在Matlab中查找绘图中的点

Question

我有以下图表和创建该图表的数据文件.我想让Matlab为我找到以下几点:

1. [y,x]表示100%线所示的峰值
2. [x]表示曲线穿过y = 0线的位置
3. [x]其中y为第1部分中50%和20%的峰值.

是否有任何人们都知道的附加工具或软件包可以帮助我实现这一目标？我需要为一组图表做这个,所以合理自动化的东西是理想的.

我当然可以在Matlab中编写编程和计算部分,这只是能够加载数据文件,将其与曲线或函数匹配,并找到各种[x,y]坐标.

Answer 1

好的,这就行了.据我所知,Matlab中没有任何一个例行程序来做你想做的事情; 你必须自己制作一个.有几点需要注意:

• 显而易见,线性插值数据最容易做,应该没有问题

• 使用单个多项式插值也不是太难,尽管还有一些细节要处理.寻找峰值应该是业务的第一顺序,其涉及找到衍生物的根(roots例如,使用).找到峰值后,通过将多项式偏移此量,找到所有所需级别(0%,20%,50%)的多项式的根.

• 使用三次样条(spline)是最复杂的.对于具有完整立方体的所有子区间,应重复上面针对一般多项式概述的例程,同时考虑到最大值也可能位于子区间的边界的可能性,并且发现的任何根和极值可能位于区间之外立方有效的地方(也不要忘记使用的x-offsets spline).

这是我对所有3种方法的实现:

%% Initialize
% ---------------------------

clc
clear all

% Create some bogus data
n = 25;

f = @(x) cos(x) .* sin(4*x/pi) + 0.5*rand(size(x));
x = sort( 2*pi * rand(n,1));
y = f(x);


%% Linear interpolation
% ---------------------------

% y peak
[y_peak, ind] = max(y);
x_peak = x(ind);

% y == 0%, 20%, 50%
lims = [0 20 50];
X = cell(size(lims));
Y = cell(size(lims));
for p = 1:numel(lims)

    % the current level line to solve for
    lim = y_peak*lims(p)/100;

    % points before and after passing through the current limit
    after    = (circshift(y<lim,1) & y>lim) | (circshift(y>lim,1) & y<lim);
    after(1) = false;
    before   = circshift(after,-1);

    xx = [x(before) x(after)];
    yy = [y(before) y(after)];

    % interpolate and insert new data
    new_X = x(before) - (y(before)-lim).*diff(xx,[],2)./diff(yy,[],2);
    X{p} = new_X;
    Y{p} = lim * ones(size(new_X));

end

% make a plot to verify
figure(1), clf, hold on
plot(x,y, 'r') % (this also plots the interpolation in this case)

plot(x_peak,y_peak, 'k.') % the peak

plot(X{1},Y{1}, 'r.') % the 0%  intersects
plot(X{2},Y{2}, 'g.') % the 20% intersects
plot(X{3},Y{3}, 'b.') % the 50% intersects

% finish plot
xlabel('X'), ylabel('Y'), title('Linear interpolation')
legend(...
    'Real data / interpolation',...
    'peak',...
    '0% intersects',...
    '20% intersects',...
    '50% intersects',...
    'location', 'southeast')



%% Cubic splines
% ---------------------------

% Find cubic splines interpolation
pp = spline(x,y);

% Finding the peak requires finding the maxima of all cubics in all
% intervals. This means evaluating the value of the interpolation on 
% the bounds of each interval, finding the roots of the derivative and
% evaluating the interpolation on those roots: 

coefs = pp.coefs;
derivCoefs = bsxfun(@times, [3 2 1], coefs(:,1:3));
LB = pp.breaks(1:end-1).'; % lower bounds of all intervals
UB = pp.breaks(2:end).';   % upper bounds of all intervals

% rename for clarity
a = derivCoefs(:,1);
b = derivCoefs(:,2);  
c = derivCoefs(:,3); 

% collect and limits x-data
x_extrema = [...
    LB, UB,...     
    LB + (-b + sqrt(b.*b - 4.*a.*c))./2./a,... % NOTE: data is offset by LB
    LB + (-b - sqrt(b.*b - 4.*a.*c))./2./a,... % NOTE: data is offset by LB
    ];

x_extrema = x_extrema(imag(x_extrema) == 0);
x_extrema = x_extrema( x_extrema >= min(x(:)) & x_extrema <= max(x(:)) );

% NOW find the peak
[y_peak, ind] = max(ppval(pp, x_extrema(:)));
x_peak = x_extrema(ind);

% y == 0%, 20% and 50%
lims = [0 20 50];
X = cell(size(lims));
Y = cell(size(lims));    
for p = 1:numel(lims)

