Switched to IPython 3.0 format.
Here's hoping everything still works on nbviewer!
This commit is contained in:
Roger Labbe
parent
65cb22d07c
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00_Preface.ipynb
1123
00_Preface.ipynb
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01_g-h_filter.ipynb
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6860
04_Gaussians.ipynb
6860
04_Gaussians.ipynb
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22568
05_Kalman_Filters.ipynb
22568
05_Kalman_Filters.ipynb
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3873
13_Smoothing.ipynb
3873
13_Smoothing.ipynb
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" white-space: nowrap;\n",
|
||||
" } \n",
|
||||
" h3 {\n",
|
||||
" font-family: 'Open sans',verdana,arial,sans-serif;\n",
|
||||
" }\n",
|
||||
" .text_cell_render h3 {\n",
|
||||
" font-weight: 300;\n",
|
||||
" font-size: 18pt;\n",
|
||||
" line-height: 100%;\n",
|
||||
" color:#d77c0c;\n",
|
||||
" margin-bottom: 0.5em;\n",
|
||||
" margin-top: 2em;\n",
|
||||
" display: block;\n",
|
||||
" white-space: nowrap;\n",
|
||||
" }\n",
|
||||
" h4 {\n",
|
||||
" font-family: 'Open sans',verdana,arial,sans-serif;\n",
|
||||
" }\n",
|
||||
" .text_cell_render h4 {\n",
|
||||
" font-weight: 300;\n",
|
||||
" font-size: 16pt;\n",
|
||||
" color:#d77c0c;\n",
|
||||
" margin-bottom: 0.5em;\n",
|
||||
" margin-top: 0.5em;\n",
|
||||
" display: block;\n",
|
||||
" white-space: nowrap;\n",
|
||||
" }\n",
|
||||
" h5 {\n",
|
||||
" font-family: 'Open sans',verdana,arial,sans-serif;\n",
|
||||
" }\n",
|
||||
" .text_cell_render h5 {\n",
|
||||
" font-weight: 300;\n",
|
||||
" font-style: normal;\n",
|
||||
" color: #1d3b84;\n",
|
||||
" font-size: 16pt;\n",
|
||||
" margin-bottom: 0em;\n",
|
||||
" margin-top: 1.5em;\n",
|
||||
" display: block;\n",
|
||||
" white-space: nowrap;\n",
|
||||
" }\n",
|
||||
" div.text_cell_render{\n",
|
||||
" font-family: 'Arimo',verdana,arial,sans-serif;\n",
|
||||
" line-height: 135%;\n",
|
||||
" font-size: 125%;\n",
|
||||
" width:750px;\n",
|
||||
" margin-left:auto;\n",
|
||||
" margin-right:auto;\n",
|
||||
" text-align:justify;\n",
|
||||
" text-justify:inter-word;\n",
|
||||
" }\n",
|
||||
" div.output_subarea.output_text.output_pyout {\n",
|
||||
" overflow-x: auto;\n",
|
||||
" overflow-y: scroll;\n",
|
||||
" max-height: 50000px;\n",
|
||||
" }\n",
|
||||
" div.output_subarea.output_stream.output_stdout.output_text {\n",
|
||||
" overflow-x: auto;\n",
|
||||
" overflow-y: scroll;\n",
|
||||
" max-height: 50000px;\n",
|
||||
" }\n",
|
||||
" code{\n",
|
||||
" font-size: 70%;\n",
|
||||
" }\n",
|
||||
" .rendered_html code{\n",
|
||||
" background-color: transparent;\n",
|
||||
" }\n",
|
||||
" ul{\n",
|
||||
" margin: 2em;\n",
|
||||
" }\n",
|
||||
" ul li{\n",
|
||||
" padding-left: 0.5em; \n",
|
||||
" margin-bottom: 0.5em; \n",
|
||||
" margin-top: 0.5em; \n",
|
||||
" }\n",
|
||||
" ul li li{\n",
|
||||
" padding-left: 0.2em; \n",
|
||||
" margin-bottom: 0.2em; \n",
|
||||
" margin-top: 0.2em; \n",
|
||||
" }\n",
|
||||
" ol{\n",
|
||||
" margin: 2em;\n",
|
||||
" }\n",
|
||||
" ol li{\n",
|
||||
" padding-left: 0.5em; \n",
|
||||
" margin-bottom: 0.5em; \n",
|
||||
" margin-top: 0.5em; \n",
|
||||
" }\n",
|
||||
" ul li{\n",
|
||||
" padding-left: 0.5em; \n",
|
||||
" margin-bottom: 0.5em; \n",
|
||||
" margin-top: 0.2em; \n",
|
||||
" }\n",
|
||||
" a:link{\n",
|
||||
" font-weight: bold;\n",
|
||||
" color:#447adb;\n",
|
||||
" }\n",
|
||||
" a:visited{\n",
|
||||
" font-weight: bold;\n",
|
||||
" color: #1d3b84;\n",
|
||||
" }\n",
|
||||
" a:hover{\n",
|
||||
" font-weight: bold;\n",
|
||||
" color: #1d3b84;\n",
|
||||
" }\n",
|
||||
" a:focus{\n",
|
||||
" font-weight: bold;\n",
|
||||
" color:#447adb;\n",
|
||||
" }\n",
|
||||
" a:active{\n",
|
||||
" font-weight: bold;\n",
|
||||
" color:#447adb;\n",
|
||||
" }\n",
|
||||
" .rendered_html :link {\n",
|
||||
" text-decoration: underline; \n",
|
||||
" }\n",
|
||||
" .rendered_html :hover {\n",
|
||||
" text-decoration: none; \n",
|
||||
" }\n",
|
||||
" .rendered_html :visited {\n",
|
||||
" text-decoration: none;\n",
|
||||
" }\n",
|
||||
" .rendered_html :focus {\n",
|
||||
" text-decoration: none;\n",
|
||||
" }\n",
|
||||
" .rendered_html :active {\n",
|
||||
" text-decoration: none;\n",
|
||||
" }\n",
|
||||
" .warning{\n",
|
||||
" color: rgb( 240, 20, 20 )\n",
|
||||
" } \n",
|
||||
" hr {\n",
|
||||
" color: #f3f3f3;\n",
|
||||
" background-color: #f3f3f3;\n",
|
||||
" height: 1px;\n",
|
||||
" }\n",
|
||||
" blockquote{\n",
|
||||
" display:block;\n",
|
||||
" background: #fcfcfc;\n",
|
||||
" border-left: 5px solid #c76c0c;\n",
|
||||
" font-family: 'Open sans',verdana,arial,sans-serif;\n",
|
||||
" width:680px;\n",
|
||||
" padding: 10px 10px 10px 10px;\n",
|
||||
" text-align:justify;\n",
|
||||
" text-justify:inter-word;\n",
|
||||
" }\n",
|
||||
" blockquote p {\n",
|
||||
" margin-bottom: 0;\n",
|
||||
" line-height: 125%;\n",
|
||||
" font-size: 100%;\n",
|
||||
" }\n",
|
||||
"</style>\n",
|
||||
"<script>\n",
|
||||
" MathJax.Hub.Config({\n",
|
||||
" TeX: {\n",
|
||||
" extensions: [\"AMSmath.js\"]\n",
|
||||
" },\n",
|
||||
" tex2jax: {\n",
|
||||
" inlineMath: [ ['$','$'], [\"\\\\(\",\"\\\\)\"] ],\n",
|
||||
" displayMath: [ ['$$','$$'], [\"\\\\[\",\"\\\\]\"] ]\n",
|
||||
" },\n",
|
||||
" displayAlign: 'center', // Change this to 'center' to center equations.\n",
|
||||
" \"HTML-CSS\": {\n",
