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| 91 | </tr> | 91 | </tr> |
| 92 | </table><p>As·before,·we·neglect·the·denominator,·which·has·no·effect·upon·the·maximisation·problem,·and·assume·a·uniform·prior·<img·src="images/img-1192.png"·alt="$\mathrm{P}\left(·\sigma·_\mathrm·{data}·|·\left\{··\mathbf{x}_·i·\right\}··\right)$"·style="vertical-align:-5px;· | 92 | </table><p>As·before,·we·neglect·the·denominator,·which·has·no·effect·upon·the·maximisation·problem,·and·assume·a·uniform·prior·<img·src="images/img-1192.png"·alt="$\mathrm{P}\left(·\sigma·_\mathrm·{data}·|·\left\{··\mathbf{x}_·i·\right\}··\right)$"·style="vertical-align:-5px;· |
| 93 | ·····································width:110px;· | 93 | ·····································width:110px;· |
| 94 | ·····································height:19px"·class="math·gen"·/>.·This·reduces·the·problem·to·the·maximisation·of·<img·src="images/img-1193.png"·alt="$\mathrm{P}\left(·\left\{··f_·i·\right\}··|·\sigma·_\mathrm·{data},·\left\{··\mathbf{x}_·i·\right\}··\right)$"·style="vertical-align:-5px;· | 94 | ·····································height:19px"·class="math·gen"·/>.·This·reduces·the·problem·to·the·maximisation·of·<img·src="images/img-1193.png"·alt="$\mathrm{P}\left(·\left\{··f_·i·\right\}··|·\sigma·_\mathrm·{data},·\left\{··\mathbf{x}_·i·\right\}··\right)$"·style="vertical-align:-5px;· |
| 95 | ·····································width:151px;· | 95 | ·····································width:151px;· |
| 96 | ·····································height:19px"·class="math·gen"·/>,·which·we·may·write·as·a·marginalised·probability·distribution·over·<img·src="images/img-1134.png"·alt="$\mathbf{u}$"·style="vertical-align:0px;· | 96 | ·····································height:19px"·class="math·gen"·/>,·which·we·may·write·as·a·marginalised·probability·distribution·over·<img·src="images/img-1134.png"·alt="$\mathbf{u}$"·style="vertical-align:0px;· |
| 97 | ·····································width:12px;· | 97 | ·····································width:12px;· |
| 98 | ·····································height:8px"·class="math·gen"·/>:·</p><table·id="<plasTeX.TeXFragment·object·at·0xffff | 98 | ·····································height:8px"·class="math·gen"·/>:·</p><table·id="<plasTeX.TeXFragment·object·at·0xffff7bc05460>"·cellpadding="7"·width="100%"·cellspacing="0"·class="eqnarray"> |
| 99 | <tr·id="a0000001991"> | 99 | <tr·id="a0000001991"> |
| 100 | ···· | 100 | ···· |
| 101 | ····<td·style="width:40%"> </td> | 101 | ····<td·style="width:40%"> </td> |
| 102 | ···· | 102 | ···· |
| 103 | ···· | 103 | ···· |
| 104 | ········<td·style="vertical-align:middle;····································text-align:right"><img·src="images/img-1194.png"·alt="$\displaystyle··\label{eqa:p_·f_·given_·sigma}·\mathrm{P}\left(·\left\{··f_·i·\right\}··|·\sigma·_\mathrm·{data},·\left\{··\mathbf{x}_·i·\right\}··\right)·=·\idotsint·_{-\infty·}^{\infty·}··$"·style="vertical-align:-16px;·width:256px;·····················height:42px"·class="math·gen"·/></td> | 104 | ········<td·style="vertical-align:middle;····································text-align:right"><img·src="images/img-1194.png"·alt="$\displaystyle··\label{eqa:p_·f_·given_·sigma}·\mathrm{P}\left(·\left\{··f_·i·\right\}··|·\sigma·_\mathrm·{data},·\left\{··\mathbf{x}_·i·\right\}··\right)·=·\idotsint·_{-\infty·}^{\infty·}··$"·style="vertical-align:-16px;·width:256px;·····················height:42px"·class="math·gen"·/></td> |
| 105 | ···· | 105 | ···· |
