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Experiments_of_statistical_tests_for_piecewise_stationary_bandit.py
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Experiments_of_statistical_tests_for_piecewise_stationary_bandit.py
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# coding: utf-8
# # Table of Contents
# <p><div class="lev1 toc-item"><a href="#Requirements-and-helper-functions" data-toc-modified-id="Requirements-and-helper-functions-1"><span class="toc-item-num">1 </span>Requirements and helper functions</a></div><div class="lev2 toc-item"><a href="#Requirements" data-toc-modified-id="Requirements-11"><span class="toc-item-num">1.1 </span>Requirements</a></div><div class="lev2 toc-item"><a href="#Mathematical-notations-for-stationary-problems" data-toc-modified-id="Mathematical-notations-for-stationary-problems-12"><span class="toc-item-num">1.2 </span>Mathematical notations for stationary problems</a></div><div class="lev2 toc-item"><a href="#Generating-stationary-data" data-toc-modified-id="Generating-stationary-data-13"><span class="toc-item-num">1.3 </span>Generating stationary data</a></div><div class="lev2 toc-item"><a href="#Mathematical-notations-for-piecewise-stationary-problems" data-toc-modified-id="Mathematical-notations-for-piecewise-stationary-problems-14"><span class="toc-item-num">1.4 </span>Mathematical notations for piecewise stationary problems</a></div><div class="lev2 toc-item"><a href="#Generating-fake-piecewise-stationary-data" data-toc-modified-id="Generating-fake-piecewise-stationary-data-15"><span class="toc-item-num">1.5 </span>Generating fake piecewise stationary data</a></div><div class="lev1 toc-item"><a href="#Python-implementations-of-some-statistical-tests" data-toc-modified-id="Python-implementations-of-some-statistical-tests-2"><span class="toc-item-num">2 </span>Python implementations of some statistical tests</a></div><div class="lev2 toc-item"><a href="#Monitored" data-toc-modified-id="Monitored-21"><span class="toc-item-num">2.1 </span><code>Monitored</code></a></div><div class="lev2 toc-item"><a href="#CUSUM" data-toc-modified-id="CUSUM-22"><span class="toc-item-num">2.2 </span><code>CUSUM</code></a></div><div class="lev2 toc-item"><a href="#PHT" data-toc-modified-id="PHT-23"><span class="toc-item-num">2.3 </span><code>PHT</code></a></div><div class="lev2 toc-item"><a href="#Gaussian-GLR" data-toc-modified-id="Gaussian-GLR-24"><span class="toc-item-num">2.4 </span><code>Gaussian GLR</code></a></div><div class="lev2 toc-item"><a href="#Bernoulli-GLR" data-toc-modified-id="Bernoulli-GLR-25"><span class="toc-item-num">2.5 </span><code>Bernoulli GLR</code></a></div><div class="lev2 toc-item"><a href="#Sub-Gaussian-GLR" data-toc-modified-id="Sub-Gaussian-GLR-26"><span class="toc-item-num">2.6 </span><code>Sub-Gaussian GLR</code></a></div><div class="lev2 toc-item"><a href="#List-of-all-Python-algorithms" data-toc-modified-id="List-of-all-Python-algorithms-27"><span class="toc-item-num">2.7 </span>List of all Python algorithms</a></div><div class="lev1 toc-item"><a href="#Comparing-the-different-implementations" data-toc-modified-id="Comparing-the-different-implementations-3"><span class="toc-item-num">3 </span>Comparing the different implementations</a></div><div class="lev2 toc-item"><a href="#Generating-some-toy-data" data-toc-modified-id="Generating-some-toy-data-31"><span class="toc-item-num">3.1 </span>Generating some toy data</a></div><div class="lev2 toc-item"><a href="#Checking-time-efficiency" data-toc-modified-id="Checking-time-efficiency-32"><span class="toc-item-num">3.2 </span>Checking time efficiency</a></div><div class="lev2 toc-item"><a href="#Checking-detection-delay" data-toc-modified-id="Checking-detection-delay-33"><span class="toc-item-num">3.3 </span>Checking detection delay</a></div><div class="lev2 toc-item"><a href="#Checking-false-alarm-probabilities" data-toc-modified-id="Checking-false-alarm-probabilities-34"><span class="toc-item-num">3.4 </span>Checking false alarm probabilities</a></div><div class="lev2 toc-item"><a href="#Checking-missed-detection-probabilities" data-toc-modified-id="Checking-missed-detection-probabilities-35"><span class="toc-item-num">3.5 </span>Checking missed detection probabilities</a></div><div class="lev1 toc-item"><a href="#More-simulations-and-some-plots" data-toc-modified-id="More-simulations-and-some-plots-4"><span class="toc-item-num">4 </span>More simulations and some plots</a></div><div class="lev2 toc-item"><a href="#Run-a-check-for-a-grid-of-values" data-toc-modified-id="Run-a-check-for-a-grid-of-values-41"><span class="toc-item-num">4.1 </span>Run a check for a grid of values</a></div><div class="lev2 toc-item"><a href="#A-version-using-joblib.Parallel-to-use-multi-core-computations" data-toc-modified-id="A-version-using-joblib.Parallel-to-use-multi-core-computations-42"><span class="toc-item-num">4.2 </span>A version using <code>joblib.Parallel</code> to use multi-core computations</a></div><div class="lev2 toc-item"><a href="#Checking-on-a-small-grid-of-values" data-toc-modified-id="Checking-on-a-small-grid-of-values-43"><span class="toc-item-num">4.3 </span>Checking on a small grid of values</a></div><div class="lev2 toc-item"><a href="#Plotting-the-result-as-a-2D-image" data-toc-modified-id="Plotting-the-result-as-a-2D-image-44"><span class="toc-item-num">4.4 </span>Plotting the result as a 2D image</a></div><div class="lev3 toc-item"><a href="#First-example" data-toc-modified-id="First-example-441"><span class="toc-item-num">4.4.1 </span>First example</a></div><div class="lev4 toc-item"><a href="#For-Monitored" data-toc-modified-id="For-Monitored-4411"><span