|
||||
" styles: {'.MathJax_Display': {\"margin\": 4}}\n",
|
||||
" }\n",
|
||||
" });\n",
|
||||
"</script>\n"
|
||||
],
|
||||
"metadata": {},
|
||||
"output_type": "pyout",
|
||||
"prompt_number": 1,
|
||||
"text": [
|
||||
"<IPython.core.display.HTML at 0x7f1f6b0279e8>"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 1
|
||||
},
|
||||
{
|
||||
"cell_type": "heading",
|
||||
"level": 1,
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Symbology"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"This is just notes at this point. \n",
|
||||
"\n",
|
||||
"### State\n",
|
||||
"\n",
|
||||
"$x$ (Brookner, Zarchan, Brown)\n",
|
||||
"\n",
|
||||
"$\\underline{x}$ Gelb)\n",
|
||||
"\n",
|
||||
"### State at step n\n",
|
||||
"\n",
|
||||
"$x_n$ (Brookner)\n",
|
||||
"\n",
|
||||
"$x_k$ (Brown, Zarchan)\n",
|
||||
"\n",
|
||||
"$\\underline{x}_k$ (Gelb)\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"### Prediction\n",
|
||||
"\n",
|
||||
"$x^-$\n",
|
||||
"\n",
|
||||
"$x_{n,n-1}$ (Brookner) \n",
|
||||
"\n",
|
||||
"$x_{k+1,k}$\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"## measurement\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"$x^*$\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"Y_n (Brookner)\n",
|
||||
"\n",
|
||||
"##control transition Matrix\n",
|
||||
"\n",
|
||||
"$G$ (Zarchan)\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"Not used (Brookner)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"#Nomenclature\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"## Equations\n",
|
||||
"### Brookner\n",
|
||||
"\n",
|
||||
"$$\n",
|
||||
"\\begin{aligned}\n",
|
||||
"X^*_{n+1,n} &= \\Phi X^*_{n,n} \\\\\n",
|
||||
"X^*_{n,n} &= X^*_{n,n-1} +H_n(Y_n - MX^*_{n,n-1}) \\\\\n",
|
||||
"H_n &= S^*_{n,n-1}M^T[R_n + MS^*_{n,n-1}M^T]^{-1} \\\\\n",
|
||||
"S^*_{n,n-1} &= \\Phi S^*_{n-1,n-1}\\Phi^T + Q_n \\\\\n",
|
||||
"S^*_{n-1,n-1} &= (I-H_{n-1}M)S^*_{n-1,n-2}\n",
|
||||
"\\end{aligned}$$\n",
|
||||
"\n",
|
||||
"### Gelb\n",
|
||||
"\n",
|
||||
"$$\n",
|
||||
"\\begin{aligned}\n",
|
||||
"\\underline{\\hat{x}}_k(-) &= \\Phi_{k-1} \\underline{\\hat{x}}_{k-1}(+) \\\\\n",
|
||||
"\\underline{\\hat{x}}_k(+) &= \\underline{\\hat{x}}_k(-) +K_k[Z_k - H_k\\underline{\\hat{x}}_k(-)] \\\\\n",
|
||||
"K_k &= P_k(-)H_k^T[H_kP_k(-)H_k^T + R_k]^{-1}\\\\\n",
|
||||
"P_k(+) &= \\Phi_{k-1} P_{k-1}(+)\\Phi_{k-1}^T + Q_{k-1} \\\\\n",
|
||||
"P_k(-) &= (I-K_kH_k)P_k(-)\n",
|
||||
"\\end{aligned}$$\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"### Brown\n",
|
||||
"\n",
|
||||
"$$\n",
|
||||
"\\begin{aligned}\n",
|
||||
"\\hat{\\textbf{x}}^-_{k+1} &= \\mathbf{\\phi}_{k}\\hat{\\textbf{x}}_{k} \\\\\n",
|
||||
"\\hat{\\textbf{x}}_k &= \\hat{\\textbf{x}}^-_k +\\textbf{K}_k[\\textbf{z}_k - \\textbf{H}_k\\hat{\\textbf{}x}^-_k] \\\\\n",
|
||||
"\\textbf{K}_k &= \\textbf{P}^-_k\\textbf{H}_k^T[\\textbf{H}_k\\textbf{P}^-_k\\textbf{H}_k^T + \\textbf{R}_k]^{-1}\\\\\n",
|
||||
"\\textbf{P}^-_{k+1} &= \\mathbf{\\phi}_k \\textbf{P}_k\\mathbf{\\phi}_k^T + \\textbf{Q}_{k} \\\\\n",
|
||||
"\\mathbf{P}_k &= (\\mathbf{I}-\\mathbf{K}_k\\mathbf{H}_k)\\mathbf{P}^-_k\n",
|
||||
"\\end{aligned}$$\n",
|
||||
"## \n",
|
||||
"\n",
|
||||
"### Zarchan\n",
|
||||
"\n",
|
||||
"$$\n",
|
||||
"\\begin{aligned}\n",
|
||||
"\\hat{x}_{k} &= \\Phi_{k}\\hat{x}_{k-1} + G_ku_{k-1} + K_k[z_k - H\\Phi_{k}\\hat{x}_{k-1} - HG_ku_{k-1} ] \\\\\n",
|
||||
"M_{k} &= \\Phi_k P_{k-1}\\phi_k^T + Q_{k} \\\\\n",
|
||||
"K_k &= M_kH^T[HM_kH^T + R_k]^{-1}\\\\\n",
|
||||
"P_k &= (I-K_kH)M_k\n",
|
||||
"\\end{aligned}$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"### Wikipedia\n",
|
||||
"$$\n",
|
||||
"\\begin{aligned}\n",
|
||||
"\\hat{\\textbf{x}}_{k\\mid k-1} &= \\textbf{F}_{k}\\hat{\\textbf{x}}_{k-1\\mid k-1} + \\textbf{B}_{k} \\textbf{u}_{k} \\\\\n",
|
||||
"\\textbf{P}_{k\\mid k-1} &= \\textbf{F}_{k} \\textbf{P}_{k-1\\mid k-1} \\textbf{F}_{k}^{\\text{T}} + \\textbf{Q}_{k}\\\\\n",
|
||||
"\\tilde{\\textbf{y}}_k &= \\textbf{z}_k - \\textbf{H}_k\\hat{\\textbf{x}}_{k\\mid k-1} \\\\\n",
|
||||
"\\textbf{S}_k &= \\textbf{H}_k \\textbf{P}_{k\\mid k-1} \\textbf{H}_k^\\text{T} + \\textbf{R}_k \\\\\n",
|
||||
"\\textbf{K}_k &= \\textbf{P}_{k\\mid k-1}\\textbf{H}_k^\\text{T}\\textbf{S}_k^{-1} \\\\\n",
|
||||
"\\hat{\\textbf{x}}_{k\\mid k} &= \\hat{\\textbf{x}}_{k\\mid k-1} + \\textbf{K}_k\\tilde{\\textbf{y}}_k \\\\\n",
|
||||
"\\textbf{P}_{k|k} &= (I - \\textbf{K}_k \\textbf{H}_k) \\textbf{P}_{k|k-1}\n",
|
||||
"\\end{aligned}$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"### Labbe\n",
|
||||
"\n",
|
||||
"$$\n",
|
||||
"\\begin{aligned}\n",
|
||||
"\\hat{\\textbf{x}}^-_{k+1} &= \\mathbf{F}_{k}\\hat{\\textbf{x}}_{k} + \\mathbf{B}_k\\mathbf{u}_k \\\\\n",
|
||||
"\\textbf{P}^-_{k+1} &= \\mathbf{F}_k \\textbf{P}_k\\mathbf{F}_k^T + \\textbf{Q}_{k} \\\\\n",
|
||||
"\\textbf{y}_k &= \\textbf{z}_k - \\textbf{H}_k\\hat{\\textbf{}x}^-_k \\\\\n",
|
||||
"\\mathbf{S}_k &= \\textbf{H}_k\\textbf{P}^-_k\\textbf{H}_k^T + \\textbf{R}_k \\\\\n",
|
||||
"\\textbf{K}_k &= \\textbf{P}^-_k\\textbf{H}_k^T\\mathbf{S}_k^{-1} \\\\\n",
|
||||
"\\hat{\\textbf{x}}_k &= \\hat{\\textbf{x}}^-_k +\\textbf{K}_k\\textbf{y} \\\\\n",
|
||||
"\\mathbf{P}_k &= (\\mathbf{I}-\\mathbf{K}_k\\mathbf{H}_k)\\mathbf{P}^-_k\n",
|
||||
"\\end{aligned}$$\n",
|
||||
"## \n",
|
||||
"\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": []
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"metadata": {}
