├── .gitignore
├── README.md
├── docs
├── .gitignore
├── BONE-SSM.png
├── bone-examples.png
├── bone-m1.png
├── bone-m2.png
├── bone-m3.png
├── day-ahead-dataset.png
├── day-ahead-forecast.png
├── day-ahead-results.png
├── index.css
├── index.css.map
├── index.html
├── index.jinja2
├── index.scss
├── qr-code.png
├── rlpr-tnoise-rl-log-posterior.png
└── rlpr-wolf-tnoise-rl-log-posterior.png
├── figures
└── empty
├── notebooks
├── 01-day-ahead-forecast-nnet.ipynb
├── 02-concept-drift-clf.ipynb
├── 03-drift-and-jump-clf.ipynb
├── 04-bernoulli-bandits.ipynb
├── 05-piecewise-polynomial-regression.ipynb
├── 06-t-distributed-nonstat-linear-regression.ipynb
├── 07-segmentation-as-regression.ipynb
├── 08-sde.ipynb
└── slides
│ ├── 01-sequential-regression.ipynb
│ └── moons_data.py
└── outputs
└── empty
/.gitignore:
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1 | .DS_Store
2 |
3 |
4 | # Additional ignore files
5 | *.xlsx
6 | *.mp4
7 | *.gif
8 | *.png
9 |
10 | # Byte-compiled / optimized / DLL files
11 | __pycache__/
12 | *.py[cod]
13 | *$py.class
14 |
15 | # C extensions
16 | *.so
17 |
18 | # Distribution / packaging
19 | .Python
20 | build/
21 | develop-eggs/
22 | dist/
23 | downloads/
24 | eggs/
25 | .eggs/
26 | lib/
27 | lib64/
28 | parts/
29 | sdist/
30 | var/
31 | wheels/
32 | share/python-wheels/
33 | *.egg-info/
34 | .installed.cfg
35 | *.egg
36 | MANIFEST
37 |
38 | # PyInstaller
39 | # Usually these files are written by a python script from a template
40 | # before PyInstaller builds the exe, so as to inject date/other infos into it.
41 | *.manifest
42 | *.spec
43 |
44 | # Installer logs
45 | pip-log.txt
46 | pip-delete-this-directory.txt
47 |
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50 | .tox/
51 | .nox/
52 | .coverage
53 | .coverage.*
54 | .cache
55 | nosetests.xml
56 | coverage.xml
57 | *.cover
58 | *.py,cover
59 | .hypothesis/
60 | .pytest_cache/
61 | cover/
62 |
63 | # Translations
64 | *.mo
65 | *.pot
66 |
67 | # Django stuff:
68 | *.log
69 | local_settings.py
70 | db.sqlite3
71 | db.sqlite3-journal
72 |
73 | # Flask stuff:
74 | instance/
75 | .webassets-cache
76 |
77 | # Scrapy stuff:
78 | .scrapy
79 |
80 | # Sphinx documentation
81 | docs/_build/
82 |
83 | # PyBuilder
84 | .pybuilder/
85 | target/
86 |
87 | # Jupyter Notebook
88 | .ipynb_checkpoints
89 |
90 | # IPython
91 | profile_default/
92 | ipython_config.py
93 |
94 | # pyenv
95 | # For a library or package, you might want to ignore these files since the code is
96 | # intended to run in multiple environments; otherwise, check them in:
97 | # .python-version
98 |
99 | # pipenv
100 | # According to pypa/pipenv#598, it is recommended to include Pipfile.lock in version control.
101 | # However, in case of collaboration, if having platform-specific dependencies or dependencies
102 | # having no cross-platform support, pipenv may install dependencies that don't work, or not
103 | # install all needed dependencies.
104 | #Pipfile.lock
105 |
106 | # poetry
107 | # Similar to Pipfile.lock, it is generally recommended to include poetry.lock in version control.
108 | # This is especially recommended for binary packages to ensure reproducibility, and is more
109 | # commonly ignored for libraries.
110 | # https://python-poetry.org/docs/basic-usage/#commit-your-poetrylock-file-to-version-control
111 | #poetry.lock
112 |
113 | # pdm
114 | # Similar to Pipfile.lock, it is generally recommended to include pdm.lock in version control.
115 | #pdm.lock
116 | # pdm stores project-wide configurations in .pdm.toml, but it is recommended to not include it
117 | # in version control.
