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Linear Approximation in Frequency Domain

Linear Approximation in Frequency Domain

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.

Linear Approximation in Time Domain

Linear Approximation in Time Domain

Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length, the...

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Related Experiment Video

Updated: Jun 15, 2026

Network Analysis of the Default Mode Network Using Functional Connectivity MRI in Temporal Lobe Epilepsy

Network Analysis of the Default Mode Network Using Functional Connectivity MRI in Temporal Lobe Epilepsy

Published on: August 5, 2014

Dimensionality reduction of fMRI time series data using locally linear embedding.

Peter Mannfolk¹, Ronnie Wirestam, Markus Nilsson

¹Department of Medical Radiation Physics, Clinical Sciences, Lund University, Barngatan 2B, 22185, Lund, Sweden. peter.mannfolk@med.lu.se

Magma (New York, N.Y.)

|March 16, 2010

Summary

Locally linear embedding (LLE) effectively analyzes functional MRI (fMRI) data by uncovering non-linear relationships, outperforming principal component analysis (PCA) in identifying both task-related and resting-state networks.

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Area of Science:

Neuroimaging
Data Science
Computational Neuroscience

Background:

Data-driven methods are valuable for fMRI analysis when a priori models are absent.
fMRI data, including resting-state, exhibit non-linear properties often overlooked by linear models.

Purpose of the Study:

Introduce the non-linear Locally Linear Embedding (LLE) algorithm for fMRI time series dimensionality reduction.
Evaluate LLE's efficacy in capturing non-linear dynamics within fMRI data.

Main Methods:

Optimized and tested LLE using simulated and volunteer fMRI data for task-evoked responses.
Compared LLE as a preprocessing step for Independent Component Analysis (ICA) against Principal Component Analysis (PCA).
Assessed LLE-ICA against PCA-ICA and non-linear PCA-ICA on data with known non-linear properties and a resting-state dataset.

Main Results:

LLE successfully identified task-related components and known resting-state networks, performing comparably to PCA.
LLE demonstrated superior ability to separate non-linearly modulated sources in a low-dimensional subspace compared to PCA.
LLE outperformed non-linear PCA when applied to the same target dimensionality.

Conclusions:

Locally Linear Embedding (LLE) shows promise for fMRI data analysis.
LLE offers potential advantages over PCA by effectively identifying non-linear relationships in fMRI data.