|
||||
"source": [
|
||||
"#format the book\n",
|
||||
"%matplotlib inline\n",
|
||||
"from __future__ import division, print_function\n",
|
||||
"import matplotlib.pyplot as plt\n",
|
||||
"import book_format\n",
|
||||
"book_format.load_style()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Symbology"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"This is just notes at this point. \n",
|
||||
"\n",
|
||||
"### State\n",
|
||||
"\n",
|
||||
"$x$ (Brookner, Zarchan, Brown)\n",
|
||||
"\n",
|
||||
"$\\underline{x}$ Gelb)\n",
|
||||
"\n",
|
||||
"### State at step n\n",
|
||||
"\n",
|
||||
"$x_n$ (Brookner)\n",
|
||||
"\n",
|
||||
"$x_k$ (Brown, Zarchan)\n",
|
||||
"\n",
|
||||
"$\\underline{x}_k$ (Gelb)\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"### Prediction\n",
|
||||
"\n",
|
||||
"$x^-$\n",
|
||||
"\n",
|
||||
"$x_{n,n-1}$ (Brookner) \n",
|
||||
"\n",
|
||||
"$x_{k+1,k}$\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"## measurement\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"$x^*$\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"Y_n (Brookner)\n",
|
||||
"\n",
|
||||
"##control transition Matrix\n",
|
||||
"\n",
|
||||
"$G$ (Zarchan)\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"Not used (Brookner)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"#Nomenclature\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"## Equations\n",
|
||||
"### Brookner\n",
|
||||
"\n",
|
||||
"$$\n",
|
||||
"\\begin{aligned}\n",
|
||||
"X^*_{n+1,n} &= \\Phi X^*_{n,n} \\\\\n",
|
||||
"X^*_{n,n} &= X^*_{n,n-1} +H_n(Y_n - MX^*_{n,n-1}) \\\\\n",
|
||||
"H_n &= S^*_{n,n-1}M^T[R_n + MS^*_{n,n-1}M^T]^{-1} \\\\\n",
|
||||
"S^*_{n,n-1} &= \\Phi S^*_{n-1,n-1}\\Phi^T + Q_n \\\\\n",
|
||||
"S^*_{n-1,n-1} &= (I-H_{n-1}M)S^*_{n-1,n-2}\n",
|
||||
"\\end{aligned}$$\n",
|
||||
"\n",
|
||||
"### Gelb\n",
|
||||
"\n",
|
||||
"$$\n",
|
||||
"\\begin{aligned}\n",
|
||||
"\\underline{\\hat{x}}_k(-) &= \\Phi_{k-1} \\underline{\\hat{x}}_{k-1}(+) \\\\\n",
|
||||
"\\underline{\\hat{x}}_k(+) &= \\underline{\\hat{x}}_k(-) +K_k[Z_k - H_k\\underline{\\hat{x}}_k(-)] \\\\\n",
|
||||
"K_k &= P_k(-)H_k^T[H_kP_k(-)H_k^T + R_k]^{-1}\\\\\n",
|
||||
"P_k(+) &= \\Phi_{k-1} P_{k-1}(+)\\Phi_{k-1}^T + Q_{k-1} \\\\\n",
|
||||
"P_k(-) &= (I-K_kH_k)P_k(-)\n",
|
||||
"\\end{aligned}$$\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"### Brown\n",
|
||||
"\n",
|
||||
"$$\n",
|
||||
"\\begin{aligned}\n",
|
||||
"\\hat{\\textbf{x}}^-_{k+1} &= \\mathbf{\\phi}_{k}\\hat{\\textbf{x}}_{k} \\\\\n",
|
||||
"\\hat{\\textbf{x}}_k &= \\hat{\\textbf{x}}^-_k +\\textbf{K}_k[\\textbf{z}_k - \\textbf{H}_k\\hat{\\textbf{}x}^-_k] \\\\\n",
|
||||
"\\textbf{K}_k &= \\textbf{P}^-_k\\textbf{H}_k^T[\\textbf{H}_k\\textbf{P}^-_k\\textbf{H}_k^T + \\textbf{R}_k]^{-1}\\\\\n",
|
||||
"\\textbf{P}^-_{k+1} &= \\mathbf{\\phi}_k \\textbf{P}_k\\mathbf{\\phi}_k^T + \\textbf{Q}_{k} \\\\\n",
|
||||
"\\mathbf{P}_k &= (\\mathbf{I}-\\mathbf{K}_k\\mathbf{H}_k)\\mathbf{P}^-_k\n",
|
||||
"\\end{aligned}$$\n",
|
||||
"## \n",
|
||||
"\n",
|
||||
"### Zarchan\n",
|
||||
"\n",
|
||||
"$$\n",
|
||||
"\\begin{aligned}\n",
|
||||
"\\hat{x}_{k} &= \\Phi_{k}\\hat{x}_{k-1} + G_ku_{k-1} + K_k[z_k - H\\Phi_{k}\\hat{x}_{k-1} - HG_ku_{k-1} ] \\\\\n",
|
||||
"M_{k} &= \\Phi_k P_{k-1}\\phi_k^T + Q_{k} \\\\\n",
|
||||
"K_k &= M_kH^T[HM_kH^T + R_k]^{-1}\\\\\n",
|
||||
"P_k &= (I-K_kH)M_k\n",
|
||||
"\\end{aligned}$$"
|
||||
]
|
||||
},
|
||||
{
|
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"metadata": {},
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"source": [
|
||||
"### Wikipedia\n",
|
||||
"$$\n",
|
||||
"\\begin{aligned}\n",
|
||||
"\\hat{\\textbf{x}}_{k\\mid k-1} &= \\textbf{F}_{k}\\hat{\\textbf{x}}_{k-1\\mid k-1} + \\textbf{B}_{k} \\textbf{u}_{k} \\\\\n",
|
||||
"\\textbf{P}_{k\\mid k-1} &= \\textbf{F}_{k} \\textbf{P}_{k-1\\mid k-1} \\textbf{F}_{k}^{\\text{T}} + \\textbf{Q}_{k}\\\\\n",
|
||||
"\\tilde{\\textbf{y}}_k &= \\textbf{z}_k - \\textbf{H}_k\\hat{\\textbf{x}}_{k\\mid k-1} \\\\\n",
|
||||
"\\textbf{S}_k &= \\textbf{H}_k \\textbf{P}_{k\\mid k-1} \\textbf{H}_k^\\text{T} + \\textbf{R}_k \\\\\n",
|
||||
"\\textbf{K}_k &= \\textbf{P}_{k\\mid k-1}\\textbf{H}_k^\\text{T}\\textbf{S}_k^{-1} \\\\\n",
|
||||
"\\hat{\\textbf{x}}_{k\\mid k} &= \\hat{\\textbf{x}}_{k\\mid k-1} + \\textbf{K}_k\\tilde{\\textbf{y}}_k \\\\\n",
|
||||
"\\textbf{P}_{k|k} &= (I - \\textbf{K}_k \\textbf{H}_k) \\textbf{P}_{k|k-1}\n",
|
||||
"\\end{aligned}$$"
|
||||
]
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},
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"source": [
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||||
"### Labbe\n",
|
||||
"\n",
|
||||
"$$\n",
|
||||
"\\begin{aligned}\n",
|
||||
"\\hat{\\textbf{x}}^-_{k+1} &= \\mathbf{F}_{k}\\hat{\\textbf{x}}_{k} + \\mathbf{B}_k\\mathbf{u}_k \\\\\n",
|
||||
"\\textbf{P}^-_{k+1} &= \\mathbf{F}_k \\textbf{P}_k\\mathbf{F}_k^T + \\textbf{Q}_{k} \\\\\n",
|
||||
"\\textbf{y}_k &= \\textbf{z}_k - \\textbf{H}_k\\hat{\\textbf{}x}^-_k \\\\\n",
|
||||
"\\mathbf{S}_k &= \\textbf{H}_k\\textbf{P}^-_k\\textbf{H}_k^T + \\textbf{R}_k \\\\\n",
|
||||
"\\textbf{K}_k &= \\textbf{P}^-_k\\textbf{H}_k^T\\mathbf{S}_k^{-1} \\\\\n",
|
||||
"\\hat{\\textbf{x}}_k &= \\hat{\\textbf{x}}^-_k +\\textbf{K}_k\\textbf{y} \\\\\n",
|
||||
"\\mathbf{P}_k &= (\\mathbf{I}-\\mathbf{K}_k\\mathbf{H}_k)\\mathbf{P}^-_k\n",
|
||||
"\\end{aligned}$$\n",
|
||||
"## \n",
|
||||
"\n"
|
||||
