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 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)

2022-09-20 14:23:36.018870: 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.
2022-09-20 14:23:36.165947: 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
2022-09-20 14:23:36.165970: 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.
2022-09-20 14:23:36.195922: 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
2022-09-20 14:23:37.051908: 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
2022-09-20 14:23:37.051976: 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
2022-09-20 14:23:37.051985: 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.
/home/circleci/project/gpflow/experimental/utils.py:42: UserWarning: You're calling gpflow.experimental.check_shapes.decorator.check_shapes which is considered *experimental*. Expect: breaking changes, poor documentation, and bugs.
  warn(
/home/circleci/project/gpflow/experimental/utils.py:42: UserWarning: You're calling gpflow.experimental.check_shapes.inheritance.inherit_check_shapes which is considered *experimental*. Expect: breaking changes, poor documentation, and bugs.
  warn(

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)

/home/circleci/project/gpflow/experimental/utils.py:42: UserWarning: You're calling gpflow.experimental.check_shapes.checker.ShapeChecker.__init__ which is considered *experimental*. Expect: breaking changes, poor documentation, and bugs.
  warn(
2022-09-20 14:23:40.363492: 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
2022-09-20 14:23:40.363517: W tensorflow/stream_executor/cuda/cuda_driver.cc:263] failed call to cuInit: UNKNOWN ERROR (303)
2022-09-20 14:23:40.363537: I tensorflow/stream_executor/cuda/cuda_diagnostics.cc:156] kernel driver does not appear to be running on this host (38e97909228e): /proc/driver/nvidia/version does not exist
2022-09-20 14:23:40.364309: 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)