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 (\(\sigma\)) to blend smoothly between different kernels. For example, a single change-point kernel is defined by:

\begin{equation} \textrm{cov}(f(x), f(y)) = k_1(x, y)\cdot\bar{\sigma}(x, y) + k_2(x, y)\cdot\sigma(x, y) \end{equation}

where \(\sigma(x, y) = \sigma(x)\cdot\sigma(y)\) and \(\bar{\sigma}(x, y) = (1 - \sigma(x))\cdot(1 - \sigma(y))\). The sigmoid (\(\sigma\)) is parameterized by a location (\(l\)) and a width (\(w\)).

import matplotlib.pyplot as plt
import numpy as np

import gpflow

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)

2023-05-03 23:14:58.151229: I tensorflow/core/platform/cpu_feature_guard.cc:193] 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.
2023-05-03 23:14:58.296969: 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
2023-05-03 23:14:58.297016: 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.
2023-05-03 23:14:58.327844: E tensorflow/stream_executor/cuda/cuda_blas.cc:2981] Unable to register cuBLAS factory: Attempting to register factory for plugin cuBLAS when one has already been registered
2023-05-03 23:14:59.053923: W tensorflow/stream_executor/platform/default/dso_loader.cc:64] Could not load dynamic library 'libnvinfer.so.7'; dlerror: libnvinfer.so.7: cannot open shared object file: No such file or directory
2023-05-03 23:14:59.053986: W tensorflow/stream_executor/platform/default/dso_loader.cc:64] Could not load dynamic library 'libnvinfer_plugin.so.7'; dlerror: libnvinfer_plugin.so.7: cannot open shared object file: No such file or directory
2023-05-03 23:14:59.053995: W tensorflow/compiler/tf2tensorrt/utils/py_utils.cc:38] TF-TRT Warning: Cannot dlopen some TensorRT libraries. If you would like to use Nvidia GPU with TensorRT, please make sure the missing libraries mentioned above are installed properly.

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.

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)

2023-05-03 23:15:01.966820: 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
2023-05-03 23:15:01.966843: W tensorflow/stream_executor/cuda/cuda_driver.cc:263] failed call to cuInit: UNKNOWN ERROR (303)
2023-05-03 23:15:01.966863: I tensorflow/stream_executor/cuda/cuda_diagnostics.cc:156] kernel driver does not appear to be running on this host (87d90438e0b5): /proc/driver/nvidia/version does not exist
2023-05-03 23:15:01.967112: I tensorflow/core/platform/cpu_feature_guard.cc:193] 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.

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

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)

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

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)