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@@ -306,53 +306,6 @@
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"So $\\alpha_{b} \\gt 0$ for all cases when the correction is necessary i.e. the existence of $H_{\\infty}$ filter is proven.\n"
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]
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},
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- {
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- "cell_type": "markdown",
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- "metadata": {},
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- "source": [
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- "# Application to Sigma Point Kalman Filters\n",
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- "\n",
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- "In Sigma Point Kalman Filters (SPKF, see [**[Merwe2004]**](#merwe)) Weighted Statistical Linear Regression technique is used to approximate nonlinear process and measurement functions:\n",
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- "\n",
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- "$\\mathbf{y} = g(\\mathbf{x}) = \\mathbf{A} \\mathbf{x} + \\mathbf{b} + \\mathbf{e}$,\n",
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- "\n",
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- "$\\mathbf{P}_{ee} = \\mathbf{P}_{yy} - \\mathbf{A} \\mathbf{P}_{xx} \\mathbf{A}^{\\top}$\n",
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- "\n",
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- "where: \n",
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- "\n",
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- "$\\mathbf{e}$ is an approximation error, \n",
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- "\n",
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- "$\\mathbf{A} = \\mathbf{P}_{xy}^{\\top} \\mathbf{P}_{xx}^{-1}$, \n",
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- "\n",
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- "$\\mathbf{b} = \\mathbf{\\bar{y}} - \\mathbf{A} \\mathbf{\\bar{x}}$,\n",
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- "\n",
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- "$\\mathbf{P}_{xx} = \\displaystyle\\sum_{i} {w}_{ci} \\left( \\mathbf{\\chi}_{i} - \\mathbf{\\bar{x}} \\right) \\left( \\mathbf{\\chi}_{i} - \\mathbf{\\bar{x}} \\right)$,\n",
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- "\n",
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- "$\\mathbf{P}_{yy} = \\displaystyle\\sum_{i} {w}_{ci} \\left( \\mathbf{\\gamma}_{i} - \\mathbf{\\bar{y}} \\right) \\left( \\mathbf{\\gamma}_{i} - \\mathbf{\\bar{y}} \\right)$,\n",
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- "\n",
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- "$\\mathbf{P}_{xy} = \\displaystyle\\sum_{i} {w}_{ci} \\left( \\mathbf{\\chi}_{i} - \\mathbf{\\bar{x}} \\right) \\left( \\mathbf{\\gamma}_{i} - \\mathbf{\\bar{y}} \\right)$,\n",
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- "\n",
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- "$\\mathbf{\\gamma}_{i} = g(\\mathbf{\\chi}_{i})$\n",
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- "\n",
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- "$\\mathbf{\\bar{x}} = \\displaystyle\\sum_{i} {w}_{mi} \\mathbf{\\chi}_{i}$,\n",
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- "\n",
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- "$\\mathbf{\\bar{y}} = \\displaystyle\\sum_{i} {w}_{mi} \\mathbf{\\gamma}_{i}$,\n",
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- "\n",
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- "${w}_{ci}$ are covariation weghts, ${w}_{mi}$ are mean weights.\n",
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- "\n",
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- "This means that approximation errors of measurement may be treated as a part of additive noise and so we show that in SPKF we can use the following approximation of $\\mathbf{S}_{k}$:\n",
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- "\n",
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- "$\\mathbf{S}_{k} = \\mathbf{H}_{k} \\mathbf{P}_{k|k-1} \\mathbf{H}_{k}^{\\top} + \\mathbf{\\tilde{R}}_{k}$, \n",
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- "\n",
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- "where \n",
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- "\n",
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- "$\\mathbf{H}_{k} = \\mathbf{P}_{xz, k}^{\\top} \\mathbf{P}_{xx, k}^{-1}$,\n",
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- "\n",
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- "$\\mathbf{\\tilde{R}}_{k} = \\mathbf{R}_{k} + \\mathbf{P}_{ee, k}$,\n",
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- "\n",
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- "and so we can use the above adaptive correction in case of SPKF divergence."
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- ]
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- },
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{
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"cell_type": "markdown",
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"metadata": {},
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@@ -361,9 +314,7 @@
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"\n",
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"<a name=\"banavar\"></a>**\\[Banavar1992\\]** R. Banavar, “A game theoretic approach to linear dynamic estimation”, Doctoral Dissertation, University of Texas at Austin, May 1992.\n",
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"\n",
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- "<a name=\"horn_et_al\"></a>**\\[Horn et al\\]** R.A. Horn, C.R Johnson, Charles R. “Matrix Analysis (2nd ed.).”, Cambridge University Press, 2013, ISBN 978-0-521-38632-6.\n",
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- "\n",
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- "<a name=\"merwe\"></a>**\\[Merwe2004\\]** R. van der Merwe, \"Sigma-Point Kalman Filters for ProbabilisticInference in Dynamic State-Space Models\", PhD Thesis, OGI School of Science & Engineering, Oregon Health & Science University, USA.\n"
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+ "<a name=\"horn_et_al\"></a>**\\[Horn et al\\]** R.A. Horn, C.R Johnson, Charles R. “Matrix Analysis (2nd ed.).”, Cambridge University Press, 2013, ISBN 978-0-521-38632-6.\n"
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]
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},
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{
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shkolnick-kun