118 | # https://pdm.fming.dev/latest/usage/project/#working-with-version-control
119 | .pdm.toml
120 | .pdm-python
121 | .pdm-build/
122 |
123 | # PEP 582; used by e.g. github.com/David-OConnor/pyflow and github.com/pdm-project/pdm
124 | __pypackages__/
125 |
126 | # Celery stuff
127 | celerybeat-schedule
128 | celerybeat.pid
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131 | *.sage.py
132 |
133 | # Environments
134 | .env
135 | .venv
136 | env/
137 | venv/
138 | ENV/
139 | env.bak/
140 | venv.bak/
141 |
142 | # Spyder project settings
143 | .spyderproject
144 | .spyproject
145 |
146 | # Rope project settings
147 | .ropeproject
148 |
149 | # mkdocs documentation
150 | /site
151 |
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153 | .mypy_cache/
154 | .dmypy.json
155 | dmypy.json
156 |
157 | # Pyre type checker
158 | .pyre/
159 |
160 | # pytype static type analyzer
161 | .pytype/
162 |
163 | # Cython debug symbols
164 | cython_debug/
165 |
166 | # PyCharm
167 | # JetBrains specific template is maintained in a separate JetBrains.gitignore that can
168 | # be found at https://github.com/github/gitignore/blob/main/Global/JetBrains.gitignore
169 | # and can be added to the global gitignore or merged into this file. For a more nuclear
170 | # option (not recommended) you can uncomment the following to ignore the entire idea folder.
171 | #.idea/
172 |
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/README.md:
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1 | # BONE: Bayesian online learning in non-stationary environments
2 |
3 | **Paper** 📄: https://arxiv.org/abs/2411.10153
4 |
5 | 
6 |
7 |
8 | ## Installation
9 | To run the experiments, make sure to have installed `jax>=0.4.2`,
10 | [rebayes-mini](https://github.com/gerdm/rebayes-mini/tree/main),
11 | [flax](https://github.com/google/flax),
12 | and the [BayesianOptimization](https://github.com/bayesian-optimization/BayesianOptimization) package:
13 |
14 | ```bash
15 | pip install git+https://github.com/gerdm/rebayes-mini.git
16 | pip install flax
17 | pip install bayesian-optimization
18 | ```
19 |
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1 | !*
2 |
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/docs/index.css:
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1 | @charset "UTF-8";
2 | /*
3 | Copyright © 2020 Clément Pit-Claudel
4 | https://github.com/cpitclaudel/academic-poster-template
5 |
6 | Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the “Software”), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:
7 |
8 | The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
9 |
10 | THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
11 | */
12 | a {
13 | color: inherit;
14 | }
15 |
16 | address {
17 | font-style: normal;
18 | }
19 |
20 | h1, h2, h3 {
21 | margin: 0;
22 | }
23 |
24 | pre, code, samp, .monospace {
25 | font-family: "Fira Code", monospace;
26 | }
27 |
28 | b {
29 | display: block;
30 | text-align: center;
31 | font-weight: 500;
32 | }
33 |
34 | @media all and (min-width: 80rem) {
35 | body {
36 | display: grid;
37 | grid-template-rows: auto 1fr auto;
38 | }
39 | body > header, body > footer {
40 | grid-auto-flow: column;
41 | justify-content: space-between;
42 | }
43 | body > header {
44 | grid-template-columns: auto 1fr auto;
45 | }
46 | main {
47 | align-content: flex-start;
48 | display: flex;
49 | flex-direction: column;
50 | flex-wrap: wrap;
51 | overflow: auto;
52 | padding: 0.5rem;
53 | }
54 | main::after {
55 | content: " ";
56 | display: block;
57 | flex-basis: 100%;
58 | width: 0.5rem;
59 | }
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61 | flex-grow: 1;
62 | margin: 0.5rem;
63 | max-width: 40rem;
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309 | .center {
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1 |
2 |
3 |
4 |
5 |
6 | BONE — generalised Bayesian online learning in non-stationary environments
8 |
26 |
27 |
29 |
30 |
BONE
31 |
A unifying framework for generalised Bayesian online learning in non-stationary environments
32 |
33 | G. Duran-Martina,b ,
34 | A.Y. Shestopaloffa ,
35 | L. Sanchez-Betancourtb,c ,
36 | K. Murphyd .
37 | a Queen Mary University ,
39 | b Oxford-Man Institute of Quantitative Finance ,
40 | c University of Oxford ,
41 | d Google Deepmind .
42 |
43 |
44 | TMRL ,
45 | March 13, 2025
46 |
47 |
51 |
52 |
53 |
56 |
57 |
58 | The BONE framework allows the classification and development of new online learning algorithms
59 | in non-stationary environments.
60 |
61 |
62 |
63 | Our framework encompasses methods in
64 | the filtering literature (e.g., Kalman filter),
65 | the segmentation literature (e.g., Bayesian online changepoint detection),
66 | the continual learning literature (e.g., variational continual learning),
67 | the contextual bandit literature (e.g., Bayesian neural bandits), and
68 | the forecasting literature (e.g., Bayesian ensemble models).
69 |
70 |
71 |
73 | State-space model (SSM) induced by BONE.
74 |
75 |
76 |
77 |
78 |
79 | BONE components
80 |
81 |
82 |
83 | (M.1) A model for observations (conditioned on features \({\bf x}_t\)) — \(h(\theta, {\bf x}_t)\)
84 | (M.2) An auxiliary variable to track regime changes — \(\psi_t\)
85 |
86 | (M.3) A model for prior beliefs (conditioned on \(\psi_t\) and data \(\cal D_{1:t}\)) —
87 | \(\pi(\theta; \psi_t, {\cal D}_{1:t-1})\)
88 |
89 |
90 | (A.1) An algorithm for weighting choices of \(\theta\) —
91 | \(q(\theta; \psi_t, {\cal D}_{1:t})\)
92 |
93 |
94 | (A.1) An algorithm for weighting choices of \(\psi_t\) —
95 | \(\nu(\psi_t, {\cal D}_{1:t})\)
96 |
97 |
98 |
99 | The expected posterior predictive under BONE is
100 | $$
101 | \begin{aligned}
102 | &\hat{\bf y}_t
103 | = \sum_{\psi_t \in \Psi_t} \nu(\psi_t, {\cal D}_{1:t})\,
104 | \int q(\theta; \psi_t, {\cal D}_{1:t})\,
105 | h(\theta, {\bf x}_t)\,d\theta,\\
106 | &\\
107 | &q(\theta; \psi_t, {\cal D}_{1:t})
108 | \propto \exp(-\ell_t(\theta))\,\pi(\theta; \psi_t, {\cal D}_{1:t-1}).
109 | \end{aligned}
110 | $$
111 |
112 |
113 |
114 |
115 |
116 | (M.1) Model for observations
117 |
118 |
119 |
120 | Represents the function that maps the features \({\bf x}_t\) to the observations \({\bf y}_t\).
121 |
122 |
123 | Choice is dependent on the nature of the data and the environment.
124 |
125 |
126 | Simple linear models
127 | $$
128 | \begin{aligned}
129 | &\text{(Regression)}\\
130 | &h(\theta, {\bf x}_t) = \theta^\intercal\,{\bf x}_t\\
131 | &\\
132 | &\text{(Classification)}\\
133 | &h(\theta, {\bf x}_t) = \sigma(\theta^\intercal\,{\bf x}_t)
134 | \end{aligned}
135 | $$
136 |
137 |
138 | Extensions to complex models
139 |
140 | Neural networks: For non-linear relationships and complex patterns (e.g., CIFAR-100 dataset).
141 | Time-series models: to model temporal dependencies (e.g., LSTM, ARIMA).
142 |
143 |
144 |
146 | Choice of measurement model \(h(\theta, {\bf x}_t)\) in BONE.
147 |
148 |
149 |
150 |
151 |
152 |
153 |
154 |
155 |
156 | (M.2) Choice of auxiliary variable
157 |
158 |
159 | Encodes our belief about what a "regime" is.
160 |
161 |
162 |
163 | name
164 | values
165 | cardinality
166 |
167 |
168 | C\(\{c\}\)
170 | \(1\)
171 |
172 |
173 | CPT\(2^{ \{0, 1, \ldots, t\} }\)
175 | \(2^t\)
176 |
177 |
178 | CPP\([0,1]\)
180 | \(\infty\)
181 |
182 |
183 | CPL\(\{0,1\}^t\)
185 | \(2^t\)
186 |
187 |
188 | CPV\((0,1)^t\)
190 | \(\infty\)
191 |
192 |
193 | ME\(\{1, \ldots, K\}\)
195 | \(K\)
196 |
197 |
198 | RL\(\{0, 1, \ldots, t\}\)
200 | \(t\)
201 |
202 |
203 | RLCC\(\{\{0, t\}, \ldots, \{t, 0\}\}\)
205 | \(2 + t(t+1)/2\)
206 |
207 |
208 |
209 | C is constant,
210 | CPT is changepoint timestep,
211 | CPP is changepoint probability,
212 | CPL is changepoint location,
213 | CPV is changepoint probability vector,
214 | ME is mixture of experts,
215 | RL is runlength,
216 | RLCC is runlength with changepoint count.
217 |
218 |
219 |
221 | Choice of auxiliary variable \(\psi_t\).
222 |
223 |
224 |
225 |
226 |
227 |
228 |
229 | (M.3) Model for conditional priors
230 |
231 |
232 | Modify prior beliefs about the model parameters based on past data and the auxiliary variable.
233 |
234 |
235 | In the Gaussian case, the conditional prior modifies the mean and covariance
236 | $$
237 | \begin{aligned}
238 | &\pi(\theta_t;\, \psi_t,\, {\cal D}_{1:t-1}) \\
239 | &={\cal N}\big(\theta_t\vert g_{t}(\psi_t, {\cal D}_{1:t-1}), G_{t}(\psi_t, {\cal D}_{1:t-1})\big).
240 | \end{aligned}
241 | $$
242 |
243 | Examples
244 |
245 | C-static: constant auxvar with static updates.
246 | $$
247 | \begin{aligned}
248 | &g_{t}(\psi_t, {\cal D}_{1:t-1}) = \mu_{t-1},\\
249 | &G_{t}(\psi_t, {\cal D}_{1:t-1}) = \Sigma_{t-1}.
250 | \end{aligned}
251 | $$
252 |
253 |
255 | RL-PR: runlength with prior reset
256 | $$
257 | \begin{aligned}
258 | &g_{t}(\psi_t, {\cal D}_{1:t-1}) = \mu_{0}{\mathbb 1}(\psi_t = 0) + \mu_{(\psi_{t-1})}{\mathbb 1}(\psi_t > 0),\\
259 | &G_{t}(\psi_t, {\cal D}_{1:t-1}) = \Sigma_{0}{\mathbb 1}(\psi_t = 0) + \Sigma_{(\psi_{t-1})}{\mathbb 1}(\psi_t > 0).
260 | \end{aligned}
261 | $$
262 | The subscript \((\psi_t)\) denotes the posterior mean and covariance using the last \(\psi_t > 0\) observations.
263 |
264 |
266 | RL-OUPR: Runlength with OU prior discount or prior reset
267 | $$
268 | g_{t}(\psi_t, {\cal D}_{1:t-1}) =
269 | \begin{cases}
270 | \mu_0\,(1 - \nu_t(\psi_t)) + \mu_{(\psi_{t-1})}\,\nu_t(\psi_t) & \text{if } \psi_t > 0,\\
271 | \mu_0 & \text{if } \psi_t = 0.
272 | \end{cases}
273 | $$
274 |
275 |
276 |
278 | Choice of auxiliary variable \(\psi_t\).
279 |
280 |
281 |
282 |
283 |
284 |
285 |
286 | (A.1 / A.2) Algorithm for posterior update for parameters \(\theta\) and auxiliary variables \(\psi_t\)
287 |
288 |
289 | $$
290 | \begin{aligned}
291 | q(\theta; \psi_t, {\cal D}_{1:t}) & \text{ posterior update}\\
292 | \nu(\psi_t, {\cal D}_{1:t}) & \text{ weight update}
293 | \end{aligned}
294 | $$
295 |
296 | Variational Bayes (VB)
297 | Conjugate updates
298 | Particle filters
299 | Linearised updates
300 | Replay-buffer stochastic gradient descent
301 |
302 |
303 |
304 |
305 |
306 |
307 |
308 | Examples in the literature
309 |
310 |
311 |
313 | Examples of BONE in the literature.
314 |
315 |
316 |
317 |
318 |
319 |
320 |
321 | Experiment: Forecasting under distribution shift
322 |
323 |
324 | We evaluate members of the BONE framework on electricity load forecasting during the Covid-19 pandemic.
325 | Our choice of measurement model h (M.1) is a two-hidden layer multilayered perceptron (MLP) with four units per
326 | layer and a ReLU activation function.
327 | Features are lagged by one hour.
328 |
329 |
330 |
332 | Forecasting under distribution shift:
333 | changes in the output \({\bf y}_t\) (electricity load) are not
334 | reflected in the input \({\bf x}_t\) (weather features).
335 |
336 |
337 |
338 |
339 |
340 |
343 |
344 |
347 | Top: Forecasting results.
348 | Bottom: 5-day mean absolute error (MAE).
349 |
350 |
351 |
352 |
353 |
354 |
355 |
356 |
357 | github.com/gerdm/BONE
358 |
359 |
360 |
361 |
362 |
364 |
365 |
366 |
367 |
368 |
369 |
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/docs/index.jinja2:
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1 | {% extends "base/base.jinja2" %}
2 |
3 |
4 | {% set poster_title = "WoLF" %}
5 | {% set venue = "Unpublished" %}
6 | {% set poster_subtitle = "Outlier-robust Kalman Filtering through generalises Bayes" %}
7 | {% set webpage_title = poster_title + " — " + poster_subtitle %}
8 | {% set project_url = "https://github.com/gerdm/weighted-likelihood-filter" %}
9 |
10 | {% set pub_datetime_iso = "2020-09-08" %}
11 | {% set pub_date = "September 8, 2020" %}
12 |
13 |
14 | {% block authors %}
15 | {# Put authors here, with one link per author. #}
16 | G. Duran-Martina,c ,
17 | M. Altamiranob ,
18 | A.Y. Shestopaloffa ,
19 | L. Sanchez-Betancourtc,d ,
20 | J. Knoblauchb ,
21 | M. Jonese ,
22 | F.X. Briolb ,
23 | K. Murphyf .
24 | {% endblock %}
25 |
26 | {% block affiliations %}
27 | a Queen Mary University ,
28 | b University College London ,
29 | c Oxford-Man Institute of Quantitative Finance ,
30 | d University of Oxford ,
31 | e University of Colorado Boulder ,
32 | f Google Deepmind .
33 | {% endblock %}
34 |
35 |
36 | {% block contents %}
37 |
38 |
39 |
40 |
41 |
42 | The weighted-observation likelihood filter (WoLF) is a provably robust variant of the Kalman filter (KF)
43 |
44 | in the presence of outliers and misspecified measurement models.
45 | We show its capabilities in
46 | tracking problems (via the KF),
47 | online learning of neural neworks (via the extended KF),
48 | and data assimilation (via the ensemble KF).
49 |
50 |
51 |
52 |
53 |
55 | The dotted blue line shows the KF posterior mean estimate and
56 | the solid orange line shows the WoLF posterior mean estimate.
57 |
58 |
59 |
60 |
61 |
62 |
63 |
64 | Problem statement (linear setting) Consider the state-space model (SSM)
66 |
67 | $$
68 | \def\R{\mathbb{R}}
69 | \def\bm#1{\boldsymbol #1}
70 | \begin{aligned}
71 | {\bm\theta_t} &= {\bf F}_t\bm\theta_{t-1} + \bm\phi_t,\\
72 | {\bm y}_t &= {\bf H}_t\bm\theta_t + \bm\varphi_t,
73 | \end{aligned}
74 | $$
75 |
76 |
77 | with \(\bm\theta_t\in\R^p\) the (latent) state vector, \({\bm y}_t\in\R^d\) the (observed) measurement vector, \({\bf F}_t\in\R^{p\times p}\) the state transition matrix, \({\bf H}_t\in\R^{d\times p}\), \(\bm\phi_t\) a zero-mean Gaussian-distributed random vector with known covariance matrix \({\bf Q}_t\), and \(\bm\varphi_t\) any zero-mean random vector representing the measurement noise.
78 |
79 |
80 |
81 | We determine either \(\mathbb{E}[\bm\theta_t \vert \bm y_{1:t}]\) or \(\mathbb{E}[\bm y_{t+1} \vert \bm y_{1:t}]\) recursively
82 | by applying the predict and modified update equations.
83 |
84 |
85 | $$
86 | \begin{aligned}
87 | q(\bm\theta_t \vert \bm y_{1:t-1})
88 | &= \int p(\bm\theta_t \vert \bm\theta_{t-1})
89 | q(\bm\theta_{t-1} \vert \bm y_{1:t-1}) d\bm\theta_{t-1}\\
90 | q(\bm\theta_t \vert \bm y_{1:t})
91 | &\propto
92 | \exp(-\ell_t(\bm\theta_t))
93 | q(\bm\theta_t \vert \bm y_{1:t-1})
94 | \end{aligned}
95 | $$
96 |
97 |
98 | with \(\ell_t(\bm\theta) = -W\left(\bm y_{t}, \hat{\bm y}_t\right)\,\log p(\bm y_t \vert \bm\theta_t)\), and
99 | \(W: \mathbb{R}^{d}\times\mathbb{R}^d \to \mathbb{R}\) the weighting function.
100 |
101 |
102 |
103 |
104 | Weighted likelihood filter (WoLF)
106 | If the SSM follows linear dynamics, the predict and update equations are
107 |
108 |
109 | Require// predict step
113 |
114 | Require// update step
121 |
122 |
123 |
124 |
125 |
126 | Two choices of weighting functions:
127 |
128 | The inverse multi-quadratic (IMQ) —
129 | a compensation-based weighting function, and
130 | The thresholded Mahalanobis distance (TMD) —
131 | a detect-and-reject weighting function.
132 |
133 |
134 |
135 | Inverse multi-quadratic weighting function (IMQ)
136 | $$
137 | W\left(\bm y_{t}, \hat{\bm y}_t\right) =\left(1+\frac{||\bm y_{t}-\hat{\bm y}_{t}||_2^2}{c^{2}}\right)^{-1/2}
138 | $$
139 | with \(c > 0\).
140 |
141 |
142 |
143 | Thresholded Mahalanobis-based weighting function (TMD)
144 | $$
145 | W\left(\bm y_{t}, \hat{\bm y}_t\right) =
146 | \begin{cases}
147 | 1 & \text{if } \|{\bf R}_t^{-1/2}(\bm y_t - \hat{\bm y}_t)\|_2^2 \leq c\\
148 | 0 & \text{otherwise}
149 | \end{cases}
150 | $$
151 | with \(c > 0\).
152 |
153 |
154 |
155 |
156 |
157 |
158 |
159 |
160 | Definition
169 |
170 |
171 | Definition: Outlier-robust filter
177 |
178 |
179 | Theorem
183 |
184 |
185 | Remarks
186 | The Kalman filter is not outlier-robust.
187 | Filters with IMQ and TMD weighting function are outlier-robust.
188 |
189 |
190 |
191 |
193 | PIF for the 2d-tracking problem.
194 | The last measurement \(\bm y_t\) is replaced by \(\bm y_t^c = \bm y_t + \bm\epsilon\),
195 | where \(\bm\epsilon \in [-5, 5]^2\).
196 |
197 |
198 |
199 |
200 |
201 |
202 |
203 |
204 | Compared to variational-Bayes (VB) methods, which require multiple iterations to converge,
205 | WoLF has an equivalent computational cost to the Kalman filter.
206 |
207 |
208 |
209 |
210 |
211 | Method
212 | Time
213 | #HP
214 | Ref
215 |
216 |
217 |
218 |
219 | KF
220 | \(O(p^3)\)
221 | 0
222 | Kalman1960
223 |
224 |
225 | KF-B
226 | \(O(I\,p^3)\)
227 | 3
228 | Wang2018
229 |
230 |
231 | KF-IW
232 | \(O(I\,p^3)\)
233 | 2
234 | Agamennoni2012
235 |
236 |
237 | OGD
238 | \(O(I\, p^2)\)
239 | 2
240 | Bencomo2023
241 |
242 |
243 | WoLF-IMQ
244 | \(O(p^3)\)
245 | 1
246 | (Ours)
247 |
248 |
249 | WoLF-TMD
250 | \(O(p^3)\)
251 | 1
252 | (Ours)
253 |
254 |
255 |
256 | Below, \(I\) is the number of inner iterations,
257 | \(p\) is the dimension of the state vector,
258 | and #HP is the number of hyperparameters.
259 |
260 |
261 |
262 |
263 |
264 | Experiment: Kalman filter (KF) 2d tracking
266 |
267 | Linear SSM with \({\bf Q}_t = q\) \({\bf I}_4\), \({\bf R}_t = r\,{\bf I}_2\),
268 |
269 |
270 | $$
271 | \begin{aligned}
272 | {\bf F}_t &=
273 | \begin{pmatrix}
274 | 1 & 0 & \Delta & 0\\
275 | 0 & 1 & 0 & \Delta \\
276 | 0 & 0 & 1 & 0 \\
277 | 0 & 0 & 0 & 1
278 | \end{pmatrix}, &
279 | {\bf H}_t &= \begin{pmatrix}
280 | 1 & 0 & 0 & 0\\
281 | 0 & 1 & 0 & 0
282 | \end{pmatrix},
283 | \end{aligned}
284 | $$
285 |
286 |
287 | \(\Delta = 0.1\) is the sampling rate, \(q = 0.10\) is the system noise, \(r = 10\) is the measurement noise, and \({\bf I}_K\) is a \(K\times K\) identity matrix.
288 |
289 |
290 | We measure the RMSE of the posterior mean estimate of the state components.
291 |
292 |
293 | Student variant
294 |
295 | Measurements are sampled according to
296 |
297 | $$
298 | \begin{aligned}
299 | p(\bm y_t \vert \bm\theta_t)
300 | &= \text{St}({\bm y_t\,\vert\,{\bf H}_t\bm\theta_t,\,{\bf R}_t,\nu_t})\\
301 | &= \int \mathcal{N}\left(\bm y_t, {\bf H}_t\bm\theta_t, {\bf R}_t/\tau\right)
302 | \text{Gam}(\tau \vert \nu_t / 2, \nu_t / 2) d\tau.
303 | \end{aligned}
304 | $$
305 |
306 | Mixture variant
307 |
308 | Measurements are sampled according to
309 |
310 | $$
311 | \begin{aligned}
312 | p(\bm y_t \vert \bm\theta_t)
313 | &= {\cal N}\left(\bm y_t \vert {\bf m}_t, {\bf R}_t\right),\\
314 | {\bf m}_t &=
315 | \begin{cases}
316 | {\bf H}_t\,\bm\theta_t & \text{w.p.}\ 1 - p_\epsilon,\\
317 | 2\,{\bf H}_t\,\bm\theta_t & \text{w.p.}\ p_\epsilon,
318 | \end{cases}
319 | \end{aligned}
320 | $$
321 |
322 | with \(\epsilon = 0.05\).
323 |
324 |
325 |
328 |
329 |
332 | Sample runs
333 |
334 |
335 |
336 |
337 |
338 |
339 |
340 |
Mean slowdown rate over KF
343 |
344 |
345 |
346 | Method
347 | Student
348 | Mixture
349 |
350 |
351 |
352 |
353 | KF-B
354 | 2.0x
355 | 3.7x
356 |
357 |
358 | KF-IW
359 | 1.2x
360 | 5.3x
361 |
362 |
363 | WoLF-IMQ (ours)
364 | 1.0x
365 | 1.0x
366 |
367 |
368 | WoLF-TMD (ours)
369 | 1.0x
370 | 1.0x
371 |
372 |
373 |
374 |
375 |
376 |
377 |
378 |
379 | Experiment: extended Kalman filter (EKF) Online learning of UCI datasets
381 |
382 | Online training of neural networks on a corrupted version of the tabular UCI datasets.
383 | We consider a multilayered perceptron (MLP) with twenty hidden units, two hidden layers, and a real-valued output unit.
384 | We evaluate the median squared error (RMedSE) between the true and predicted output.
385 |
386 | The SSM is
387 | $$
388 | \begin{aligned}
389 | {\bm\theta_t} &= \bm\theta_{t-1} + \bm\phi_t\\
390 | {\bm y}_t &= h_t(\bm\theta_t) + \bm\varphi_t,
391 | \end{aligned}
392 | $$
393 |
398 |
399 | We estimate \( \mathbb{E}[\bm\theta_t \vert \bm y_{1:t}] \) via the extended Kalman filter (EKF) — one measurement, one parameter update.
400 | We modify the measurement mean with
401 |
402 | $$
403 | \mathbb{E}[\bm y_t \vert \bm\theta_t]
404 | \approx {\bf H}_t(\bm\theta_t - \bm\mu_{t | t-1}) + h_t(\bm\mu_{t | t-1}) =: \bar{\bm y}_t.
405 | $$
406 |
407 |
408 |
409 |
410 |
411 |
412 |
413 |
415 | RMedSE versus time per step, relative to online gradient descent (OGD), acrross corrupted UCI datasets.
416 |
417 |
418 |
419 |
420 |
421 |
422 | Experiment: ensemble Kalman filter (EnKF) Data assimilation
424 | The SSM is
425 | $$
426 | \begin{aligned}
427 | \dot{\bm\theta}_{s,i} &= \Big(\bm\theta_{s, i+1} - \bm\theta_{s, i-2}\Big) \bm\theta_{s, i-1} - \bm\theta_{s,i} + \bm\phi_{s,i},\\
428 | {\bm y}_{s,i} &=
429 | \begin{cases}
430 | \bm\theta_{s,i} + \bm\varphi_{s,i} & \text{w.p. } 1 - p_\epsilon,\\
431 | 100 & \text{w.p. } p_\epsilon.
432 | \end{cases}
433 | \end{aligned}
434 | $$
435 |
436 | Here, \(\bm\theta_{s,k}\) is the value of the state component \(k\) at step \(s\),
437 | \(\bm\phi_{s,i} \sim {\cal N}(8, 1)\), \(\bm\varphi_{s,i} \sim {\cal N}(0, 1)\), \(p_\epsilon = 0.001\),
438 | \(i = 1, \ldots, d\), \(s = 1, \ldots, S\), with \(S \gg 1\) the number of steps, and
439 | \(\bm\theta_{s, d + k} = \bm\theta_{s, k}\), \(\bm\theta_{s, -k} = \bm\theta_{s, d - k}\).
440 |
441 | We consider the metric
442 | \(L_t = \sqrt{\frac{1}{d}(\bm\theta_t - \bm\mu_t)^\intercal (\bm\theta_t - \bm\mu_t)}\).
443 |
444 |
445 |
446 |
447 |
448 |
449 |
450 |
452 |
453 |
454 |
455 |
457 | Bootstrap estimate of \(L_T\) over 20 runs and 500 samples
458 | as a function of the \(c\) hyperparameter.
459 |
460 |
461 |
462 |
463 |
464 |
465 | {% endblock %}
466 |
467 |
468 | {% block footer_left %}
469 |
470 | github.com/gerdm/weighted-likelihood-filter
471 |
472 | {% endblock %}
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/docs/index.scss:
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1 | @use "base/base.scss" as *;
2 |
3 | //////////////////
4 | // Tango colors //
5 | //////////////////
6 |
7 | $tango-blue-l: #729fcf;
8 | $tango-blue-m: #3465a4;
9 | $tango-blue-d: #204a87;
10 |
11 | $tango-aluminum-1: #eeeeec;
12 | $tango-aluminum-2: #d3d7cf;
13 | $tango-aluminum-3: #babdb6;
14 | $tango-aluminum-4: #888a85;
15 | $tango-aluminum-5: #555753;
16 | $tango-aluminum-6: #2e3436;
17 |
18 | @include base(
19 | $color-title: $tango-aluminum-1,
20 | $color-accent: $tango-aluminum-3,
21 | $color-bg-light: white,
22 | $color-bg-dark: $tango-aluminum-5,
23 | $color-header-bg: $tango-aluminum-6,
24 | $color-header-text: white
25 | );
26 |
27 |
28 |
29 | body{
30 | font-family: 'DM Sans Condensed', 'DM Sans', sans-serif;
31 | }
32 |
33 | article{
34 | > header{
35 | > h3{
36 | font-weight: 800;
37 | }
38 | }
39 | }
40 |
41 | html { font-size: 1rem }
42 |
43 |
44 | .row {
45 | display: flex;
46 | }
47 |
48 | /* Create three equal columns that sits next to each other */
49 | .column {
50 | flex: 50%;
51 | padding: 10px;
52 | }
53 |
54 | $grid-gap: 2.0rem;
55 |
56 | @media all and (min-width: 60rem) {
57 | main > * {
58 | flex-grow: 1;
59 | margin: calc($grid-gap / 2);
60 | max-width: 40rem;
61 | min-width: 25rem;
62 | width: calc(100% / 4 - $grid-gap);
63 | }
64 | }
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/docs/qr-code.png:
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/docs/rlpr-tnoise-rl-log-posterior.png:
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/docs/rlpr-wolf-tnoise-rl-log-posterior.png:
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/notebooks/slides/moons_data.py:
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1 | """
2 | Prepcocessing and data augmentation for the datasets.
