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Fix equation labels
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‎notebooks/5.0-Visualization-and-modeling.ipynb‎

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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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"## Linear fits\n",
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{
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"cell_type": "markdown",
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"metadata": {
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"### Explicit formulas for uniform uncertainty\n",
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"which we can differentiate to find the extremal conditions\n",
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"\n",
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"\\begin{align}\n",
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"\\left.\\frac{\\partial{\\chi^2}}{\\partial m}\\right|_{\\hat{m},\\hat{c}} &= -\\frac{2}{\\alpha^2}\\sum_{k=1}^N[y_k - (\\hat{m}x_k + \\hat{c})]x_k = 0,\\\\\n",
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"\\left.\\frac{\\partial{\\chi^2}}{\\partial c}\\right|_{\\hat{m},\\hat{c}} &= -\\frac{2}{\\alpha^2}\\sum_{k=1}^N[y_k - (\\hat{m}x_k + \\hat{c})] = 0.\n",
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"\\left.\\frac{\\partial{\\chi^2}}{\\partial m}\\right|_{\\hat{m},\\hat{c}} &= -\\frac{2}{\\alpha^2}\\sum_{k=1}^N[y_k - (\\hat{m}x_k + \\hat{c})]x_k = 0,\\label{eq:extremal-m}\\tag{4}\\\\\n",
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"\\left.\\frac{\\partial{\\chi^2}}{\\partial c}\\right|_{\\hat{m},\\hat{c}} &= -\\frac{2}{\\alpha^2}\\sum_{k=1}^N[y_k - (\\hat{m}x_k + \\hat{c})] = 0.\\label{eq:extremal-c}\\tag{5}\n",
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"\\end{align}\n",
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"\n",
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"Solving this system of equations for $\\hat{m}$ and $\\hat{c}$ give *MU* Eqs. (5.1) and (5.2) (rewritten for consistency with the notation here),\n",
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"\n",
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"\\begin{align}\n",
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"\\hat{c} &= \\frac{\\left(\\sum_k x_k^2\\right)\\left(\\sum_k y_k\\right) - \\left(\\sum_k x_k\\right)\\left(\\sum_k x_k y_k\\right)}{\\Delta}, \\label{eq:c-hat}\\tag{4}\\\\\n",
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"\\hat{m} &= \\frac{N\\sum_k x_k y_k - \\left(\\sum_k x_k\\right)\\left(\\sum_k y_k\\right)}{\\Delta},\\quad\\text{with}\\\\\n",
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"\\Delta &= N\\sum_k x_k^2 - \\left(\\sum_k x_k\\right)^2. \\label{eq:m-hat}\\tag{5}\n",
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"\\hat{c} &= \\frac{\\left(\\sum_k x_k^2\\right)\\left(\\sum_k y_k\\right) - \\left(\\sum_k x_k\\right)\\left(\\sum_k x_k y_k\\right)}{\\Delta}, \\label{eq:c-hat}\\tag{6}\\\\\n",
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"\\hat{m} &= \\frac{N\\sum_k x_k y_k - \\left(\\sum_k x_k\\right)\\left(\\sum_k y_k\\right)}{\\Delta},\\quad\\text{with} \\label{eq:m-hat}\\tag{7}\\\\\n",
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"\\Delta &= N\\sum_k x_k^2 - \\left(\\sum_k x_k\\right)^2. \\label{eq:delta}\\tag{8}\n",
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"\\end{align}\n",
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"\n",
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"These equations express $\\hat{m}$ and $\\hat{c}$ *as unique, explicit functions of the data* that will *automatically* minimize $\\chi^2$ in Eq. (\\ref{eq:lsquniform}) for *any* values of $\\{x_k, y_k\\}$. This is one of the chief advantages of a linear fit: the least-squares fit parameters may be determined unambiguously from the data, without a search algorithm. It is important to recognize, though, that these equations will yield values even when $\\{x_k, y_k\\}$ do *not* satisfy a linear relationship, so we must always evaluate the quality of the fit before assigning much significance to the best-fit parameters obtained from it.\n",
@@ -872,34 +866,34 @@
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"Now that we have expressed $\\hat{m}$ and $\\hat{c}$ as functions of $\\{x_k, y_k\\}$, we can use standard error propagation to compute the uncertainties in $\\hat{m}$ and $\\hat{c}$. To do this, we must treat each measurement $y_k$ as an independent variable in the functions $\\hat{m}$ and $\\hat{c}$, all with standard error $\\alpha$. Recall that we have assumed the $x_k$ variables as known, so we do not need to propagate their uncertainty.\n",
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"\n",
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"\\begin{align}\n",
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"\\alpha_\\hat{m}^2 &= \\sum_{l = 1}^N\\left(\\frac{\\partial\\hat{m}}{\\partial y_l}\\right)^2\\alpha^2 = \\frac{\\alpha^2}{\\Delta^2}\\sum_{l = 1}^N\\left(Nx_l - \\sum_{k=1}^N x_k\\right)^2 = \\frac{N}{\\Delta}\\alpha^2,\\\\\n",
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"\\alpha_\\hat{c}^2 &= \\sum_{l = 1}^N\\left(\\frac{\\partial\\hat{c}}{\\partial y_l}\\right)^2\\alpha^2 = \\frac{\\alpha^2}{\\Delta^2}\\sum_{l = 1}^N\\left(\\sum_{k=1}^N x_k^2 - x_l\\sum_{k=1}^N x_k\\right)^2 = \\frac{\\sum_{k=1}^N x_k^2}{\\Delta}\\alpha^2.\n",