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"[Table of Contents](http://nbviewer.ipython.org/github/rlabbe/Kalman-and-Bayesian-Filters-in-Python/blob/master/table_of_contents.ipynb)"
|
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||||
"<center><h1>Kalman and Bayesian Filters in Python</h1></center>\n",
|
||||
"<center><a href =\"http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/toc.ipynb\">Table of Contents</a></center>"
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||||
"#Introduction\n",
|
||||
"##### Version 0.0\n",
|
||||
"\n",
|
||||
"**Author's note - this is obsolete, read the preface instead.**\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"The Kalman filter was introduced to the world via papers published in 1958 and 1960 by Rudolph E Kalman. This work built on work by Nobert Wiener. Kalman's early papers were extremely abstract, but researchers quickly realized that the papers described a very practical technique to filter noisy data. From then until now it has been an ongoing topic of research, and there are many books and papers devoted not only to the basics, but many specializations and extensions to the technique. If you are reading this, you have likely come across some of them.\n",
|
||||
"\n",
|
||||
"If you are like me, you probably find them nearly impenetrable. I find that almost all start with very abstract math, assume familiarity with notation and naming conventions that I haven't seen before, and focus heavily on proof rather than exposition and teaching. This is perhaps understandable, but it is a regrettable situation and not necessary. \n",
|
||||
"\n",
|
||||
"After struggling through this material for some time, things finally began making sense. A majority of my 'aha' moments were due to implementing and experimenting with various simple filters. What to make of an equation like $K_k=P_{k}^{-}H^{T}_{k}[H_k P^{-}_{k} H^{T}_{k} + R_k]^{-1}$ is initially puzzing. This is especially true when it pops out as the result of two to three pages of linear algrebra, and each variable is given only a very abstract, mathematically rigorous definition. One book I have doesn't bother to define $R_k$ despite using it throughout its 4 pages of derivation of K. If instead I tell you that K is just a scaling factor for choosing how much of a measurement and how much of a prediction to use in the filter, what K is becomes obvious (although perhaps the computation is still a bit mysterious). After implementing a few Kalman filters for toy problems and varying the values of various matrices and constants I developed both an intuitive and fairly deep understanding of how Kalman filters work. This knowledge is indispensible; it is trivial to code the handful of linear equations for a Kalman filter, and they never change. I do mean a handful - you can implement the simplest Kalman filter in 10 lines of code. \n",
|
||||
" "
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"** this needs a lot of editting. bored with it for now.**"
|
||||
]
|
||||
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|
||||
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||||
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||||
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||||
"source": [
|
||||
"While you do need to know some basic probability and some very basic linear algebra to understand and implement Kalman filters, by and large you really do not need to understand the complicated, multi-page derivations. The end result of all the math is a small handful of equations which you can use to perform very sophisticated filtering, smoothing, and tracking. Implementing the basic equations is never difficult. Kalman filter design is much more an art than a science - implementers spend a few minutes writing the basic equations, and then a lot of time tuning the filter for their specific problem. \n",
|
||||
"\n",
|
||||
"I compare this to a student learning the equation for the volume of a sphere: $V(r) = \\frac{4}{3}\\pi r^3$ A student can use this equation without even understanding the functional notation $V(r)$, and certainly they do not need to be able to derive the equation via calculus. Eventually, in some domains, knowledge of the calculus will become useful, but an enormous amount of work can be done by only knowing the equation and how to apply it. Also, it is often useful to understand facts about a sphere, such as it the shape that encloses the greatest volume with the smallest surface area. You do not have to know how to prove that to make use of that information to explain the reason for why bubbles are round, or to economically fence an area for livestock. \n",
|
||||
"\n",
|
||||
"I argue the same is largely true with Kalman filters. In this book I will not prove the Kalman filter equations are correct, nor will I derive them. I will instead strive to build in you an physical, intuitive understanding of what the equations mean, and how they interact. Of course, if you are trying to navigate a spaceship to Mars you will require a more sophisticated understanding than I provide, and you will not view this as a useful resource. But the rest of people, who perhaps wants to track heads in video, or control a little hobby robot, or smooth some data, I think this approach should provide a lot of insight into how to create Kalman filters, and provide enough background that the standard texts are now approachable."