class="toc-item-num">4.4.1.1 </span>For <code>Monitored</code></a></div><div class="lev4 toc-item"><a href="#For-CUSUM" data-toc-modified-id="For-CUSUM-4412"><span class="toc-item-num">4.4.1.2 </span>For <code>CUSUM</code></a></div><div class="lev3 toc-item"><a href="#Second-example" data-toc-modified-id="Second-example-442"><span class="toc-item-num">4.4.2 </span>Second example</a></div><div class="lev4 toc-item"><a href="#For-Monitored" data-toc-modified-id="For-Monitored-4421"><span class="toc-item-num">4.4.2.1 </span>For <code>Monitored</code></a></div><div class="lev4 toc-item"><a href="#For-Monitored-for-Gaussian-data" data-toc-modified-id="For-Monitored-for-Gaussian-data-4422"><span class="toc-item-num">4.4.2.2 </span>For <code>Monitored</code> for Gaussian data</a></div><div class="lev4 toc-item"><a href="#For-CUSUM" data-toc-modified-id="For-CUSUM-4423"><span class="toc-item-num">4.4.2.3 </span>For <code>CUSUM</code></a></div><div class="lev4 toc-item"><a href="#For-PHT" data-toc-modified-id="For-PHT-4424"><span class="toc-item-num">4.4.2.4 </span>For <code>PHT</code></a></div><div class="lev4 toc-item"><a href="#For-Bernoulli-GLR" data-toc-modified-id="For-Bernoulli-GLR-4425"><span class="toc-item-num">4.4.2.5 </span>For <code>Bernoulli GLR</code></a></div><div class="lev4 toc-item"><a href="#For-Gaussian-GLR" data-toc-modified-id="For-Gaussian-GLR-4426"><span class="toc-item-num">4.4.2.6 </span>For <code>Gaussian GLR</code></a></div><div class="lev4 toc-item"><a href="#For-Sub-Gaussian-GLR" data-toc-modified-id="For-Sub-Gaussian-GLR-4427"><span class="toc-item-num">4.4.2.7 </span>For <code>Sub-Gaussian GLR</code></a></div><div class="lev1 toc-item"><a href="#Exploring-the-parameters-of-change-point-detection-algorithms:-how-to-tune-them?" data-toc-modified-id="Exploring-the-parameters-of-change-point-detection-algorithms:-how-to-tune-them?-5"><span class="toc-item-num">5 </span>Exploring the parameters of change point detection algorithms: how to tune them?</a></div><div class="lev2 toc-item"><a href="#A-simple-problem-function" data-toc-modified-id="A-simple-problem-function-51"><span class="toc-item-num">5.1 </span>A simple problem function</a></div><div class="lev2 toc-item"><a href="#A-generic-function" data-toc-modified-id="A-generic-function-52"><span class="toc-item-num">5.2 </span>A generic function</a></div><div class="lev2 toc-item"><a href="#Plotting-the-result-as-a-1D-plot" data-toc-modified-id="Plotting-the-result-as-a-1D-plot-53"><span class="toc-item-num">5.3 </span>Plotting the result as a 1D plot</a></div><div class="lev2 toc-item"><a href="#Experiments-for-Monitored" data-toc-modified-id="Experiments-for-Monitored-54"><span class="toc-item-num">5.4 </span>Experiments for <code>Monitored</code></a></div><div class="lev2 toc-item"><a href="#Experiments-for-Bernoulli-GLR" data-toc-modified-id="Experiments-for-Bernoulli-GLR-55"><span class="toc-item-num">5.5 </span>Experiments for <code>Bernoulli GLR</code></a></div><div class="lev2 toc-item"><a href="#Experiments-for-Gaussian-GLR" data-toc-modified-id="Experiments-for-Gaussian-GLR-56"><span class="toc-item-num">5.6 </span>Experiments for <code>Gaussian GLR</code></a></div><div class="lev2 toc-item"><a href="#Experiments-for-CUSUM" data-toc-modified-id="Experiments-for-CUSUM-57"><span class="toc-item-num">5.7 </span>Experiments for <code>CUSUM</code></a></div><div class="lev2 toc-item"><a href="#Experiments-for-Sub-Gaussian-GLR" data-toc-modified-id="Experiments-for-Sub-Gaussian-GLR-58"><span class="toc-item-num">5.8 </span>Experiments for <code>Sub-Gaussian GLR</code></a></div><div class="lev2 toc-item"><a href="#Other-experiments" data-toc-modified-id="Other-experiments-59"><span class="toc-item-num">5.9 </span>Other experiments</a></div><div class="lev1 toc-item"><a href="#Conclusions" data-toc-modified-id="Conclusions-6"><span class="toc-item-num">6 </span>Conclusions</a></div>
# # Requirements and helper functions
# ## Requirements
#
# This notebook requires to have [`numpy`](https://www.numpy.org/) and [`matplotlib`](https://matplotlib.org/) installed.
# One function needs a function from [`scipy.special`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.comb.html#scipy.special.comb).
# [`joblib`](https://joblib.readthedocs.io/en/latest/) is used to have parallel computations (at the end).
#
# The bottleneck performance of the main functions are very simple functions, for which we can write efficient versions using either [`numba.jit`](https://numba.pydata.org/numba-doc/latest/reference/jit-compilation.html#numba.jit) or [`cython`](https://cython.readthedocs.io/en/latest/src/quickstart/build.html#jupyter-notebook).
# In[3]:
get_ipython().system('pip3 install watermark numpy scipy matplotlib joblib numba cython')
get_ipython().run_line_magic('load_ext', 'watermark')
get_ipython().run_line_magic('watermark', '-v -m -p numpy,scipy,matplotlib,joblib,numba,cython -a "Lilian Besson and Emilie Kaufmann"')
# In[4]:
import numpy as np
# In[5]:
try:
from tqdm import tqdm_notebook as tqdm
except:
def tqdm(iterator, *args, **kwargs):
return iterator
# ## Mathematical notations for stationary problems
#
# We consider $K \geq 1$ arms, which are distributions $\nu_k$.
# We focus on Bernoulli distributions, which are characterized by their means, $\nu_k = \mathcal{B}(\mu_k)$ for $\mu_k\in[0,1]$.
# A stationary bandit problem is defined here by the vector $[\mu_1,\dots,\mu_K]$.
#
# For a fixed problem and a *horizon* $T\in\mathbb{N}$, $T\geq1$, we draw samples from the $K$ distributions to get *data*: $\forall t, r_k(t) \sim \nu_k$, ie, $\mathbb{P}(r_k(t) = 1) = \mu_k$ and $r_k(t) \in \{0,1\}$.
