如何使用线性最小二乘拟合拟合两个重叠的高斯？

Question

First off, I created a jupyter notebook to show the code properly. Please take a look . Second, I'm using linear least squares fitting rather than non-linear least squares fitting for speed. The problem I'm trying to solves involves fitting a large number of gaussians on a spectrum image for electron microscopy, and our current nlls algorithm is slow. It requires fitting within bounds, which is why we are not using numpy.linalg.lstsq

I'm trying to use scipy.optimize.lsq_linear to do a simple fit of two gaussian components ( g1 and g2 ) with fixed sigma and centre components, to a 2D array G that is a list of different mixes of those two gaussians.

First off, defining the gaussian:

def gaussian(x, mu, sig):
    return np.exp(-np.power(x - mu, 2.) / (2 * np.power(sig, 2.)))

Then creating the components:

x = np.linspace(-3, 3, 100)
g1 = gaussian(x, -0.5, 0.5)
g2 = gaussian(x, 0.5, 0.5)

And then creating two sets of data to fit to by mixing the components:

G1 = 5*g1 + 8*g2
G2 = 2*g1 + 2*g2
G = np.stack([G1,G2]).T 

I take the transpose of the stack because the lsq_linear function requires the data to be of shape (m, n) where (m,) is the shape of the component to be fit.

I fit the components one by one through

g1_res = lsq_linear(G, g1)
g2_res = lsq_linear(G, g2)

which should give me the weight of the component for each mix in g1_res["x"] in the documentation . Multiplying the weights with the components returns:

g1_fit = [g1*fit for fit in g1_res["x"]]
g2_fit = [g2*fit for fit in g2_res["x"]]

However, the fit (first mix is shown below) is incorrect. Can someone help me figure out what I'm doing wrong?

plt.plot(x,G.T[0], label="mixed_signal")
plt.plot(x,g1_fit[0], label="gaussian 1")
plt.plot(x,g2_fit[0], label="gaussian 2")
plt.legend()

Answer 1

You mixed something up. You are fitting all measured spectra using 1 component. Instead, you want to fit each spectrum using both components. Therefore, the things need to be swapped:

g = np.stack([g1,g2]).T

g1_res = lsq_linear(g, G1)
g2_res = lsq_linear(g, G2)



g1_res["x"]
# --> array([ 5.,  8.])
g2_res["x"]
# --> array([ 2.,  2.])

By the way, if you have many spectra and the shape parameters of the gaussian (as given here [+-0.5,0.5]) are always the same but not known or the shape is the same but does not even correspond ideally to gaussian, then you might want to try Principal Component Analysis. There are already Python packages for it.