2D OptimizationΒΆ

[12]:

import numpy as np
import matplotlib.pyplot as plt
plt.rcParams.update({"font.size": 16})
from matplotlib import cm
import pandas as pd
from scipy.optimize import minimize

[13]:

df = pd.read_csv("https://raw.githubusercontent.com/roualdes/data/master/bike.csv")
\[loss(\mu, \sigma | x) = N * \log{\sigma} + \frac{1}{2\sigma^2} * \sum_{n=1}^N (y_n - \mu)^2\]
[14]:

data = {
    "N": df["cnt"].size,
    "y": df["cnt"]
}

def loss(theta, data):
    one = data["N"] * np.log(theta[1])
    two = 0.5 * np.sum((data["y"] - theta[0]) ** 2) / theta[1] ** 2
    return one + two

[15]:

mn = np.mean(data["y"])
sd = np.std(data["y"])
[mn, sd]

[15]:

[4504.3488372093025, 1935.8859561152221]

[16]:

mu_min, mu_max = 3500, 5500
sigma_min, sigma_max = 1500, 3000
mu = np.linspace(mu_min, mu_max, 100)
sigma = np.linspace(sigma_min, sigma_max, 100)
m, s = np.meshgrid(mu, sigma)

[17]:

z = [loss([m, s], data) for m,s in zip(m.flatten(), s.flatten())]
l = np.asarray(z).reshape(np.shape(m))

[18]:

plt.contour(m, s, l, 9, colors = "black");
plt.imshow(l, extent = [mu_min, mu_max, sigma_min, sigma_max],
           origin = "lower",
           cmap = cm.coolwarm,
           alpha = 0.5);
plt.colorbar();
plt.scatter([mn], [sd],
           color = "black", marker = "*");
plt.xlabel("$\mu$");
plt.ylabel("$\sigma$");
../_images/notebooks_two_d_optimization_8_0.png 
[19]:

minimize(loss, np.random.exponential(size = 2),
         args = (data),
         bounds = [(-np.inf, np.inf), (0, np.inf)],
         method = "L-BFGS-B")

[19]:

      fun: 5897.942183502328
 hess_inv: <2x2 LbfgsInvHessProduct with dtype=float64>
      jac: array([0., 0.])
  message: 'CONVERGENCE: NORM_OF_PROJECTED_GRADIENT_<=_PGTOL'
     nfev: 168
      nit: 48
     njev: 56
   status: 0
  success: True
        x: array([4504.26791768, 1935.84450478])

[20]:

[mn, sd]

[20]:

[4504.3488372093025, 1935.8859561152221]