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| 181 | ·····································height:21px"·class="math·gen"·/>.·In·the·small-angle·approximation,·such·that·<img·src="images/img-0402.png"·alt="$\sin·\theta·\approx·\theta·$"·style="vertical-align:0px;· | 181 | ·····································height:21px"·class="math·gen"·/>.·In·the·small-angle·approximation,·such·that·<img·src="images/img-0402.png"·alt="$\sin·\theta·\approx·\theta·$"·style="vertical-align:0px;· |
| 182 | ·····································width:67px;· | 182 | ·····································width:67px;· |
| 183 | ·····································height:13px"·class="math·gen"·/>,·it·reduces·to·the·equation·for·simple·harmonic·motion,·with·the·solution·</p></td> | 183 | ·····································height:13px"·class="math·gen"·/>,·it·reduces·to·the·equation·for·simple·harmonic·motion,·with·the·solution·</p></td> |
| 184 | </tr><tr> | 184 | </tr><tr> |
| 185 | ···· | 185 | ···· |
| 186 | ····<td·style="text-align:left;·border-right:1px·solid·black;·border-left:1px·solid·black"><table·id="<plasTeX.TeXFragment·object·at·0xffff | 186 | ····<td·style="text-align:left;·border-right:1px·solid·black;·border-left:1px·solid·black"><table·id="<plasTeX.TeXFragment·object·at·0xffff7ec786d0>"·class="equation"·width="100%"·cellspacing="0"·cellpadding="7"> |
| 187 | <tr> | 187 | <tr> |
| 188 | ···· | 188 | ···· |
| 189 | ····<td·style="width:40%"> </td> | 189 | ····<td·style="width:40%"> </td> |
| 190 | ····<td><img·src="images/img-0403.png"·alt="\begin{equation}··\theta·_\mathrm·{approx}=\omega·\sin·\left(\sqrt·{\frac{g}{l}}t\right).·\label{eq:pendulum_·approx}·\end{equation}"·style="width:188px;· | 190 | ····<td><img·src="images/img-0403.png"·alt="\begin{equation}··\theta·_\mathrm·{approx}=\omega·\sin·\left(\sqrt·{\frac{g}{l}}t\right).·\label{eq:pendulum_·approx}·\end{equation}"·style="width:188px;· |
| 191 | ····························height:45px"·class="math·gen"·/></td> | 191 | ····························height:45px"·class="math·gen"·/></td> |
| 192 | ···· | 192 | ···· |
| 193 | ····<td·style="width:40%"> </td> | 193 | ····<td·style="width:40%"> </td> |
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| 203 | ····<td·style="text-align:left;·border-right:1px·solid·black;·border-left:1px·solid·black"><p>A·more·exact·solution·requires·integration·of·the·second-order·differential·equation·of·motion·including·the·<img·src="images/img-0404.png"·alt="$\sin·\theta·$"·style="vertical-align:0px;· | 203 | ····<td·style="text-align:left;·border-right:1px·solid·black;·border-left:1px·solid·black"><p>A·more·exact·solution·requires·integration·of·the·second-order·differential·equation·of·motion·including·the·<img·src="images/img-0404.png"·alt="$\sin·\theta·$"·style="vertical-align:0px;· |
| 204 | ·····································width:34px;· | 204 | ·····································width:34px;· |
| 205 | ·····································height:13px"·class="math·gen"·/>·term.·This·integral·cannot·be·done·analytically,·but·the·solution·can·be·written·in·the·form·</p></td> | 205 | ·····································height:13px"·class="math·gen"·/>·term.·This·integral·cannot·be·done·analytically,·but·the·solution·can·be·written·in·the·form·</p></td> |
| 206 | </tr><tr> | 206 | </tr><tr> |
| 207 | ···· | 207 | ···· |
| 208 | ····<td·style="text-align:left;·border-right:1px·solid·black;·border-left:1px·solid·black"><table·id="<plasTeX.TeXFragment·object·at·0xffff | 208 | ····<td·style="text-align:left;·border-right:1px·solid·black;·border-left:1px·solid·black"><table·id="<plasTeX.TeXFragment·object·at·0xffff7ec78b48>"·class="equation"·width="100%"·cellspacing="0"·cellpadding="7"> |