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"\\alpha_\\hat{m}^2 &= \\sum_{l = 1}^N\\left(\\frac{\\partial\\hat{m}}{\\partial y_l}\\right)^2\\alpha^2 = \\frac{\\alpha^2}{\\Delta^2}\\sum_{l = 1}^N\\left(Nx_l - \\sum_{k=1}^N x_k\\right)^2 = \\frac{N}{\\Delta}\\alpha^2, \\label{eq:alpha-m-hat-sq}\\tag{9}\\\\\n",
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"\\alpha_\\hat{c}^2 &= \\sum_{l = 1}^N\\left(\\frac{\\partial\\hat{c}}{\\partial y_l}\\right)^2\\alpha^2 = \\frac{\\alpha^2}{\\Delta^2}\\sum_{l = 1}^N\\left(\\sum_{k=1}^N x_k^2 - x_l\\sum_{k=1}^N x_k\\right)^2 = \\frac{\\sum_{k=1}^N x_k^2}{\\Delta}\\alpha^2.\\label{eq:alpha-c-hat-sq}\\tag{10}\n",
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"\\end{align}\n",
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"\n",
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"Taking the square root of these expressions, we have\n",
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"\n",
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"\\begin{align}\n",
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"\\alpha_{\\hat{m}} &= \\alpha\\sqrt{\\frac{N}{\\Delta}}, & \\alpha_{\\hat{c}} &= \\alpha\\sqrt{\\frac{\\sum_{k=1}^N x_k^2}{\\Delta}}, \\label{eq:alpha-m-c}\\tag{6}\n",
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"\\alpha_{\\hat{m}} &= \\alpha\\sqrt{\\frac{N}{\\Delta}}, & \\alpha_{\\hat{c}} &= \\alpha\\sqrt{\\frac{\\sum_{k=1}^N x_k^2}{\\Delta}}, \\label{eq:alpha-m-c}\\tag{11}\n",
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"\\end{align}\n",
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"\n",
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"which are equivalent to *MU* Eqs. (5.3) and (5.4) if we substitute $\\alpha = \\alpha_\\text{CU}$, where\n",
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"\n",
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"$$\n",
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"\\alpha_{CU} = \\sqrt{\\frac{1}{N-2}\\sum_{k=1}^{N}[y_k - (\\hat{m}x_k + \\hat{c})]^2} \\label{eq:alpha-CU}\\tag{7}\n",
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"\\alpha_{CU} = \\sqrt{\\frac{1}{N-2}\\sum_{k=1}^{N}[y_k - (\\hat{m}x_k + \\hat{c})]^2} \\label{eq:alpha-CU}\\tag{12}\n",
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"$$\n",
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"\n",
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"is known as the *common uncertainty*. The distinction here is that $\\alpha$ is assumed to be *known*, just as we assume $\\{x_k\\}$ known, while we determine $\\alpha_\\text{CU}$ from the same data that we use to obtain the fit. \n",
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"\n",
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"Just as the data will vary from one set of measurements to another, so will $\\alpha_\\text{CU}$, while $\\alpha$ remains fixed. If your estimate for $\\alpha$ is accurate, then the expectation of $\\alpha_\\text{CU}$ (ie, the average over $N\\rightarrow\\infty$ data sets) will be\n",
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"\n",
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"$$\n",
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"\\left\\langle \\alpha_\\text{CU}^2\\right\\rangle = \\alpha^2,\n",
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"\\left\\langle \\alpha_\\text{CU}^2\\right\\rangle = \\alpha^2,\\label{eq:E-alpha-CU}\\tag{13}\n",
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"$$\n",
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"\n",
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"so the difference between $\\alpha_\\text{CU}$ and $\\alpha$ is usually small. But if your estimate for $\\alpha$ is *inaccurate*, then the difference between $\\alpha_\\text{CU}$ and $\\alpha$ can be large, so comparing the two provides a good consistency check. The standard way to make this comparison is to compute\n",
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"\n",
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"$$\n",
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"\\chi^2_\\text{min} = \\chi^2(\\hat{m}, \\hat{c}) = \\frac{1}{\\alpha^2}\\sum_{k=1}^N[y_k - (\\hat{m}x_k + \\hat{c})]^2 = \\frac{\\alpha_{CU}^2}{\\alpha^2}(N-2),\n",
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"\\chi^2_\\text{min} = \\chi^2(\\hat{m}, \\hat{c}) = \\frac{1}{\\alpha^2}\\sum_{k=1}^N[y_k - (\\hat{m}x_k + \\hat{c})]^2 = \\frac{\\alpha_{CU}^2}{\\alpha^2}(N-2),\\label{eq:chi2}\\tag{14}\n",
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"$$\n",
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"\n",
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"and check that $\\chi^2_\\text{min}\\approx N - 2$. We compare to $N-2$ here because there are two free fit parameters, $\\hat{m}$ and $\\hat{c}$—in general, for a fit with $p$ fit parameters, we would compare to $N-p$. For more sophisticated ways to evaluate a fit, see the [Residual analysis](#Residual-analysis) section below.\n",
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"execution_count": null,
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{
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"### Exercise 2\n",
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"### Linear fits with `polyfit`\n",
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"### Exercise 3\n",
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"## Residual analysis\n",
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"### Exercise 4\n",
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