→ least-squares/

least-squares/least-squares.py

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