| 209 | <tr> | 209 | <tr> |
| 210 | ···· | 210 | ···· |
| 211 | ····<td·style="width:40%"> </td> | 211 | ····<td·style="width:40%"> </td> |
| 212 | ····<td><img·src="images/img-0405.png"·alt="\begin{equation}··\theta·_\mathrm·{exact}(t)·=·2\sin·^{-1}\left[·k\,·\mathrm{sn}\left(\sqrt·{\frac{g}{l}}t,k\right)\right].·\label{eq:pendulum_·exact}·\end{equation}"·style="width:286px;· | 212 | ····<td><img·src="images/img-0405.png"·alt="\begin{equation}··\theta·_\mathrm·{exact}(t)·=·2\sin·^{-1}\left[·k\,·\mathrm{sn}\left(\sqrt·{\frac{g}{l}}t,k\right)\right].·\label{eq:pendulum_·exact}·\end{equation}"·style="width:286px;· |
| 213 | ····························height:45px"·class="math·gen"·/></td> | 213 | ····························height:45px"·class="math·gen"·/></td> |
| 214 | ···· | 214 | ···· |
| 215 | ····<td·style="width:40%"> </td> | 215 | ····<td·style="width:40%"> </td> |
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| 67 | ·····································width:19px;· | 67 | ·····································width:19px;· |
| 68 | ·····································height:10px"·class="math·gen"·/>-dimensional·Gaussian·distribution·around·<img·src="images/img-1141.png"·alt="$\mathbf{u}^0$"·style="vertical-align:0px;· | 68 | ·····································height:10px"·class="math·gen"·/>-dimensional·Gaussian·distribution·around·<img·src="images/img-1141.png"·alt="$\mathbf{u}^0$"·style="vertical-align:0px;· |
| 69 | ·····································width:19px;· | 69 | ·····································width:19px;· |
| 70 | ·····································height:16px"·class="math·gen"·/>.·Taking·a·Taylor·expansion·of·<img·src="images/img-1142.png"·alt="$L(\mathbf{u})$"·style="vertical-align:-5px;· | 70 | ·····································height:16px"·class="math·gen"·/>.·Taking·a·Taylor·expansion·of·<img·src="images/img-1142.png"·alt="$L(\mathbf{u})$"·style="vertical-align:-5px;· |
| 71 | ·····································width:37px;· | 71 | ·····································width:37px;· |
| 72 | ·····································height:19px"·class="math·gen"·/>·about·<img·src="images/img-1141.png"·alt="$\mathbf{u}^0$"·style="vertical-align:0px;· | 72 | ·····································height:19px"·class="math·gen"·/>·about·<img·src="images/img-1141.png"·alt="$\mathbf{u}^0$"·style="vertical-align:0px;· |
| 73 | ·····································width:19px;· | 73 | ·····································width:19px;· |
| 74 | ·····································height:16px"·class="math·gen"·/>,·we·can·write:·</p><table·id="<plasTeX.TeXFragment·object·at·0xffff | 74 | ·····································height:16px"·class="math·gen"·/>,·we·can·write:·</p><table·id="<plasTeX.TeXFragment·object·at·0xffff7bca7460>"·cellpadding="7"·width="100%"·cellspacing="0"·class="eqnarray"> |
| 75 | <tr·id="a0000001968"> | 75 | <tr·id="a0000001968"> |
| 76 | ···· | 76 | ···· |
| 77 | ····<td·style="width:40%"> </td> | 77 | ····<td·style="width:40%"> </td> |
| 78 | ···· | 78 | ···· |
| 79 | ···· | 79 | ···· |
| 80 | ········<td·style="vertical-align:middle;····································text-align:right"><img·src="images/img-1143.png"·alt="$\displaystyle··L(\mathbf{u})··$"·style="vertical-align:-5px;·width:37px;·····················height:19px"·class="math·gen"·/></td> | 80 | ········<td·style="vertical-align:middle;····································text-align:right"><img·src="images/img-1143.png"·alt="$\displaystyle··L(\mathbf{u})··$"·style="vertical-align:-5px;·width:37px;·····················height:19px"·class="math·gen"·/></td> |
| 81 | ···· | 81 | ···· |