|
||||
]
|
||||
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|
||||
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|
||||
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|
||||
"metadata": {},
|
||||
"source": [
|
||||
"#### Prerequisites\n",
|
||||
"\n",
|
||||
"While Kalman filters are used in many domains, they were initially created to solve problems with missle tracking and navigation. Most of my examples will draw from physical examples such as these. However, I do this not just from a sense of history, but because I believe that helps the student form strong pysical intuitions about what the filter is doing. For example, we will learn how the Kalman filter creates and uses something called \"hidden variables\". This is normally presented in a highly abstract manner. However, it is actually quite simple. Suppose I know your position at several points in time. From that I can calculate your velocity even though I don't have any sensor that directly measures your velocity. In a Kalman filter, if I have a position sensor, the filter will generate the hidden, or unobserved variable velocity. A lot of seemingly arcane terminology suddenly becomes concrete and clear. So you should have taken a basic physics course and understand equations like $d = \\frac{a}{2} t^2 + v_0 t + d_0$\n",
|
||||
"\n",
|
||||
"You will need some basic calculus - you should be able to integrate velocity to get distance, or take the derivative of the distance equation above to get the velocit equation. You should be familiar with trigonometry.\n",
|
||||
"\n",
|
||||
"Kalman filtering depends heavily on statistics. I have a basic Kalman filtering textbook that devotes several chapters to statistics before even trying to discuss Kalman filtering. Again, I feel like this is a place where you can get a long way with a little information. So I will assume that you have had some exposure to probability. If you have seen done calculus you probably know how to perform basic probability computations. I will provide some remedial math for gaussian distributions, but I will move fairly quickly through it. \n",
|
||||
"\n",
|
||||
"Finally, you will probably want to have some exposure to linear algebra. I will assume that you understand matrices, arrays, and simple operations like matrix multiplication. In the more difficult moments we will use things like LU and Cholesky decomposition, and touch on things like eigenvalues and eigenvectors, but you really do not need to be able to do math at that level. \n",
|
||||
"\n",
|
||||
"I am writing this in IPython Notebook. I use Python because of its excellent mathematics library in the form of numpy and scipy, and its strength as a general purpose programming language. Kalman filter books that do include code invariably use Matlab. This makes sense, as any professional engineer with have access to it, and long practice with it. However, I am addressing the hobbiest, and most cannot afford to, or do not want to buy Matlab. Finally, the interactive aspect of mixing code, results, and text in one document is irrestible to me. I have read texts and papers with intriguiging graphs, and I struggle to understand how the graph was generated. With IPython Notebook nothing is hidden - the math is all revealed. If you wonder 'what would happen if I changed this parameter' - you can just type in a new value and see the results. Finally, because of the facilities of numpy and scipy you do not have to code or understand details of linear algebra - when the Kalman filter equations state that we need to find the Cholesky decomposition we will just call *numpy.linalg.cholesky*. I will spend a few words explaining what that means when we get to it, but to be honest you do not need to understand it to use the code. \n",
|
||||
"\n",
|
||||
"Since this is a book on doing Kalman filtering with Python I will expect that you know Python; I do not attempt to teach it. With that said, it is a very easy language to learn, and it reads much like pseudocode. If you are more familiar with another language I do not think you will have any major difficulties reading the source code. I purposefully restrict the code to the more basic features of Python to accomodate people with varying skill levels. If you use this code in your own work, feel free to use more advanced Python facilities if it strikes your fancy.\n",
|
||||
"\n",
|
||||
"So that is a fair amount of prerequisites, but I think you will already have most of them if you are seriously trying to solve a problem where Kalman filters are useful. "
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"#Introduction\n",
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"##### Version 0.0\n",
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"\n",
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"**Author's note - this is obsolete, read the preface instead.**\n",
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"\n",
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"\n",
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"The Kalman filter was introduced to the world via papers published in 1958 and 1960 by Rudolph E Kalman. This work built on work by Nobert Wiener. Kalman's early papers were extremely abstract, but researchers quickly realized that the papers described a very practical technique to filter noisy data. From then until now it has been an ongoing topic of research, and there are many books and papers devoted not only to the basics, but many specializations and extensions to the technique. If you are reading this, you have likely come across some of them.\n",
|
||||
"\n",
|
||||
"If you are like me, you probably find them nearly impenetrable. I find that almost all start with very abstract math, assume familiarity with notation and naming conventions that I haven't seen before, and focus heavily on proof rather than exposition and teaching. This is perhaps understandable, but it is a regrettable situation and not necessary. \n",
|
||||
"\n",
|
||||
"After struggling through this material for some time, things finally began making sense. A majority of my 'aha' moments were due to implementing and experimenting with various simple filters. What to make of an equation like $K_k=P_{k}^{-}H^{T}_{k}[H_k P^{-}_{k} H^{T}_{k} + R_k]^{-1}$ is initially puzzing. This is especially true when it pops out as the result of two to three pages of linear algrebra, and each variable is given only a very abstract, mathematically rigorous definition. One book I have doesn't bother to define $R_k$ despite using it throughout its 4 pages of derivation of K. If instead I tell you that K is just a scaling factor for choosing how much of a measurement and how much of a prediction to use in the filter, what K is becomes obvious (although perhaps the computation is still a bit mysterious). After implementing a few Kalman filters for toy problems and varying the values of various matrices and constants I developed both an intuitive and fairly deep understanding of how Kalman filters work. This knowledge is indispensible; it is trivial to code the handful of linear equations for a Kalman filter, and they never change. I do mean a handful - you can implement the simplest Kalman filter in 10 lines of code. \n",
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"** this needs a lot of editting. bored with it for now.**"
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"While you do need to know some basic probability and some very basic linear algebra to understand and implement Kalman filters, by and large you really do not need to understand the complicated, multi-page derivations. The end result of all the math is a small handful of equations which you can use to perform very sophisticated filtering, smoothing, and tracking. Implementing the basic equations is never difficult. Kalman filter design is much more an art than a science - implementers spend a few minutes writing the basic equations, and then a lot of time tuning the filter for their specific problem. \n",
|
||||
"\n",
|
||||
"I compare this to a student learning the equation for the volume of a sphere: $V(r) = \\frac{4}{3}\\pi r^3$ A student can use this equation without even understanding the functional notation $V(r)$, and certainly they do not need to be able to derive the equation via calculus. Eventually, in some domains, knowledge of the calculus will become useful, but an enormous amount of work can be done by only knowing the equation and how to apply it. Also, it is often useful to understand facts about a sphere, such as it the shape that encloses the greatest volume with the smallest surface area. You do not have to know how to prove that to make use of that information to explain the reason for why bubbles are round, or to economically fence an area for livestock. \n",
|
||||
"\n",
|
||||
"I argue the same is largely true with Kalman filters. In this book I will not prove the Kalman filter equations are correct, nor will I derive them. I will instead strive to build in you an physical, intuitive understanding of what the equations mean, and how they interact. Of course, if you are trying to navigate a spaceship to Mars you will require a more sophisticated understanding than I provide, and you will not view this as a useful resource. But the rest of people, who perhaps wants to track heads in video, or control a little hobby robot, or smooth some data, I think this approach should provide a lot of insight into how to create Kalman filters, and provide enough background that the standard texts are now approachable."
