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#!/usr/bin/python2.7
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from __future__ import division, print_function
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from sympy import Symbol, diff, solve, lambdify, simplify
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import matplotlib.pyplot as plt
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import numpy as np
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# Model parameters: We look for a line y = b1*x + b2.
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b1 = Symbol('b1')
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b2 = Symbol('b2')
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# Data points
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data = [(1, 14), (2, 13), (3, 12), (4, 10), (5, 9), (7, 8), (9, 5)]
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# S is the function to minimize:
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#
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# For each data point the vertical error/residual is x*b1 + b2 - y. We want to
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# minimize the sum of the squared residuals (least squares).
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S = sum((p[0] * b1 + b2 - p[1]) ** 2 for p in data)
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S = simplify(S)
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print("Function to minimize: S = {}".format(S))
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# Minimize S by setting its partial derivatives to zero.
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d1 = diff(S, b1)
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d2 = diff(S, b2)
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solutions = solve([d1, d2], [b1, b2])
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print("S is minimal for b1 = {}, b2 = {}".format(solutions[b1], solutions[b2]))
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# Construct fitted line from the solutions
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x = Symbol('x')
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fitted_line = solutions[b1] * x + solutions[b2]
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print("Fitted line: y = {}".format(fitted_line))
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# Construct something we can plot with matplotlib
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fitted_line_func = lambdify(x, fitted_line, modules=['numpy'])
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x_plot = np.linspace(min(p[0] for p in data),
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max(p[0] for p in data), 100)
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# Plot data points and fitted line
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plt.scatter([p[0] for p in data], [p[1] for p in data], marker="+")
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plt.plot(x_plot, fitted_line_func(x_plot), 'r')
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plt.show()
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