I’m trying to optimize a function based on discrete data sets I got in the lab.

Spectrum_1 and Spectrum_2 are experimental data sets with length N. Those two arrays contain values taken for each lambda (wavelength) value in lambdas array.

So:

len(Spectrum_1) == len(Spectrum_2) == len(lambdas)

Each Spectrum_1 and Spectrum_2 have the symbolic form

Spectrum = G(T,lambda) * E(lambda)

E(lambda) is unknown, but it is known to not vary with T.

G(T, lambda) is a known function of T and lambda (it is a continuous and defined function). It is the Planck blackbody radiation equation in regards to wavelength to be more specific.

E_1 and E_2 should be equal:

E_1 = Spectrum_1/np.array([G(T_1, lamda) for lamda in lambdas])
E_2 = Spectrum_2/np.array([G(T_2, lamda) for lamda in lambdas])

T_1 and T_2 are unknown but I know they are within 400 to 1000 range (both). T_1 and T_2 are two scalar values.

Knowing that, I need to minimize:

np.sum(np.array([(E_1[i] - E_2[i])**2 for i in lamdas]))

Or at least that’s what I think I should minimize. Ideally (E_1[i]-E_2[i])==0 but that won’t be the case granted the experimental data in Spectrum_1 and _2 contain noise and distortions due to atmospheric transmission.

I’m not very familiar with optimizing with multiple unknown variables (T_1 and T_2) in Python. I could brute test millions of combinations of T_1 and T_2 I suppose, but I wish to do it correctly.
I wonder if anybody could help me.

I hear scipy.optimize could do it for me, but many methods ask for Jacobian and Hessian and I’m unsure how to proceed given I have experimental data (Spectrum_1 and Spectrum_2) and I’m not dealing with continuous/smooth functions.