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| 113 | </tr> | 113 | </tr> |
| 114 | </table><p>Since·the·logarithm·of·a·Gaussian·distribution·is·a·parabola,·the·quadratic·terms·in·the·above·expansion·encode·the·Gaussian·component·of·the·probability·distribution·<img·src="images/img-1135.png"·alt="$\mathrm{P}\left(·\mathbf{u}·|·\left\{··\mathbf{x}_·i,·f_·i,·\sigma·_·i·\right\}··\right)$"·style="vertical-align:-5px;· | 114 | </table><p>Since·the·logarithm·of·a·Gaussian·distribution·is·a·parabola,·the·quadratic·terms·in·the·above·expansion·encode·the·Gaussian·component·of·the·probability·distribution·<img·src="images/img-1135.png"·alt="$\mathrm{P}\left(·\mathbf{u}·|·\left\{··\mathbf{x}_·i,·f_·i,·\sigma·_·i·\right\}··\right)$"·style="vertical-align:-5px;· |
| 115 | ·····································width:129px;· | 115 | ·····································width:129px;· |
| 116 | ·····································height:19px"·class="math·gen"·/>·about·<img·src="images/img-1141.png"·alt="$\mathbf{u}^0$"·style="vertical-align:0px;· | 116 | ·····································height:19px"·class="math·gen"·/>·about·<img·src="images/img-1141.png"·alt="$\mathbf{u}^0$"·style="vertical-align:0px;· |
| 117 | ·····································width:19px;· | 117 | ·····································width:19px;· |
| 118 | ·····································height:16px"·class="math·gen"·/>.<a·href="#a0000001970"·class="footnote"><sup·class="footnotemark">1</sup></a>·We·may·write·the·sum·of·these·terms,·which·we·denote·<img·src="images/img-1146.png"·alt="$Q$"·style="vertical-align:-4px;· | 118 | ·····································height:16px"·class="math·gen"·/>.<a·href="#a0000001970"·class="footnote"><sup·class="footnotemark">1</sup></a>·We·may·write·the·sum·of·these·terms,·which·we·denote·<img·src="images/img-1146.png"·alt="$Q$"·style="vertical-align:-4px;· |
| 119 | ·····································width:14px;· | 119 | ·····································width:14px;· |
| 120 | ·····································height:17px"·class="math·gen"·/>,·in·matrix·form:·</p><table·id="<plasTeX.TeXFragment·object·at·0xffff | 120 | ·····································height:17px"·class="math·gen"·/>,·in·matrix·form:·</p><table·id="<plasTeX.TeXFragment·object·at·0xffff7bcc70b8>"·class="equation"·width="100%"·cellspacing="0"·cellpadding="7"> |
| 121 | <tr> | 121 | <tr> |
| 122 | ···· | 122 | ···· |
| 123 | ····<td·style="width:40%"> </td> | 123 | ····<td·style="width:40%"> </td> |
| 124 | ····<td><img·src="images/img-1147.png"·alt="\begin{equation}··Q·=·\frac{1}{2}·\left(\mathbf{u}·-·\mathbf{u}^0\right)^\mathbf·{T}·\mathbf{A}·\left(\mathbf{u}·-·\mathbf{u}^0\right)·\label{eqn:Q_·vector}·\end{equation}"·style="width:226px;· | 124 | ····<td><img·src="images/img-1147.png"·alt="\begin{equation}··Q·=·\frac{1}{2}·\left(\mathbf{u}·-·\mathbf{u}^0\right)^\mathbf·{T}·\mathbf{A}·\left(\mathbf{u}·-·\mathbf{u}^0\right)·\label{eqn:Q_·vector}·\end{equation}"·style="width:226px;· |
| 125 | ····························height:37px"·class="math·gen"·/></td> | 125 | ····························height:37px"·class="math·gen"·/></td> |
| 126 | ···· | 126 | ···· |
| 127 | ····<td·style="width:40%"> </td> | 127 | ····<td·style="width:40%"> </td> |
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| 55 | </span> | 55 | </span> |
| 56 | <hr·/> | 56 | <hr·/> |
| 57 | </div> | 57 | </div> |
| 58 | <div><h1·id="a0000000300">C.4·The·covariance·matrix</h1> | 58 | <div><h1·id="a0000000300">C.4·The·covariance·matrix</h1> |