# ## Generating stationary data
#
# Here we give some examples of stationary problems and examples of data we can draw from them.
# In[13]:
def bernoulli_samples(means, horizon=1000):
if np.size(means) == 1:
return np.random.binomial(1, means, size=horizon)
else:
results = np.zeros((np.size(means), horizon))
for i, mean in enumerate(means):
results[i] = np.random.binomial(1, mean, size=horizon)
return results
# In[14]:
problem1 = [0.5]
bernoulli_samples(problem1, horizon=20)
# In[27]:
get_ipython().run_line_magic('timeit', 'bernoulli_samples(problem1, horizon=1000)')
# Now for Gaussian data:
# In[49]:
sigma = 0.25 # Bernoulli are 1/4-sub Gaussian too!
# In[50]:
def gaussian_samples(means, horizon=1000, sigma=sigma):
if np.size(means) == 1:
return np.random.normal(loc=means, scale=sigma, size=horizon)
else:
results = np.zeros((np.size(means), horizon))
for i, mean in enumerate(means):
results[i] = np.random.normal(loc=mean, scale=sigma, size=horizon)
return results
# In[51]:
gaussian_samples(problem1, horizon=20)
# In[52]:
get_ipython().run_line_magic('timeit', 'gaussian_samples(problem1, horizon=1000)')
# For bandit problem with $K \geq 2$ arms, the *goal* is to design an online learning algorithm that roughly do the following:
#
# - For time $t=1$ to $t=T$ (unknown horizon)
# 1. Algorithm $A$ decide to draw arm $A(t) \in\{1,\dots,K\}$,
# 2. Get the reward $r(t) = r_{A(t)}(t) \sim \nu_{A(t)}$ from the (Bernoulli) distribution of that arm,
# 3. Give this observation of reward $r(t)$ coming from arm $A(t)$ to the algorithm,
# 4. Update internal state of the algorithm
#
# An algorithm is efficient if it obtains a high (expected) sum reward, ie, $\sum_{t=1}^T r(t)$.
#
# Note that I don't focus on bandit algorithm here.
# In[9]:
problem2 = [0.1, 0.5, 0.9]
bernoulli_samples(problem2, horizon=20)
# In[10]:
problem2 = [0.1, 0.5, 0.9]
gaussian_samples(problem2, horizon=20)
# For instance on these data, the best arm is clearly the third one, with expected reward of $\mu^* = \max_k \mu_k = 0.9$.
# ## Mathematical notations for piecewise stationary problems
#
# Now we fix the horizon $T\in\mathbb{N}$, $T\geq1$ and we also consider a set of $\Upsilon_T$ *break points*, $\tau_1,\dots,\tau_{\Upsilon_T} \in\{1,\dots,T\}$. We denote $\tau_0 = 0$ and $\tau_{\Upsilon_T+1} = T$ for convenience of notations.
# We can assume that breakpoints are far "enough" from each other, for instance that there exists an integer $N\in\mathbb{N},N\geq1$ such that $\min_{i=0}^{\Upsilon_T} \tau_{i+1} - \tau_i \geq N K$. That is, on each *stationary interval*, a uniform sampling of the $K$ arms gives at least $N$ samples by arm.
#
# Now, in any stationary interval $[\tau_i + 1, \tau_{i+1}]$, the $K \geq 1$ arms are distributions $\nu_k^{(i)}$.
# We focus on Bernoulli distributions, which are characterized by their means, $\nu_k^{(i)} := \mathcal{B}(\mu_k^{(i)})$ for $\mu_k^{(i)}\in[0,1]$.
# A piecewise stationary bandit problem is defined here by the vector $[\mu_k^{(i)}]_{1\leq k \leq K, 1 \leq i \leq \Upsilon_T}$.
#
# For a fixed problem and a *horizon* $T\in\mathbb{N}$, $T\geq1$, we draw samples from the $K$ distributions to get *data*: $\forall t, r_k(t) \sim \nu_k^{(i)}$ for $i$ the unique index of stationary interval such that $t\in[\tau_i + 1, \tau_{i+1}]$.
# ## Generating fake piecewise stationary data
#
# The format to define piecewise stationary problem will be the following. It is compact but generic!
#
# The first example considers a unique arm, with 2 breakpoints uniformly spaced.
# - On the first interval, for instance from $t=1$ to $t=500$, that is $\tau_1 = 500$, $\mu_1^{(1)} = 0.1$,
# - On the second interval, for instance from $t=501$ to $t=1000$, that is $\tau_2 = 100$, $\mu_1^{(2)} = 0.5$,
# - On the third interval, for instance from $t=1001$ to $t=1500$, that $\mu_1^{(3)} = 0.9$.
# In[53]:
# With 1 arm only!
problem_piecewise_0 = lambda horizon: {
"listOfMeans": [
[0.1], # 0 to 499
[0.5], # 500 to 999
[0.8], # 1000 to 1499
],
"changePoints": [
int(0 * horizon / 1500.0),
int(500 * horizon / 1500.0),
int(1000 * horizon / 1500.0),
],
}
# In[54]:
# With 2 arms
problem_piecewise_1 = lambda horizon: {
"listOfMeans": [
[0.1, 0.2], # 0 to 399
[0.1, 0.3], # 400 to 799
[0.5, 0.3], # 800 to 1199
[0.4, 0.3], # 1200 to 1599
[0.3, 0.9], # 1600 to end
],
"changePoints": [
int(0 * horizon / 2000.0),
int(400 * horizon / 2000.0),
int(800 * horizon / 2000.0),
int(1200 * horizon / 2000.0),
int(1600 * horizon / 2000.0),
],
}
# In[55]:
# With 3 arms
problem_piecewise_2 = lambda horizon: {
"listOfMeans": [
[0.2, 0.5, 0.9], # 0 to 399
[0.2, 0.2, 0.9], # 400 to 799
[0.2, 0.2, 0.1], # 800 to 1199
[0.7, 0.2, 0.1], # 1200 to 1599
[0.7, 0.5, 0.1], # 1600 to end
],
"changePoints": [
int(0 * horizon / 2000.0),
int(400 * horizon / 2000.0),
int(800 * horizon / 2000.0),
int(1200 * horizon / 2000.0),
int(1600 * horizon / 2000.0),
],
}
# In[56]:
# With 3 arms
problem_piecewise_3 = lambda horizon: {
"listOfMeans": [
[0.4, 0.5, 0.9], # 0 to 399
[0.5, 0.4, 0.7], # 400 to 799
[0.6, 0.3, 0.5], # 800 to 1199
[0.7, 0.2, 0.3], # 1200 to 1599
[0.8, 0.1, 0.1], # 1600 to end
],
"changePoints": [
int(0 * horizon / 2000.0),
int(400 * horizon / 2000.0),
int(800 * horizon / 2000.0),
int(1200 * horizon / 2000.0),
int(1600 * horizon / 2000.0),
],
}
# Now we can write a utility function that transform this compact representation into a full list of means.
# In[57]:
def getFullHistoryOfMeans(problem, horizon=2000):
"""Return the vector of mean of the arms, for a piece-wise stationary MAB.
