Curve fitting: temperature as a function of month of the year

We have the min and max temperatures in Alaska for each months of the year. We would like to find a function to describe this yearly evolution.

For this, we will fit a periodic function.

The data

import numpy as np

temp_max = np.array([17, 19, 21, 28, 33, 38, 37, 37, 31, 23, 19, 18])
temp_min = np.array([-62, -59, -56, -46, -32, -18, -9, -13, -25, -46, -52, -58])

import matplotlib.pyplot as plt

months = np.arange(12)
plt.plot(months, temp_max, "ro")
plt.plot(months, temp_min, "bo")
plt.xlabel("Month")
plt.ylabel("Min and max temperature")

Text(35.472222222222214, 0.5, 'Min and max temperature')

Fitting it to a periodic function

import scipy as sp


def yearly_temps(times, avg, ampl, time_offset):
    return avg + ampl * np.cos((times + time_offset) * 2 * np.pi / times.max())


res_max, cov_max = sp.optimize.curve_fit(yearly_temps, months, temp_max, [20, 10, 0])
res_min, cov_min = sp.optimize.curve_fit(yearly_temps, months, temp_min, [-40, 20, 0])

Plotting the fit

days = np.linspace(0, 12, num=365)

plt.figure()
plt.plot(months, temp_max, "ro")
plt.plot(days, yearly_temps(days, *res_max), "r-")
plt.plot(months, temp_min, "bo")
plt.plot(days, yearly_temps(days, *res_min), "b-")
plt.xlabel("Month")
plt.ylabel(r"Temperature ($^\circ$C)")

plt.show()