Do you need a robust method for parameter estimation of Gaussians within MATLAB? You have come to the right place!
This algorithm demonstrates the fundamentals of multiscale parameter estimation (msPE) by using it for estimation of the parameters of noisy Gaussians signals. All parameters of the Gaussian are computed from algebraic expressions. As demonstrated in the articles given below, even better parameter estimates can be achieved my means of a simple multivariate optimization, where msPE results serve as starting values.
msPE can be applied to real-life signals effortlessly with only a small number of open parameters. As Gaussian functions appear in many contexts in science and engineering, there are many potential applications. The processing of electrocardiography (ECG) signals has already been conducted sucessfully .
This algorithm was tested using MATLAB R2018b and R2019b on Linux and MacOS. The only dependency is the Wavelet Toolbox from which we use the cwt() command. Additionally, we use the tight_subplot() script by P. Kumpulainen for visualizations and the crossing() script by S. Brueckner for zero-crossing detection.
Start the file run_me.m from within MATLAB. Here you can define the signals to be processed. Available options are:
- Number of Gaussian signals
- Selection of Gaussian parameters
- Linearly increasing
- Random
- Type of noise
- Normally distributed noise
- Uniformly distributed noise
- Correlated (pink) noise
- Signal-to-noise ratio [dB]
Subsequently, the algorithm processes each signal using msPE (shown here: Gaussian signal corrupted by pink noise)
After processing each signal, a summary is given. Individual results can be opened by clicking the respective index.
If you want to customize msPE to your signals, we provide quickstart.m which can serve as a starting point (30 lines
of code only). Just replace the Gaussian signal with your own data. The only open parameter is the number of scales. If
msPE fails, we suggest to reduce the number of scales.
clear all; close all; format('compact');
addpath("scripts/");
addpath("external scripts/");
% create a Gaussian
t = [-5:0.01:5]; % time axis
A = 5; % amplitude
B = 0; % baseline level
sigma = 1; % width
mu = 0; % shift on time axis
signal = B + A .* exp(-((t-mu).^2 / (2*sigma.^2))); % Gaussian
signal = signal + randn(size(signal)); % Add noise
% apply it to msPE
[res, ~, ~] = mspe(signal, t, 256); % 256 = number scales
% average results over all scales;
A_mspe = median(res.A_mspe(:), 'omitnan');
B_mspe = median(res.B_mspe(:), 'omitnan');
sigma_mspe = median(res.sigma_mspe(:), 'omitnan');
mu_mspe = median(res.mu_mspe(:), 'omitnan');
% generate signal with parameters from msPE
signal_mspe = B_mspe + A_mspe .* exp(-((t-mu_mspe).^2 / (2*sigma_mspe.^2)));
% show results
figure; hold on; box on;
plot(t, signal, 'r.');
plot(t, signal_mspe, 'Color', 'k', 'LineWidth', 2);
legend('Data', 'Gaussian estimated by msPE');
@ARTICLE{8579214,
author={M. {Kukuk} and N. {Spicher}},
journal={IEEE Transactions on Signal Processing},
title={Parameter Estimation Based on Scale-Dependent Algebraic Expressions and Scale-Space Fitting},
year={2019},
volume={67},
number={6},
pages={1431-1446}}
Nicolai Spicher (Website)
Peter L. Reichertz Institute for Medical Informatics
Technical University of Braunschweig
Prof. Dr. Markus Kukuk (Website)
Department of Computer Science
University of Applied Sciences and Arts Dortmund