|
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||||
"#### Prerequisites\n",
|
||||
"\n",
|
||||
"While Kalman filters are used in many domains, they were initially created to solve problems with missle tracking and navigation. Most of my examples will draw from physical examples such as these. However, I do this not just from a sense of history, but because I believe that helps the student form strong pysical intuitions about what the filter is doing. For example, we will learn how the Kalman filter creates and uses something called \"hidden variables\". This is normally presented in a highly abstract manner. However, it is actually quite simple. Suppose I know your position at several points in time. From that I can calculate your velocity even though I don't have any sensor that directly measures your velocity. In a Kalman filter, if I have a position sensor, the filter will generate the hidden, or unobserved variable velocity. A lot of seemingly arcane terminology suddenly becomes concrete and clear. So you should have taken a basic physics course and understand equations like $d = \\frac{a}{2} t^2 + v_0 t + d_0$\n",
|
||||
"\n",
|
||||
"You will need some basic calculus - you should be able to integrate velocity to get distance, or take the derivative of the distance equation above to get the velocit equation. You should be familiar with trigonometry.\n",
|
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"\n",
|
||||
"Kalman filtering depends heavily on statistics. I have a basic Kalman filtering textbook that devotes several chapters to statistics before even trying to discuss Kalman filtering. Again, I feel like this is a place where you can get a long way with a little information. So I will assume that you have had some exposure to probability. If you have seen done calculus you probably know how to perform basic probability computations. I will provide some remedial math for gaussian distributions, but I will move fairly quickly through it. \n",
|
||||
"\n",
|
||||
"Finally, you will probably want to have some exposure to linear algebra. I will assume that you understand matrices, arrays, and simple operations like matrix multiplication. In the more difficult moments we will use things like LU and Cholesky decomposition, and touch on things like eigenvalues and eigenvectors, but you really do not need to be able to do math at that level. \n",
|
||||
"\n",
|
||||
"I am writing this in IPython Notebook. I use Python because of its excellent mathematics library in the form of numpy and scipy, and its strength as a general purpose programming language. Kalman filter books that do include code invariably use Matlab. This makes sense, as any professional engineer with have access to it, and long practice with it. However, I am addressing the hobbiest, and most cannot afford to, or do not want to buy Matlab. Finally, the interactive aspect of mixing code, results, and text in one document is irrestible to me. I have read texts and papers with intriguiging graphs, and I struggle to understand how the graph was generated. With IPython Notebook nothing is hidden - the math is all revealed. If you wonder 'what would happen if I changed this parameter' - you can just type in a new value and see the results. Finally, because of the facilities of numpy and scipy you do not have to code or understand details of linear algebra - when the Kalman filter equations state that we need to find the Cholesky decomposition we will just call *numpy.linalg.cholesky*. I will spend a few words explaining what that means when we get to it, but to be honest you do not need to understand it to use the code. \n",
|
||||
"\n",
|
||||
"Since this is a book on doing Kalman filtering with Python I will expect that you know Python; I do not attempt to teach it. With that said, it is a very easy language to learn, and it reads much like pseudocode. If you are more familiar with another language I do not think you will have any major difficulties reading the source code. I purposefully restrict the code to the more basic features of Python to accomodate people with varying skill levels. If you use this code in your own work, feel free to use more advanced Python facilities if it strikes your fancy.\n",
|
||||
"\n",
|
||||
"So that is a fair amount of prerequisites, but I think you will already have most of them if you are seriously trying to solve a problem where Kalman filters are useful. "
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||||
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|
||||
"source": [
|
||||
"#format the book\n",
|
||||
"import book_format\n",
|
||||
"book_format.load_style()"
|
||||
]
|
||||
}
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"kernelspec": {
|
||||
"display_name": "Python 3",
|
||||
"language": "python",
|
||||
"name": "python3"
|
||||
},
|
||||
"language_info": {
|
||||
"codemirror_mode": {
|
||||
"name": "ipython",
|
||||
"version": 3
|
||||
},
|
||||
"file_extension": ".py",
|
||||
"mimetype": "text/x-python",
|
||||
"name": "python",
|
||||
"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython3",
|
||||
"version": "3.4.2"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 0
|
||||
}
|
||||
|
|
|
|||
|
|
@ -30,6 +30,10 @@ All of the filters used in this book as well as others not in this book are impl
|
|||
|
||||
Downloading the book
|
||||
-----
|
||||
|
||||
** Breaking change: I have upgraded to IPython 3.0. This release alters the notebook format (.ipynb) files. IPython 2.4 can read the files, but not write them. I apologize if you are using an earlier version, but this is an unavoidable change and I'd rather change now instead of later. This will not affect you if you are reading online, only if you are running the notebooks on your local computer. Please note that this has nothing to do with *Python 3 - you can run Python 2.7 in IPython 3 so far as I know**.
|
||||
|
||||
|
||||
However, this book is intended to be interactive and I recommend using it in that form. If you install IPython on your computer and then clone this book you will be able to run all of the code in the book yourself. You can perform experiments, see how filters react to different data, see how different filters react to the same data, and so on. I find this sort of immediate feedback both vital and invigorating. You do not have to wonder "what happens if". Try it and see!
|
||||
|
||||
The github pages for this project are at https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python You can clone it to your hard drive with the command
|
||||
|
|
|
|||
|
|
@ -38,7 +38,7 @@ def equal_axis():
|
|||
plt.axis('equal')
|
||||
|
||||
def reset_axis():
|
||||
pylab.rcParams['figure.figsize'] = 12, 6
|
||||
pylab.rcParams['figure.figsize'] = 11, 5.5
|
||||
|
||||
def set_figsize(x, y):
|
||||
pylab.rcParams['figure.figsize'] = x, y
|
||||
|
|
|
|||
|
|
@ -1,27 +1,21 @@
|
|||
% Inherit from report
|
||||
((* extends 'report.tplx' *))
|
||||
|
||||
%===============================================================================
|
||||
% Latex Book
|
||||
%===============================================================================
|
||||
|
||||
|
||||
|
||||
|
||||
((* block docclass *))
|
||||
\documentclass{book}
|
||||
\setcounter{chapter}{-1}
|
||||
((* endblock docclass *))
|
||||
|
||||
((* block preamble *))
|
||||
((* endblock preamble *))
|
||||
|
||||
((* block title *))
|
||||
\title{Kalman and Bayesian Filters in Python}
|
||||
\author{Roger R Labbe Jr}
|
||||
\date{}
|
||||
\setcounter{secnumdepth}{3}
|
||||
\setcounter{tocdepth}{3}
|
||||
((* endblock title *))
|
||||
|
||||
((* block abstract *))\tableofcontents((* endblock abstract *))
|
||||
|
||||
% Define block headings
|
||||
% Note: latex will only number headings that aren't starred
|
||||
% (i.e. \subsection , but not \subsection* )
|
||||
|
||||
((* block h1 -*))\chapter((* endblock h1 -*))
|
||||
((* block h2 -*))\section((* endblock h2 -*))
|
||||
((* block h3 -*))\subsection((* endblock h3 -*))
|
||||
((* block h4 -*))\subsubsection((* endblock h4 -*))
|
||||
((* block h5 -*))\paragraph((* endblock h5 -*))
|
||||
((* block h6 -*))\subparagraph((* endblock h6 -*))
|
||||
|
||||
((* block markdowncell scoped *))
|
||||
((( cell.source | citation2latex | strip_files_prefix | markdown2latex(extra_args=["--chapters"]) )))
|
||||
((* endblock markdowncell *))
|
||||
|
|
@ -0,0 +1,27 @@
|
|||
((* extends 'report.tplx' *))
|
||||
|
||||
%===============================================================================
|
||||
% Latex Book
|
||||
%===============================================================================
|
||||
|
||||
|
||||
((* block title *))
|
||||
\title{Kalman and Bayesian Filters in Python}
|
||||
\author{Roger R Labbe Jr}
|
||||
\date{}
|
||||
\setcounter{secnumdepth}{3}
|
||||
\setcounter{tocdepth}{3}
|
||||
((* endblock title *))
|
||||
|
||||
((* block abstract *))\tableofcontents((* endblock abstract *))
|
||||
|
||||
% Define block headings
|
||||
% Note: latex will only number headings that aren't starred
|
||||
% (i.e. \subsection , but not \subsection* )
|
||||
|
||||
((* block h1 -*))\chapter((* endblock h1 -*))
|
||||
((* block h2 -*))\section((* endblock h2 -*))
|
||||
((* block h3 -*))\subsection((* endblock h3 -*))
|
||||
((* block h4 -*))\subsubsection((* endblock h4 -*))
|
||||
((* block h5 -*))\paragraph((* endblock h5 -*))
|
||||
((* block h6 -*))\subparagraph((* endblock h6 -*))
|
||||
|
|
@ -5,7 +5,7 @@ echo "merging book..."
