# Finding a best-fit straight line

A straight-line best fit is just a special case of a polynomial least-squares fit (with deg=1). Consider the following data giving the absorbance over a path length of 55 mm of UV light at 280 nm, $A$, by a protein as a function of the concentration, $[\mathrm{P}]$:

$[P] / \mathrm{\mu g/mL}$$A$
02.287
203.528
404.336
806.909
1208.274
18012.855
26016.085
40024.797
80049.058
150089.400

We expect the absorbance to be linearly related to the protein concentration: $A = m[\mathrm{P}] + A_0$ where $A_0$ is the absorbance in the absence of protein (for example, due to the solvent and experimental components).

import numpy as np
import pylab
Polynomial = np.polynomial.Polynomial

# The data: conc = [P] and absorbance, A
conc = np.array([0, 20, 40, 80, 120, 180, 260, 400, 800, 1500])
A = np.array([2.287, 3.528, 4.336, 6.909, 8.274, 12.855, 16.085, 24.797,
49.058, 89.400])

cmin, cmax = min(conc), max(conc)
pfit, stats = Polynomial.fit(conc, A, 1, full=True, window=(cmin, cmax),
domain=(cmin, cmax))

print('Raw fit results:', pfit, stats, sep='\n')

A0, m = pfit
resid, rank, sing_val, rcond = stats
rms = np.sqrt(resid/len(A))

print('Fit: A = {:.3f}[P] + {:.3f}'.format(m, A0),
'(rms residual = {:.4f})'.format(rms))

pylab.plot(conc, A, 'o', color='k')
pylab.plot(conc, pfit(conc), color='k')
pylab.xlabel('[P] /$\mathrm{\mu g\cdot mL^{-1}}$')
pylab.ylabel('Absorbance')
pylab.show()


The output shows a good straight-line fit to the data:

Raw fit results:
poly([ 1.92896129  0.0583057 ])
[array([ 2.47932733]), 2, array([ 1.26633786,  0.62959385]), 2.2204460492503131e-15]
Fit: A = 0.058[P] + 1.929 (rms residual = 0.4979) 