Change points#

Joseph Hall (October 2019)

This notebook demonstrates the use of the ChangePoints kernel, which can be used to describe one-dimensional functions that contain a number of change-points, or regime changes. The kernel makes use of sigmoids (σ) to blend smoothly between different kernels. For example, a single change-point kernel is defined by:

cov(f(x),f(y))=k1(x,y)σ¯(x,y)+k2(x,y)σ(x,y)

where σ(x,y)=σ(x)σ(y) and σ¯(x,y)=(1σ(x))(1σ(y)). The sigmoid (σ) is parameterized by a location (l) and a width (w).

[1]:
import gpflow
import numpy as np
import matplotlib.pyplot as plt

np.random.seed(123)  # for reproducibility of this notebook

plt.style.use("ggplot")
%matplotlib inline


def plotkernelsample(k, ax, xmin=-3, xmax=3, title=None):
    xx = np.linspace(xmin, xmax, 100)[:, None]
    ax.plot(xx, np.random.multivariate_normal(np.zeros(100), k(xx), 3).T)
    ax.set_title(title)
2022-05-10 10:48:08.769177: W tensorflow/stream_executor/platform/default/dso_loader.cc:64] Could not load dynamic library 'libcudart.so.11.0'; dlerror: libcudart.so.11.0: cannot open shared object file: No such file or directory
2022-05-10 10:48:08.769201: I tensorflow/stream_executor/cuda/cudart_stub.cc:29] Ignore above cudart dlerror if you do not have a GPU set up on your machine.

We demonstrate the use of the kernel by drawing a number of samples from different parameterizations. Firstly, a simple single change-point between two kernels of differing lengthscales.

[2]:
np.random.seed(1)

base_k1 = gpflow.kernels.Matern32(lengthscales=0.2)
base_k2 = gpflow.kernels.Matern32(lengthscales=2.0)
k = gpflow.kernels.ChangePoints([base_k1, base_k2], [0.0], steepness=5.0)

f, ax = plt.subplots(1, 1, figsize=(10, 3))
plotkernelsample(k, ax)
2022-05-10 10:48:11.940242: W tensorflow/stream_executor/platform/default/dso_loader.cc:64] Could not load dynamic library 'libcuda.so.1'; dlerror: libcuda.so.1: cannot open shared object file: No such file or directory
2022-05-10 10:48:11.940273: W tensorflow/stream_executor/cuda/cuda_driver.cc:269] failed call to cuInit: UNKNOWN ERROR (303)
2022-05-10 10:48:11.940290: I tensorflow/stream_executor/cuda/cuda_diagnostics.cc:156] kernel driver does not appear to be running on this host (49c966262641): /proc/driver/nvidia/version does not exist
2022-05-10 10:48:11.940542: I tensorflow/core/platform/cpu_feature_guard.cc:151] This TensorFlow binary is optimized with oneAPI Deep Neural Network Library (oneDNN) to use the following CPU instructions in performance-critical operations:  AVX2 AVX512F FMA
To enable them in other operations, rebuild TensorFlow with the appropriate compiler flags.
../../_images/notebooks_advanced_changepoints_3_1.png

Secondly, an implementation of a “change window” in which we change from one kernel to another, then back to the original.

[3]:
np.random.seed(3)

base_k1 = gpflow.kernels.Matern32(lengthscales=0.3)
base_k2 = gpflow.kernels.Constant()
k = gpflow.kernels.ChangePoints([base_k1, base_k2, base_k1], locations=[-1, 1], steepness=10.0)

f, ax = plt.subplots(1, 1, figsize=(10, 3))
plotkernelsample(k, ax)
../../_images/notebooks_advanced_changepoints_5_0.png

And finally, allowing different change-points to occur more or less abruptly by defining different steepness parameters.

[4]:
np.random.seed(2)

base_k1 = gpflow.kernels.Matern32(lengthscales=0.3)
base_k2 = gpflow.kernels.Constant()
k = gpflow.kernels.ChangePoints(
    [base_k1, base_k2, base_k1], locations=[-1, 1], steepness=[5.0, 50.0]
)

f, ax = plt.subplots(1, 1, figsize=(10, 3))
plotkernelsample(k, ax)
../../_images/notebooks_advanced_changepoints_7_0.png