|
|||
python merge_book.py > Kalman_and_Bayesian_Filters_in_Python.ipynb
|
||||
|
||||
echo "creating pdf..."
|
||||
ipython nbconvert --to latex --template book --post PDF Kalman_and_Bayesian_Filters_in_Python.ipynb
|
||||
ipython nbconvert --to PDF --template book Kalman_and_Bayesian_Filters_in_Python.ipynb
|
||||
|
||||
mv Kalman_and_Bayesian_Filters_in_Python.pdf ..
|
||||
echo "done."
|
||||
|
|
|
|||
|
|
@ -5,7 +5,7 @@ echo "merging book..."
|
|||
ipython short_merge_book.py > short.ipynb
|
||||
|
||||
echo "creating pdf..."
|
||||
ipython nbconvert --to latex --template book --post PDF short.ipynb
|
||||
ipython nbconvert --to PDF --template book short.ipynb
|
||||
|
||||
echo "done."
|
||||
|
||||
|
|
|
|||
|
|
@ -1,148 +1,158 @@
|
|||
{
|
||||
"metadata": {
|
||||
"name": "",
|
||||
"signature": "sha256:737a9a5aedce2bcd7e5a588bb46086800990f104bd246468e69e3098bc002ade"
|
||||
},
|
||||
"nbformat": 3,
|
||||
"nbformat_minor": 0,
|
||||
"worksheets": [
|
||||
"cells": [
|
||||
{
|
||||
"cells": [
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"<center><h1>Kalman and Bayesian Filters in Python</h1></center>\n",
|
||||
"<p>\n",
|
||||
" <p>\n",
|
||||
"Table of Contents\n",
|
||||
"-----\n",
|
||||
"\n",
|
||||
"[**Preface**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/00_Preface.ipynb)\n",
|
||||
" \n",
|
||||
"Motivation behind writing the book. How to download and read the book. Requirements for IPython Notebook and Python. github links.\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 1: The g-h Filter**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/01_g-h_filter.ipynb)\n",
|
||||
"\n",
|
||||
"Intuitive introduction to the g-h filter, which is a family of filters that includes the Kalman filter. Not filler - once you understand this chapter you will understand the concepts behind the Kalman filter. \n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 2: The Discrete Bayes Filter**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/02_Discrete_Bayes.ipynb)\n",
|
||||
"\n",
|
||||
"Introduces the Discrete Bayes Filter. From this you will learn the probabilistic reasoning that underpins the Kalman filter in an easy to digest form.\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 3: Least Squares Filter**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/03_Least_Squares_Filters.ipynb)\n",
|
||||
"\n",
|
||||
"Introduces the least squares filter in batch and recursive forms. I've not made a start on authoring this yet.\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 4: Gaussian Probabilities**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/04_Gaussians.ipynb)\n",
|
||||
"\n",
|
||||
"Introduces using Gaussians to represent beliefs in the Bayesian sense. Gaussians allow us to implement the algorithms used in the Discrete Bayes Filter to work in continuous domains.\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 5: One Dimensional Kalman Filters**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/05_Kalman_Filters.ipynb)\n",
|
||||
"\n",
|
||||
"Implements a Kalman filter by modifying the Discrete Bayesian Filter to use Gaussians. This is a full featured Kalman filter, albeit only useful for 1D problems. \n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 6: Multivariate Kalman Filter**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/06_Multivariate_Kalman_Filters.ipynb)\n",
|
||||
"\n",
|
||||
"We extend the Kalman filter developed in the previous chapter to the full, generalized filter. \n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 7: Kalman Filter Math**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/07_Kalman_Filter_Math.ipynb)\n",
|
||||
"\n",
|
||||
"We gotten about as far as we can without forming a strong mathematical foundation. This chapter is optional, especially the first time, but if you intend to write robust, numerically stable filters, or to read the literature, you will need to know this. \n",
|
||||
"\n",
|
||||
"*This still needs a lot of work. *\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 8: Designing Kalman Filters**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/08_Designing_Kalman_Filters.ipynb)\n",
|
||||
"\n",
|
||||
"Building on material in Chapter 6, walks you through the design of several Kalman filters. Discusses, but does not solve issues like numerical stability.\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 9: Nonlinear Filtering**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/09_Nonlinear_Filtering.ipynb)\n",
|
||||
"\n",
|
||||
"Kalman filter as covered only work for linear problems. Here I introduce the problems that nonlinear systems pose to the filter, and briefly discuss the various algorithms that we will be learning in subsequent chapters which work with nonlinear systems.\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 10: Unscented Kalman Filters**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/10_Unscented_Kalman_Filter.ipynb)\n",
|
||||
"\n",
|
||||
"Unscented Kalman filters (UKF) are a recent development in Kalman filter theory. They allow you to filter nonlinear problems without requiring a closed form solution like the Extended Kalman filter requires.\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 11: Extended Kalman Filters**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/11_Extended_Kalman_Filters.ipynb)\n",
|
||||
"\n",
|
||||
"Kalman filter as covered only work for linear problems. Extended Kalman filters (EKF) are the most common approach to linearizing non-linear problems.\n",
|
||||
"\n",
|
||||
"*Still very early going on this chapter.*\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 12: Designing Nonlinear Kalman Filters**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/12_Designing_Nonlinear_Kalman_Filters.ipynb)\n",
|
||||
"\n",
|
||||
"Works through some examples of the design of Kalman filters for nonlinear problems. *This is still very much a work in progress.*\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 13: Smoothing**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/13_Smoothing.ipynb)\n",
|
||||
"\n",
|
||||
"Kalman filters are recursive, and thus very suitable for real time filtering. However, they work well for post-processing data. We discuss some common approaches.\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 14: Adaptive Filtering**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/14_Adaptive_Filtering.ipynb)\n",
|
||||
" \n",
|
||||
"Kalman filters assume a single process model, but manuevering targets typically need to be described by several different process models. Adaptive filtering uses several techniques to allow the Kalman filter to adapt to the changing behavior of the target.\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 15: H-Infinity Filters**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/15_HInfinity_Filters.ipynb)\n",
|
||||
" \n",
|
||||
"Describes the $H_\\infty$ filter. \n",
|
||||
"\n",
|
||||
"*I have code that implements the filter, but no supporting text yet.*\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 16: Ensemble Kalman Filters**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/16_Ensemble_Kalman_Filters.ipynb)\n",
|
||||
"\n",
|
||||
"Discusses the ensemble Kalman Filter, which uses a Monte Carlo approach to deal with very large Kalman filter states in nonlinear systems.\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter XX: Numerical Stability**](not implemented)\n",
|
||||
"\n",
|
||||
"EKF and UKF are linear approximations of nonlinear problems. Unless programmed carefully, they are not numerically stable. We discuss some common approaches to this problem.\n",
|
||||
"\n",
|
||||
"*This chapter is not started. I'm likely to rearrange where this material goes - this is just a placeholder.*\n",
|
||||
"\n",
|
||||
"[**Chapter XX: Particle Filters**](not implemented)\n",
|
||||
" \n",
|
||||
"Particle filters uses a Monte Carlo technique to filter. \n",
|
||||
"\n",
|
||||
"*This is not implemented, and I have not decided if I want to make it part of this book or not.*\n",
|
||||
" \n",
|
||||
"\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Appendix: Installation, Python, NumPy, and filterpy**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/Appendix_A_Installation.ipynb)\n",
|
||||
"\n",
|
||||
"Brief introduction of Python and how it is used in this book. Description of the companion\n",
|
||||
"library filterpy. \n",
|
||||
" \n",
|
||||
"\n",
|
||||
"[**Appendix: Symbols and Notations**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/Appendix_B_Symbols_and_Notations.ipynb)\n",
|
||||
"\n",
|
||||
"Symbols and notations used in this book. Comparison with notations used in the literature.\n",
|
||||
"\n",
|
||||
