Analysis of the End of the Night Period :: Wavelets in Action: Examples and Case Studies (Wavelet Toolbox)

Wavelet Toolbox

Analysis of the End of the Night Period

The shape of the curve during the end of the night is a slow descent, globally smooth, but locally highly irregular. One can hardly distinguish two successive local extrema in the vicinity of time t = 1600 and t = 1625. The approximation A₅is quite good except at these two modes.

The accuracy of the approximation can be explained by the fact that there remains only a low-frequency signal corrupted by noises. The massive and simultaneous changes of personal electrical appliances are absent.

The details D₁, D₂, and D₃ show the kind of variation and have, roughly speaking, similar shape and mean value. They contain the local short period irregularities caused by noises, and the inspection of D₂ and D₃ allows you to detect the local minimum around t =1625.

The details D₄ and D₅ exhibit the slope changes of the regular part of the signal, and A₄ and A₅ are piecewise linear.

In conclusion, none of the time scales brings a significant contribution sufficiently different from the noise level, and no additional correction is needed. The retained approximation is A₄ or A₅.

All the figures in this paragraph are generated using the graphical user interface tools, but the user can also process the analysis using the command line mode. The following example corresponds to a command line equivalent for producing the figure below.

% Load the original 1-D signal, decompose, reconstruct details in 
% original time and plot.
% Load the signal. 
load leleccum; s = leleccum;

% Decompose the signal s at level 5 using the wavelet db3. 
w = 'db3'; 
[c,l] = wavedec(s,5,w);

% Reconstruct the details using the decomposition structure. 
for i = 1:5
   D(i,:) = wrcoef('d',c,l,w,i);
end

Note This loop replaces five separate wrcoef statements defining the details. The variable D contains the five details.

% Avoid edge effects by suppressing edge values and plot. 
tt = 1+100:length(s)-100; 
subplot(6,1,1); plot(tt,s(tt),'r'); 
title('Electrical Signal and Details'); 
for i = 1:5, subplot(6,1,i+1); plot(tt,D(5-i+1,tt),'g'); end

Analysis of the Midday Period Suggestions for Further Analysis