File: plot_gpr_co2.py

package info (click to toggle)
scikit-learn 0.23.2-5
  • links: PTS, VCS
  • area: main
  • in suites: bookworm, bullseye, sid
  • size: 21,892 kB
  • sloc: python: 132,020; cpp: 5,765; javascript: 2,201; ansic: 831; makefile: 213; sh: 44
file content (156 lines) | stat: -rw-r--r-- 6,357 bytes parent folder | download
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
"""
========================================================
Gaussian process regression (GPR) on Mauna Loa CO2 data.
========================================================

This example is based on Section 5.4.3 of "Gaussian Processes for Machine
Learning" [RW2006]. It illustrates an example of complex kernel engineering and
hyperparameter optimization using gradient ascent on the
log-marginal-likelihood. The data consists of the monthly average atmospheric
CO2 concentrations (in parts per million by volume (ppmv)) collected at the
Mauna Loa Observatory in Hawaii, between 1958 and 2001. The objective is to
model the CO2 concentration as a function of the time t.

The kernel is composed of several terms that are responsible for explaining
different properties of the signal:

- a long term, smooth rising trend is to be explained by an RBF kernel. The
  RBF kernel with a large length-scale enforces this component to be smooth;
  it is not enforced that the trend is rising which leaves this choice to the
  GP. The specific length-scale and the amplitude are free hyperparameters.

- a seasonal component, which is to be explained by the periodic
  ExpSineSquared kernel with a fixed periodicity of 1 year. The length-scale
  of this periodic component, controlling its smoothness, is a free parameter.
  In order to allow decaying away from exact periodicity, the product with an
  RBF kernel is taken. The length-scale of this RBF component controls the
  decay time and is a further free parameter.

- smaller, medium term irregularities are to be explained by a
  RationalQuadratic kernel component, whose length-scale and alpha parameter,
  which determines the diffuseness of the length-scales, are to be determined.
  According to [RW2006], these irregularities can better be explained by
  a RationalQuadratic than an RBF kernel component, probably because it can
  accommodate several length-scales.

- a "noise" term, consisting of an RBF kernel contribution, which shall
  explain the correlated noise components such as local weather phenomena,
  and a WhiteKernel contribution for the white noise. The relative amplitudes
  and the RBF's length scale are further free parameters.

Maximizing the log-marginal-likelihood after subtracting the target's mean
yields the following kernel with an LML of -83.214::

   34.4**2 * RBF(length_scale=41.8)
   + 3.27**2 * RBF(length_scale=180) * ExpSineSquared(length_scale=1.44,
                                                      periodicity=1)
   + 0.446**2 * RationalQuadratic(alpha=17.7, length_scale=0.957)
   + 0.197**2 * RBF(length_scale=0.138) + WhiteKernel(noise_level=0.0336)

Thus, most of the target signal (34.4ppm) is explained by a long-term rising
trend (length-scale 41.8 years). The periodic component has an amplitude of
3.27ppm, a decay time of 180 years and a length-scale of 1.44. The long decay
time indicates that we have a locally very close to periodic seasonal
component. The correlated noise has an amplitude of 0.197ppm with a length
scale of 0.138 years and a white-noise contribution of 0.197ppm. Thus, the
overall noise level is very small, indicating that the data can be very well
explained by the model. The figure shows also that the model makes very
confident predictions until around 2015.
"""
# Authors: Jan Hendrik Metzen <jhm@informatik.uni-bremen.de>
#
# License: BSD 3 clause


import numpy as np

from matplotlib import pyplot as plt
from sklearn.datasets import fetch_openml
from sklearn.gaussian_process import GaussianProcessRegressor
from sklearn.gaussian_process.kernels \
    import RBF, WhiteKernel, RationalQuadratic, ExpSineSquared

print(__doc__)


def load_mauna_loa_atmospheric_co2():
    ml_data = fetch_openml(data_id=41187)
    months = []
    ppmv_sums = []
    counts = []

    y = ml_data.data[:, 0]
    m = ml_data.data[:, 1]
    month_float = y + (m - 1) / 12
    ppmvs = ml_data.target

    for month, ppmv in zip(month_float, ppmvs):
        if not months or month != months[-1]:
            months.append(month)
            ppmv_sums.append(ppmv)
            counts.append(1)
        else:
            # aggregate monthly sum to produce average
            ppmv_sums[-1] += ppmv
            counts[-1] += 1

    months = np.asarray(months).reshape(-1, 1)
    avg_ppmvs = np.asarray(ppmv_sums) / counts
    return months, avg_ppmvs


X, y = load_mauna_loa_atmospheric_co2()

# Kernel with parameters given in GPML book
k1 = 66.0**2 * RBF(length_scale=67.0)  # long term smooth rising trend
k2 = 2.4**2 * RBF(length_scale=90.0) \
    * ExpSineSquared(length_scale=1.3, periodicity=1.0)  # seasonal component
# medium term irregularity
k3 = 0.66**2 \
    * RationalQuadratic(length_scale=1.2, alpha=0.78)
k4 = 0.18**2 * RBF(length_scale=0.134) \
    + WhiteKernel(noise_level=0.19**2)  # noise terms
kernel_gpml = k1 + k2 + k3 + k4

gp = GaussianProcessRegressor(kernel=kernel_gpml, alpha=0,
                              optimizer=None, normalize_y=True)
gp.fit(X, y)

print("GPML kernel: %s" % gp.kernel_)
print("Log-marginal-likelihood: %.3f"
      % gp.log_marginal_likelihood(gp.kernel_.theta))

# Kernel with optimized parameters
k1 = 50.0**2 * RBF(length_scale=50.0)  # long term smooth rising trend
k2 = 2.0**2 * RBF(length_scale=100.0) \
    * ExpSineSquared(length_scale=1.0, periodicity=1.0,
                     periodicity_bounds="fixed")  # seasonal component
# medium term irregularities
k3 = 0.5**2 * RationalQuadratic(length_scale=1.0, alpha=1.0)
k4 = 0.1**2 * RBF(length_scale=0.1) \
    + WhiteKernel(noise_level=0.1**2,
                  noise_level_bounds=(1e-3, np.inf))  # noise terms
kernel = k1 + k2 + k3 + k4

gp = GaussianProcessRegressor(kernel=kernel, alpha=0,
                              normalize_y=True)
gp.fit(X, y)

print("\nLearned kernel: %s" % gp.kernel_)
print("Log-marginal-likelihood: %.3f"
      % gp.log_marginal_likelihood(gp.kernel_.theta))

X_ = np.linspace(X.min(), X.max() + 30, 1000)[:, np.newaxis]
y_pred, y_std = gp.predict(X_, return_std=True)

# Illustration
plt.scatter(X, y, c='k')
plt.plot(X_, y_pred)
plt.fill_between(X_[:, 0], y_pred - y_std, y_pred + y_std,
                 alpha=0.5, color='k')
plt.xlim(X_.min(), X_.max())
plt.xlabel("Year")
plt.ylabel(r"CO$_2$ in ppm")
plt.title(r"Atmospheric CO$_2$ concentration at Mauna Loa")
plt.tight_layout()
plt.show()