"*Still just a collection of notes at this point.*\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"### Github repository\n",
|
||||
"http://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {}
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"<center><h1>Kalman and Bayesian Filters in Python</h1></center>\n",
|
||||
"<p>\n",
|
||||
" <p>\n",
|
||||
"Table of Contents\n",
|
||||
"-----\n",
|
||||
"\n",
|
||||
"[**Preface**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/00_Preface.ipynb)\n",
|
||||
" \n",
|
||||
"Motivation behind writing the book. How to download and read the book. Requirements for IPython Notebook and Python. github links.\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 1: The g-h Filter**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/01_g-h_filter.ipynb)\n",
|
||||
"\n",
|
||||
"Intuitive introduction to the g-h filter, which is a family of filters that includes the Kalman filter. Not filler - once you understand this chapter you will understand the concepts behind the Kalman filter. \n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 2: The Discrete Bayes Filter**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/02_Discrete_Bayes.ipynb)\n",
|
||||
"\n",
|
||||
"Introduces the Discrete Bayes Filter. From this you will learn the probabilistic reasoning that underpins the Kalman filter in an easy to digest form.\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 3: Least Squares Filter**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/03_Least_Squares_Filters.ipynb)\n",
|
||||
"\n",
|
||||
"Introduces the least squares filter in batch and recursive forms. I've not made a start on authoring this yet.\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 4: Gaussian Probabilities**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/04_Gaussians.ipynb)\n",
|
||||
"\n",
|
||||
"Introduces using Gaussians to represent beliefs in the Bayesian sense. Gaussians allow us to implement the algorithms used in the Discrete Bayes Filter to work in continuous domains.\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 5: One Dimensional Kalman Filters**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/05_Kalman_Filters.ipynb)\n",
|
||||
"\n",
|
||||
"Implements a Kalman filter by modifying the Discrete Bayesian Filter to use Gaussians. This is a full featured Kalman filter, albeit only useful for 1D problems. \n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 6: Multivariate Kalman Filter**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/06_Multivariate_Kalman_Filters.ipynb)\n",
|
||||
"\n",
|
||||
"We extend the Kalman filter developed in the previous chapter to the full, generalized filter. \n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 7: Kalman Filter Math**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/07_Kalman_Filter_Math.ipynb)\n",
|
||||
"\n",
|
||||
"We gotten about as far as we can without forming a strong mathematical foundation. This chapter is optional, especially the first time, but if you intend to write robust, numerically stable filters, or to read the literature, you will need to know this. \n",
|
||||
"\n",
|
||||
"*This still needs a lot of work. *\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 8: Designing Kalman Filters**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/08_Designing_Kalman_Filters.ipynb)\n",
|
||||
"\n",
|
||||
"Building on material in Chapter 6, walks you through the design of several Kalman filters. Discusses, but does not solve issues like numerical stability.\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 9: Nonlinear Filtering**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/09_Nonlinear_Filtering.ipynb)\n",
|
||||
"\n",
|
||||
"Kalman filter as covered only work for linear problems. Here I introduce the problems that nonlinear systems pose to the filter, and briefly discuss the various algorithms that we will be learning in subsequent chapters which work with nonlinear systems.\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 10: Unscented Kalman Filters**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/10_Unscented_Kalman_Filter.ipynb)\n",
|
||||
"\n",
|
||||
"Unscented Kalman filters (UKF) are a recent development in Kalman filter theory. They allow you to filter nonlinear problems without requiring a closed form solution like the Extended Kalman filter requires.\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 11: Extended Kalman Filters**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/11_Extended_Kalman_Filters.ipynb)\n",
|
||||
"\n",
|
||||
"Kalman filter as covered only work for linear problems. Extended Kalman filters (EKF) are the most common approach to linearizing non-linear problems.\n",
|
||||
"\n",
|
||||
"*Still very early going on this chapter.*\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 12: Designing Nonlinear Kalman Filters**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/12_Designing_Nonlinear_Kalman_Filters.ipynb)\n",
|
||||
"\n",
|
||||
"Works through some examples of the design of Kalman filters for nonlinear problems. *This is still very much a work in progress.*\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 13: Smoothing**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/13_Smoothing.ipynb)\n",
|
||||
"\n",
|
||||
"Kalman filters are recursive, and thus very suitable for real time filtering. However, they work well for post-processing data. We discuss some common approaches.\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 14: Adaptive Filtering**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/14_Adaptive_Filtering.ipynb)\n",
|
||||
" \n",
|
||||
"Kalman filters assume a single process model, but manuevering targets typically need to be described by several different process models. Adaptive filtering uses several techniques to allow the Kalman filter to adapt to the changing behavior of the target.\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 15: H-Infinity Filters**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/15_HInfinity_Filters.ipynb)\n",
|
||||
" \n",
|
||||
"Describes the $H_\\infty$ filter. \n",
|
||||
"\n",
|
||||
"*I have code that implements the filter, but no supporting text yet.*\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter 16: Ensemble Kalman Filters**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/16_Ensemble_Kalman_Filters.ipynb)\n",
|
||||
"\n",
|
||||
"Discusses the ensemble Kalman Filter, which uses a Monte Carlo approach to deal with very large Kalman filter states in nonlinear systems.\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Chapter XX: Numerical Stability**](not implemented)\n",
|
||||
"\n",
|
||||
"EKF and UKF are linear approximations of nonlinear problems. Unless programmed carefully, they are not numerically stable. We discuss some common approaches to this problem.\n",
|
||||
"\n",
|
||||
"*This chapter is not started. I'm likely to rearrange where this material goes - this is just a placeholder.*\n",
|
||||
"\n",
|
||||
"[**Chapter XX: Particle Filters**](not implemented)\n",
|
||||
" \n",
|
||||
"Particle filters uses a Monte Carlo technique to filter. \n",
|
||||
"\n",
|
||||
"*This is not implemented, and I have not decided if I want to make it part of this book or not.*\n",
|
||||
" \n",
|
||||
"\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Appendix: Installation, Python, NumPy, and filterpy**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/Appendix_A_Installation.ipynb)\n",
|
||||
"\n",
|
||||
"Brief introduction of Python and how it is used in this book. Description of the companion\n",
|
||||
"library filterpy. \n",
|
||||
" \n",
|
||||
"\n",
|
||||
"[**Appendix: Symbols and Notations**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/Appendix_B_Symbols_and_Notations.ipynb)\n",
|
||||
"\n",
|
||||
"Symbols and notations used in this book. Comparison with notations used in the literature.\n",
|
||||
"\n",
|
||||
"*Still just a collection of notes at this point.*\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"### Github repository\n",
|
||||
"http://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python\n"
|
||||
]
|
||||
}
|
||||
]
|
||||
}
|
||||
],
|
||||
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|
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|
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|
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|
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|
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|
||||
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|
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|
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|
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|
|
|
|||
Loading…
Reference in New Issue