometrics.