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\right]^{\rm{T}}`. # The kernel weight vector :math:`\mathbf{w}` is the solution to a set of linear equations in the # RKHS. Unlike the particle filter, there are no non-negativity or normalisation constraints. +# +# .. image:: ../../_static/KME.png +# :width: 800 +# :alt: Illustration of Kernel Mean Embedding from data space to kernel feature space +# +# This figure represents the KME of the conditional distribution :math:`p(X|\mathbf{Y})` is +# embedded as a point in kernel feature space as +# :math:`\mu_{X|y} = \int_{\mathcal{X}}\phi_x(x) d P(x|y)`. +# Given the training data sampled from :math:`P(X, Y)`, the empirical KME of :math:`P(X|y)` is +# approximated as a linear operation in RKHS, i.e., +# :math:`\hat{\mu}_{X|y}=\hat{C}_{X|Y}\phi_y(y)=\Phi\mathbf{w}`. # %% # AKKF based on KME # ^^^^^^^^^^^^^^^^^ +# # The AKKF draws inspiration from the concepts of KME and Kalman Filter to address tracking # problems in nonlinear systems. Unlike the particle filter, the AKKF approximates the posterior # PDF using particles and weights, but not in the state space. 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z$_i%8;&{f$zTaE={QSQS(O-l0Fa-1oz8ye6`b#V|J-i`dKA8#dQw@c#h!sp>!g literal 0 HcmV?d00001 diff --git a/docs/tutorials/filters/AKKF.py b/docs/tutorials/filters/AKKF.py index 0f42a072f..31463ed0a 100644 --- a/docs/tutorials/filters/AKKF.py +++ b/docs/tutorials/filters/AKKF.py @@ -187,6 +187,10 @@ # mean vector and covariance matrix at time :math:`k-1`, respectively. # The transition matrix :math:`\Gamma_{k}` represents the change of sample representation, and # :math:`{V}_{k}` represents the finite matrix representation of the transition residual matrix. +# +# .. image:: ../../_static/AKKf_flow_diagram.png +# :width: 800 +# :alt: Flow diagram of the AKKF # %% # Nearly-constant velocity & Bearing-only Tracking (BOT) example From ac3c4035818ae2c2ee36bbc742e6531a58b394bb Mon Sep 17 00:00:00 2001 From: James Wright Date: Wed, 17 Jul 2024 14:38:33 +0100 Subject: [PATCH 3/4] Add missing description sections to AKKF tutorial --- docs/tutorials/filters/AKKF.py | 40 ++++++++++++++++++++++++++++++++++ 1 file changed, 40 insertions(+) diff --git a/docs/tutorials/filters/AKKF.py b/docs/tutorials/filters/AKKF.py index 31463ed0a..64bd45aea 100644 --- a/docs/tutorials/filters/AKKF.py +++ b/docs/tutorials/filters/AKKF.py @@ -188,9 +188,49 @@ # The transition matrix :math:`\Gamma_{k}` represents the change of sample representation, and # :math:`{V}_{k}` represents the finite matrix representation of the transition residual matrix. # +# Updater uses prediction +# """"""""""""""""""""""" +# +# :math:`{\color{green}\blacksquare}` In the measurement space, the measurement particles are +# generated according to the measurement model as: +# +# .. math:: +# \mathbf{y}_k^{\{i\}} = \mathtt{h}\left({\mathbf{x}}_{k}^{\{i\}}, +# \mathbf{v}_{k}^{\{i\}} \right). +# +# :math:`{\color{orange}\blacksquare}` In the RKHS, :math:`{\mathbf{y}}_{k}^{\{i=1:M_{\text{A}}\}}` +# are mapped as feature mappings :math:`\Upsilon_k`. The posterior kernel weight vector and +# covariance matrix are updated as: +# +# .. math:: +# \mathbf{w}^{+}_{k} &= \mathbf{w}^{-}_{k} + +# Q_k\left(\mathbf{g}_{\mathbf{yy}_k} - G_\mathbf{yy}\mathbf{w}^{-}_{k}\right)\\ +# S_k^{+} &= S_k^{-} - Q_k G_\mathbf{yy}S_k^{-} +# +# +# Here, :math:`G_\mathbf{yy} = \Upsilon^\mathrm{T}_k\Upsilon_k`, +# :math:`\mathbf{g}_{\mathbf{yy}_k}=\Upsilon_k^\mathrm{T}\phi_\mathbf{y}(\mathbf{y}_k)` +# represents