Adaptive-Degree Polynomial Filter (Savitzky-Golay Filter)

Obtain the polynomial degree selected in the adaptive-degree polynomial filter.

Description

function polynomial_degree=adpf(data_frame)

Adaptive-Degree Polynomial Filter (Savitzky-Golay Filter)

Commonly, the degree of the fitting polynomial for the Savitzky-Golay filter is fixed. However, the polynomial degree can vary according to the sum of squares of fitting residuals and the statistical testing to obtain the adaptive-degree polynomial filter.

This function can be used to obtain the polynomial degree (output: polynomial_degree) selected in the adaptive-degree polynomial filter (Savitzky-Golay filter) for the data subset of length 2M+1 (intput: data_frame).

Usage:
Run Pkmi_Calculate first to obtain the numerical table of gram polynomials so as to accelerate the calculation.

Examples:
%Examples:
adpf_demo %A demo of data smoothing with the adaptive-degree polynomial filter
Fig_1 %Obtain figure 1 of Ref. [1]
Fig_2 %Obtain figure 2 of Ref. [1]

Author:
      Jianwen Luo <[e-mail], [e-mail]> 2004-10-26
      Department of Biomedical Engineering, Department of Electrical Engineering
      Tsinghua University, Beijing 100084, P. R. China

References:
[1] P. Barak,
"Smoothing and differentiation by an adaptive-degree polynomial filter," Analytical Chemistry, vol. 67, pp. 2758-2762, 1995.
[2] M. Jakubowska and W. W. Kubiak,
"Adaptive-degree polynomial filter for voltammetric signals," Analytica Chimica Acta, vol. 512, pp. 241-250, 2004.
[3] A. Savitzky and M. J. E. Golay,
"Smoothing and Differentiation of Data by Simplified Least Squares Procedures," Analytical Chemistry, vol. 36, pp. 1627-1639, 1964.
[4]P. A. Gorry,
"General Least-Squares Smoothing and Differentiation by the Convolution (Savitzky-Golay) Method," Analytical Chemistry, vol. 62, pp. 570-573, 1990.
[5] J. W. Luo, K. Ying, P. He and J. Bai,
¡°Properties of Savitzky-Golay digital differentiators,¡±
Digital Signal Processing, in press
[6] http://www.mathworks.com/matlabcentral/fileexchange/loadFileList.do?objectType=search&criteria=savitzky+or+sgolayfilt&Search=Go
[7] http://uk.cambridge.org/chemistry/resources/delevie/

• Acknowledgements: This submission has inspired the following:; z-transform of 1D & 2D Savitzky-Golay Smoothing and Differentiation Filter
• MATLAB release: MATLAB 7.0.1 (R14SP1)

[Jianwen Luo]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Revision:

if N==3
    polynomial_degree=1;%%%%%%%%%%%%revised from 2
else
    k_start=1;% 0/1 for smoothing
    k_end=N-2;%%%%%%%%%%%revised from N-1-2
    k=k_start;
    p_fit=polyfit(x,y,k);%polynomial degree = k_start
    res_odd=sum((y-polyval(p_fit,x)).^2);
    
    Fxs=[];
    F_alphas=[];
    while k<k_end%%%%%%%%%%%%%revised from k<=k_end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Demo:
global Pkmi_array
if exist('Pkmi_array.mat','file')
    load Pkmi_array
else
    error('Please run Pkmi_Calculate first to obtain the numerical table of gram polynomials!')
end

A=1;
x0s=[20 40 60 90];
ws=[1 2 4 8]*2;
dx=min(ws)/5;

x=0:dx:120;
y=0;
for k=1:length(x0s)
    x0=x0s(k);
    w=ws(k);
    y1=A*exp(-4*log(2)*(x-x0).^2/w^2);
    y=y+y1;
end

yn=y+randn(size(y))*0.2;

figure
plot(x,y,'r');
hold on
plot(x,yn,'m')
title('data');

m=10;
hs=zeros(2*m,2*m+1);
for deg=1:2*m
    hs(deg,1:2*m+1)=sgsdf_smooth_closed_form(deg,m);
end

degs=[];
y_adpf=[];
for k=m+1:length(yn)-m
    k
    y1=yn(k-m:k+m);
    deg=adpf(y1);
    degs=[degs deg];
    h=hs(deg,:);
    
    y_adpf=[y_adpf block_oper(y1,h)];
end

figure
plot(x,y,'r')
hold on
plot(x(m+1:end-m),y_adpf);
title('adaptive degree')

figure
stem(x(m+1:end-m),degs);
title('polynomial degree');

for deg=1:2:7
    h=hs(deg,:);
    y_smoothed=block_oper(yn,h);
    figure
    plot(x,y,'r');
    hold on
    plot(x(m+1:end-m),y_smoothed);
    title(sprintf('polynomial degree = %d',deg));
end

[Jianwen Luo]

See all the contributions about the Savitzky-Goaly filter in MATLAB Central File Exchange at http://www.mathworks.com/matlabcentral/fileexchange/loadFileList.do?objectType=search&criteria=savitzky+or+sgolayfilt&Search=Go

[nabhan-1 A. HAMDON]