Least square method in python?

Question

I have these values:

T_values = (222, 284, 308.5, 333, 358, 411, 477, 518, 880, 1080, 1259) (x values)

C/(3Nk)_values = (0.1282, 0.2308, 0.2650, 0.3120 , 0.3547, 0.4530, 0.5556, 0.6154, 0.8932, 0.9103, 0.9316) (y values)

I know they follow the model:

C/(3Nk)=(h*w/(k*T))**2*(exp(h*w/(k*T)))/(exp(h*w/(k*T)-1))**2

I also know that k=1.38*10**(-23) and h=6.626*10**(-34). I have to find the w that best describes the measurement data. I'd like to solve this using the least square method in python, however I don't really understand how this works. Can anyone help me?

Answer 1

This answer provides a walk-through on using Python to determine fitting parameters for a general exponential pattern. See also a related posts on linearization techniques and using the lmfit library.

Data Cleaning

First, let's input and organize the sampling data as numpy arrays, which will later help with computation and clarity.

import matplotlib.pyplot as plt
import scipy.optimize as opt
import numpy as np


#% matplotlib inline

# DATA ------------------------------------------------------------------------
T_values = np.array([222, 284, 308.5, 333, 358, 411, 477, 518, 880, 1080, 1259])
C_values = np.array([0.1282, 0.2308, 0.2650, 0.3120 , 0.3547, 0.4530, 0.5556, 0.6154, 0.8932, 0.9103, 0.9316])

x_samp = T_values
y_samp = C_values    

There are many curve fitting functions in scipy and numpy and each is used differently, e.g. scipy.optimize.leastsq and scipy.optimize.least_squares. For simplicity, we will use scipy.optimize.curve_fit, but it is difficult to find an optimized regression curve without selecting reasonable starting parameters. A simple technique will later be demonstrated on selecting starting parameters.

Review

First, although the OP provided an expected fitting equation, we will approach the problem of using Python to curve fit by reviewing the general equation for an exponential function:

Now we build this general function, which will be used a few times:

# GENERAL EQUATION ------------------------------------------------------------
def func(x, A, c, d):
    return A*np.exp(c*x) + d

Trends

• amplitude: a small A gives a small amplitude
• shape: a small c controls the shape by flattening the "knee" of the curve
• position: d sets the y-intercept
• orientation: a negative A flips the curve across a horizontal axis; a negative c flips the curve across a vertical axis

The latter trends are illustrated below, highlighting the control (black line) compared to a line with a varied parameter (red line):

Selecting Initial Parameters

Using the latter trends, let us next look at the data and try to emulate the curve by adjusting these parameters. For demonstration, we plot several trial equations against our data:

# SURVEY ----------------------------------------------------------------------
# Plotting Sampling Data
plt.plot(x_samp, y_samp, "ko", label="Data")

x_lin = np.linspace(0, x_samp.max(), 50)                   # a number line, 50 evenly spaced digits between 0 and max

# Trials
A, c, d = -1, -1e-2, 1
y_trial1 = func(x_lin,  A,     c, d)
y_trial2 = func(x_lin, -1, -1e-3, 1)
y_trial3 = func(x_lin, -1, -3e-3, 1)

plt.plot(x_lin, y_trial1, "--", label="Trial 1")
plt.plot(x_lin, y_trial2, "--", label="Trial 2")
plt.plot(x_lin, y_trial3, "--", label="Trial 3")
plt.legend()

From simple trial and error, we can approximate the shape, amplitude, position and orientation of the curve better. For instance, we know the first two parameters (A and c) must be negative. We also have a reasonable guess for the order of magnitude for c.

Computing Estimated Parameters

We will now use the parameters of the best trial for our initial guesses:

# REGRESSION ------------------------------------------------------------------
p0 = [-1, -3e-3, 1]                                        # guessed params
w, _ = opt.curve_fit(func, x_samp, y_samp, p0=p0)     
print("Estimated Parameters", w)  

# Model
y_model = func(x_lin, *w)

# PLOT ------------------------------------------------------------------------
# Visualize data and fitted curves
plt.plot(x_samp, y_samp, "ko", label="Data")
plt.plot(x_lin, y_model, "k--", label="Fit")
plt.title("Least squares regression")
plt.legend(loc="upper left")

# Estimated Parameters [-1.66301087 -0.0026884   1.00995394]

How Does this Work?

curve_fit is one of many optimization functions offered by scipy. Given an initial value, the resulting estimated parameters are iteratively refined so that the resulting curve minimizes the residual error, or difference between the fitted line and sampling data. A better guess reduces the number of iterations and speeds up the result. With these estimated parameters for the fitted curve, one can now calculate the specific coefficients for a particular equation (a final exercise left to the OP).