the kernel vector of the measurement at time :math:`k`, :math:`Q_k` is the kernel +# Kalman gain operator. +# +# Proposal generated in updater +# """"""""""""""""""""""""""""" +# +# :math:`{\color{blue}\blacksquare}` The AKKF replaces :math:`X_{k} = \mathbf{x}_k^{\{i=1:M\}}` +# with new weighted proposal particles :math:`\tilde{X}_{k} = \tilde{\mathbf{x}}_k^{\{i=1:M\}}`, +# where +# +# .. math:: +# \tilde{\mathbf{x}}_k^{\{i=1:M_{\text{A}}\}} &\sim +# \mathcal{N}\left(\mathbb{E}\left(X_{k}\right), \mathrm{Cov}\left(X_{k}\right)\right)\\ +# \mathbb{E}\left(X_{k}\right) &= X_{k}\mathbf{w}^{+}_{k}\\ +# \mathrm{Cov}\left(X_{k}\right) &= X_{k}S^{+}_{k} X_{k}^{\rm{T}}. +# # .. image:: ../../_static/AKKf_flow_diagram.png # :width: 800 # :alt: Flow diagram of the AKKF +# +# Figure 2: Flow diagram of the AKKF # %% # Nearly-constant velocity & Bearing-only Tracking (BOT) example From be495802a3f9e6b987f5b55f39ee79a8bb278bc3 Mon Sep 17 00:00:00 2001 From: James Wright Date: Wed, 17 Jul 2024 14:41:43 +0100 Subject: [PATCH 4/4] Update figure referencing --- docs/tutorials/filters/AKKF.py | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/docs/tutorials/filters/AKKF.py b/docs/tutorials/filters/AKKF.py index 64bd45aea..404554f85 100644 --- a/docs/tutorials/filters/AKKF.py +++ b/docs/tutorials/filters/AKKF.py @@ -80,7 +80,7 @@ # :math:`\Upsilon := \left[\phi_{\mathbf{y}}(\mathbf{y}^{\{1\}}),\dots, # \phi_{\mathbf{y}}(\mathbf{y}^{\{M\}})\right]`. # The estimate of the conditional embedding operator :math:`\hat{\mathcal{C}}_{X|Y}` is obtained -# as a linear regression in the RKHS, as illustrated in Fig. 1. +# as a linear regression in the RKHS, as illustrated in figure 1. # Then, the empirical KME of the conditional distribution, i.e., # :math:`p(\mathbf{x}\mid\mathbf{y})\xrightarrow{\text{KME}} \hat{\mu}_{X|\mathbf{y}}`, # is calculated by a linear algebra operation as: @@ -113,8 +113,8 @@ # :width: 800 # :alt: Illustration of Kernel Mean Embedding from data space to kernel feature space # -# This figure represents the KME of the conditional distribution :math:`p(X|\mathbf{Y})` is -# embedded as a point in kernel feature space as +# Figure 1: This figure represents the KME of the conditional distribution :math:`p(X|\mathbf{Y})` +# is embedded as a point in kernel feature space as # :math:`\mu_{X|y} = \int_{\mathcal{X}}\phi_x(x) d P(x|y)`. # Given the training data sampled from :math:`P(X, Y)`, the empirical KME of :math:`P(X|y)` is # approximated as a linear operation in RKHS, i.e., @@ -157,7 +157,7 @@ # Implement the AKKF # ^^^^^^^^^^^^^^^^^^ # -# The AKKF consists of three modules, as depicted in Fig. 2: a predictor that utilises both prior +# The AKKF consists of three modules, as depicted in figure 2: a predictor that utilises both prior # and proposal information at time :math:`k-1` to update the prior state particles and predict the # kernel weight mean and covariance at time :math:`k`, an updater that employs the predicted # values to update the kernel weight and covariance, and an updater that generates the proposal @@ -226,7 +226,7 @@ # \mathbb{E}\left(X_{k}\right) &= X_{k}\mathbf{w}^{+}_{k}\\ # \mathrm{Cov}\left(X_{k}\right) &= X_{k}S^{+}_{k} X_{k}^{\rm{T}}. # -# .. image:: ../../_static/AKKf_flow_diagram.png +# .. image:: ../../_static/AKKF_flow_diagram.png # :width: 800 # :alt: Flow diagram of the AKKF #