3 | """
4 | import re
5 | import io
6 | import os
7 | import jax
8 | import chex
9 | import zipfile
10 | import requests
11 | import numpy as np
12 | import pandas as pd
13 | import jax.numpy as jnp
14 | import jax.random as jr
15 | from jax import vmap
16 |
17 | from sklearn.datasets import make_moons
18 |
19 |
20 | def make_showdown_moons(n_train, n_test, n_train_warmup, n_test_warmup, noise, seed=314):
21 | np.random.seed(seed)
22 | train = make_moons(n_samples=n_train, noise=noise)
23 | test = make_moons(n_samples=n_test, noise=noise)
24 | warmup_train = make_moons(n_samples=n_train_warmup, noise=noise)
25 | warmup_test = make_moons(n_samples=n_test_warmup, noise=noise)
26 |
27 | train = jax.tree_map(jnp.array, train)
28 | test = jax.tree_map(jnp.array, test)
29 | warmup_train = jax.tree_map(jnp.array, warmup_train)
30 | warmup_test = jax.tree_map(jnp.array, warmup_test)
31 |
32 | return train, test, warmup_train, warmup_test
33 |
34 | def _rotation_matrix(angle):
35 | """
36 | Create a rotation matrix that rotates the
37 | space 'angle'-radians.
38 | """
39 | R = np.array([
40 | [np.cos(angle), -np.sin(angle)],
41 | [np.sin(angle), np.cos(angle)]
42 | ])
43 | return R
44 |
45 |
46 | def _make_rotating_moons(radians, n_samples=100, **kwargs):
47 | """
48 | Make two interleaving half circles rotated by 'radians' radians
49 |
50 | Parameters
51 | ----------
52 | radians: float
53 | Angle of rotation
54 | n_samples: int
55 | Number of samples
56 | **kwargs:
57 | Extra arguments passed to the `make_moons` function
58 | """
59 | X, y = make_moons(n_samples=n_samples, **kwargs)
60 | X = jnp.einsum("nm,mk->nk", X, _rotation_matrix(radians))
61 | return X, y
62 |
63 |
64 | def make_rotating_moons(n_train, n_test, n_rotations, min_angle=0, max_angle=360, seed=314, **kwargs):
65 | """
66 | n_train: int
67 | Number of training samples per rotation
68 | n_test: int
69 | Number of test samples per rotation
70 | n_rotations: int
71 | Number of rotations
72 | """
73 | np.random.seed(seed)
74 | n_samples = n_train + n_test
75 | min_rad = np.deg2rad(min_angle)
76 | max_rad = np.deg2rad(max_angle)
77 |
78 | radians = np.linspace(min_rad, max_rad, n_rotations)
79 | X_train_all, y_train_all, rads_train_all = [], [], []
80 | X_test_all, y_test_all, rads_test_all = [], [], []
81 | for rad in radians:
82 | X, y = _make_rotating_moons(rad, n_samples=n_samples, **kwargs)
83 | rads = jnp.ones(n_samples) * rad
84 |
85 | X_train = X[:n_train]
86 | y_train = y[:n_train]
87 | rad_train = rads[:n_train]
88 |
89 | X_test = X[n_train:]
90 | y_test = y[n_train:]
91 | rad_test = rads[n_train:]
92 |
93 | X_train_all.append(X_train)
94 | y_train_all.append(y_train)
95 | rads_train_all.append(rad_train)
96 |
97 | X_test_all.append(X_test)
98 | y_test_all.append(y_test)
99 | rads_test_all.append(rad_test)
100 |
101 | X_train_all = jnp.concatenate(X_train_all, axis=0)
102 | y_train_all = jnp.concatenate(y_train_all, axis=0)
103 | rads_train_all = jnp.concatenate(rads_train_all, axis=0)
104 | X_test_all = jnp.concatenate(X_test_all, axis=0)
105 | y_test_all = jnp.concatenate(y_test_all, axis=0)
106 | rads_test_all = jnp.concatenate(rads_test_all, axis=0)
107 |
108 | train = (X_train_all, y_train_all, rads_train_all)
109 | test = (X_test_all, y_test_all, rads_test_all)
110 |
111 | return train, test
112 |
113 |
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