| 59 | <p>·<a·name="a0000001974"·id="a0000001974"></a>·</p><p>The·terms·of·the·covariance·matrix·<img·src="images/img-1159.png"·alt="$V_{ij}$"·style="vertical-align:-5px;· | 59 | <p>·<a·name="a0000001974"·id="a0000001974"></a>·</p><p>The·terms·of·the·covariance·matrix·<img·src="images/img-1159.png"·alt="$V_{ij}$"·style="vertical-align:-5px;· |
| 60 | ·····································width:21px;· | 60 | ·····································width:21px;· |
| 61 | ·····································height:18px"·class="math·gen"·/>·are·defined·by:·</p><table·id="<plasTeX.TeXFragment·object·at·0xffff | 61 | ·····································height:18px"·class="math·gen"·/>·are·defined·by:·</p><table·id="<plasTeX.TeXFragment·object·at·0xffff7bc58db8>"·class="equation"·width="100%"·cellspacing="0"·cellpadding="7"> |
| 62 | <tr> | 62 | <tr> |
| 63 | ···· | 63 | ···· |
| 64 | ····<td·style="width:40%"> </td> | 64 | ····<td·style="width:40%"> </td> |
| 65 | ····<td><img·src="images/img-1160.png"·alt="\begin{equation}··V_{ij}·=·\left<·\left(u_·i·-·u^0_·i\right)·\left(u_·j·-·u^0_·j\right)·\right>·\label{eqn:def_·covar}·\end{equation}"·style="width:213px;· | 65 | ····<td><img·src="images/img-1160.png"·alt="\begin{equation}··V_{ij}·=·\left<·\left(u_·i·-·u^0_·i\right)·\left(u_·j·-·u^0_·j\right)·\right>·\label{eqn:def_·covar}·\end{equation}"·style="width:213px;· |
| 66 | ····························height:24px"·class="math·gen"·/></td> | 66 | ····························height:24px"·class="math·gen"·/></td> |
| 67 | ···· | 67 | ···· |
| 68 | ····<td·style="width:40%"> </td> | 68 | ····<td·style="width:40%"> </td> |
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| 181 | ····························height:45px"·class="math·gen"·/></td> | 181 | ····························height:45px"·class="math·gen"·/></td> |
| 182 | ···· | 182 | ···· |
| 183 | ····<td·style="width:40%"> </td> | 183 | ····<td·style="width:40%"> </td> |
| 184 | ····<td·class="eqnnum"·style="width:20%"><span>(<span>C.11</span>)</span></td> | 184 | ····<td·class="eqnnum"·style="width:20%"><span>(<span>C.11</span>)</span></td> |
| 185 | </tr> | 185 | </tr> |
| 186 | </table><p>which·we·may·identify·as·equalling·<img·src="images/img-1159.png"·alt="$V_{ij}$"·style="vertical-align:-5px;· | 186 | </table><p>which·we·may·identify·as·equalling·<img·src="images/img-1159.png"·alt="$V_{ij}$"·style="vertical-align:-5px;· |
| 187 | ·····································width:21px;· | 187 | ·····································width:21px;· |
| 188 | ·····································height:18px"·class="math·gen"·/>:·</p><table·id="<plasTeX.TeXFragment·object·at·0xffff | 188 | ·····································height:18px"·class="math·gen"·/>:·</p><table·id="<plasTeX.TeXFragment·object·at·0xffff7bcd3b48>"·cellpadding="7"·width="100%"·cellspacing="0"·class="eqnarray"> |
| 189 | <tr·id="a0000001982"> | 189 | <tr·id="a0000001982"> |
| 190 | ···· | 190 | ···· |
| 191 | ····<td·style="width:40%"> </td> | 191 | ····<td·style="width:40%"> </td> |
| 192 | ···· | 192 | ···· |
| 193 | ···· | 193 | ···· |
| 194 | ········<td·style="vertical-align:middle;····································text-align:right"><img·src="images/img-1177.png"·alt="$\displaystyle··\label{eqa:v_·zrelate}·V_{ij}··$"·style="vertical-align:-5px;·width:21px;·····················height:18px"·class="math·gen"·/></td> | 194 | ········<td·style="vertical-align:middle;····································text-align:right"><img·src="images/img-1177.png"·alt="$\displaystyle··\label{eqa:v_·zrelate}·V_{ij}··$"·style="vertical-align:-5px;·width:21px;·····················height:18px"·class="math·gen"·/></td> |
| 195 | ···· | 195 | ···· |