    % the current level line to solve for
    lim = y_peak * lims(p)/100;

    % find all 3 roots of all cubics
    R = NaN(size(coefs,1), 3); 
    for ii = 1:size(coefs,1) 

        % offset coefficients to find the right intersects
        C = coefs(ii,:);
        C(end) = C(end)-lim;

        % NOTE: data is offset by LB
        Rr = roots(C) + LB(ii); 

        % prune roots
        Rr( imag(Rr)~=0 ) = NaN;
        Rr( Rr <= LB(ii) | Rr >= UB(ii) ) = NaN;
        % insert results
        R(ii,:) = Rr;
    end

    % now evaluate and save all valid points    
    X{p} = R(~isnan(R));
    Y{p} = ppval(pp, X{p});

end

% as a sanity check, plot everything 
xx = linspace(min(x(:)), max(x(:)), 20*numel(x));
yy = ppval(pp, xx);

figure(2), clf, hold on

plot(x,y, 'r') % the actual data
plot(xx,yy) % the cubic-splines interpolation 

plot(x_peak,y_peak, 'k.') % the peak

plot(X{1},Y{1}, 'r.') % the 0%  intersects
plot(X{2},Y{2}, 'g.') % the 20% intersects
plot(X{3},Y{3}, 'b.') % the 50% intersects

% finish plot
xlabel('X'), ylabel('Y'), title('Cubic splines interpolation')
legend(...
    'Real data',...
    'interpolation',...
    'peak',...
    '0% intersects',...
    '20% intersects',...
    '50% intersects',...
    'location', 'southeast')


%% (N-1)th degree polynomial
% ---------------------------

% Find best interpolating polynomial
coefs = bsxfun(@power, x, n-1:-1:0) \ y;
% (alternatively, you can use polyfit() to do this, but this is faster)

% To find the peak, we'll have to find the roots of the derivative: 
derivCoefs = (n-1:-1:1).' .* coefs(1:end-1);
Rderiv = roots(derivCoefs);
Rderiv = Rderiv(imag(Rderiv) == 0);
Rderiv = Rderiv(Rderiv >= min(x(:)) &  Rderiv <= max(x(:)));

[y_peak, ind] = max(polyval(coefs, Rderiv));
x_peak = Rderiv(ind);

% y == 0%, 20%, 50%
lims = [0 20 50];
X = cell(size(lims));
Y = cell(size(lims));
for p = 1:numel(lims)

    % the current level line to solve for
    lim = y_peak * lims(p)/100;

    % offset coefficients as to find the right intersects
    C = coefs;
    C(end) = C(end)-lim;

    % find and prune roots
    R = roots(C);
    R = R(imag(R) == 0);
    R = R(R>min(x(:)) & R<max(x(:)));

    % evaluate polynomial at these roots to get actual data
    X{p} = R;
    Y{p} = polyval(coefs, R);

end

% as a sanity check, plot everything 
xx = linspace(min(x(:)), max(x(:)), 20*numel(x));
yy = polyval(coefs, xx);

figure(3), clf, hold on

plot(x,y, 'r') % the actual data
plot(xx,yy) % the cubic-splines interpolation 

plot(x_peak,y_peak, 'k.') % the peak

plot(X{1},Y{1}, 'r.') % the 0%  intersects
plot(X{2},Y{2}, 'g.') % the 20% intersects
plot(X{3},Y{3}, 'b.') % the 50% intersects

% finish plot
xlabel('X'), ylabel('Y'), title('(N-1)th degree polynomial')
legend(...
    'Real data',...
    'interpolation',...
    'peak',...
    '0% intersects',...
    '20% intersects',...
    '50% intersects',...
    'location', 'southeast')

这导致了这三个图:

(注意在(N-1)次多项式中出现了问题; 20%的交叉点在结束时都是错误的.因此,在复制粘贴之前,请更彻底地检查所有内容:)

正如我之前所说的,正如您可以清楚地看到的那样,如果底层数据不适合,使用单个多项式进行插值通常会引入很多问题.此外,正如您可以从这些图中清楚地看到的那样,插值方法非常强烈地影响交叉点的位置 - 最重要的是,您至少要了解哪些模型是您数据的基础.

对于一般情况,三次样条通常是最好的方法.但是,这是一种通用方法,它将为您(以及您的出版物的读者)提供对数据准确性的错误认知.使用三次样条曲线可以首先了解相交是什么以及它们的行为方式,但是一旦真实模型变得更加清晰,就会回过头来重新审视分析.当然,当用于通过数据创建更平滑,更"视觉上吸引人"的曲线时,当然不发布三次样条曲线:)