Using cumulative areas as an error threshold

Question

Jack 2023-9-28

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评论： Voss 2023-10-5

Hello!

I have a function r(t), which in this case is a logistic function, but for all intents and purposes could be any function r(t).

My goal is to begin at the first value of r and compare it to the second value of r. If the area formed by the triangle under the curve is above a certain threshold then I keep the second value of r. If it is below, then I move to the third value of r and compare the area of the triangle now formed between the first and third r values to the threshold. I repeat until I find the value of r that is above the threshold. For the parameters below, this r value is the 75th index.

I then begin the process again between the 75th and 76th index and so on and so on. My final result is therefore a logical array that is true when r is above the threshold.

I have written a looping procedure that carries this out successfully, but my instinct is telling me there should be a more efficient solution, perhaps involving some loop-less cumulative function. Or a clever use of a step function...

Would appreciate any suggestions / ideas!

r0 = 3; % start r

r1 = 6; % end r

t0 = 5; % halfway point

k = 2; % exponential growth

r_ = @(t) 1./(1+exp(-k*(t-t0))) * (r1-r0) + r0;

t = linspace(0,15,301);

r = r_(t);

dt = mean(diff(t));

dr = abs(diff(r));

threshold = 5e-3;

A_ = 0;

lgc = false;

for idr = 1:numel(dr)

A = dt*(dr(idr))/2;

A_ = A_ + A;

if A_ >= threshold

lgc(idr) = true;

A_ = 0;

else

lgc(idr) = false;

end

lgc = [true lgc];

hold on

plot(t,r)

plot(t(lgc),r(lgc),'x')

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Answer 1

Voss 2023-9-28

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https://ww2.mathworks.cn/matlabcentral/answers/2027059-using-cumulative-areas-as-an-error-threshold#answer_1321369

在 MATLAB Online 中打开

You can use cumsum to perform the cumulative sum over all dr at once. Then loop over sections of that cumulative sum, finding the first index where it exceeds threshold each time and subtracting off that element. You still have a loop, but it has fewer iterations.

r0 = 3; % start r

r1 = 6; % end r

t0 = 5; % halfway point

k = 2; % exponential growth

r_ = @(t) 1./(1+exp(-k*(t-t0))) * (r1-r0) + r0;

t = linspace(0,15,301);

r = r_(t);

dt = mean(diff(t));

dr = abs(diff(r));

threshold = 5e-3;

lgc = [true false(1,numel(dr))];

A = dt*cumsum(dr)/2;

while true

idx = find(A >= threshold, 1);

if isempty(idx)

break

end

lgc(idx+1) = true;

A = A-A(idx);

end

hold on

plot(t,r)

plot(t(lgc),r(lgc),'x')

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Voss 2023-9-29

编辑：Voss 2023-9-29

在 MATLAB Online 中打开

Not with cumsum, but you can do that using the expression for A below.

r-r.' is the difference between each element of r and every other element of r (including itself), giving a matrix of those differences.

toeplitz(0:numel(r)-1) gives you a matrix of integers with zeros along the main diagonal, ones along the 1 and -1 diagonals (one away from main), twos along the +/-2 diagonals, etc. This matrix is essentially the lengths of the base of each triangle divided by dt, i.e., it's j-i when you want to calculate |r(j)-r(i)|*dt*(j-i)/2 .

Then triu is used to keep only the upper triangular section of the matrix so that j>=i, i.e., only considering differences that look forward not backward.

Once the matrix of areas A is calculated, you can search through it like in my method before, except now you must jump rows basically. Starting with the first row, find the index where the area exceeds the threshold. Then move to that row to find the next one, etc.

r0 = 3; % start r

r1 = 6; % end r

t0 = 5; % halfway point

k = 2; % exponential growth

r_ = @(t) 1./(1+exp(-k*(t-t0))) * (r1-r0) + r0;

t = linspace(0,15,301);

r = r_(t);

dt = mean(diff(t));

threshold = 5e-3;

A = triu(dt*abs(r-r.')/2.*toeplitz(0:numel(r)-1));

lgc = [true false(1,numel(r)-1)];

row = 1;

while true

idx = find(A(row,:) >= threshold, 1);

if isempty(idx)

break

end

lgc(idx+1) = true;

row = idx;

end

hold on

plot(t,r)

plot(t(lgc),r(lgc),'x')

Jack 2023-10-4

I have one more question actually.

Is this adaptable to changing from the area of the triangle to the area under the curve (within the triangle)? I.e. the hypotenuse of the triangle from before becomes the curve between the two points, whereas the opposite and adjacent sides of the triangle stay the same and its now that area you calculate.

Perhaps some way of using cumtrapz, trapz or integral...

Voss 2023-10-5

在 MATLAB Online 中打开

Maybe something like this. See if it seems right.

r0 = 3; % start r

r1 = 6; % end r

t0 = 5; % halfway point

k = 2; % exponential growth

r_ = @(t) 1./(1+exp(-k*(t-t0))) * (r1-r0) + r0;

t = linspace(0,15,301);

r = r_(t);

dt = mean(diff(t));

cA = cumtrapz(t,r-r(1));

dA = cA-cA.';

threshold = 5e-3;

lgc = [true false(1,numel(r)-1)];

row = 1;

while true

idx = find(dA(row,:) >= threshold, 1);

if isempty(idx)

break

end

dA = dA-dA(row,idx);

lgc(idx+1) = true;

row = idx;

end

hold on

plot(t,r)

plot(t(lgc),r(lgc),'x')

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