- It is a numpy array of shape (nbArms, horizon).
"""
pb = problem(horizon)
listOfMeans, changePoints = pb['listOfMeans'], pb['changePoints']
nbArms = len(listOfMeans[0])
if horizon is None:
horizon = np.max(changePoints)
meansOfArms = np.ones((nbArms, horizon))
for armId in range(nbArms):
nbChangePoint = 0
for t in range(horizon):
if nbChangePoint < len(changePoints) - 1 and t >= changePoints[nbChangePoint + 1]:
nbChangePoint += 1
meansOfArms[armId][t] = listOfMeans[nbChangePoint][armId]
return meansOfArms
# For examples :
# In[16]:
getFullHistoryOfMeans(problem_piecewise_0, horizon=50)
# In[17]:
getFullHistoryOfMeans(problem_piecewise_1, horizon=50)
# In[18]:
getFullHistoryOfMeans(problem_piecewise_2, horizon=50)
# In[19]:
getFullHistoryOfMeans(problem_piecewise_3, horizon=50)
# And now we need to be able to generate samples from such distributions.
# In[58]:
def piecewise_bernoulli_samples(problem, horizon=1000):
fullMeans = getFullHistoryOfMeans(problem, horizon=horizon)
nbArms, horizon = np.shape(fullMeans)
results = np.zeros((nbArms, horizon))
for i in range(nbArms):
mean_i = fullMeans[i, :]
for t in range(horizon):
mean_i_t = max(0, min(1, mean_i[t])) # crop to [0, 1] !
results[i, t] = np.random.binomial(1, mean_i_t)
return results
# In[59]:
def piecewise_gaussian_samples(problem, horizon=1000, sigma=sigma):
fullMeans = getFullHistoryOfMeans(problem, horizon=horizon)
nbArms, horizon = np.shape(fullMeans)
results = np.zeros((nbArms, horizon))
for i in range(nbArms):
mean_i = fullMeans[i, :]
for t in range(horizon):
mean_i_t = mean_i[t]
results[i, t] = np.random.normal(loc=mean_i_t, scale=sigma, size=1)
return results
# Examples:
# In[22]:
getFullHistoryOfMeans(problem_piecewise_0, horizon=100)
piecewise_bernoulli_samples(problem_piecewise_0, horizon=100)
# In[23]:
piecewise_gaussian_samples(problem_piecewise_0, horizon=100)
# We easily spot the (approximate) location of the breakpoint!
#
# Another example:
# In[24]:
piecewise_bernoulli_samples(problem_piecewise_1, horizon=100)
# In[25]:
piecewise_gaussian_samples(problem_piecewise_1, horizon=20)
# ----
# # Python implementations of some statistical tests
#
# I will implement here the following statistical tests.
# I give a link to the implementation of the correspond bandit policy in my framework [`SMPyBandits`](https://smpybandits.github.io/)
#
# - Monitored (based on a McDiarmid inequality), for Monitored-UCB or [`M-UCB`](),
# - CUSUM, for [`CUSUM-UCB`](https://smpybandits.github.io/docs/Policies.CD_UCB.html?highlight=cusum#Policies.CD_UCB.CUSUM_IndexPolicy),
# - PHT, for [`PHT-UCB`](https://smpybandits.github.io/docs/Policies.CD_UCB.html?highlight=cusum#Policies.CD_UCB.PHT_IndexPolicy),
# - Gaussian GLR, for [`GaussianGLR-UCB`](https://smpybandits.github.io/docs/Policies.CD_UCB.html?highlight=glr#Policies.CD_UCB.GaussianGLR_IndexPolicy),
# - Bernoulli GLR, for [`BernoulliGLR-UCB`](https://smpybandits.github.io/docs/Policies.CD_UCB.html?highlight=glr#Policies.CD_UCB.BernoulliGLR_IndexPolicy).
# In[60]:
class ChangePointDetector(object):
def __init__(self, **kwargs):
self._kwargs = kwargs
for key, value in kwargs.items():
setattr(self, key, value)
def __str__(self):
return f"{self.__class__.__name__}{f'({repr(self._kwargs)})' if self._kwargs else ''}"
def detect(self, all_data, t):
raise NotImplementedError
# Having classes is simply to be able to pretty print the algorithms when they have parameters:
# In[27]:
print(ChangePointDetector())
# In[28]:
print(ChangePointDetector(w=10, b=1))
# ## `Monitored`
#
# It uses a McDiarmid inequality. For a (pair) window size $w\in\mathbb{N}$ and a threshold $b\in\mathbb{R}^+$.
# At time $t$, if there is at least $w$ data in the data vector $(X_i)_i$, then let $Y$ denote the last $w$ data.
# A change is detected if
# $$ |\sum_{i=w/2+1}^{w} Y_i - \sum_{i=1}^{w/2} Y_i | > b ? $$
# In[64]:
NB_ARMS = 1
WINDOW_SIZE = 80
# In[65]:
import numba
# In[88]:
class Monitored(ChangePointDetector):
def __init__(self, window_size=WINDOW_SIZE, threshold_b=None):
super().__init__(window_size=window_size, threshold_b=threshold_b)
def __str__(self):
if self.threshold_b:
return f"Monitored($w={self.window_size:.3g}$, $b={self.threshold_b:.3g}$)"
else:
latexname = r"\sqrt{\frac{w}{2} \log(2 T^2)}"
return f"Monitored($w={self.window_size:.3g}$, $b={latexname}$)"
def detect(self, all_data, t):
r""" A change is detected for the current arm if the following test is true:
.. math:: |\sum_{i=w/2+1}^{w} Y_i - \sum_{i=1}^{w/2} Y_i | > b ?
- where :math:`Y_i` is the i-th data in the latest w data from this arm (ie, :math:`X_k(t)` for :math:`t = n_k - w + 1` to :math:`t = n_k` current number of samples from arm k).
- where :attr:`threshold_b` is the threshold b of the test, and :attr:`window_size` is the window-size w.
