Least squares smoothing

This example illustrates smooth a noisy ECG waveform by least squares. This approach regularizes the energy of the second-order derivative.

Ivan Selesnick
selesi@poly.edu

Contents

Start

clear
close all

Load data

load data.txt;
whos

y = data;         % data value
N = length(y);

  Name        Size            Bytes  Class     Attributes

  data      300x1              2400  double              

Display data

figure(1)
clf
plot(y)
title('Data')

Smoothing (degree = 2)

D is the second-order difference matrix. It approximates the second-order derivative. In order to exploit fast banded solvers in Matlab, we define D as a sparse matrix using 'spdiags'.

e = ones(N, 1);
D = spdiags([e -2*e e], 0:2, N-2, N);

Observe the first and last corners of D. (D is too big to display, so we show the first and last corners only.)

First corner of D:

full(D(1:5, 1:5))

ans =

     1    -2     1     0     0
     0     1    -2     1     0
     0     0     1    -2     1
     0     0     0     1    -2
     0     0     0     0     1

Last corner of D:

full(D(end-4:end, end-4:end))

ans =

     1     0     0     0     0
    -2     1     0     0     0
     1    -2     1     0     0
     0     1    -2     1     0
     0     0     1    -2     1

Solve the least square problem

lam = 50;
F = speye(N) + lam * D' * D;            % F is a banded matrix
x = F \ y;                              % Matlab uses a fast solver for banded systems)

figure(1)
plot(x)

Published with MATLAB® 7.8