Integral average of y values with respect to x values

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matnewbie 2015-10-4

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评论： Star Strider 2015-10-7

scattered plot.png

I have two vectors (x and y) of the same size with x having more than one value in the y direction. For example, x is [0.02 0.02 0.03 0.03 0.03 0.04 0.04 0.05 0.05 0.05] and y is [0.23 0.40 0.12 0.16 0.09 0.50 0.02 0.33 0.10 0.08]. I want to convert the attached scattered plot of the two vectors to a simple continuous curve, by evaluating the integral average of y values with respect to x values. Take into account that the x vector is already ordered from small to large values. How could I do this? Thank you in advance.

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Image Analyst 2015-10-5

"I have two vectors (x and y) of the same size with x having more than one value in the y direction" Huh? Does it have one more, or are x and y the same length? Not sure what you're saying. Are you saying that for any given x value, it may be in the x array more than once, with each occurrence of that x value having a different y value stored at corresponding indexes in the y array?

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Answer 1

Star Strider 2015-10-4

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I’m not exactly certain what you want, but one approach would be to use the cumtrapz function:

x = [0.02 0.02 0.03 0.03 0.03 0.04 0.04 0.05 0.05 0.05];
y = [0.23 0.40 0.12 0.16 0.09 0.50 0.02 0.33 0.10 0.08];
I = cumtrapz(x,y);                      % Integrated Vector Of ‘y’ w.r.t. ‘x’
figure(1)
plot(x, I)
grid

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arich82 2015-10-5

编辑：arich82 2015-10-5

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[editted to show something analogous to a lagging filter, in addition to the special case for the right-hand boundary]

I think a variation on the accumarray method using histc should work around this, by computing the average over linearly-spaced intervals:

nbins = 4;   % analogous to inverse of filter length
edges = [linspace(min(x), max(x), nbins), inf]; % inf padded
[n, bin] = histc(x(:), edges);
offset = diff(edges(1:2))/2; % distance from edge to center of bin
c = edges(1:end - 1) + offset;
wgts = n(1:end - 1); % n(end) counts the number of infs
sums = accumarray(bin, y);
avgs = sums./wgts;
plot(c, avgs)

Note that the histc is strictly 'less-than' on the right-hand edge. For the discrete case, the above seems to match the expected result, but for a quasi-continuous case, any of the data points lying on max(x) would get grouped into their own bin for no apparent reason. An alternative, which would slightly expand the range of the final bin to include the rhs-boundary (making that bin slightly larger), could be

nbins = 3;   % actually, number of intervals between min & max
edges = linspace(min(x), max(x), nbins + 1);
edges(end) = inf; % make last bin open ended; cf. doc HISTC
% the rest is the same as before, i.e.
% [n, bin] = histc(...

This is equvialent to nudging the the right-most edge of the last bin, e.g. edge(end) = max(x) + eps(max(x)), since no data lies beyond the max (but writing inf is cleaner). It's trivial to modify how it handles the edge-case, but this seems like a quick-and-dirty way to get a lumped average over a fixed-width interval...

matnewbie 2015-10-5

编辑：matnewbie 2015-10-5

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input.csv

You can find the "input.csv" file with the data attached, and here is the code I use to load and sort the data.

clc,close all,clear all
fileID=fopen('input.csv');
C=textscan(fileID,'%f,%f,%f,%f,%f,%f,%f,%f');
A=cell2mat(C);   % loads the values in A
fclose(fileID);
N=size(A,1);
grid_size=3e-4; %size of grid element
Rcol=0.0105; %radius
X=A(:,7)-Rcol-grid_size; %translation of x coordinate
Y=A(:,8)-Rcol-grid_size; %translation of y coordinate
r=sqrt(X.^2+Y.^2); %radial position
axial_vel=A(:,2); %velocity component in the axial direction
% Cutting off values outside the region of interest
tol=1e-8; %tolerance for velocity cutoff
j=1;
for i=1:N
    if axial_vel(i)>tol
        R(j)=r(i);
        V(j)=axial_vel(i);
        j=j+1;
    end
end
figure
plot(R,V,'r.')
[Rsorted, SortIndex] = sort(R);
Vsorted = V(SortIndex);
figure
plot(Rsorted,Vsorted,'b.')

The idea is to obtain a smoothed continuous curve for the integral average... I tried the filter function, but I can't get what I need.

Star Strider 2015-10-5

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I would experiment with the filter order and use filtfilt if you have the Signal Processing Toolbox. This gave what appeared to me to be close to what you want:

N = 50;                                 % Length Of Filter
b = ones(1,N);                          % Filter Coefficients
a = N;                                  % Filter Coefficients
sf = filtfilt(b, a, Vsorted);           % Filter Data

I experimented with filter lengths ranging from 50 to 200 and it seemed to give the result you illustrated. It is not possible to design a typical low-pass filter because your data are not regularly sampled. It would be necessary to interpolate your data (using interp1 if your data meet its requriements) to a regularly-sampled time base (here Rsorted) to design a filter for it. That could make the curve smoother, but I can’t promise that it would be.

The first step after interpolation would be to do a fft in order to determine what frequencies you want to exclude as noise. When you decide on a cutoff frequency for your low-pass filter, I would then use the filter design procedure I outlined here.

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Answer 2

arich82 2015-10-5

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编辑：arich82 2015-10-5

在 MATLAB Online 中打开

[edited to fix typo in comments, and clean-up useless clutter]

Perhaps try combining the filter solution with the accumarray (or filtfilt, if you have the appropriate toolbox, which I do not):

x = Rsorted(:).';
y = Vsorted(:).';
% compute average y for each unique x
[c, ic, ia] = unique(x);
sums = accumarray(ia, y);
wgts = accumarray(ia, ones(size(y))); % weights, i.e. number of reps
avgs = sums./wgts;
% interpolate data on regular grid
n = numel(c); % or some other number
xq = linspace(c(1), c(end), n);
yq = interp1(c, avgs, xq);
% filter gridded data
windowSize = 50;
a = windowSize;
b = ones(1, windowSize);
yf = filter(b, a, yq);
% plot results
hold all;
plot(xq, yf, 'r-', 'LineWidth', 3)

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matnewbie 2015-10-7

Yes, you're right: with filtfilt the results are more similar to those obtained by Star Strider (magenta line in the plot).

Star Strider 2015-10-7

In my latest Comment (5 Oct 2015 at 21:09), I used filtfilt.

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Integral average of y values with respect to x values

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