"""
data = all_data[:t]
# don't try to detect change if there is not enough data!
if len(data) < self.window_size:
return False
# compute parameters
horizon = len(all_data)
threshold_b = self.threshold_b
if threshold_b is None:
threshold_b = np.sqrt(self.window_size/2 * np.log(2 * NB_ARMS * horizon**2))
last_w_data = data[-self.window_size:]
sum_first_half = np.sum(last_w_data[:self.window_size//2])
sum_second_half = np.sum(last_w_data[self.window_size//2:])
return abs(sum_first_half - sum_second_half) > threshold_b
# ## `CUSUM`
#
# The two-sided CUSUM algorithm, from [Page, 1954], works like this:
#
# - For each *data* k, compute:
#
# $$
# s_k^- = (y_k - \hat{u}_0 - \varepsilon) 1(k > M),\\
# s_k^+ = (\hat{u}_0 - y_k - \varepsilon) 1(k > M),\\
# g_k^+ = \max(0, g_{k-1}^+ + s_k^+),\\
# g_k^- = \max(0, g_{k-1}^- + s_k^-).
# $$
#
# - The change is detected if $\max(g_k^+, g_k^-) > h$, where $h=$`threshold_h` is the threshold of the test,
# - And $\hat{u}_0 = \frac{1}{M} \sum_{k=1}^{M} y_k$ is the mean of the first M samples, where M is `M` the min number of observation between change points.
# In[67]:
#: Precision of the test.
EPSILON = 0.5
#: Default value of :math:`\lambda`.
LAMBDA = 1
#: Hypothesis on the speed of changes: between two change points, there is at least :math:`M * K` time steps, where K is the number of arms, and M is this constant.
MIN_NUMBER_OF_OBSERVATION_BETWEEN_CHANGE_POINT = 50
MAX_NB_RANDOM_EVENTS = 1
# In[68]:
from scipy.special import comb
# In[89]:
def compute_h__CUSUM(horizon,
verbose=False,
M=MIN_NUMBER_OF_OBSERVATION_BETWEEN_CHANGE_POINT,
max_nb_random_events=MAX_NB_RANDOM_EVENTS,
nbArms=1,
epsilon=EPSILON,
lmbda=LAMBDA,
):
r""" Compute the values :math:`C_1^+, C_1^-, C_1, C_2, h` from the formulas in Theorem 2 and Corollary 2 in the paper."""
T = int(max(1, horizon))
UpsilonT = int(max(1, max_nb_random_events))
K = int(max(1, nbArms))
C1_minus = np.log(((4 * epsilon) / (1-epsilon)**2) * comb(M, int(np.floor(2 * epsilon * M))) * (2 * epsilon)**M + 1)
C1_plus = np.log(((4 * epsilon) / (1+epsilon)**2) * comb(M, int(np.ceil(2 * epsilon * M))) * (2 * epsilon)**M + 1)
C1 = min(C1_minus, C1_plus)
if C1 == 0: C1 = 1 # FIXME
h = 1/C1 * np.log(T / UpsilonT)
return h
# In[151]:
class CUSUM(ChangePointDetector):
def __init__(self,
epsilon=EPSILON,
M=MIN_NUMBER_OF_OBSERVATION_BETWEEN_CHANGE_POINT,
threshold_h=None,
):
assert 0 < epsilon < 1, f"Error: epsilon for CUSUM must be in (0, 1) but is {epsilon}."
super().__init__(epsilon=epsilon, M=M, threshold_h=threshold_h)
def __str__(self):
if self.threshold_h:
return fr"CUSUM($\varepsilon={self.epsilon:.3g}$, $M={self.M}$, $h={self.threshold_h:.3g}$)"
else:
return fr"CUSUM($\varepsilon={self.epsilon:.3g}$, $M={self.M}$, $h=$'auto')"
def detect(self, all_data, t):
r""" Detect a change in the current arm, using the two-sided CUSUM algorithm [Page, 1954].
- For each *data* k, compute:
.. math::
s_k^- &= (y_k - \hat{u}_0 - \varepsilon) 1(k > M),\\
s_k^+ &= (\hat{u}_0 - y_k - \varepsilon) 1(k > M),\\
g_k^+ &= \max(0, g_{k-1}^+ + s_k^+),\\
g_k^- &= \max(0, g_{k-1}^- + s_k^-).
- The change is detected if :math:`\max(g_k^+, g_k^-) > h`, where :attr:`threshold_h` is the threshold of the test,
- And :math:`\hat{u}_0 = \frac{1}{M} \sum_{k=1}^{M} y_k` is the mean of the first M samples, where M is :attr:`M` the min number of observation between change points.
"""
data = all_data[:t]
# compute parameters
horizon = len(all_data)
threshold_h = self.threshold_h
if self.threshold_h is None:
threshold_h = compute_h__CUSUM(horizon, self.M, 1, epsilon=self.epsilon)
gp, gm = 0, 0
# First we use the first M samples to calculate the average :math:`\hat{u_0}`.
u0hat = np.mean(data[:self.M])
for k in range(self.M + 1, len(data)):
y_k = data[k]
sp = u0hat - y_k - self.epsilon # no need to multiply by (k > self.M)
sm = y_k - u0hat - self.epsilon # no need to multiply by (k > self.M)
gp = max(0, gp + sp)
gm = max(0, gm + sm)
if max(gp, gm) >= threshold_h:
return True
return False
# ## `PHT`
#
# The two-sided CUSUM algorithm, from [Hinkley, 1971], works like this:
#
# - For each *data* k, compute:
#
# $$
# s_k^- = y_k - \hat{y}_k - \varepsilon,\\
# s_k^+ = \hat{y}_k - y_k - \varepsilon,\\
# g_k^+ = \max(0, g_{k-1}^+ + s_k^+),\\
# g_k^- = \max(0, g_{k-1}^- + s_k^-).
# $$
#
# - The change is detected if $\max(g_k^+, g_k^-) > h$, where $h=$`threshold_h` is the threshold of the test,
# - And $\hat{y}_k = \frac{1}{k} \sum_{s=1}^{k} y_s$ is the mean of the first k samples.
# In[152]:
class PHT(ChangePointDetector):
def __init__(self,
epsilon=EPSILON,
M=MIN_NUMBER_OF_OBSERVATION_BETWEEN_CHANGE_POINT,
threshold_h=None,
):
assert 0 < epsilon < 1, f"Error: epsilon for CUSUM must be in (0, 1) but is {epsilon}."
super().__init__(epsilon=epsilon, M=M, threshold_h=threshold_h)
def __str__(self):
if self.threshold_h:
return fr"PHT($\varepsilon={self.epsilon:.3g}$, $M={self.M}$, $h={self.threshold_h:.3g}$)"
else:
return fr"PHT($\varepsilon={self.epsilon:.3g}$, $M={self.M}$, $h=$'auto')"
def detect(self, all_data, t):
r""" Detect a change in the current arm, using the two-sided PHT algorithm [Hinkley, 1971].
- For each *data* k, compute:
.. math::
s_k^- &= y_k - \hat{y}_k - \varepsilon,\\
s_k^+ &= \hat{y}_k - y_k - \varepsilon,\\
g_k^+ &= \max(0, g_{k-1}^+ + s_k^+),\\
g_k^- &= \max(0, g_{k-1}^- + s_k^-).
- The change is detected if :math:`\max(g_k^+, g_k^-) > h`, where :attr:`threshold_h` is the threshold of the test,
- And :math:`\hat{y}_k = \frac{1}{k} \sum_{s=1}^{k} y_s` is the mean of the first k samples.
"""
data = all_data[:t]
# compute parameters
horizon = len(all_data)
threshold_h = self.threshold_h
if threshold_h is None:
threshold_h = compute_h__CUSUM(horizon, self.M, 1, epsilon=self.epsilon)
gp, gm = 0, 0
y_k_hat = 0
# First we use the first M samples to calculate the average :math:`\hat{u_0}`.
for k, y_k in enumerate(data):
y_k_hat = (k * y_k_hat + y_k) / (k + 1) # XXX smart formula to update the mean!
sp = y_k_hat - y_k - self.epsilon
sm = y_k - y_k_hat - self.epsilon
gp = max(0, gp + sp)
gm = max(0, gm + sm)
if max(gp, gm) >= threshold_h:
return True
return False
# ---
# ## `Gaussian GLR`
#
# The Generalized Likelihood Ratio test (GLR) works with a one-dimensional exponential family, for which we have a function `kl` such that if $\mu_1,\mu_2$ are the means of two distributions $\nu_1,\nu_2$, then $\mathrm{KL}(\mathcal{D}(\nu_1), \mathcal{D}(\nu_1))=$ `kl` $(\mu_1,\mu_2)$.
#
# - For each *time step* $s$ between $t_0=0$ and $t$, compute:
# $$G^{\mathcal{N}_1}_{t_0:s:t} = (s-t_0+1) \mathrm{kl}(\mu_{t_0,s}, \mu_{t_0,t}) + (t-s) \mathrm{kl}(\mu_{s+1,t}, \mu_{t_0,t}).$$
#
# - The change is detected if there is a time $s$ such that $G^{\mathcal{N}_1}_{t_0:s:t} > b(t_0, s, t, \delta)$, where $b(t_0, s, t, \delta)=$ `threshold_h` is the threshold of the test,
# - And $\mu_{a,b} = \frac{1}{b-a+1} \sum_{s=a}^{b} y_s$ is the mean of the samples between $a$ and $b$.
#
# The threshold is computed as:
# $$ b(t_0, s, t, \delta):= \left(1 + \frac{1}{t - t_0 + 1}\right) 2 \log\left(\frac{2 (t - t_0) \sqrt{(t - t_0) + 2}}{\delta}\right).$$
#
# Another threshold we want to check is the following:
# $$ b(t_0, s, t, \delta):= \log\left(\frac{(s - t_0 + 1) (t - s)}{\delta}\right).$$
# In[124]:
from math import log, isinf
def compute_c__GLR_0(t0, s, t, horizon=None, delta=None):
r""" Compute the values :math:`c` from the corollary of of Theorem 2 from ["Sequential change-point detection: Laplace concentration of scan statistics and non-asymptotic delay bounds", O.-A. Maillard, 2018].
- The threshold is computed as:
.. math:: h := \left(1 + \frac{1}{t - t_0 + 1}\right) 2 \log\left(\frac{2 (t - t_0) \sqrt{(t - t_0) + 2}}{\delta}\right).
"""
if delta is None:
T = int(max(1, horizon))
delta = 1.0 / T
t_m_t0 = abs(t - t0)
c = (1 + (1 / (t_m_t0 + 1.0))) * 2 * log((2 * t_m_t0 * np.sqrt(t_m_t0 + 2)) / delta)
if c < 0 and isinf(c): c = float('+inf')
return c
# In[125]:
from math import log, isinf
def compute_c__GLR(t0, s, t, horizon=None, delta=None):
r""" Compute the values :math:`c` from the corollary of of Theorem 2 from ["Sequential change-point detection: Laplace concentration of scan statistics and non-asymptotic delay bounds", O.-A. Maillard, 2018].
- The threshold is computed as:
.. math:: h := \log\left(\frac{(s - t_0 + 1) (t - s)}{\delta}\right).
"""
if delta is None:
T = int(max(1, horizon))
delta = 1.0 / T
arg = (s - t0 + 1) * (t - s) / delta
if arg <= 0: c = float('+inf')
else: c = log(arg)
return c
# For Gaussian distributions of known variance, the Kullback-Leibler divergence is easy to compute:
#
# Kullback-Leibler divergence for Gaussian distributions of means $x$ and $y$ and variances $\sigma^2_x = \sigma^2_y$, $\nu_1 = \mathcal{N}(x, \sigma_x^2)$ and $\nu_2 = \mathcal{N}(y, \sigma_x^2)$ is:
#
# $$\mathrm{KL}(\nu_1, \nu_2) = \frac{(x - y)^2}{2 \sigma_y^2} + \frac{1}{2}\left( \frac{\sigma_x^2}{\sigma_y^2} - 1 \log\left(\frac{\sigma_x^2}{\sigma_y^2}\right) \right).$$
# In[126]:
def klGauss(x, y, sig2x=0.25):
r""" Kullback-Leibler divergence for Gaussian distributions of means ``x`` and ``y`` and variances ``sig2x`` and ``sig2y``, :math:`\nu_1 = \mathcal{N}(x, \sigma_x^2)` and :math:`\nu_2 = \mathcal{N}(y, \sigma_x^2)`:
.. math:: \mathrm{KL}(\nu_1, \nu_2) = \frac{(x - y)^2}{2 \sigma_y^2} + \frac{1}{2}\left( \frac{\sigma_x^2}{\sigma_y^2} - 1 \log\left(\frac{\sigma_x^2}{\sigma_y^2}\right) \right).
See https://en.wikipedia.org/wiki/Normal_distribution#Other_properties
- sig2y = sig2x (same variance).
"""
return (x - y) ** 2 / (2. * sig2x)
# In[127]:
class GaussianGLR(ChangePointDetector):
def __init__(self, mult_threshold_h=1, delta=None):
super().__init__(mult_threshold_h=mult_threshold_h, delta=delta)
def __str__(self):
return r"Gaussian-GLR($h_0={}$, $\delta={}$)".format(
f"{self.mult_threshold_h:.3g}" if self.mult_threshold_h is not None else 'auto',
f"{self.delta:.3g}" if self.delta is not None else 'auto',
)
def detect(self, all_data, t):
r""" Detect a change in the current arm, using the Generalized Likelihood Ratio test (GLR) and the :attr:`kl` function.
- For each *time step* :math:`s` between :math:`t_0=0` and :math:`t`, compute:
.. math::
G^{\mathcal{N}_1}_{t_0:s:t} = (s-t_0+1)(t-s) \mathrm{kl}(\mu_{s+1,t}, \mu_{t_0,s}) / (t-t_0+1).
- The change is detected if there is a time :math:`s` such that :math:`G^{\mathcal{N}_1}_{t_0:s:t} > h`, where :attr:`threshold_h` is the threshold of the test,
- And :math:`\mu_{a,b} = \frac{1}{b-a+1} \sum_{s=a}^{b} y_s` is the mean of the samples between :math:`a` and :math:`b`.
"""
data = all_data[:t]
t0 = 0
horizon = len(all_data)
# compute parameters
mean_all = np.mean(data[t0 : t+1])
mean_before = 0
mean_after = mean_all
for s in range(t0, t):
# DONE okay this is efficient we don't compute the same means too many times!
y = data[s]
mean_before = (s * mean_before + y) / (s + 1)
mean_after = ((t + 1 - s + t0) * mean_after - y) / (t - s + t0)
kl_before = klGauss(mean_before, mean_all)
kl_after = klGauss(mean_after, mean_all)
threshold_h = self.mult_threshold_h * compute_c__GLR(t0, s, t, horizon=horizon, delta=self.delta)
glr = (s - t0 + 1) * kl_before + (t - s) * kl_after
if glr >= threshold_h:
return True
return False
# ## `Bernoulli GLR`
#
# The same GLR algorithm but using the Bernoulli KL, given by:
#
# $$\mathrm{KL}(\mathcal{B}(x), \mathcal{B}(y)) = x \log(\frac{x}{y}) + (1-x) \log(\frac{1-x}{1-y}).$$
# In[257]:
import cython
get_ipython().run_line_magic('load_ext', 'cython')
# In[265]:
def klBern(x: float, y: float) -> float:
r""" Kullback-Leibler divergence for Bernoulli distributions. https://en.wikipedia.org/wiki/Bernoulli_distribution#Kullback.E2.80.93Leibler_divergence
.. math:: \mathrm{KL}(\mathcal{B}(x), \mathcal{B}(y)) = x \log(\frac{x}{y}) + (1-x) \log(\frac{1-x}{1-y})."""
x = min(max(x, 1e-6), 1 - 1e-6)
y = min(max(y, 1e-6), 1 - 1e-6)
return x * log(x / y) + (1 - x) * log((1 - x) / (1 - y))
# In[259]:
get_ipython().run_line_magic('timeit', 'klBern(np.random.random(), np.random.random())')
# In[260]:
get_ipython().run_cell_magic('cython', '--annotate', 'from libc.math cimport log\neps = 1e-6 #: Threshold value: everything in [0, 1] is truncated to [eps, 1 - eps]\n\ndef klBern_cython(float x, float y) -> float:\n r""" Kullback-Leibler divergence for Bernoulli distributions. https://en.wikipedia.org/wiki/Bernoulli_distribution#Kullback.E2.80.93Leibler_divergence\n\n .. math:: \\mathrm{KL}(\\mathcal{B}(x), \\mathcal{B}(y)) = x \\log(\\frac{x}{y}) + (1-x) \\log(\\frac{1-x}{1-y})."""\n x = min(max(x, 1e-6), 1 - 1e-6)\n y = min(max(y, 1e-6), 1 - 1e-6)\n return x * log(x / y) + (1 - x) * log((1 - x) / (1 - y))')
# In[261]:
get_ipython().run_line_magic('timeit', 'klBern_cython(np.random.random(), np.random.random())')
# Now the class, with this optimized kl function.
# In[262]:
class BernoulliGLR(ChangePointDetector):
def __init__(self, mult_threshold_h=1, delta=None):
super().__init__(mult_threshold_h=mult_threshold_h, delta=delta)
def __str__(self):
return r"Bernoulli-GLR($h_0={}$, $\delta={}$)".format(
f"{self.mult_threshold_h:.3g}" if self.mult_threshold_h is not None else 'auto',
f"{self.delta:.3g}" if self.delta is not None else 'auto',
)
def detect(self, all_data, t):
r""" Detect a change in the current arm, using the Generalized Likelihood Ratio test (GLR) and the :attr:`kl` function.
- For each *time step* :math:`s` between :math:`t_0=0` and :math:`t`, compute:
.. math::
G^{\mathcal{N}_1}_{t_0:s:t} = (s-t_0+1)(t-s) \mathrm{kl}(\mu_{s+1,t}, \mu_{t_0,s}) / (t-t_0+1).
- The change is detected if there is a time :math:`s` such that :math:`G^{\mathcal{N}_1}_{t_0:s:t} > h`, where :attr:`threshold_h` is the threshold of the test,
- And :math:`\mu_{a,b} = \frac{1}{b-a+1} \sum_{s=a}^{b} y_s` is the mean of the samples between :math:`a` and :math:`b`.
"""
data = all_data[:t]
t0 = 0
horizon = len(all_data)
# compute parameters
mean_all = np.mean(data[t0 : t+1])
mean_before = 0
mean_after = mean_all
for s in range(t0, t):
# DONE okay this is efficient we don't compute the same means too many times!
y = data[s]
mean_before = (s * mean_before + y) / (s + 1)
mean_after = ((t + 1 - s + t0) * mean_after - y) / (t - s + t0)
kl_before = klBern(mean_before, mean_all)
kl_after = klBern(mean_after, mean_all)
threshold_h = self.mult_threshold_h * compute_c__GLR(t0, s, t, horizon=horizon, delta=self.delta)
glr = (s - t0 + 1) * kl_before + (t - s) * kl_after
if glr >= threshold_h:
return True
return False
# ## `Sub-Gaussian GLR`
#
# A slightly different GLR algorithm for non-parametric sub-Gaussian distributions.
# We assume the distributions $\nu^1$ and $\nu^2$ to be $\sigma^2$-sub Gaussian, for a known value of $\sigma\in\mathbb{R}^+$, and if we consider a confidence level $\delta\in(0,1)$ (typically, it is set to $\frac{1}{T}$ if the horizon $T$ is known, or $\delta=\delta_t=\frac{1}{t^2}$ to have $\sum_{t=1}{T} \delta_t < +\infty$).
#
# Then we consider the following test: the non-parametric sub-Gaussian Generalized Likelihood Ratio test (GLR) works like this:
#
# - For each *time step* $s$ between $t_0=0$ and $t$, compute:
# $$G^{\text{sub-}\sigma}_{t_0:s:t} = |\mu_{t_0,s} - \mu_{s+1,t}|.$$
#
# - The change is detected if there is a time $s$ such that $G^{\text{sub-}\sigma}_{t_0:s:t} > b_{t_0}(s,t,\delta)$, where $b_{t_0}(s,t,\delta)$ is the threshold of the test,
# - And $\mu_{a,b} = \frac{1}{b-a+1} \sum_{s=a}^{b} y_s$ is the mean of the samples between $a$ and $b$.
#
# The threshold is computed as either the "joint" variant:
# $$b^{\text{joint}}_{t_0}(s,t,\delta) := \sigma \sqrt{ \left(\frac{1}{s-t_0+1} + \frac{1}{t-s}\right) \left(1 + \frac{1}{t-t_0+1}\right) 2 \log\left( \frac{2(t-t_0)\sqrt{t-t_0+2}}{\delta} \right)}.$$
# or the "disjoint" variant:
#
# $$b^{\text{disjoint}}_{t_0}(s,t,\delta) := \sqrt{2} \sigma \sqrt{
# \frac{1 + \frac{1}{s - t_0 + 1}}{s - t_0 + 1} \log\left( \frac{4 \sqrt{s - t_0 + 2}}{\delta}\right)
# } + \sqrt{
# \frac{1 + \frac{1}{t - s + 1}}{t - s + 1} \log\left( \frac{4 (t - t_0) \sqrt{t - s + 1}}{\delta}\right)
# }.$$
#
# In[129]:
# Default confidence level?
DELTA = 0.01
# By default, assume distributions are 0.25-sub Gaussian, like Bernoulli
# or any distributions with support on [0,1]
SIGMA = 0.25
# In[130]:
# Whether to use the joint or disjoint threshold function
JOINT = True
# In[159]:
from math import log, sqrt
def threshold_SubGaussianGLR_joint(t0, s, t, delta=DELTA, sigma=SIGMA):
return sigma * sqrt(
(1.0 / (s - t0 + 1) + 1.0/(t - s)) * (1.0 + 1.0/(t - t0+1))
* 2 * max(0, log(( 2 * (t - t0) * sqrt(t - t0 + 2)) / delta ))
)
# In[160]:
from math import log, sqrt
def threshold_SubGaussianGLR_disjoint(t0, s, t, delta=DELTA, sigma=SIGMA):
return np.sqrt(2) * sigma * (sqrt(
((1.0 + (1.0 / (s - t0 + 1))) / (s - t0 + 1)) * max(0, log( (4 * sqrt(s - t0 + 2)) / delta ))
) + sqrt(
((1.0 + (1.0 / (t - s + 1))) / (t - s + 1)) * max(0, log( (4 * (t - t0) * sqrt(t - s + 1)) / delta ))
))
# In[161]:
def threshold_SubGaussianGLR(t0, s, t, delta=DELTA, sigma=SIGMA, joint=JOINT):
if joint:
return threshold_SubGaussianGLR_joint(t0, s, t, delta, sigma=sigma)
else:
return threshold_SubGaussianGLR_disjoint(t0, s, t, delta, sigma=sigma)
# And now we can write the CD algorithm:
# In[162]:
class SubGaussianGLR(ChangePointDetector):
def __init__(self, delta=DELTA, sigma=SIGMA, joint=JOINT):
super().__init__(delta=delta, sigma=sigma, joint=joint)
def __str__(self):
return fr"SubGaussian-GLR($\delta=${self.delta:.3g}, $\sigma=${self.sigma:.3g}, {'joint' if self.joint else 'disjoint'})"
def detect(self, all_data, t):
r""" Detect a change in the current arm, using the non-parametric sub-Gaussian Generalized Likelihood Ratio test (GLR) works like this:
- For each *time step* :math:`s` between :math:`t_0=0` and :math:`t`, compute:
.. math:: G^{\text{sub-}\sigma}_{t_0:s:t} = |\mu_{t_0,s} - \mu_{s+1,t}|.
- The change is detected if there is a time :math:`s` such that :math:`G^{\text{sub-}\sigma}_{t_0:s:t} > b_{t_0}(s,t,\delta)`, where :math:`b_{t_0}(s,t,\delta)` is the threshold of the test,
The threshold is computed as:
.. math:: b_{t_0}(s,t,\delta) := \sigma \sqrt{ \left(\frac{1}{s-t_0+1} + \frac{1}{t-s}\right) \left(1 + \frac{1}{t-t_0+1}\right) 2 \log\left( \frac{2(t-t_0)\sqrt{t-t_0+2}}{\delta} \right)}.
- And :math:`\mu_{a,b} = \frac{1}{b-a+1} \sum_{s=a}^{b} y_s` is the mean of the samples between :math:`a` and :math:`b`.
"""
data = all_data[:t]
t0 = 0
horizon = len(all_data)
delta = self.delta
if delta is None:
delta = 1.0 / max(1, horizon)
mean_before = 0
mean_after = np.mean(data[t0 : t+1])
for s in range(t0, t):
# DONE okay this is efficient we don't compute the same means too many times!
y = data[s]
mean_before = (s * mean_before + y) / (s + 1)
mean_after = ((t + 1 - s + t0) * mean_after - y) / (t - s + t0)
# compute threshold
threshold = threshold_SubGaussianGLR(t0, s, t, delta=delta, sigma=self.sigma, joint=self.joint)
glr = abs(mean_before - mean_after)
if glr >= threshold:
# print(f"DEBUG: t0 = {t0}, t = {t}, s = {s}, horizon = {horizon}, delta = {delta}, threshold = {threshold} and mu(s+1, t) = {mu(s+1, t)}, and mu(t0, s) = {mu(t0, s)}, and and glr = {glr}.")
return True
return False
# ## List of all Python algorithms
# In[135]: