diff --git a/doc/src/LectureNotes/_build/.doctrees/chapter1.doctree b/doc/src/LectureNotes/_build/.doctrees/chapter1.doctree new file mode 100644 index 000000000..eeb601dca Binary files /dev/null and b/doc/src/LectureNotes/_build/.doctrees/chapter1.doctree differ diff --git a/doc/src/LectureNotes/_build/.doctrees/chapter2.doctree b/doc/src/LectureNotes/_build/.doctrees/chapter2.doctree new file mode 100644 index 000000000..6a273686b Binary files /dev/null and b/doc/src/LectureNotes/_build/.doctrees/chapter2.doctree differ diff --git a/doc/src/LectureNotes/_build/.doctrees/chapter3.doctree b/doc/src/LectureNotes/_build/.doctrees/chapter3.doctree new file mode 100644 index 000000000..14b23d1e0 Binary files /dev/null and b/doc/src/LectureNotes/_build/.doctrees/chapter3.doctree differ diff --git a/doc/src/LectureNotes/_build/.doctrees/chapter4.doctree b/doc/src/LectureNotes/_build/.doctrees/chapter4.doctree new file mode 100644 index 000000000..374dbd1bd Binary files /dev/null and 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similarity index 99% rename from doc/src/LectureNotes/_build/jupyter_execute/introduction.ipynb rename to doc/src/LectureNotes/_build/jupyter_execute/chapter1.ipynb index 9eb624962..7b8b8ccb1 100644 --- a/doc/src/LectureNotes/_build/jupyter_execute/introduction.ipynb +++ b/doc/src/LectureNotes/_build/jupyter_execute/chapter1.ipynb @@ -447,9 +447,10 @@ ":titlesonly:\n", "\n", "\n", - "gettingstarted.ipynb\n", - "regression.ipynb\n", - "logistic.ipynb\n", + "chapter2\n", + "chapter3\n", + "chapter4\n", + "chapter5\n", "```\n" ] } diff --git a/doc/src/LectureNotes/_build/jupyter_execute/introduction.txt b/doc/src/LectureNotes/_build/jupyter_execute/chapter1.txt similarity index 99% rename from doc/src/LectureNotes/_build/jupyter_execute/introduction.txt rename to doc/src/LectureNotes/_build/jupyter_execute/chapter1.txt index dd97f9c8c..9f0fc835a 100644 --- a/doc/src/LectureNotes/_build/jupyter_execute/introduction.txt +++ b/doc/src/LectureNotes/_build/jupyter_execute/chapter1.txt @@ -436,7 +436,8 @@ society. :titlesonly: -gettingstarted.ipynb -regression.ipynb -logistic.ipynb +chapter2 +chapter3 +chapter4 +chapter5 ``` diff --git a/doc/src/LectureNotes/_build/jupyter_execute/gettingstarted.ipynb b/doc/src/LectureNotes/_build/jupyter_execute/chapter2.ipynb similarity index 92% rename from doc/src/LectureNotes/_build/jupyter_execute/gettingstarted.ipynb rename to doc/src/LectureNotes/_build/jupyter_execute/chapter2.ipynb index 705ebbb6f..d80d66bf9 100644 --- a/doc/src/LectureNotes/_build/jupyter_execute/gettingstarted.ipynb +++ b/doc/src/LectureNotes/_build/jupyter_execute/chapter2.ipynb @@ -449,9 +449,9 @@ "name": "stdout", "output_type": "stream", "text": [ - "[ 0.48917724 -0.43782011 1.48192392 2.25192475 -0.05455206 -0.68134058\n", - " -0.11602272 0.66295579 0.4799895 0.52405335]\n", - "-0.437820110993849\n" + "[ 0.66109153 0.97132954 -0.91470448 1.81588452 0.49665195 0.77698678\n", + " 1.04576423 1.65832096 -0.25613699 -0.7554783 ]\n", + "0.971329541323762\n" ] } ], @@ -806,26 +806,26 @@ "name": "stdout", "output_type": "stream", "text": [ - "[[0.58695697 0.26052142 0.14110843 0.74356519 0.20479382 0.07526258\n", - " 0.86862247 0.86613041 0.08332556 0.96366793]\n", - " [0.94558515 0.79716346 0.68245511 0.34262527 0.82136273 0.7050853\n", - " 0.53146547 0.7064895 0.07048963 0.69034727]\n", - " [0.49126259 0.30355312 0.05233209 0.86899932 0.13256945 0.44525314\n", - " 0.58698083 0.10149771 0.89831811 0.14480922]\n", - " [0.15768243 0.65091856 0.80481376 0.68506764 0.07613184 0.56283625\n", - " 0.81314443 0.98202227 0.33982193 0.03551219]\n", - " [0.94931184 0.48115403 0.11278354 0.88971209 0.81385201 0.64257017\n", - " 0.90409915 0.33969279 0.09492318 0.88539763]\n", - " [0.18975396 0.16404114 0.59567791 0.99452361 0.92938817 0.9646773\n", - " 0.1462189 0.95940328 0.86272261 0.26091217]\n", - " [0.22807853 0.50767509 0.67349029 0.95174387 0.96480061 0.75037774\n", - " 0.10062502 0.27772682 0.88819491 0.71719298]\n", - " [0.22080447 0.19928524 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[0.23509059 0.79672774 0.81910307 0.53487394 0.50992632 0.112086\n", + " 0.55751711 0.10662789 0.70911163 0.59072491]\n", + " [0.0259775 0.26815865 0.5090044 0.41809535 0.32232515 0.72743704\n", + " 0.32110496 0.04202295 0.18265714 0.70084503]\n", + " [0.07736581 0.57300172 0.97099592 0.62357997 0.36579921 0.37062279\n", + " 0.74683978 0.59433006 0.35540585 0.97384662]\n", + " [0.25205634 0.7601454 0.40639052 0.24504215 0.21745329 0.95788719\n", + " 0.7490677 0.29789413 0.81839129 0.93822723]\n", + " [0.6302074 0.47072129 0.16215217 0.53540566 0.91760002 0.14146861\n", + " 0.37860338 0.8099108 0.07897476 0.69254202]]\n" ] } ], @@ -917,13 +917,13 @@ "name": "stdout", "output_type": "stream", "text": [ - "0.10286439863835316\n", - "4.149835827233691\n", - "0.47221917888337617\n", - "[[0.74868647 2.17169586 1.71154957]\n", - " [2.17169586 7.49304629 5.02306261]\n", - " [1.71154957 5.02306261 6.91687749]]\n", - "[12.86269572 0.09473336 2.20118117]\n" + "-0.13489611340279922\n", + "3.7349937337703607\n", + "-0.6150223388222948\n", + "[[ 1.15822308 3.44928776 3.99889831]\n", + " [ 3.44928776 11.11151113 12.25819175]\n", + " [ 3.99889831 12.25819175 21.09855273]]\n", + "[30.28557034 0.07629455 3.00642205]\n" ] } ], @@ -973,7 +973,7 @@ }, "metadata": { "filenames": { - "image/png": "/Users/hjensen/Teaching/FYS-STK4150/doc/src/LectureNotes/_build/jupyter_execute/gettingstarted_42_1.png" + "image/png": "/Users/hjensen/Teaching/FYS-STK4150/doc/src/LectureNotes/_build/jupyter_execute/chapter2_42_1.png" }, "needs_background": "light" }, @@ -1841,7 +1841,7 @@ }, "metadata": { "filenames": { - "image/png": "/Users/hjensen/Teaching/FYS-STK4150/doc/src/LectureNotes/_build/jupyter_execute/gettingstarted_54_2.png" + "image/png": "/Users/hjensen/Teaching/FYS-STK4150/doc/src/LectureNotes/_build/jupyter_execute/chapter2_54_2.png" } }, "output_type": "display_data" @@ -1855,7 +1855,7 @@ }, "metadata": { "filenames": { - "image/png": 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\n", + "image/png": 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Released under CC Attribution-NonCommercial 4.0 license\n", + "\n", + "\n", + "\n", + "\n", + "## To do list\n", + "\n", + "* add math about MVN and define MLE and other quantities\n", + "\n", + "* rewrite about covariance matrix\n", + "\n", + "* add KL theorem\n", + "\n", + "## Domains and probabilities\n", + "Consider the following simple example, namely the tossing of two dice, resulting in the following possible values" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\{2,3,4,5,6,7,8,9,10,11,12\\}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "These values are called the *domain*. \n", + "To this domain we have the corresponding *probabilities*" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\{1/36,2/36/,3/36,4/36,5/36,6/36,5/36,4/36,3/36,2/36,1/36\\}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Tossing the dice\n", + "The numbers in the domain are the outcomes of the physical process of tossing say two dice.\n", + "We cannot tell beforehand whether the outcome is 3 or 5 or any other number in this domain.\n", + "This defines the randomness of the outcome, or unexpectedness or any other synonimous word which\n", + "encompasses the uncertitude of the final outcome. \n", + "\n", + "The only thing we can tell beforehand\n", + "is that say the outcome 2 has a certain probability. \n", + "If our favorite hobby is to spend an hour every evening throwing dice and \n", + "registering the sequence of outcomes, we will note that the numbers in the above domain" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\{2,3,4,5,6,7,8,9,10,11,12\\},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "appear in a random order. After 11 throws the results may look like" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\{10,8,6,3,6,9,11,8,12,4,5\\}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Stochastic variables\n", + "\n", + "**Random variables are characterized by a domain which contains all possible values that the random value may take. This domain has a corresponding probability distribution function(PDF)**.\n", + "\n", + "\n", + "\n", + "## Stochastic variables and the main concepts, the discrete case\n", + "There are two main concepts associated with a stochastic variable. The\n", + "*domain* is the set $\\mathbb D = \\{x\\}$ of all accessible values\n", + "the variable can assume, so that $X \\in \\mathbb D$. An example of a\n", + "discrete domain is the set of six different numbers that we may get by\n", + "throwing of a dice, $x\\in\\{1,\\,2,\\,3,\\,4,\\,5,\\,6\\}$.\n", + "\n", + "The *probability distribution function (PDF)* is a function\n", + "$p(x)$ on the domain which, in the discrete case, gives us the\n", + "probability or relative frequency with which these values of $X$\n", + "occur" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "p(x) = \\mathrm{Prob}(X=x).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Stochastic variables and the main concepts, the continuous case\n", + "In the continuous case, the PDF does not directly depict the\n", + "actual probability. Instead we define the probability for the\n", + "stochastic variable to assume any value on an infinitesimal interval\n", + "around $x$ to be $p(x)dx$. The continuous function $p(x)$ then gives us\n", + "the *density* of the probability rather than the probability\n", + "itself. The probability for a stochastic variable to assume any value\n", + "on a non-infinitesimal interval $[a,\\,b]$ is then just the integral" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathrm{Prob}(a\\leq X\\leq b) = \\int_a^b p(x)dx.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Qualitatively speaking, a stochastic variable represents the values of\n", + "numbers chosen as if by chance from some specified PDF so that the\n", + "selection of a large set of these numbers reproduces this PDF.\n", + "\n", + "\n", + "\n", + "## The cumulative probability\n", + "Of interest to us is the *cumulative probability\n", + "distribution function* (**CDF**), $P(x)$, which is just the probability\n", + "for a stochastic variable $X$ to assume any value less than $x$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "P(x)=\\mathrm{Prob(}X\\leq x\\mathrm{)} =\n", + "\\int_{-\\infty}^x p(x^{\\prime})dx^{\\prime}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The relation between a CDF and its corresponding PDF is then" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "p(x) = \\frac{d}{dx}P(x).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Properties of PDFs\n", + "\n", + "There are two properties that all PDFs must satisfy. The first one is\n", + "positivity (assuming that the PDF is normalized)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "0 \\leq p(x) \\leq 1.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Naturally, it would be nonsensical for any of the values of the domain\n", + "to occur with a probability greater than $1$ or less than $0$. Also,\n", + "the PDF must be normalized. That is, all the probabilities must add up\n", + "to unity. The probability of \"anything\" to happen is always unity. For\n", + "both discrete and continuous PDFs, this condition is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{align*}\n", + "\\sum_{x_i\\in\\mathbb D} p(x_i) & = 1,\\\\\n", + "\\int_{x\\in\\mathbb D} p(x)\\,dx & = 1.\n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Important distributions, the uniform distribution\n", + "The first one\n", + "is the most basic PDF; namely the uniform distribution" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + "p(x) = \\frac{1}{b-a}\\theta(x-a)\\theta(b-x).\n", + "\\label{eq:unifromPDF} \\tag{1}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "For $a=0$ and $b=1$ we have" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{array}{ll}\n", + "p(x)dx = dx & \\in [0,1].\n", + "\\end{array}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The latter distribution is used to generate random numbers. For other PDFs, one needs normally a mapping from this distribution to say for example the exponential distribution.\n", + "\n", + "\n", + "\n", + "## Gaussian distribution\n", + "The second one is the Gaussian Distribution" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "p(x) = \\frac{1}{\\sigma\\sqrt{2\\pi}} \\exp{(-\\frac{(x-\\mu)^2}{2\\sigma^2})},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "with mean value $\\mu$ and standard deviation $\\sigma$. If $\\mu=0$ and $\\sigma=1$, it is normally called the **standard normal distribution**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "p(x) = \\frac{1}{\\sqrt{2\\pi}} \\exp{(-\\frac{x^2}{2})},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The following simple Python code plots the above distribution for different values of $\\mu$ and $\\sigma$." + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "
" + ] + }, + "metadata": { + "filenames": { + "image/png": "/Users/hjensen/Teaching/FYS-STK4150/doc/src/LectureNotes/_build/jupyter_execute/chapter3_29_0.png" + }, + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "%matplotlib inline\n", + "\n", + "import numpy as np\n", + "from math import acos, exp, sqrt\n", + "from matplotlib import pyplot as plt\n", + "from matplotlib import rc, rcParams\n", + "import matplotlib.units as units\n", + "import matplotlib.ticker as ticker\n", + "rc('text',usetex=True)\n", + "rc('font',**{'family':'serif','serif':['Gaussian distribution']})\n", + "font = {'family' : 'serif',\n", + " 'color' : 'darkred',\n", + " 'weight' : 'normal',\n", + " 'size' : 16,\n", + " }\n", + "pi = acos(-1.0)\n", + "mu0 = 0.0\n", + "sigma0 = 1.0\n", + "mu1= 1.0\n", + "sigma1 = 2.0\n", + "mu2 = 2.0\n", + "sigma2 = 4.0\n", + "\n", + "x = np.linspace(-20.0, 20.0)\n", + "v0 = np.exp(-(x*x-2*x*mu0+mu0*mu0)/(2*sigma0*sigma0))/sqrt(2*pi*sigma0*sigma0)\n", + "v1 = np.exp(-(x*x-2*x*mu1+mu1*mu1)/(2*sigma1*sigma1))/sqrt(2*pi*sigma1*sigma1)\n", + "v2 = np.exp(-(x*x-2*x*mu2+mu2*mu2)/(2*sigma2*sigma2))/sqrt(2*pi*sigma2*sigma2)\n", + "plt.plot(x, v0, 'b-', x, v1, 'r-', x, v2, 'g-')\n", + "plt.title(r'{\\bf Gaussian distributions}', fontsize=20)\n", + "plt.text(-19, 0.3, r'Parameters: $\\mu = 0$, $\\sigma = 1$', fontdict=font)\n", + "plt.text(-19, 0.18, r'Parameters: $\\mu = 1$, $\\sigma = 2$', fontdict=font)\n", + "plt.text(-19, 0.08, r'Parameters: $\\mu = 2$, $\\sigma = 4$', fontdict=font)\n", + "plt.xlabel(r'$x$',fontsize=20)\n", + "plt.ylabel(r'$p(x)$ [MeV]',fontsize=20)\n", + "\n", + "# Tweak spacing to prevent clipping of ylabel \n", + "plt.subplots_adjust(left=0.15)\n", + "plt.savefig('gaussian.pdf', format='pdf')\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Exponential distribution\n", + "Another important distribution in science is the exponential distribution" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "p(x) = \\alpha\\exp{-(\\alpha x)}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Expectation values\n", + "Let $h(x)$ be an arbitrary continuous function on the domain of the stochastic\n", + "variable $X$ whose PDF is $p(x)$. We define the *expectation value*\n", + "of $h$ with respect to $p$ as follows" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + "\\langle h \\rangle_X \\equiv \\int\\! h(x)p(x)\\,dx\n", + "\\label{eq:expectation_value_of_h_wrt_p} \\tag{2}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Whenever the PDF is known implicitly, like in this case, we will drop\n", + "the index $X$ for clarity. \n", + "A particularly useful class of special expectation values are the\n", + "*moments*. The $n$-th moment of the PDF $p$ is defined as\n", + "follows" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\langle x^n \\rangle \\equiv \\int\\! x^n p(x)\\,dx\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Stochastic variables and the main concepts, mean values\n", + "The zero-th moment $\\langle 1\\rangle$ is just the normalization condition of\n", + "$p$. The first moment, $\\langle x\\rangle$, is called the *mean* of $p$\n", + "and often denoted by the letter $\\mu$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\langle x\\rangle = \\mu \\equiv \\int x p(x)dx,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "for a continuous distribution and" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\langle x\\rangle = \\mu \\equiv \\sum_{i=1}^N x_i p(x_i),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "for a discrete distribution. \n", + "Qualitatively it represents the centroid or the average value of the\n", + "PDF and is therefore simply called the expectation value of $p(x)$.\n", + "\n", + "\n", + "\n", + "## Stochastic variables and the main concepts, central moments, the variance\n", + "\n", + "A special version of the moments is the set of *central moments*, the n-th central moment defined as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\langle (x-\\langle x\\rangle )^n\\rangle \\equiv \\int\\! (x-\\langle x\\rangle)^n p(x)\\,dx\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The zero-th and first central moments are both trivial, equal $1$ and\n", + "$0$, respectively. But the second central moment, known as the\n", + "*variance* of $p$, is of particular interest. For the stochastic\n", + "variable $X$, the variance is denoted as $\\sigma^2_X$ or $\\mathrm{Var}(X)$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{align*}\n", + "\\sigma^2_X &=\\mathrm{Var}(X) = \\langle (x-\\langle x\\rangle)^2\\rangle =\n", + "\\int (x-\\langle x\\rangle)^2 p(x)dx\\\\\n", + "& = \\int\\left(x^2 - 2 x \\langle x\\rangle^{2} +\\langle x\\rangle^2\\right)p(x)dx\\\\\n", + "& = \\langle x^2\\rangle - 2 \\langle x\\rangle\\langle x\\rangle + \\langle x\\rangle^2\\\\\n", + "& = \\langle x^2 \\rangle - \\langle x\\rangle^2\n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The square root of the variance, $\\sigma =\\sqrt{\\langle (x-\\langle x\\rangle)^2\\rangle}$ is called the \n", + "**standard deviation** of $p$. It is the RMS (root-mean-square)\n", + "value of the deviation of the PDF from its mean value, interpreted\n", + "qualitatively as the \"spread\" of $p$ around its mean.\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Probability Distribution Functions\n", + "\n", + "The following table collects properties of probability distribution functions.\n", + "In our notation we reserve the label $p(x)$ for the probability of a certain event,\n", + "while $P(x)$ is the cumulative probability. \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "
Discrete PDF Continuous PDF
Domain $\\left\\{x_1, x_2, x_3, \\dots, x_N\\right\\}$ $[a,b]$
Probability $p(x_i)$ $p(x)dx$
Cumulative $P_i=\\sum_{l=1}^ip(x_l)$ $P(x)=\\int_a^xp(t)dt$
Positivity $0 \\le p(x_i) \\le 1$ $p(x) \\ge 0$
Positivity $0 \\le P_i \\le 1$ $0 \\le P(x) \\le 1$
Monotonic $P_i \\ge P_j$ if $x_i \\ge x_j$ $P(x_i) \\ge P(x_j)$ if $x_i \\ge x_j$
Normalization $P_N=1$ $P(b)=1$
\n", + "\n", + "\n", + "\n", + "\n", + "## Probability Distribution Functions\n", + "With a PDF we can compute expectation values of selected quantities such as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\langle x^k\\rangle=\\sum_{i=1}^{N}x_i^kp(x_i),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "if we have a discrete PDF or" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\langle x^k\\rangle=\\int_a^b x^kp(x)dx,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "in the case of a continuous PDF. We have already defined the mean value $\\mu$\n", + "and the variance $\\sigma^2$.\n", + "\n", + "\n", + "\n", + "## The three famous Probability Distribution Functions\n", + "\n", + "There are at least three PDFs which one may encounter. These are the\n", + "\n", + "**Uniform distribution**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "p(x)=\\frac{1}{b-a}\\Theta(x-a)\\Theta(b-x),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "yielding probabilities different from zero in the interval $[a,b]$.\n", + "\n", + "**The exponential distribution**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "p(x)=\\alpha \\exp{(-\\alpha x)},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "yielding probabilities different from zero in the interval $[0,\\infty)$ and with mean value" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mu = \\int_0^{\\infty}xp(x)dx=\\int_0^{\\infty}x\\alpha \\exp{(-\\alpha x)}dx=\\frac{1}{\\alpha},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "with variance" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\sigma^2=\\int_0^{\\infty}x^2p(x)dx-\\mu^2 = \\frac{1}{\\alpha^2}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Probability Distribution Functions, the normal distribution\n", + "Finally, we have the so-called univariate normal distribution, or just the **normal distribution**" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "p(x)=\\frac{1}{b\\sqrt{2\\pi}}\\exp{\\left(-\\frac{(x-a)^2}{2b^2}\\right)}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "with probabilities different from zero in the interval $(-\\infty,\\infty)$.\n", + "The integral $\\int_{-\\infty}^{\\infty}\\exp{\\left(-(x^2\\right)}dx$ appears in many calculations, its value\n", + "is $\\sqrt{\\pi}$, a result we will need when we compute the mean value and the variance.\n", + "The mean value is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mu = \\int_0^{\\infty}xp(x)dx=\\frac{1}{b\\sqrt{2\\pi}}\\int_{-\\infty}^{\\infty}x \\exp{\\left(-\\frac{(x-a)^2}{2b^2}\\right)}dx,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which becomes with a suitable change of variables" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mu =\\frac{1}{b\\sqrt{2\\pi}}\\int_{-\\infty}^{\\infty}b\\sqrt{2}(a+b\\sqrt{2}y)\\exp{-y^2}dy=a.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Probability Distribution Functions, the normal distribution\n", + "Similarly, the variance becomes" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\sigma^2 = \\frac{1}{b\\sqrt{2\\pi}}\\int_{-\\infty}^{\\infty}(x-\\mu)^2 \\exp{\\left(-\\frac{(x-a)^2}{2b^2}\\right)}dx,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and inserting the mean value and performing a variable change we obtain" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\sigma^2 = \\frac{1}{b\\sqrt{2\\pi}}\\int_{-\\infty}^{\\infty}b\\sqrt{2}(b\\sqrt{2}y)^2\\exp{\\left(-y^2\\right)}dy=\n", + "\\frac{2b^2}{\\sqrt{\\pi}}\\int_{-\\infty}^{\\infty}y^2\\exp{\\left(-y^2\\right)}dy,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and performing a final integration by parts we obtain the well-known result $\\sigma^2=b^2$.\n", + "It is useful to introduce the standard normal distribution as well, defined by $\\mu=a=0$, viz. a distribution\n", + "centered around zero and with a variance $\\sigma^2=1$, leading to" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " p(x)=\\frac{1}{\\sqrt{2\\pi}}\\exp{\\left(-\\frac{x^2}{2}\\right)}.\n", + "\\label{_auto1} \\tag{3}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Probability Distribution Functions, the cumulative distribution\n", + "\n", + "The exponential and uniform distributions have simple cumulative functions,\n", + "whereas the normal distribution does not, being proportional to the so-called\n", + "error function $erf(x)$, given by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "P(x) = \\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^x\\exp{\\left(-\\frac{t^2}{2}\\right)}dt,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which is difficult to evaluate in a quick way.\n", + "\n", + "\n", + "\n", + "\n", + "## Probability Distribution Functions, other important distribution\n", + "\n", + "Some other PDFs which one encounters often in the natural sciences are the binomial distribution" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "p(x) = \\left(\\begin{array}{c} n \\\\ x\\end{array}\\right)y^x(1-y)^{n-x} \\hspace{0.5cm}x=0,1,\\dots,n,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $y$ is the probability for a specific event, such as the tossing of a coin or moving left or right\n", + "in case of a random walker. Note that $x$ is a discrete stochastic variable. \n", + "\n", + "The sequence of binomial trials is characterized by the following definitions\n", + "\n", + " * Every experiment is thought to consist of $N$ independent trials.\n", + "\n", + " * In every independent trial one registers if a specific situation happens or not, such as the jump to the left or right of a random walker.\n", + "\n", + " * The probability for every outcome in a single trial has the same value, for example the outcome of tossing (either heads or tails) a coin is always $1/2$.\n", + "\n", + "\n", + "\n", + "## Probability Distribution Functions, the binomial distribution\n", + "\n", + "In order to compute the mean and variance we need to recall Newton's binomial\n", + "formula" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "(a+b)^m=\\sum_{n=0}^m \\left(\\begin{array}{c} m \\\\ n\\end{array}\\right)a^nb^{m-n},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which can be used to show that" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\sum_{x=0}^n\\left(\\begin{array}{c} n \\\\ x\\end{array}\\right)y^x(1-y)^{n-x} = (y+1-y)^n = 1,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "the PDF is normalized to one. \n", + "The mean value is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mu = \\sum_{x=0}^n x\\left(\\begin{array}{c} n \\\\ x\\end{array}\\right)y^x(1-y)^{n-x} =\n", + "\\sum_{x=0}^n x\\frac{n!}{x!(n-x)!}y^x(1-y)^{n-x},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "resulting in" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mu = \n", + "\\sum_{x=0}^n x\\frac{(n-1)!}{(x-1)!(n-1-(x-1))!}y^{x-1}(1-y)^{n-1-(x-1)},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which we rewrite as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mu=ny\\sum_{\\nu=0}^n\\left(\\begin{array}{c} n-1 \\\\ \\nu\\end{array}\\right)y^{\\nu}(1-y)^{n-1-\\nu} =ny(y+1-y)^{n-1}=ny.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The variance is slightly trickier to get. It reads $\\sigma^2=ny(1-y)$. \n", + "\n", + "## Probability Distribution Functions, Poisson's distribution\n", + "\n", + "Another important distribution with discrete stochastic variables $x$ is \n", + "the Poisson model, which resembles the exponential distribution and reads" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "p(x) = \\frac{\\lambda^x}{x!} e^{-\\lambda} \\hspace{0.5cm}x=0,1,\\dots,;\\lambda > 0.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In this case both the mean value and the variance are easier to calculate," + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mu = \\sum_{x=0}^{\\infty} x \\frac{\\lambda^x}{x!} e^{-\\lambda} = \\lambda e^{-\\lambda}\\sum_{x=1}^{\\infty}\n", + "\\frac{\\lambda^{x-1}}{(x-1)!}=\\lambda,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and the variance is $\\sigma^2=\\lambda$.\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Probability Distribution Functions, Poisson's distribution\n", + "An example of applications of the Poisson distribution could be the counting\n", + "of the number of $\\alpha$-particles emitted from a radioactive source in a given time interval.\n", + "In the limit of $n\\rightarrow \\infty$ and for small probabilities $y$, the binomial distribution\n", + "approaches the Poisson distribution. Setting $\\lambda = ny$, with $y$ the probability for an event in\n", + "the binomial distribution we can show that" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\lim_{n\\rightarrow \\infty}\\left(\\begin{array}{c} n \\\\ x\\end{array}\\right)y^x(1-y)^{n-x} e^{-\\lambda}=\\sum_{x=1}^{\\infty}\\frac{\\lambda^x}{x!} e^{-\\lambda}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Meet the covariance!\n", + "An important quantity in a statistical analysis is the so-called covariance. \n", + "\n", + "Consider the set $\\{X_i\\}$ of $n$\n", + "stochastic variables (not necessarily uncorrelated) with the\n", + "multivariate PDF $P(x_1,\\dots,x_n)$. The *covariance* of two\n", + "of the stochastic variables, $X_i$ and $X_j$, is defined as follows" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + "\\mathrm{Cov}(X_i,\\,X_j) = \\langle (x_i-\\langle x_i\\rangle)(x_j-\\langle x_j\\rangle)\\rangle \n", + "\\label{_auto2} \\tag{4}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation} \n", + "=\\int\\cdots\\int (x_i-\\langle x_i\\rangle)(x_j-\\langle x_j\\rangle)P(x_1,\\dots,x_n)\\,dx_1\\dots dx_n,\n", + "\\label{eq:def_covariance} \\tag{5}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "with" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\langle x_i\\rangle =\n", + "\\int\\cdots\\int x_i P(x_1,\\dots,x_n)\\,dx_1\\dots dx_n.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Meet the covariance in matrix disguise\n", + "If we consider the above covariance as a matrix" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "C_{ij} =\\mathrm{Cov}(X_i,\\,X_j),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "then the diagonal elements are just the familiar\n", + "variances, $C_{ii} = \\mathrm{Cov}(X_i,\\,X_i) = \\mathrm{Var}(X_i)$. It turns out that\n", + "all the off-diagonal elements are zero if the stochastic variables are\n", + "uncorrelated.\n", + "\n", + "\n", + "\n", + "## Covariance" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "1.3311287509199639\n", + "[[14.31659816 13.99268172 19.21873233 14.42933266 18.41189432 16.6094975\n", + " 11.00141571 4.4034975 14.34205163 7.63509233]\n", + " [13.99268172 13.67609397 18.78390394 14.10286557 17.99532083 16.23370367\n", + " 10.75250606 4.30386732 14.0175593 7.46234655]\n", + " [19.21873233 18.78390394 25.79940208 19.37006815 24.71629535 22.29674138\n", + " 14.76840108 5.91129532 19.2529013 10.24941779]\n", + " [14.42933266 14.10286557 19.37006815 14.54295487 18.55687678 16.74028719\n", + " 11.08804516 4.43817236 14.45498656 7.69521404]\n", + " [18.41189432 17.99532083 24.71629535 18.55687678 23.67865945 21.36068284\n", + " 14.14839622 5.66312819 18.44462881 9.81912823]\n", + " [16.6094975 16.23370367 22.29674138 16.74028719 21.36068284 19.26962007\n", + " 12.76336631 5.1087472 16.63902752 8.85790364]\n", + " [11.00141571 10.75250606 14.76840108 11.08804516 14.14839622 12.76336631\n", + " 8.45390408 3.38381409 11.02097512 5.86709383]\n", + " [ 4.4034975 4.30386732 5.91129532 4.43817236 5.66312819 5.1087472\n", + " 3.38381409 1.35442722 4.41132648 2.34840076]\n", + " [14.34205163 14.0175593 19.2529013 14.45498656 18.44462881 16.63902752\n", + " 11.02097512 4.41132648 14.36755036 7.64866676]\n", + " [ 7.63509233 7.46234655 10.24941779 7.69521404 9.81912823 8.85790364\n", + " 5.86709383 2.34840076 7.64866676 4.07182169]]\n" + ] + } + ], + "source": [ + "# Importing various packages\n", + "from math import exp, sqrt\n", + "from random import random, seed\n", + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "\n", + "def covariance(x, y, n):\n", + " sum = 0.0\n", + " mean_x = np.mean(x)\n", + " mean_y = np.mean(y)\n", + " for i in range(0, n):\n", + " sum += (x[(i)]-mean_x)*(y[i]-mean_y)\n", + " return sum/n\n", + "\n", + "n = 10\n", + "\n", + "x=np.random.normal(size=n)\n", + "y = 4+3*x+np.random.normal(size=n)\n", + "covxy = covariance(x,y,n)\n", + "print(covxy)\n", + "z = np.vstack((x, y))\n", + "c = np.cov(z.T)\n", + "\n", + "print(c)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Meet the covariance, uncorrelated events\n", + "\n", + "Consider the stochastic variables $X_i$ and $X_j$, ($i\\neq j$). We have" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{align*}\n", + "Cov(X_i,\\,X_j) &= \\langle (x_i-\\langle x_i\\rangle)(x_j-\\langle x_j\\rangle)\\rangle\\\\\n", + "&=\\langle x_i x_j - x_i\\langle x_j\\rangle - \\langle x_i\\rangle x_j + \\langle x_i\\rangle\\langle x_j\\rangle\\rangle\\\\\n", + "&=\\langle x_i x_j\\rangle - \\langle x_i\\langle x_j\\rangle\\rangle - \\langle \\langle x_i\\rangle x_j \\rangle +\n", + "\\langle \\langle x_i\\rangle\\langle x_j\\rangle\\rangle \\\\\n", + "&=\\langle x_i x_j\\rangle - \\langle x_i\\rangle\\langle x_j\\rangle - \\langle x_i\\rangle\\langle x_j\\rangle +\n", + "\\langle x_i\\rangle\\langle x_j\\rangle \\\\\n", + "&=\\langle x_i x_j\\rangle - \\langle x_i\\rangle\\langle x_j\\rangle\n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "If $X_i$ and $X_j$ are independent (assuming $i \\neq j$), we have that" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\langle x_i x_j\\rangle = \\langle x_i\\rangle\\langle x_j\\rangle,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "leading to" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "Cov(X_i, X_j) = 0 \\hspace{0.1cm} (i\\neq j).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Numerical experiments and the covariance\n", + "\n", + "Now that we have constructed an idealized mathematical framework, let\n", + "us try to apply it to empirical observations. Examples of relevant\n", + "physical phenomena may be spontaneous decays of nuclei, or a purely\n", + "mathematical set of numbers produced by some deterministic\n", + "mechanism. It is the latter we will deal with, using so-called pseudo-random\n", + "number generators. In general our observations will contain only a limited set of\n", + "observables. We remind the reader that\n", + "a *stochastic process* is a process that produces sequentially a\n", + "chain of values" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\{x_1, x_2,\\dots\\,x_k,\\dots\\}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Numerical experiments and the covariance\n", + "We will call these\n", + "values our *measurements* and the entire set as our measured\n", + "*sample*. The action of measuring all the elements of a sample\n", + "we will call a stochastic *experiment* (since, operationally,\n", + "they are often associated with results of empirical observation of\n", + "some physical or mathematical phenomena; precisely an experiment). We\n", + "assume that these values are distributed according to some \n", + "PDF $p_X^{\\phantom X}(x)$, where $X$ is just the formal symbol for the\n", + "stochastic variable whose PDF is $p_X^{\\phantom X}(x)$. Instead of\n", + "trying to determine the full distribution $p$ we are often only\n", + "interested in finding the few lowest moments, like the mean\n", + "$\\mu_X^{\\phantom X}$ and the variance $\\sigma_X^{\\phantom X}$.\n", + "\n", + "\n", + "\n", + "\n", + "## Numerical experiments and the covariance, actual situations\n", + "In practical situations however, a sample is always of finite size. Let that\n", + "size be $n$. The expectation value of a sample $\\alpha$, the **sample mean**, is then defined as follows" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\langle x_{\\alpha} \\rangle \\equiv \\frac{1}{n}\\sum_{k=1}^n x_{\\alpha,k}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The *sample variance* is:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathrm{Var}(x) \\equiv \\frac{1}{n}\\sum_{k=1}^n (x_{\\alpha,k} - \\langle x_{\\alpha} \\rangle)^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "with its square root being the *standard deviation of the sample*.\n", + "\n", + "\n", + "\n", + "\n", + "## Numerical experiments and the covariance, our observables\n", + "You can think of the above observables as a set of quantities which define\n", + "a given experiment. This experiment is then repeated several times, say $m$ times.\n", + "The total average is then" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + "\\langle X_m \\rangle= \\frac{1}{m}\\sum_{\\alpha=1}^mx_{\\alpha}=\\frac{1}{mn}\\sum_{\\alpha, k} x_{\\alpha,k},\n", + "\\label{eq:exptmean} \\tag{6}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where the last sums end at $m$ and $n$.\n", + "The total variance is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\sigma^2_m= \\frac{1}{mn^2}\\sum_{\\alpha=1}^m(\\langle x_{\\alpha} \\rangle-\\langle X_m \\rangle)^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which we rewrite as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + "\\sigma^2_m=\\frac{1}{m}\\sum_{\\alpha=1}^m\\sum_{kl=1}^n (x_{\\alpha,k}-\\langle X_m \\rangle)(x_{\\alpha,l}-\\langle X_m \\rangle).\n", + "\\label{eq:exptvariance} \\tag{7}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Numerical experiments and the covariance, the sample variance\n", + "\n", + "We define also the sample variance $\\sigma^2$ of all $mn$ individual experiments as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + "\\sigma^2=\\frac{1}{mn}\\sum_{\\alpha=1}^m\\sum_{k=1}^n (x_{\\alpha,k}-\\langle X_m \\rangle)^2.\n", + "\\label{eq:sampleexptvariance} \\tag{8}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "These quantities, being known experimental values or the results from our calculations, \n", + "may differ, in some cases\n", + "significantly, from the similarly named\n", + "exact values for the mean value $\\mu_X$, the variance $\\mathrm{Var}(X)$\n", + "and the covariance $\\mathrm{Cov}(X,Y)$.\n", + "\n", + "\n", + "\n", + "## Numerical experiments and the covariance, central limit theorem\n", + "\n", + "The central limit theorem states that the PDF $\\tilde{p}(z)$ of\n", + "the average of $m$ random values corresponding to a PDF $p(x)$ \n", + "is a normal distribution whose mean is the \n", + "mean value of the PDF $p(x)$ and whose variance is the variance\n", + "of the PDF $p(x)$ divided by $m$, the number of values used to compute $z$.\n", + "\n", + "The central limit theorem leads then to the well-known expression for the\n", + "standard deviation, given by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\sigma_m=\n", + "\\frac{\\sigma}{\\sqrt{m}}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In many cases the above estimate for the standard deviation, in particular if correlations are strong, may be too simplistic. We need therefore a more precise defintion of the error and the variance in our results.\n", + "\n", + "\n", + "\n", + "## Definition of Correlation Functions and Standard Deviation\n", + "Our estimate of the true average $\\mu_{X}$ is the sample mean $\\langle X_m \\rangle$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mu_{X}^{\\phantom X} \\approx X_m=\\frac{1}{mn}\\sum_{\\alpha=1}^m\\sum_{k=1}^n x_{\\alpha,k}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can then use Eq. ([7](#eq:exptvariance))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\sigma^2_m=\\frac{1}{mn^2}\\sum_{\\alpha=1}^m\\sum_{kl=1}^n (x_{\\alpha,k}-\\langle X_m \\rangle)(x_{\\alpha,l}-\\langle X_m \\rangle),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and rewrite it as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\sigma^2_m=\\frac{\\sigma^2}{n}+\\frac{2}{mn^2}\\sum_{\\alpha=1}^m\\sum_{k\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + "\\kappa_d = \\frac{f_d}{\\sigma^2}\n", + "\\label{eq:autocorrelformal} \\tag{9}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which gives us a useful measure of the correlation pair correlation\n", + "starting always at $1$ for $d=0$.\n", + "\n", + "\n", + "\n", + "## Definition of Correlation Functions and Standard Deviation, sample variance\n", + "\n", + "The sample variance of the $mn$ experiments can now be\n", + "written in terms of the autocorrelation function" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + "\\sigma_m^2=\\frac{\\sigma^2}{n}+\\frac{2}{n}\\cdot\\sigma^2\\sum_{d=1}^{n-1}\n", + "\\frac{f_d}{\\sigma^2}=\\left(1+2\\sum_{d=1}^{n-1}\\kappa_d\\right)\\frac{1}{n}\\sigma^2=\\frac{\\tau}{n}\\cdot\\sigma^2\n", + "\\label{eq:error_estimate_corr_time} \\tag{10}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and we see that $\\sigma_m$ can be expressed in terms of the\n", + "uncorrelated sample variance times a correction factor $\\tau$ which\n", + "accounts for the correlation between measurements. We call this\n", + "correction factor the *autocorrelation time*" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + "\\tau = 1+2\\sum_{d=1}^{n-1}\\kappa_d\n", + "\\label{eq:autocorrelation_time} \\tag{11}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "\n", + "For a correlation free experiment, $\\tau$\n", + "equals 1.\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Definition of Correlation Functions and Standard Deviation\n", + "From the point of view of\n", + "Eq. ([10](#eq:error_estimate_corr_time)) we can interpret a sequential\n", + "correlation as an effective reduction of the number of measurements by\n", + "a factor $\\tau$. The effective number of measurements becomes" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "n_\\mathrm{eff} = \\frac{n}{\\tau}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "To neglect the autocorrelation time $\\tau$ will always cause our\n", + "simple uncorrelated estimate of $\\sigma_m^2\\approx \\sigma^2/n$ to\n", + "be less than the true sample error. The estimate of the error will be\n", + "too \"good\". On the other hand, the calculation of the full\n", + "autocorrelation time poses an efficiency problem if the set of\n", + "measurements is very large. The solution to this problem is given by \n", + "more practically oriented methods like the blocking technique.\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Code to compute the Covariance matrix and the Covariance" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "0.2642716177294466\n", + "4.940580327634602\n", + "1.1829871582902483\n", + "1.015877605436128 9.696167748586095 17.72713727297909\n", + "2.9616975181296192 3.4095763130714993 9.375087003275128\n", + "[[ 1.01587761 2.96169752 3.40957631]\n", + " [ 2.96169752 9.69616775 9.375087 ]\n", + " [ 3.40957631 9.375087 17.72713727]]\n", + "[24.75536438 0.07216528 3.61165297]\n" + ] + } + ], + "source": [ + "# Importing various packages\n", + "from math import exp, sqrt\n", + "from random import random, seed\n", + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "\n", + "# Sample covariance, note the factor 1/(n-1)\n", + "def covariance(x, y, n):\n", + " sum = 0.0\n", + " mean_x = np.mean(x)\n", + " mean_y = np.mean(y)\n", + " for i in range(0, n):\n", + " sum += (x[(i)]-mean_x)*(y[i]-mean_y)\n", + " return sum/(n-1.)\n", + "\n", + "n = 100\n", + "x = np.random.normal(size=n)\n", + "print(np.mean(x))\n", + "y = 4+3*x+np.random.normal(size=n)\n", + "print(np.mean(y))\n", + "z = x**3+np.random.normal(size=n)\n", + "print(np.mean(z))\n", + "covxx = covariance(x,x,n)\n", + "covyy = covariance(y,y,n)\n", + "covzz = covariance(z,z,n)\n", + "covxy = covariance(x,y,n)\n", + "covxz = covariance(x,z,n)\n", + "covyz = covariance(y,z,n)\n", + "print(covxx,covyy, covzz)\n", + "print(covxy,covxz, covyz)\n", + "w = np.vstack((x, y, z))\n", + "#print(w)\n", + "c = np.cov(w)\n", + "print(c)\n", + "#eigen = np.zeros(n)\n", + "Eigvals, Eigvecs = np.linalg.eig(c)\n", + "print(Eigvals)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Random Numbers\n", + "\n", + "Uniform deviates are just random numbers that lie within a specified range\n", + "(typically 0 to 1), with any one number in the range just as likely as any other. They\n", + "are, in other words, what you probably think random numbers are. However,\n", + "we want to distinguish uniform deviates from other sorts of random numbers, for\n", + "example numbers drawn from a normal (Gaussian) distribution of specified mean\n", + "and standard deviation. These other sorts of deviates are almost always generated by\n", + "performing appropriate operations on one or more uniform deviates, as we will see\n", + "in subsequent sections. So, a reliable source of random uniform deviates, the subject\n", + "of this section, is an essential building block for any sort of stochastic modeling\n", + "or Monte Carlo computer work.\n", + "\n", + "\n", + "\n", + "\n", + "# Random Numbers, better name: pseudo random numbers\n", + "\n", + "A disclaimer is however appropriate. It should be fairly obvious that \n", + "something as deterministic as a computer cannot generate purely random numbers.\n", + "\n", + "Numbers generated by any of the standard algorithms are in reality pseudo random\n", + "numbers, hopefully abiding to the following criteria:\n", + "\n", + " * they produce a uniform distribution in the interval [0,1].\n", + "\n", + " * correlations between random numbers are negligible\n", + "\n", + " * the period before the same sequence of random numbers is repeated is as large as possible and finally\n", + "\n", + " * the algorithm should be fast.\n", + "\n", + "\n", + "\n", + "\n", + "# Random number generator RNG\n", + " The most common random number generators are based on so-called\n", + "Linear congruential relations of the type" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "N_i=(aN_{i-1}+c) \\mathrm{MOD} (M),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which yield a number in the interval [0,1] through" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "x_i=N_i/M\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The number \n", + "$M$ is called the period and it should be as large as possible \n", + " and \n", + "$N_0$ is the starting value, or seed. The function $\\mathrm{MOD}$ means the remainder,\n", + "that is if we were to evaluate $(13)\\mathrm{MOD}(9)$, the outcome is the remainder\n", + "of the division $13/9$, namely $4$.\n", + "\n", + "\n", + "\n", + "# Random number generator RNG and periodic outputs\n", + "\n", + "The problem with such generators is that their outputs are periodic;\n", + "they \n", + "will start to repeat themselves with a period that is at most $M$. If however\n", + "the parameters $a$ and $c$ are badly chosen, the period may be even shorter.\n", + "\n", + "Consider the following example" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "N_i=(6N_{i-1}+7) \\mathrm{MOD} (5),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "with a seed $N_0=2$. This generator produces the sequence\n", + "$4,1,3,0,2,4,1,3,0,2,...\\dots$, i.e., a sequence with period $5$.\n", + "However, increasing $M$ may not guarantee a larger period as the following\n", + "example shows" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "N_i=(27N_{i-1}+11) \\mathrm{MOD} (54),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which still, with $N_0=2$, results in $11,38,11,38,11,38,\\dots$, a period of\n", + "just $2$.\n", + "\n", + "\n", + "\n", + "# Random number generator RNG and its period\n", + "Typical periods for the random generators provided in the program library \n", + "are of the order of $\\sim 10^9$ or larger. Other random number generators which have\n", + "become increasingly popular are so-called shift-register generators.\n", + "In these generators each successive number depends on many preceding\n", + "values (rather than the last values as in the linear congruential\n", + "generator).\n", + "For example, you could make a shift register generator whose $l$th \n", + "number is the sum of the $l-i$th and $l-j$th values with modulo $M$," + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "N_l=(aN_{l-i}+cN_{l-j})\\mathrm{MOD}(M).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Random number generator RNG, other examples\n", + "Such a generator again produces a sequence of pseudorandom numbers\n", + "but this time with a period much larger than $M$.\n", + "It is also possible to construct more elaborate algorithms by including\n", + "more than two past terms in the sum of each iteration.\n", + "One example is the generator of [Marsaglia and Zaman](http://dl.acm.org/citation.cfm?id=187154)\n", + "which consists of two congruential relations" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " N_l=(N_{l-3}-N_{l-1})\\mathrm{MOD}(2^{31}-69),\n", + "\\label{eq:mz1} \\tag{12}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "followed by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " N_l=(69069N_{l-1}+1013904243)\\mathrm{MOD}(2^{32}),\n", + "\\label{eq:mz2} \\tag{13}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which according to the authors has a period larger than $2^{94}$.\n", + "\n", + "\n", + "\n", + "# Random number generator RNG, other examples\n", + "Instead of using modular addition, we could use the bitwise\n", + "exclusive-OR ($\\oplus$) operation so that" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "N_l=(N_{l-i})\\oplus (N_{l-j})\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where the bitwise action of $\\oplus$ means that if $N_{l-i}=N_{l-j}$ the result is\n", + "$0$ whereas if $N_{l-i}\\ne N_{l-j}$ the result is\n", + "$1$. As an example, consider the case where $N_{l-i}=6$ and $N_{l-j}=11$. The first\n", + "one has a bit representation (using 4 bits only) which reads $0110$ whereas the \n", + "second number is $1011$. Employing the $\\oplus$ operator yields \n", + "$1101$, or $2^3+2^2+2^0=13$.\n", + "\n", + "In Fortran90, the bitwise $\\oplus$ operation is coded through the intrinsic\n", + "function $\\mathrm{IEOR}(m,n)$ where $m$ and $n$ are the input numbers, while in $C$\n", + "it is given by $m\\wedge n$.\n", + "\n", + "\n", + "\n", + "\n", + "# Random number generator RNG, RAN0\n", + "\n", + "We show here how the linear congruential algorithm can be implemented, namely" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "N_i=(aN_{i-1}) \\mathrm{MOD} (M).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "However, since $a$ and $N_{i-1}$ are integers and their multiplication \n", + "could become greater than the standard 32 bit integer, there is a trick via \n", + "Schrage's algorithm which approximates the multiplication\n", + "of large integers through the factorization" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "M=aq+r,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where we have defined" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "q=[M/a],\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "r = M\\hspace{0.1cm}\\mathrm{MOD} \\hspace{0.1cm}a.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where the brackets denote integer division. In the code below the numbers \n", + "$q$ and $r$ are chosen so that $r < q$.\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "# Random number generator RNG, RAN0\n", + "\n", + "To see how this works we note first that" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + "(aN_{i-1}) \\mathrm{MOD} (M)= (aN_{i-1}-[N_{i-1}/q]M)\\mathrm{MOD} (M),\n", + "\\label{eq:rntrick1} \\tag{14}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "since we can add or subtract any integer multiple of $M$ from $aN_{i-1}$.\n", + "The last term $[N_{i-1}/q]M\\mathrm{MOD}(M)$ is zero since the integer division \n", + "$[N_{i-1}/q]$ just yields a constant which is multiplied with $M$.\n", + "\n", + "\n", + "\n", + "\n", + "# Random number generator RNG, RAN0\n", + "We can now rewrite Eq. ([14](#eq:rntrick1)) as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + "(aN_{i-1}) \\mathrm{MOD} (M)= (aN_{i-1}-[N_{i-1}/q](aq+r))\\mathrm{MOD} (M),\n", + "\\label{eq:rntrick2} \\tag{15}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which results\n", + "in" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + "(aN_{i-1}) \\mathrm{MOD} (M)= \\left(a(N_{i-1}-[N_{i-1}/q]q)-[N_{i-1}/q]r)\\right)\\mathrm{MOD} (M),\n", + "\\label{eq:rntrick3} \\tag{16}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "yielding" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + "(aN_{i-1}) \\mathrm{MOD} (M)= \\left(a(N_{i-1}\\mathrm{MOD} (q)) -[N_{i-1}/q]r)\\right)\\mathrm{MOD} (M).\n", + "\\label{eq:rntrick4} \\tag{17}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Random number generator RNG, RAN0\n", + "The term $[N_{i-1}/q]r$ is always smaller or equal $N_{i-1}(r/q)$ and with $r < q$ we obtain always a \n", + "number smaller than $N_{i-1}$, which is smaller than $M$. \n", + "And since the number $N_{i-1}\\mathrm{MOD} (q)$ is between zero and $q-1$ then\n", + "$a(N_{i-1}\\mathrm{MOD} (q))< aq$. Combined with our definition of $q=[M/a]$ ensures that \n", + "this term is also smaller than $M$ meaning that both terms fit into a\n", + "32-bit signed integer. None of these two terms can be negative, but their difference could.\n", + "The algorithm below adds $M$ if their difference is negative.\n", + "Note that the program uses the bitwise $\\oplus$ operator to generate\n", + "the starting point for each generation of a random number. The period\n", + "of $ran0$ is $\\sim 2.1\\times 10^{9}$. A special feature of this\n", + "algorithm is that is should never be called with the initial seed \n", + "set to $0$.\n", + "\n", + "\n", + "\n", + "\n", + "# Random number generator RNG, RAN0 code" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + " /*\n", + " ** The function\n", + " ** ran0()\n", + " ** is an \"Minimal\" random number generator of Park and Miller\n", + " ** Set or reset the input value\n", + " ** idum to any integer value (except the unlikely value MASK)\n", + " ** to initialize the sequence; idum must not be altered between\n", + " ** calls for sucessive deviates in a sequence.\n", + " ** The function returns a uniform deviate between 0.0 and 1.0.\n", + " */\n", + " double ran0(long &idum)\n", + " {\n", + " const int a = 16807, m = 2147483647, q = 127773;\n", + " const int r = 2836, MASK = 123459876;\n", + " const double am = 1./m;\n", + " long k;\n", + " double ans;\n", + " idum ^= MASK;\n", + " k = (*idum)/q;\n", + " idum = a*(idum - k*q) - r*k;\n", + " // add m if negative difference\n", + " if(idum < 0) idum += m;\n", + " ans=am*(idum);\n", + " idum ^= MASK;\n", + " return ans;\n", + " } // End: function ran0() \n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Properties of Selected Random Number Generators\n", + "\n", + "As mentioned previously, the underlying PDF for the generation of\n", + "random numbers is the uniform distribution, meaning that the \n", + "probability for finding a number $x$ in the interval [0,1] is $p(x)=1$.\n", + "\n", + "A random number generator should produce numbers which are uniformly distributed\n", + "in this interval. The table shows the distribution of $N=10000$ random\n", + "numbers generated by the functions in the program library.\n", + "We note in this table that the number of points in the various\n", + "intervals $0.0-0.1$, $0.1-0.2$ etc are fairly close to $1000$, with some minor\n", + "deviations. \n", + "\n", + "Two additional measures are the standard deviation $\\sigma$ and the mean\n", + "$\\mu=\\langle x\\rangle$.\n", + "\n", + "\n", + "\n", + "## Properties of Selected Random Number Generators\n", + "For the uniform distribution, the mean value $\\mu$ is then" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mu=\\langle x\\rangle=\\frac{1}{2}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "while the standard deviation is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\sigma=\\sqrt{\\langle x^2\\rangle-\\mu^2}=\\frac{1}{\\sqrt{12}}=0.2886.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Properties of Selected Random Number Generators\n", + "The various random number generators produce results which agree rather well with\n", + "these limiting values. \n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "
$x$-bin ran0 ran1 ran2 ran3
0.0-0.1 1013 991 938 1047
0.1-0.2 1002 1009 1040 1030
0.2-0.3 989 999 1030 993
0.3-0.4 939 960 1023 937
0.4-0.5 1038 1001 1002 992
0.5-0.6 1037 1047 1009 1009
0.6-0.7 1005 989 1003 989
0.7-0.8 986 962 985 954
0.8-0.9 1000 1027 1009 1023
0.9-1.0 991 1015 961 1026
$\\mu$ 0.4997 0.5018 0.4992 0.4990
$\\sigma$ 0.2882 0.2892 0.2861 0.2915
\n", + "\n", + "\n", + "\n", + "## Simple demonstration of RNGs using python\n", + "The following simple Python code plots the distribution of the produced random numbers using the linear congruential RNG employed by Python. The trend displayed in the previous table is seen rather clearly." + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "image/png": 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" + ] + }, + "metadata": { + "filenames": { + "image/png": "/Users/hjensen/Teaching/FYS-STK4150/doc/src/LectureNotes/_build/jupyter_execute/chapter3_178_0.png" + }, + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "#!/usr/bin/env python\n", + "import numpy as np\n", + "import matplotlib.mlab as mlab\n", + "import matplotlib.pyplot as plt\n", + "import random\n", + "\n", + "# initialize the rng with a seed\n", + "random.seed() \n", + "counts = 10000\n", + "values = np.zeros(counts) \n", + "for i in range (1, counts, 1):\n", + " values[i] = random.random()\n", + "\n", + "# the histogram of the data\n", + "n, bins, patches = plt.hist(values, 10, facecolor='green')\n", + "\n", + "plt.xlabel('$x$')\n", + "plt.ylabel('Number of counts')\n", + "plt.title(r'Test of uniform distribution')\n", + "plt.axis([0, 1, 0, 1100])\n", + "plt.grid(True)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Properties of Selected Random Number Generators\n", + "Since our random numbers, which are typically generated via a linear congruential algorithm,\n", + "are never fully independent, we can then define \n", + "an important test which measures the degree of correlation, namely the so-called \n", + "auto-correlation function defined previously, see again Eq. ([9](#eq:autocorrelformal)).\n", + "We rewrite it here as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "C_k=\\frac{f_d}\n", + " {\\sigma^2},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "with $C_0=1$. Recall that \n", + "$\\sigma^2=\\langle x_i^2\\rangle-\\langle x_i\\rangle^2$ and that" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "f_d = \\frac{1}{nm}\\sum_{\\alpha=1}^m\\sum_{k=1}^{n-d}(x_{\\alpha,k}-\\langle X_m \\rangle)(x_{\\alpha,k+d}-\\langle X_m \\rangle),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The non-vanishing of $C_k$ for $k\\ne 0$ means that the random\n", + "numbers are not independent. The independence of the random numbers is crucial \n", + "in the evaluation of other expectation values. If they are not independent, our\n", + "assumption for approximating $\\sigma_N$ is no longer valid.\n", + "\n", + "\n", + "\n", + "\n", + "## Autocorrelation function\n", + "This program computes the autocorrelation function as discussed in the equation on the previous slide for random numbers generated with the normal distribution $N(0,1)$." + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "0.04229032140441689 0.9337417995235178\n" + ] + }, + { + "data": { + "image/png": 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\n", + "text/plain": [ + "
" + ] + }, + "metadata": { + "filenames": { + "image/png": "/Users/hjensen/Teaching/FYS-STK4150/doc/src/LectureNotes/_build/jupyter_execute/chapter3_184_1.png" + }, + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "# Importing various packages\n", + "from math import exp, sqrt\n", + "from random import random, seed\n", + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "\n", + "def autocovariance(x, n, k, mean_x):\n", + " sum = 0.0\n", + " for i in range(0, n-k):\n", + " sum += (x[(i+k)]-mean_x)*(x[i]-mean_x)\n", + " return sum/n\n", + "\n", + "n = 1000\n", + "x=np.random.normal(size=n)\n", + "autocor = np.zeros(n)\n", + "figaxis = np.zeros(n)\n", + "mean_x=np.mean(x)\n", + "var_x = np.var(x)\n", + "print(mean_x, var_x)\n", + "for i in range (0, n):\n", + " figaxis[i] = i\n", + " autocor[i]=(autocovariance(x, n, i, mean_x))/var_x \n", + "\n", + "plt.plot(figaxis, autocor, \"r-\")\n", + "plt.axis([0,n,-0.1, 1.0])\n", + "plt.xlabel(r'$i$')\n", + "plt.ylabel(r'$\\gamma_i$')\n", + "plt.title(r'Autocorrelation function')\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "As can be seen from the plot, the first point gives back the variance and a value of one. \n", + "For the remaining values we notice that there are still non-zero values for the auto-correlation function.\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Correlation function and which random number generators should I use\n", + "The program here computes the correlation function for one of the standard functions included with the c++ compiler." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + " // This function computes the autocorrelation function for \n", + " // the standard c++ random number generator\n", + " \n", + " #include \n", + " #include \n", + " #include \n", + " #include \n", + " using namespace std;\n", + " // output file as global variable\n", + " ofstream ofile; \n", + " \n", + " // Main function begins here \n", + " int main(int argc, char* argv[])\n", + " {\n", + " int n;\n", + " char *outfilename;\n", + " \n", + " cin >> n;\n", + " double MCint = 0.; double MCintsqr2=0.;\n", + " double invers_period = 1./RAND_MAX; // initialise the random number generator\n", + " srand(time(NULL)); // This produces the so-called seed in MC jargon\n", + " // Compute the variance and the mean value of the uniform distribution\n", + " // Compute also the specific values x for each cycle in order to be able to\n", + " // the covariance and the correlation function \n", + " // Read in output file, abort if there are too few command-line arguments\n", + " if( argc <= 2 ){\n", + " cout << \"Bad Usage: \" << argv[0] << \n", + " \t \" read also output file and number of cycles on same line\" << endl;\n", + " exit(1);\n", + " }\n", + " else{\n", + " outfilename=argv[1];\n", + " }\n", + " ofile.open(outfilename); \n", + " // Get the number of Monte-Carlo samples\n", + " n = atoi(argv[2]);\n", + " double *X; \n", + " X = new double[n];\n", + " for (int i = 0; i < n; i++){\n", + " double x = double(rand())*invers_period; \n", + " X[i] = x;\n", + " MCint += x;\n", + " MCintsqr2 += x*x;\n", + " }\n", + " double Mean = MCint/((double) n );\n", + " MCintsqr2 = MCintsqr2/((double) n );\n", + " double STDev = sqrt(MCintsqr2-Mean*Mean);\n", + " double Variance = MCintsqr2-Mean*Mean;\n", + " // Write mean value and standard deviation \n", + " cout << \" Standard deviation= \" << STDev << \" Integral = \" << Mean << endl;\n", + " \n", + " // Now we compute the autocorrelation function\n", + " double *autocor; autocor = new double[n];\n", + " for (int j = 0; j < n; j++){\n", + " double sum = 0.0;\n", + " for (int k = 0; k < (n-j); k++){\n", + " \t sum += (X[k]-Mean)*(X[k+j]-Mean); \n", + " }\n", + " autocor[j] = sum/Variance/((double) n );\n", + " ofile << setiosflags(ios::showpoint | ios::uppercase);\n", + " ofile << setw(15) << setprecision(8) << j;\n", + " ofile << setw(15) << setprecision(8) << autocor[j] << endl;\n", + " }\n", + " ofile.close(); // close output file\n", + " return 0;\n", + " } // end of main program \n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Which RNG should I use?\n", + "* C++ has a class called **random**. The [random class](http://www.cplusplus.com/reference/random/) contains a large selection of RNGs and is highly recommended. Some of these RNGs have very large periods making it thereby very safe to use these RNGs in case one is performing large calculations. In particular, the [Mersenne twister random number engine](http://www.cplusplus.com/reference/random/mersenne_twister_engine/) has a period of $2^{19937}$. \n", + "\n", + "* Add RNGs in Python\n", + "\n", + "\n", + "\n", + "\n", + "## How to use the Mersenne generator\n", + "The following part of a c++ code (from project 4) sets up the uniform distribution for $x\\in [0,1]$." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + " /*\n", + " \n", + " // You need this \n", + " #include \n", + " \n", + " // Initialize the seed and call the Mersienne algo\n", + " std::random_device rd;\n", + " std::mt19937_64 gen(rd());\n", + " // Set up the uniform distribution for x \\in [[0, 1]\n", + " std::uniform_real_distribution RandomNumberGenerator(0.0,1.0);\n", + " \n", + " // Now use the RNG\n", + " int ix = (int) (RandomNumberGenerator(gen)*NSpins);\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Why blocking?\n", + "**Statistical analysis.**\n", + "\n", + " * Monte Carlo simulations can be treated as *computer experiments*\n", + "\n", + " * The results can be analysed with the same statistical tools as we would use analysing experimental data.\n", + "\n", + " * As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors.\n", + "\n", + "A very good article which explains blocking is H. Flyvbjerg and H. G. Petersen, *Error estimates on averages of correlated data*, [Journal of Chemical Physics 91, 461-466 (1989)](http://scitation.aip.org/content/aip/journal/jcp/91/1/10.1063/1.457480).\n", + "\n", + " \n", + "\n", + "\n", + "\n", + "## Why blocking?\n", + "**Statistical analysis.**\n", + "\n", + " * As in other experiments, Monte Carlo experiments have two classes of errors:\n", + "\n", + " * Statistical errors\n", + "\n", + " * Systematical errors\n", + "\n", + "\n", + " * Statistical errors can be estimated using standard tools from statistics\n", + "\n", + " * Systematical errors are method specific and must be treated differently from case to case. (In VMC a common source is the step length or time step in importance sampling)\n", + "\n", + " \n", + "\n", + "\n", + "## Code to demonstrate the calculation of the autocorrelation function\n", + "The following code computes the autocorrelation function, the covariance and the standard deviation\n", + "for standard RNG. \n", + "The [following file](https://github.com/CompPhysics/ComputationalPhysics2/tree/gh-pages/doc/Programs/LecturePrograms/programs/Blocking/autocorrelation.cpp) gives the code." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + " // This function computes the autocorrelation function for \n", + " // the Mersenne random number generator with a uniform distribution\n", + " #include \n", + " #include \n", + " #include \n", + " #include \n", + " #include \n", + " #include \n", + " #include \n", + " #include \n", + " using namespace std;\n", + " using namespace arma;\n", + " // output file\n", + " ofstream ofile;\n", + " \n", + " // Main function begins here \n", + " int main(int argc, char* argv[])\n", + " {\n", + " int MonteCarloCycles;\n", + " string filename;\n", + " if (argc > 1) {\n", + " filename=argv[1];\n", + " MonteCarloCycles = atoi(argv[2]);\n", + " string fileout = filename;\n", + " string argument = to_string(MonteCarloCycles);\n", + " fileout.append(argument);\n", + " ofile.open(fileout);\n", + " }\n", + " \n", + " // Compute the variance and the mean value of the uniform distribution\n", + " // Compute also the specific values x for each cycle in order to be able to\n", + " // compute the covariance and the correlation function \n", + " \n", + " vec X = zeros(MonteCarloCycles);\n", + " double MCint = 0.; double MCintsqr2=0.;\n", + " std::random_device rd;\n", + " std::mt19937_64 gen(rd());\n", + " // Set up the uniform distribution for x \\in [[0, 1]\n", + " std::uniform_real_distribution RandomNumberGenerator(0.0,1.0);\n", + " for (int i = 0; i < MonteCarloCycles; i++){\n", + " double x = RandomNumberGenerator(gen); \n", + " X(i) = x;\n", + " MCint += x;\n", + " MCintsqr2 += x*x;\n", + " }\n", + " double Mean = MCint/((double) MonteCarloCycles );\n", + " MCintsqr2 = MCintsqr2/((double) MonteCarloCycles );\n", + " double STDev = sqrt(MCintsqr2-Mean*Mean);\n", + " double Variance = MCintsqr2-Mean*Mean;\n", + " // Write mean value and variance\n", + " cout << \" Sample variance= \" << Variance << \" Mean value = \" << Mean << endl;\n", + " // Now we compute the autocorrelation function\n", + " vec autocorrelation = zeros(MonteCarloCycles);\n", + " for (int j = 0; j < MonteCarloCycles; j++){\n", + " double sum = 0.0;\n", + " for (int k = 0; k < (MonteCarloCycles-j); k++){\n", + " sum += (X(k)-Mean)*(X(k+j)-Mean); \n", + " }\n", + " autocorrelation(j) = sum/Variance/((double) MonteCarloCycles );\n", + " ofile << setiosflags(ios::showpoint | ios::uppercase);\n", + " ofile << setw(15) << setprecision(8) << j;\n", + " ofile << setw(15) << setprecision(8) << autocorrelation(j) << endl;\n", + " }\n", + " // Now compute the exact covariance using the autocorrelation function\n", + " double Covariance = 0.0;\n", + " for (int j = 0; j < MonteCarloCycles; j++){\n", + " Covariance += autocorrelation(j);\n", + " }\n", + " Covariance *= 2.0/((double) MonteCarloCycles);\n", + " // Compute now the total variance, including the covariance, and obtain the standard deviation\n", + " double TotalVariance = (Variance/((double) MonteCarloCycles ))+Covariance;\n", + " cout << \"Covariance =\" << Covariance << \"Totalvariance= \" << TotalVariance << \"Sample Variance/n= \" << (Variance/((double) MonteCarloCycles )) << endl;\n", + " cout << \" STD from sample variance= \" << sqrt(Variance/((double) MonteCarloCycles )) << \" STD with covariance = \" << sqrt(TotalVariance) << endl;\n", + " \n", + " ofile.close(); // close output file\n", + " return 0;\n", + " } // end of main program \n", + " \n", + " \n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## What is blocking?\n", + "**Blocking.**\n", + "\n", + " * Say that we have a set of samples from a Monte Carlo experiment\n", + "\n", + " * Assuming (wrongly) that our samples are uncorrelated our best estimate of the standard deviation of the mean $\\langle \\mathbf{M}\\rangle$ is given by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\sigma=\\sqrt{\\frac{1}{n}\\left(\\langle \\mathbf{M}^2\\rangle-\\langle \\mathbf{M}\\rangle^2\\right)}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "* If the samples are correlated we can rewrite our results to show that" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\sigma=\\sqrt{\\frac{1+2\\tau/\\Delta t}{n}\\left(\\langle \\mathbf{M}^2\\rangle-\\langle \\mathbf{M}\\rangle^2\\right)}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $\\tau$ is the correlation time (the time between a sample and the next uncorrelated sample) and $\\Delta t$ is time between each sample\n", + "\n", + " \n", + "\n", + "## What is blocking?\n", + "**Blocking.**\n", + "\n", + " * If $\\Delta t\\gg\\tau$ our first estimate of $\\sigma$ still holds\n", + "\n", + " * Much more common that $\\Delta t<\\tau$\n", + "\n", + " * In the method of data blocking we divide the sequence of samples into blocks\n", + "\n", + " * We then take the mean $\\langle \\mathbf{M}_i\\rangle$ of block $i=1\\ldots n_{blocks}$ to calculate the total mean and variance\n", + "\n", + " * The size of each block must be so large that sample $j$ of block $i$ is not correlated with sample $j$ of block $i+1$\n", + "\n", + " * The correlation time $\\tau$ would be a good choice\n", + "\n", + "\n", + "\n", + "## What is blocking?\n", + "**Blocking.**\n", + "\n", + " * Problem: We don't know $\\tau$ or it is too expensive to compute\n", + "\n", + " * Solution: Make a plot of std. dev. as a function of blocksize\n", + "\n", + " * The estimate of std. dev. of correlated data is too low $\\to$ the error will increase with increasing block size until the blocks are uncorrelated, where we reach a plateau\n", + "\n", + " * When the std. dev. stops increasing the blocks are uncorrelated\n", + "\n", + "\n", + "\n", + "## Implementation\n", + " * Do a Monte Carlo simulation, storing all samples to file\n", + "\n", + " * Do the statistical analysis on this file, independently of your Monte Carlo program\n", + "\n", + " * Read the file into an array\n", + "\n", + " * Loop over various block sizes\n", + "\n", + " * For each block size $n_b$, loop over the array in steps of $n_b$ taking the mean of elements $i n_b,\\ldots,(i+1) n_b$\n", + "\n", + " * Take the mean and variance of the resulting array\n", + "\n", + " * Write the results for each block size to file for later\n", + " analysis\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Actual implementation with code, main function\n", + "When the file gets large, it can be useful to write your data in binary mode instead of ascii characters.\n", + "The [following python file](https://github.com/CompPhysics/MachineLearning/blob/master/doc/Programs/Sampling/analysis.py) reads data from file with the output from every Monte Carlo cycle." + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "ename": "IndentationError", + "evalue": "unexpected indent (, line 2)", + "output_type": "error", + "traceback": [ + "\u001b[0;36m File \u001b[0;32m\"\"\u001b[0;36m, line \u001b[0;32m2\u001b[0m\n\u001b[0;31m @timeFunction\u001b[0m\n\u001b[0m ^\u001b[0m\n\u001b[0;31mIndentationError\u001b[0m\u001b[0;31m:\u001b[0m unexpected indent\n" + ] + } + ], + "source": [ + "# Blocking\n", + " @timeFunction\n", + " def blocking(self, blockSizeMax = 500):\n", + " blockSizeMin = 1\n", + "\n", + " self.blockSizes = []\n", + " self.meanVec = []\n", + " self.varVec = []\n", + "\n", + " for i in range(blockSizeMin, blockSizeMax):\n", + " if(len(self.data) % i != 0):\n", + " pass#continue\n", + " blockSize = i\n", + " meanTempVec = []\n", + " varTempVec = []\n", + " startPoint = 0\n", + " endPoint = blockSize\n", + "\n", + " while endPoint <= len(self.data):\n", + " meanTempVec.append(np.average(self.data[startPoint:endPoint]))\n", + " startPoint = endPoint\n", + " endPoint += blockSize\n", + " mean, var = np.average(meanTempVec), np.var(meanTempVec)/len(meanTempVec)\n", + " self.meanVec.append(mean)\n", + " self.varVec.append(var)\n", + " self.blockSizes.append(blockSize)\n", + "\n", + " self.blockingAvg = np.average(self.meanVec[-200:])\n", + " self.blockingVar = (np.average(self.varVec[-200:]))\n", + " self.blockingStd = np.sqrt(self.blockingVar)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## The Bootstrap method\n", + "\n", + "The Bootstrap resampling method is also very popular. It is very simple:\n", + "\n", + "1. Start with your sample of measurements and compute the sample variance and the mean values\n", + "\n", + "2. Then start again but pick in a random way the numbers in the sample and recalculate the mean and the sample variance.\n", + "\n", + "3. Repeat this $K$ times.\n", + "\n", + "It can be shown, see the article by [Efron](https://projecteuclid.org/download/pdf_1/euclid.aos/1176344552)\n", + "that it produces the correct standard deviation.\n", + "\n", + "This method is very useful for small ensembles of data points. \n", + "\n", + "\n", + "## Bootstrapping\n", + "Given a set of $N$ data, assume that we are interested in some \n", + "observable $\\theta$ which may be estimated from that set. This observable can also be for example the result of a fit based on all $N$ raw data. \n", + "Let us call the value of the observable obtained from the original \n", + "data set $\\hat{\\theta}$. One recreates from the sample repeatedly \n", + "other samples by choosing randomly $N$ data out of the original set. \n", + "This costs essentially nothing, since we just recycle the original data set for the building of new sets. \n", + "\n", + "\n", + "## Bootstrapping, recipe\n", + "Let us assume we have done this $K$ times and thus have $K$ sets of $N$ \n", + "data values each. \n", + "Of course some values will enter more than once in the new sets. For each of these sets one computes the observable $\\theta$ resulting in values $\\theta_k$ with $k = 1,...,K$. Then one determines" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\tilde{\\theta} = \\frac{1}{K} \\sum_{k=1}^K \\theta_k,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "sigma^2_{\\tilde{\\theta}} = \\frac{1}{K} \\sum_{k=1}^K \\left(\\theta_k-\\tilde{\\theta}\\right)^2.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "These are estimators for $\\angle\\theta\\rangle$ and its variance. They are not unbiased and therefore \n", + "$\\tilde{\\theta}\\neq\\hat{\\theta}$ for finite K. \n", + "\n", + "The difference is called bias and gives an idea on how far away the result may be from \n", + "the true $\\angle\\theta\\rangle$. As final result for the observable one quotes $\\angle\\theta\\rangle = \\tilde{\\theta} \\pm \\sigma_{\\tilde{\\theta}}$ .\n", + "\n", + "\n", + "\n", + "## Bootstrapping, [code](https://github.com/CompPhysics/MachineLearning/blob/master/doc/Programs/Sampling/analysis.py)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + " # Bootstrap\n", + " @timeFunction\n", + " def bootstrap(self, nBoots = 1000):\n", + " bootVec = np.zeros(nBoots)\n", + " for k in range(0,nBoots):\n", + " bootVec[k] = np.average(np.random.choice(self.data, len(self.data)))\n", + " self.bootAvg = np.average(bootVec)\n", + " self.bootVar = np.var(bootVec)\n", + " self.bootStd = np.std(bootVec)\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "## Jackknife, [code](https://github.com/CompPhysics/MachineLearning/blob/master/doc/Programs/Sampling/analysis.py)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + " # Jackknife\n", + " @timeFunction\n", + " def jackknife(self):\n", + " jackknVec = np.zeros(len(self.data))\n", + " for k in range(0,len(self.data)):\n", + " jackknVec[k] = np.average(np.delete(self.data, k))\n", + " self.jackknAvg = self.avg - (len(self.data) - 1) * (np.average(jackknVec) - self.avg)\n", + " self.jackknVar = float(len(self.data) - 1) * np.var(jackknVec)\n", + " self.jackknStd = np.sqrt(self.jackknVar)\n" + ] + } + ], + "metadata": { + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.8.3" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} \ No newline at end of file diff --git a/doc/src/LectureNotes/_build/jupyter_execute/chapter3.py b/doc/src/LectureNotes/_build/jupyter_execute/chapter3.py new file mode 100644 index 000000000..acbf9b5ee --- /dev/null +++ b/doc/src/LectureNotes/_build/jupyter_execute/chapter3.py @@ -0,0 +1,1803 @@ + +# Data Analysis and Machine Learning: Elements of Probability Theory and Statistical Data Analysis + + +**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University + +Date: **Sep 20, 2020** + +Copyright 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license + + + + +## To do list + +* add math about MVN and define MLE and other quantities + +* rewrite about covariance matrix + +* add KL theorem + +## Domains and probabilities +Consider the following simple example, namely the tossing of two dice, resulting in the following possible values + +$$ +\{2,3,4,5,6,7,8,9,10,11,12\}. +$$ + +These values are called the *domain*. +To this domain we have the corresponding *probabilities* + +$$ +\{1/36,2/36/,3/36,4/36,5/36,6/36,5/36,4/36,3/36,2/36,1/36\}. +$$ + +## Tossing the dice +The numbers in the domain are the outcomes of the physical process of tossing say two dice. +We cannot tell beforehand whether the outcome is 3 or 5 or any other number in this domain. +This defines the randomness of the outcome, or unexpectedness or any other synonimous word which +encompasses the uncertitude of the final outcome. + +The only thing we can tell beforehand +is that say the outcome 2 has a certain probability. +If our favorite hobby is to spend an hour every evening throwing dice and +registering the sequence of outcomes, we will note that the numbers in the above domain + +$$ +\{2,3,4,5,6,7,8,9,10,11,12\}, +$$ + +appear in a random order. After 11 throws the results may look like + +$$ +\{10,8,6,3,6,9,11,8,12,4,5\}. +$$ + +## Stochastic variables + +**Random variables are characterized by a domain which contains all possible values that the random value may take. This domain has a corresponding probability distribution function(PDF)**. + + + +## Stochastic variables and the main concepts, the discrete case +There are two main concepts associated with a stochastic variable. The +*domain* is the set $\mathbb D = \{x\}$ of all accessible values +the variable can assume, so that $X \in \mathbb D$. An example of a +discrete domain is the set of six different numbers that we may get by +throwing of a dice, $x\in\{1,\,2,\,3,\,4,\,5,\,6\}$. + +The *probability distribution function (PDF)* is a function +$p(x)$ on the domain which, in the discrete case, gives us the +probability or relative frequency with which these values of $X$ +occur + +$$ +p(x) = \mathrm{Prob}(X=x). +$$ + +## Stochastic variables and the main concepts, the continuous case +In the continuous case, the PDF does not directly depict the +actual probability. Instead we define the probability for the +stochastic variable to assume any value on an infinitesimal interval +around $x$ to be $p(x)dx$. The continuous function $p(x)$ then gives us +the *density* of the probability rather than the probability +itself. The probability for a stochastic variable to assume any value +on a non-infinitesimal interval $[a,\,b]$ is then just the integral + +$$ +\mathrm{Prob}(a\leq X\leq b) = \int_a^b p(x)dx. +$$ + +Qualitatively speaking, a stochastic variable represents the values of +numbers chosen as if by chance from some specified PDF so that the +selection of a large set of these numbers reproduces this PDF. + + + +## The cumulative probability +Of interest to us is the *cumulative probability +distribution function* (**CDF**), $P(x)$, which is just the probability +for a stochastic variable $X$ to assume any value less than $x$ + +$$ +P(x)=\mathrm{Prob(}X\leq x\mathrm{)} = +\int_{-\infty}^x p(x^{\prime})dx^{\prime}. +$$ + +The relation between a CDF and its corresponding PDF is then + +$$ +p(x) = \frac{d}{dx}P(x). +$$ + +## Properties of PDFs + +There are two properties that all PDFs must satisfy. The first one is +positivity (assuming that the PDF is normalized) + +$$ +0 \leq p(x) \leq 1. +$$ + +Naturally, it would be nonsensical for any of the values of the domain +to occur with a probability greater than $1$ or less than $0$. Also, +the PDF must be normalized. That is, all the probabilities must add up +to unity. The probability of "anything" to happen is always unity. For +both discrete and continuous PDFs, this condition is + +$$ +\begin{align*} +\sum_{x_i\in\mathbb D} p(x_i) & = 1,\\ +\int_{x\in\mathbb D} p(x)\,dx & = 1. +\end{align*} +$$ + +## Important distributions, the uniform distribution +The first one +is the most basic PDF; namely the uniform distribution + + +
+ +$$ +\begin{equation} +p(x) = \frac{1}{b-a}\theta(x-a)\theta(b-x). +\label{eq:unifromPDF} \tag{1} +\end{equation} +$$ + +For $a=0$ and $b=1$ we have + +$$ +\begin{array}{ll} +p(x)dx = dx & \in [0,1]. +\end{array} +$$ + +The latter distribution is used to generate random numbers. For other PDFs, one needs normally a mapping from this distribution to say for example the exponential distribution. + + + +## Gaussian distribution +The second one is the Gaussian Distribution + +$$ +p(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp{(-\frac{(x-\mu)^2}{2\sigma^2})}, +$$ + +with mean value $\mu$ and standard deviation $\sigma$. If $\mu=0$ and $\sigma=1$, it is normally called the **standard normal distribution** + +$$ +p(x) = \frac{1}{\sqrt{2\pi}} \exp{(-\frac{x^2}{2})}, +$$ + +The following simple Python code plots the above distribution for different values of $\mu$ and $\sigma$. + +%matplotlib inline + +import numpy as np +from math import acos, exp, sqrt +from matplotlib import pyplot as plt +from matplotlib import rc, rcParams +import matplotlib.units as units +import matplotlib.ticker as ticker +rc('text',usetex=True) +rc('font',**{'family':'serif','serif':['Gaussian distribution']}) +font = {'family' : 'serif', + 'color' : 'darkred', + 'weight' : 'normal', + 'size' : 16, + } +pi = acos(-1.0) +mu0 = 0.0 +sigma0 = 1.0 +mu1= 1.0 +sigma1 = 2.0 +mu2 = 2.0 +sigma2 = 4.0 + +x = np.linspace(-20.0, 20.0) +v0 = np.exp(-(x*x-2*x*mu0+mu0*mu0)/(2*sigma0*sigma0))/sqrt(2*pi*sigma0*sigma0) +v1 = np.exp(-(x*x-2*x*mu1+mu1*mu1)/(2*sigma1*sigma1))/sqrt(2*pi*sigma1*sigma1) +v2 = np.exp(-(x*x-2*x*mu2+mu2*mu2)/(2*sigma2*sigma2))/sqrt(2*pi*sigma2*sigma2) +plt.plot(x, v0, 'b-', x, v1, 'r-', x, v2, 'g-') +plt.title(r'{\bf Gaussian distributions}', fontsize=20) +plt.text(-19, 0.3, r'Parameters: $\mu = 0$, $\sigma = 1$', fontdict=font) +plt.text(-19, 0.18, r'Parameters: $\mu = 1$, $\sigma = 2$', fontdict=font) +plt.text(-19, 0.08, r'Parameters: $\mu = 2$, $\sigma = 4$', fontdict=font) +plt.xlabel(r'$x$',fontsize=20) +plt.ylabel(r'$p(x)$ [MeV]',fontsize=20) + +# Tweak spacing to prevent clipping of ylabel +plt.subplots_adjust(left=0.15) +plt.savefig('gaussian.pdf', format='pdf') +plt.show() + +## Exponential distribution +Another important distribution in science is the exponential distribution + +$$ +p(x) = \alpha\exp{-(\alpha x)}. +$$ + +## Expectation values +Let $h(x)$ be an arbitrary continuous function on the domain of the stochastic +variable $X$ whose PDF is $p(x)$. We define the *expectation value* +of $h$ with respect to $p$ as follows + + +
+ +$$ +\begin{equation} +\langle h \rangle_X \equiv \int\! h(x)p(x)\,dx +\label{eq:expectation_value_of_h_wrt_p} \tag{2} +\end{equation} +$$ + +Whenever the PDF is known implicitly, like in this case, we will drop +the index $X$ for clarity. +A particularly useful class of special expectation values are the +*moments*. The $n$-th moment of the PDF $p$ is defined as +follows + +$$ +\langle x^n \rangle \equiv \int\! x^n p(x)\,dx +$$ + +## Stochastic variables and the main concepts, mean values +The zero-th moment $\langle 1\rangle$ is just the normalization condition of +$p$. The first moment, $\langle x\rangle$, is called the *mean* of $p$ +and often denoted by the letter $\mu$ + +$$ +\langle x\rangle = \mu \equiv \int x p(x)dx, +$$ + +for a continuous distribution and + +$$ +\langle x\rangle = \mu \equiv \sum_{i=1}^N x_i p(x_i), +$$ + +for a discrete distribution. +Qualitatively it represents the centroid or the average value of the +PDF and is therefore simply called the expectation value of $p(x)$. + + + +## Stochastic variables and the main concepts, central moments, the variance + +A special version of the moments is the set of *central moments*, the n-th central moment defined as + +$$ +\langle (x-\langle x\rangle )^n\rangle \equiv \int\! (x-\langle x\rangle)^n p(x)\,dx +$$ + +The zero-th and first central moments are both trivial, equal $1$ and +$0$, respectively. But the second central moment, known as the +*variance* of $p$, is of particular interest. For the stochastic +variable $X$, the variance is denoted as $\sigma^2_X$ or $\mathrm{Var}(X)$ + +$$ +\begin{align*} +\sigma^2_X &=\mathrm{Var}(X) = \langle (x-\langle x\rangle)^2\rangle = +\int (x-\langle x\rangle)^2 p(x)dx\\ +& = \int\left(x^2 - 2 x \langle x\rangle^{2} +\langle x\rangle^2\right)p(x)dx\\ +& = \langle x^2\rangle - 2 \langle x\rangle\langle x\rangle + \langle x\rangle^2\\ +& = \langle x^2 \rangle - \langle x\rangle^2 +\end{align*} +$$ + +The square root of the variance, $\sigma =\sqrt{\langle (x-\langle x\rangle)^2\rangle}$ is called the +**standard deviation** of $p$. It is the RMS (root-mean-square) +value of the deviation of the PDF from its mean value, interpreted +qualitatively as the "spread" of $p$ around its mean. + + + + + +## Probability Distribution Functions + +The following table collects properties of probability distribution functions. +In our notation we reserve the label $p(x)$ for the probability of a certain event, +while $P(x)$ is the cumulative probability. + + + + + + + + + + + + + + + +
Discrete PDF Continuous PDF
Domain $\left\{x_1, x_2, x_3, \dots, x_N\right\}$ $[a,b]$
Probability $p(x_i)$ $p(x)dx$
Cumulative $P_i=\sum_{l=1}^ip(x_l)$ $P(x)=\int_a^xp(t)dt$
Positivity $0 \le p(x_i) \le 1$ $p(x) \ge 0$
Positivity $0 \le P_i \le 1$ $0 \le P(x) \le 1$
Monotonic $P_i \ge P_j$ if $x_i \ge x_j$ $P(x_i) \ge P(x_j)$ if $x_i \ge x_j$
Normalization $P_N=1$ $P(b)=1$
+ + + + +## Probability Distribution Functions +With a PDF we can compute expectation values of selected quantities such as + +$$ +\langle x^k\rangle=\sum_{i=1}^{N}x_i^kp(x_i), +$$ + +if we have a discrete PDF or + +$$ +\langle x^k\rangle=\int_a^b x^kp(x)dx, +$$ + +in the case of a continuous PDF. We have already defined the mean value $\mu$ +and the variance $\sigma^2$. + + + +## The three famous Probability Distribution Functions + +There are at least three PDFs which one may encounter. These are the + +**Uniform distribution** + +$$ +p(x)=\frac{1}{b-a}\Theta(x-a)\Theta(b-x), +$$ + +yielding probabilities different from zero in the interval $[a,b]$. + +**The exponential distribution** + +$$ +p(x)=\alpha \exp{(-\alpha x)}, +$$ + +yielding probabilities different from zero in the interval $[0,\infty)$ and with mean value + +$$ +\mu = \int_0^{\infty}xp(x)dx=\int_0^{\infty}x\alpha \exp{(-\alpha x)}dx=\frac{1}{\alpha}, +$$ + +with variance + +$$ +\sigma^2=\int_0^{\infty}x^2p(x)dx-\mu^2 = \frac{1}{\alpha^2}. +$$ + +## Probability Distribution Functions, the normal distribution +Finally, we have the so-called univariate normal distribution, or just the **normal distribution** + +$$ +p(x)=\frac{1}{b\sqrt{2\pi}}\exp{\left(-\frac{(x-a)^2}{2b^2}\right)} +$$ + +with probabilities different from zero in the interval $(-\infty,\infty)$. +The integral $\int_{-\infty}^{\infty}\exp{\left(-(x^2\right)}dx$ appears in many calculations, its value +is $\sqrt{\pi}$, a result we will need when we compute the mean value and the variance. +The mean value is + +$$ +\mu = \int_0^{\infty}xp(x)dx=\frac{1}{b\sqrt{2\pi}}\int_{-\infty}^{\infty}x \exp{\left(-\frac{(x-a)^2}{2b^2}\right)}dx, +$$ + +which becomes with a suitable change of variables + +$$ +\mu =\frac{1}{b\sqrt{2\pi}}\int_{-\infty}^{\infty}b\sqrt{2}(a+b\sqrt{2}y)\exp{-y^2}dy=a. +$$ + +## Probability Distribution Functions, the normal distribution +Similarly, the variance becomes + +$$ +\sigma^2 = \frac{1}{b\sqrt{2\pi}}\int_{-\infty}^{\infty}(x-\mu)^2 \exp{\left(-\frac{(x-a)^2}{2b^2}\right)}dx, +$$ + +and inserting the mean value and performing a variable change we obtain + +$$ +\sigma^2 = \frac{1}{b\sqrt{2\pi}}\int_{-\infty}^{\infty}b\sqrt{2}(b\sqrt{2}y)^2\exp{\left(-y^2\right)}dy= +\frac{2b^2}{\sqrt{\pi}}\int_{-\infty}^{\infty}y^2\exp{\left(-y^2\right)}dy, +$$ + +and performing a final integration by parts we obtain the well-known result $\sigma^2=b^2$. +It is useful to introduce the standard normal distribution as well, defined by $\mu=a=0$, viz. a distribution +centered around zero and with a variance $\sigma^2=1$, leading to + + +
+ +$$ +\begin{equation} + p(x)=\frac{1}{\sqrt{2\pi}}\exp{\left(-\frac{x^2}{2}\right)}. +\label{_auto1} \tag{3} +\end{equation} +$$ + +## Probability Distribution Functions, the cumulative distribution + +The exponential and uniform distributions have simple cumulative functions, +whereas the normal distribution does not, being proportional to the so-called +error function $erf(x)$, given by + +$$ +P(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^x\exp{\left(-\frac{t^2}{2}\right)}dt, +$$ + +which is difficult to evaluate in a quick way. + + + + +## Probability Distribution Functions, other important distribution + +Some other PDFs which one encounters often in the natural sciences are the binomial distribution + +$$ +p(x) = \left(\begin{array}{c} n \\ x\end{array}\right)y^x(1-y)^{n-x} \hspace{0.5cm}x=0,1,\dots,n, +$$ + +where $y$ is the probability for a specific event, such as the tossing of a coin or moving left or right +in case of a random walker. Note that $x$ is a discrete stochastic variable. + +The sequence of binomial trials is characterized by the following definitions + + * Every experiment is thought to consist of $N$ independent trials. + + * In every independent trial one registers if a specific situation happens or not, such as the jump to the left or right of a random walker. + + * The probability for every outcome in a single trial has the same value, for example the outcome of tossing (either heads or tails) a coin is always $1/2$. + + + +## Probability Distribution Functions, the binomial distribution + +In order to compute the mean and variance we need to recall Newton's binomial +formula + +$$ +(a+b)^m=\sum_{n=0}^m \left(\begin{array}{c} m \\ n\end{array}\right)a^nb^{m-n}, +$$ + +which can be used to show that + +$$ +\sum_{x=0}^n\left(\begin{array}{c} n \\ x\end{array}\right)y^x(1-y)^{n-x} = (y+1-y)^n = 1, +$$ + +the PDF is normalized to one. +The mean value is + +$$ +\mu = \sum_{x=0}^n x\left(\begin{array}{c} n \\ x\end{array}\right)y^x(1-y)^{n-x} = +\sum_{x=0}^n x\frac{n!}{x!(n-x)!}y^x(1-y)^{n-x}, +$$ + +resulting in + +$$ +\mu = +\sum_{x=0}^n x\frac{(n-1)!}{(x-1)!(n-1-(x-1))!}y^{x-1}(1-y)^{n-1-(x-1)}, +$$ + +which we rewrite as + +$$ +\mu=ny\sum_{\nu=0}^n\left(\begin{array}{c} n-1 \\ \nu\end{array}\right)y^{\nu}(1-y)^{n-1-\nu} =ny(y+1-y)^{n-1}=ny. +$$ + +The variance is slightly trickier to get. It reads $\sigma^2=ny(1-y)$. + +## Probability Distribution Functions, Poisson's distribution + +Another important distribution with discrete stochastic variables $x$ is +the Poisson model, which resembles the exponential distribution and reads + +$$ +p(x) = \frac{\lambda^x}{x!} e^{-\lambda} \hspace{0.5cm}x=0,1,\dots,;\lambda > 0. +$$ + +In this case both the mean value and the variance are easier to calculate, + +$$ +\mu = \sum_{x=0}^{\infty} x \frac{\lambda^x}{x!} e^{-\lambda} = \lambda e^{-\lambda}\sum_{x=1}^{\infty} +\frac{\lambda^{x-1}}{(x-1)!}=\lambda, +$$ + +and the variance is $\sigma^2=\lambda$. + + + + + +## Probability Distribution Functions, Poisson's distribution +An example of applications of the Poisson distribution could be the counting +of the number of $\alpha$-particles emitted from a radioactive source in a given time interval. +In the limit of $n\rightarrow \infty$ and for small probabilities $y$, the binomial distribution +approaches the Poisson distribution. Setting $\lambda = ny$, with $y$ the probability for an event in +the binomial distribution we can show that + +$$ +\lim_{n\rightarrow \infty}\left(\begin{array}{c} n \\ x\end{array}\right)y^x(1-y)^{n-x} e^{-\lambda}=\sum_{x=1}^{\infty}\frac{\lambda^x}{x!} e^{-\lambda}. +$$ + +## Meet the covariance! +An important quantity in a statistical analysis is the so-called covariance. + +Consider the set $\{X_i\}$ of $n$ +stochastic variables (not necessarily uncorrelated) with the +multivariate PDF $P(x_1,\dots,x_n)$. The *covariance* of two +of the stochastic variables, $X_i$ and $X_j$, is defined as follows + + +
+ +$$ +\begin{equation} +\mathrm{Cov}(X_i,\,X_j) = \langle (x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)\rangle +\label{_auto2} \tag{4} +\end{equation} +$$ + + +
+ +$$ +\begin{equation} +=\int\cdots\int (x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)P(x_1,\dots,x_n)\,dx_1\dots dx_n, +\label{eq:def_covariance} \tag{5} +\end{equation} +$$ + +with + +$$ +\langle x_i\rangle = +\int\cdots\int x_i P(x_1,\dots,x_n)\,dx_1\dots dx_n. +$$ + +## Meet the covariance in matrix disguise +If we consider the above covariance as a matrix + +$$ +C_{ij} =\mathrm{Cov}(X_i,\,X_j), +$$ + +then the diagonal elements are just the familiar +variances, $C_{ii} = \mathrm{Cov}(X_i,\,X_i) = \mathrm{Var}(X_i)$. It turns out that +all the off-diagonal elements are zero if the stochastic variables are +uncorrelated. + + + +## Covariance + +# Importing various packages +from math import exp, sqrt +from random import random, seed +import numpy as np +import matplotlib.pyplot as plt + +def covariance(x, y, n): + sum = 0.0 + mean_x = np.mean(x) + mean_y = np.mean(y) + for i in range(0, n): + sum += (x[(i)]-mean_x)*(y[i]-mean_y) + return sum/n + +n = 10 + +x=np.random.normal(size=n) +y = 4+3*x+np.random.normal(size=n) +covxy = covariance(x,y,n) +print(covxy) +z = np.vstack((x, y)) +c = np.cov(z.T) + +print(c) + +## Meet the covariance, uncorrelated events + +Consider the stochastic variables $X_i$ and $X_j$, ($i\neq j$). We have + +$$ +\begin{align*} +Cov(X_i,\,X_j) &= \langle (x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)\rangle\\ +&=\langle x_i x_j - x_i\langle x_j\rangle - \langle x_i\rangle x_j + \langle x_i\rangle\langle x_j\rangle\rangle\\ +&=\langle x_i x_j\rangle - \langle x_i\langle x_j\rangle\rangle - \langle \langle x_i\rangle x_j \rangle + +\langle \langle x_i\rangle\langle x_j\rangle\rangle \\ +&=\langle x_i x_j\rangle - \langle x_i\rangle\langle x_j\rangle - \langle x_i\rangle\langle x_j\rangle + +\langle x_i\rangle\langle x_j\rangle \\ +&=\langle x_i x_j\rangle - \langle x_i\rangle\langle x_j\rangle +\end{align*} +$$ + +If $X_i$ and $X_j$ are independent (assuming $i \neq j$), we have that + +$$ +\langle x_i x_j\rangle = \langle x_i\rangle\langle x_j\rangle, +$$ + +leading to + +$$ +Cov(X_i, X_j) = 0 \hspace{0.1cm} (i\neq j). +$$ + +## Numerical experiments and the covariance + +Now that we have constructed an idealized mathematical framework, let +us try to apply it to empirical observations. Examples of relevant +physical phenomena may be spontaneous decays of nuclei, or a purely +mathematical set of numbers produced by some deterministic +mechanism. It is the latter we will deal with, using so-called pseudo-random +number generators. In general our observations will contain only a limited set of +observables. We remind the reader that +a *stochastic process* is a process that produces sequentially a +chain of values + +$$ +\{x_1, x_2,\dots\,x_k,\dots\}. +$$ + +## Numerical experiments and the covariance +We will call these +values our *measurements* and the entire set as our measured +*sample*. The action of measuring all the elements of a sample +we will call a stochastic *experiment* (since, operationally, +they are often associated with results of empirical observation of +some physical or mathematical phenomena; precisely an experiment). We +assume that these values are distributed according to some +PDF $p_X^{\phantom X}(x)$, where $X$ is just the formal symbol for the +stochastic variable whose PDF is $p_X^{\phantom X}(x)$. Instead of +trying to determine the full distribution $p$ we are often only +interested in finding the few lowest moments, like the mean +$\mu_X^{\phantom X}$ and the variance $\sigma_X^{\phantom X}$. + + + + +## Numerical experiments and the covariance, actual situations +In practical situations however, a sample is always of finite size. Let that +size be $n$. The expectation value of a sample $\alpha$, the **sample mean**, is then defined as follows + +$$ +\langle x_{\alpha} \rangle \equiv \frac{1}{n}\sum_{k=1}^n x_{\alpha,k}. +$$ + +The *sample variance* is: + +$$ +\mathrm{Var}(x) \equiv \frac{1}{n}\sum_{k=1}^n (x_{\alpha,k} - \langle x_{\alpha} \rangle)^2, +$$ + +with its square root being the *standard deviation of the sample*. + + + + +## Numerical experiments and the covariance, our observables +You can think of the above observables as a set of quantities which define +a given experiment. This experiment is then repeated several times, say $m$ times. +The total average is then + + +
+ +$$ +\begin{equation} +\langle X_m \rangle= \frac{1}{m}\sum_{\alpha=1}^mx_{\alpha}=\frac{1}{mn}\sum_{\alpha, k} x_{\alpha,k}, +\label{eq:exptmean} \tag{6} +\end{equation} +$$ + +where the last sums end at $m$ and $n$. +The total variance is + +$$ +\sigma^2_m= \frac{1}{mn^2}\sum_{\alpha=1}^m(\langle x_{\alpha} \rangle-\langle X_m \rangle)^2, +$$ + +which we rewrite as + + +
+ +$$ +\begin{equation} +\sigma^2_m=\frac{1}{m}\sum_{\alpha=1}^m\sum_{kl=1}^n (x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,l}-\langle X_m \rangle). +\label{eq:exptvariance} \tag{7} +\end{equation} +$$ + +## Numerical experiments and the covariance, the sample variance + +We define also the sample variance $\sigma^2$ of all $mn$ individual experiments as + + +
+ +$$ +\begin{equation} +\sigma^2=\frac{1}{mn}\sum_{\alpha=1}^m\sum_{k=1}^n (x_{\alpha,k}-\langle X_m \rangle)^2. +\label{eq:sampleexptvariance} \tag{8} +\end{equation} +$$ + +These quantities, being known experimental values or the results from our calculations, +may differ, in some cases +significantly, from the similarly named +exact values for the mean value $\mu_X$, the variance $\mathrm{Var}(X)$ +and the covariance $\mathrm{Cov}(X,Y)$. + + + +## Numerical experiments and the covariance, central limit theorem + +The central limit theorem states that the PDF $\tilde{p}(z)$ of +the average of $m$ random values corresponding to a PDF $p(x)$ +is a normal distribution whose mean is the +mean value of the PDF $p(x)$ and whose variance is the variance +of the PDF $p(x)$ divided by $m$, the number of values used to compute $z$. + +The central limit theorem leads then to the well-known expression for the +standard deviation, given by + +$$ +\sigma_m= +\frac{\sigma}{\sqrt{m}}. +$$ + +In many cases the above estimate for the standard deviation, in particular if correlations are strong, may be too simplistic. We need therefore a more precise defintion of the error and the variance in our results. + + + +## Definition of Correlation Functions and Standard Deviation +Our estimate of the true average $\mu_{X}$ is the sample mean $\langle X_m \rangle$ + +$$ +\mu_{X}^{\phantom X} \approx X_m=\frac{1}{mn}\sum_{\alpha=1}^m\sum_{k=1}^n x_{\alpha,k}. +$$ + +We can then use Eq. ([7](#eq:exptvariance)) + +$$ +\sigma^2_m=\frac{1}{mn^2}\sum_{\alpha=1}^m\sum_{kl=1}^n (x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,l}-\langle X_m \rangle), +$$ + +and rewrite it as + +$$ +\sigma^2_m=\frac{\sigma^2}{n}+\frac{2}{mn^2}\sum_{\alpha=1}^m\sum_{k +
+ +$$ +\begin{equation} +\kappa_d = \frac{f_d}{\sigma^2} +\label{eq:autocorrelformal} \tag{9} +\end{equation} +$$ + +which gives us a useful measure of the correlation pair correlation +starting always at $1$ for $d=0$. + + + +## Definition of Correlation Functions and Standard Deviation, sample variance + +The sample variance of the $mn$ experiments can now be +written in terms of the autocorrelation function + + +
+ +$$ +\begin{equation} +\sigma_m^2=\frac{\sigma^2}{n}+\frac{2}{n}\cdot\sigma^2\sum_{d=1}^{n-1} +\frac{f_d}{\sigma^2}=\left(1+2\sum_{d=1}^{n-1}\kappa_d\right)\frac{1}{n}\sigma^2=\frac{\tau}{n}\cdot\sigma^2 +\label{eq:error_estimate_corr_time} \tag{10} +\end{equation} +$$ + +and we see that $\sigma_m$ can be expressed in terms of the +uncorrelated sample variance times a correction factor $\tau$ which +accounts for the correlation between measurements. We call this +correction factor the *autocorrelation time* + + +
+ +$$ +\begin{equation} +\tau = 1+2\sum_{d=1}^{n-1}\kappa_d +\label{eq:autocorrelation_time} \tag{11} +\end{equation} +$$ + + + +For a correlation free experiment, $\tau$ +equals 1. + + + + + +## Definition of Correlation Functions and Standard Deviation +From the point of view of +Eq. ([10](#eq:error_estimate_corr_time)) we can interpret a sequential +correlation as an effective reduction of the number of measurements by +a factor $\tau$. The effective number of measurements becomes + +$$ +n_\mathrm{eff} = \frac{n}{\tau} +$$ + +To neglect the autocorrelation time $\tau$ will always cause our +simple uncorrelated estimate of $\sigma_m^2\approx \sigma^2/n$ to +be less than the true sample error. The estimate of the error will be +too "good". On the other hand, the calculation of the full +autocorrelation time poses an efficiency problem if the set of +measurements is very large. The solution to this problem is given by +more practically oriented methods like the blocking technique. + + + + + + +## Code to compute the Covariance matrix and the Covariance + +# Importing various packages +from math import exp, sqrt +from random import random, seed +import numpy as np +import matplotlib.pyplot as plt + +# Sample covariance, note the factor 1/(n-1) +def covariance(x, y, n): + sum = 0.0 + mean_x = np.mean(x) + mean_y = np.mean(y) + for i in range(0, n): + sum += (x[(i)]-mean_x)*(y[i]-mean_y) + return sum/(n-1.) + +n = 100 +x = np.random.normal(size=n) +print(np.mean(x)) +y = 4+3*x+np.random.normal(size=n) +print(np.mean(y)) +z = x**3+np.random.normal(size=n) +print(np.mean(z)) +covxx = covariance(x,x,n) +covyy = covariance(y,y,n) +covzz = covariance(z,z,n) +covxy = covariance(x,y,n) +covxz = covariance(x,z,n) +covyz = covariance(y,z,n) +print(covxx,covyy, covzz) +print(covxy,covxz, covyz) +w = np.vstack((x, y, z)) +#print(w) +c = np.cov(w) +print(c) +#eigen = np.zeros(n) +Eigvals, Eigvecs = np.linalg.eig(c) +print(Eigvals) + +# Random Numbers + +Uniform deviates are just random numbers that lie within a specified range +(typically 0 to 1), with any one number in the range just as likely as any other. They +are, in other words, what you probably think random numbers are. However, +we want to distinguish uniform deviates from other sorts of random numbers, for +example numbers drawn from a normal (Gaussian) distribution of specified mean +and standard deviation. These other sorts of deviates are almost always generated by +performing appropriate operations on one or more uniform deviates, as we will see +in subsequent sections. So, a reliable source of random uniform deviates, the subject +of this section, is an essential building block for any sort of stochastic modeling +or Monte Carlo computer work. + + + + +# Random Numbers, better name: pseudo random numbers + +A disclaimer is however appropriate. It should be fairly obvious that +something as deterministic as a computer cannot generate purely random numbers. + +Numbers generated by any of the standard algorithms are in reality pseudo random +numbers, hopefully abiding to the following criteria: + + * they produce a uniform distribution in the interval [0,1]. + + * correlations between random numbers are negligible + + * the period before the same sequence of random numbers is repeated is as large as possible and finally + + * the algorithm should be fast. + + + + +# Random number generator RNG + The most common random number generators are based on so-called +Linear congruential relations of the type + +$$ +N_i=(aN_{i-1}+c) \mathrm{MOD} (M), +$$ + +which yield a number in the interval [0,1] through + +$$ +x_i=N_i/M +$$ + +The number +$M$ is called the period and it should be as large as possible + and +$N_0$ is the starting value, or seed. The function $\mathrm{MOD}$ means the remainder, +that is if we were to evaluate $(13)\mathrm{MOD}(9)$, the outcome is the remainder +of the division $13/9$, namely $4$. + + + +# Random number generator RNG and periodic outputs + +The problem with such generators is that their outputs are periodic; +they +will start to repeat themselves with a period that is at most $M$. If however +the parameters $a$ and $c$ are badly chosen, the period may be even shorter. + +Consider the following example + +$$ +N_i=(6N_{i-1}+7) \mathrm{MOD} (5), +$$ + +with a seed $N_0=2$. This generator produces the sequence +$4,1,3,0,2,4,1,3,0,2,...\dots$, i.e., a sequence with period $5$. +However, increasing $M$ may not guarantee a larger period as the following +example shows + +$$ +N_i=(27N_{i-1}+11) \mathrm{MOD} (54), +$$ + +which still, with $N_0=2$, results in $11,38,11,38,11,38,\dots$, a period of +just $2$. + + + +# Random number generator RNG and its period +Typical periods for the random generators provided in the program library +are of the order of $\sim 10^9$ or larger. Other random number generators which have +become increasingly popular are so-called shift-register generators. +In these generators each successive number depends on many preceding +values (rather than the last values as in the linear congruential +generator). +For example, you could make a shift register generator whose $l$th +number is the sum of the $l-i$th and $l-j$th values with modulo $M$, + +$$ +N_l=(aN_{l-i}+cN_{l-j})\mathrm{MOD}(M). +$$ + +# Random number generator RNG, other examples +Such a generator again produces a sequence of pseudorandom numbers +but this time with a period much larger than $M$. +It is also possible to construct more elaborate algorithms by including +more than two past terms in the sum of each iteration. +One example is the generator of [Marsaglia and Zaman](http://dl.acm.org/citation.cfm?id=187154) +which consists of two congruential relations + + +
+ +$$ +\begin{equation} + N_l=(N_{l-3}-N_{l-1})\mathrm{MOD}(2^{31}-69), +\label{eq:mz1} \tag{12} +\end{equation} +$$ + +followed by + + +
+ +$$ +\begin{equation} + N_l=(69069N_{l-1}+1013904243)\mathrm{MOD}(2^{32}), +\label{eq:mz2} \tag{13} +\end{equation} +$$ + +which according to the authors has a period larger than $2^{94}$. + + + +# Random number generator RNG, other examples +Instead of using modular addition, we could use the bitwise +exclusive-OR ($\oplus$) operation so that + +$$ +N_l=(N_{l-i})\oplus (N_{l-j}) +$$ + +where the bitwise action of $\oplus$ means that if $N_{l-i}=N_{l-j}$ the result is +$0$ whereas if $N_{l-i}\ne N_{l-j}$ the result is +$1$. As an example, consider the case where $N_{l-i}=6$ and $N_{l-j}=11$. The first +one has a bit representation (using 4 bits only) which reads $0110$ whereas the +second number is $1011$. Employing the $\oplus$ operator yields +$1101$, or $2^3+2^2+2^0=13$. + +In Fortran90, the bitwise $\oplus$ operation is coded through the intrinsic +function $\mathrm{IEOR}(m,n)$ where $m$ and $n$ are the input numbers, while in $C$ +it is given by $m\wedge n$. + + + + +# Random number generator RNG, RAN0 + +We show here how the linear congruential algorithm can be implemented, namely + +$$ +N_i=(aN_{i-1}) \mathrm{MOD} (M). +$$ + +However, since $a$ and $N_{i-1}$ are integers and their multiplication +could become greater than the standard 32 bit integer, there is a trick via +Schrage's algorithm which approximates the multiplication +of large integers through the factorization + +$$ +M=aq+r, +$$ + +where we have defined + +$$ +q=[M/a], +$$ + +and + +$$ +r = M\hspace{0.1cm}\mathrm{MOD} \hspace{0.1cm}a. +$$ + +where the brackets denote integer division. In the code below the numbers +$q$ and $r$ are chosen so that $r < q$. + + + + + +# Random number generator RNG, RAN0 + +To see how this works we note first that + + +
+ +$$ +\begin{equation} +(aN_{i-1}) \mathrm{MOD} (M)= (aN_{i-1}-[N_{i-1}/q]M)\mathrm{MOD} (M), +\label{eq:rntrick1} \tag{14} +\end{equation} +$$ + +since we can add or subtract any integer multiple of $M$ from $aN_{i-1}$. +The last term $[N_{i-1}/q]M\mathrm{MOD}(M)$ is zero since the integer division +$[N_{i-1}/q]$ just yields a constant which is multiplied with $M$. + + + + +# Random number generator RNG, RAN0 +We can now rewrite Eq. ([14](#eq:rntrick1)) as + + +
+ +$$ +\begin{equation} +(aN_{i-1}) \mathrm{MOD} (M)= (aN_{i-1}-[N_{i-1}/q](aq+r))\mathrm{MOD} (M), +\label{eq:rntrick2} \tag{15} +\end{equation} +$$ + +which results +in + + +
+ +$$ +\begin{equation} +(aN_{i-1}) \mathrm{MOD} (M)= \left(a(N_{i-1}-[N_{i-1}/q]q)-[N_{i-1}/q]r)\right)\mathrm{MOD} (M), +\label{eq:rntrick3} \tag{16} +\end{equation} +$$ + +yielding + + +
+ +$$ +\begin{equation} +(aN_{i-1}) \mathrm{MOD} (M)= \left(a(N_{i-1}\mathrm{MOD} (q)) -[N_{i-1}/q]r)\right)\mathrm{MOD} (M). +\label{eq:rntrick4} \tag{17} +\end{equation} +$$ + +# Random number generator RNG, RAN0 +The term $[N_{i-1}/q]r$ is always smaller or equal $N_{i-1}(r/q)$ and with $r < q$ we obtain always a +number smaller than $N_{i-1}$, which is smaller than $M$. +And since the number $N_{i-1}\mathrm{MOD} (q)$ is between zero and $q-1$ then +$a(N_{i-1}\mathrm{MOD} (q))< aq$. Combined with our definition of $q=[M/a]$ ensures that +this term is also smaller than $M$ meaning that both terms fit into a +32-bit signed integer. None of these two terms can be negative, but their difference could. +The algorithm below adds $M$ if their difference is negative. +Note that the program uses the bitwise $\oplus$ operator to generate +the starting point for each generation of a random number. The period +of $ran0$ is $\sim 2.1\times 10^{9}$. A special feature of this +algorithm is that is should never be called with the initial seed +set to $0$. + + + + +# Random number generator RNG, RAN0 code + + /* + ** The function + ** ran0() + ** is an "Minimal" random number generator of Park and Miller + ** Set or reset the input value + ** idum to any integer value (except the unlikely value MASK) + ** to initialize the sequence; idum must not be altered between + ** calls for sucessive deviates in a sequence. + ** The function returns a uniform deviate between 0.0 and 1.0. + */ + double ran0(long &idum) + { + const int a = 16807, m = 2147483647, q = 127773; + const int r = 2836, MASK = 123459876; + const double am = 1./m; + long k; + double ans; + idum ^= MASK; + k = (*idum)/q; + idum = a*(idum - k*q) - r*k; + // add m if negative difference + if(idum < 0) idum += m; + ans=am*(idum); + idum ^= MASK; + return ans; + } // End: function ran0() + + +## Properties of Selected Random Number Generators + +As mentioned previously, the underlying PDF for the generation of +random numbers is the uniform distribution, meaning that the +probability for finding a number $x$ in the interval [0,1] is $p(x)=1$. + +A random number generator should produce numbers which are uniformly distributed +in this interval. The table shows the distribution of $N=10000$ random +numbers generated by the functions in the program library. +We note in this table that the number of points in the various +intervals $0.0-0.1$, $0.1-0.2$ etc are fairly close to $1000$, with some minor +deviations. + +Two additional measures are the standard deviation $\sigma$ and the mean +$\mu=\langle x\rangle$. + + + +## Properties of Selected Random Number Generators +For the uniform distribution, the mean value $\mu$ is then + +$$ +\mu=\langle x\rangle=\frac{1}{2} +$$ + +while the standard deviation is + +$$ +\sigma=\sqrt{\langle x^2\rangle-\mu^2}=\frac{1}{\sqrt{12}}=0.2886. +$$ + +## Properties of Selected Random Number Generators +The various random number generators produce results which agree rather well with +these limiting values. + + + + + + + + + + + + + + + + + + + +
$x$-bin ran0 ran1 ran2 ran3
0.0-0.1 1013 991 938 1047
0.1-0.2 1002 1009 1040 1030
0.2-0.3 989 999 1030 993
0.3-0.4 939 960 1023 937
0.4-0.5 1038 1001 1002 992
0.5-0.6 1037 1047 1009 1009
0.6-0.7 1005 989 1003 989
0.7-0.8 986 962 985 954
0.8-0.9 1000 1027 1009 1023
0.9-1.0 991 1015 961 1026
$\mu$ 0.4997 0.5018 0.4992 0.4990
$\sigma$ 0.2882 0.2892 0.2861 0.2915
+ + + +## Simple demonstration of RNGs using python +The following simple Python code plots the distribution of the produced random numbers using the linear congruential RNG employed by Python. The trend displayed in the previous table is seen rather clearly. + +#!/usr/bin/env python +import numpy as np +import matplotlib.mlab as mlab +import matplotlib.pyplot as plt +import random + +# initialize the rng with a seed +random.seed() +counts = 10000 +values = np.zeros(counts) +for i in range (1, counts, 1): + values[i] = random.random() + +# the histogram of the data +n, bins, patches = plt.hist(values, 10, facecolor='green') + +plt.xlabel('$x$') +plt.ylabel('Number of counts') +plt.title(r'Test of uniform distribution') +plt.axis([0, 1, 0, 1100]) +plt.grid(True) +plt.show() + +## Properties of Selected Random Number Generators +Since our random numbers, which are typically generated via a linear congruential algorithm, +are never fully independent, we can then define +an important test which measures the degree of correlation, namely the so-called +auto-correlation function defined previously, see again Eq. ([9](#eq:autocorrelformal)). +We rewrite it here as + +$$ +C_k=\frac{f_d} + {\sigma^2}, +$$ + +with $C_0=1$. Recall that +$\sigma^2=\langle x_i^2\rangle-\langle x_i\rangle^2$ and that + +$$ +f_d = \frac{1}{nm}\sum_{\alpha=1}^m\sum_{k=1}^{n-d}(x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,k+d}-\langle X_m \rangle), +$$ + +The non-vanishing of $C_k$ for $k\ne 0$ means that the random +numbers are not independent. The independence of the random numbers is crucial +in the evaluation of other expectation values. If they are not independent, our +assumption for approximating $\sigma_N$ is no longer valid. + + + + +## Autocorrelation function +This program computes the autocorrelation function as discussed in the equation on the previous slide for random numbers generated with the normal distribution $N(0,1)$. + +# Importing various packages +from math import exp, sqrt +from random import random, seed +import numpy as np +import matplotlib.pyplot as plt + +def autocovariance(x, n, k, mean_x): + sum = 0.0 + for i in range(0, n-k): + sum += (x[(i+k)]-mean_x)*(x[i]-mean_x) + return sum/n + +n = 1000 +x=np.random.normal(size=n) +autocor = np.zeros(n) +figaxis = np.zeros(n) +mean_x=np.mean(x) +var_x = np.var(x) +print(mean_x, var_x) +for i in range (0, n): + figaxis[i] = i + autocor[i]=(autocovariance(x, n, i, mean_x))/var_x + +plt.plot(figaxis, autocor, "r-") +plt.axis([0,n,-0.1, 1.0]) +plt.xlabel(r'$i$') +plt.ylabel(r'$\gamma_i$') +plt.title(r'Autocorrelation function') +plt.show() + +As can be seen from the plot, the first point gives back the variance and a value of one. +For the remaining values we notice that there are still non-zero values for the auto-correlation function. + + + + + + + + + + + + + + + + +## Correlation function and which random number generators should I use +The program here computes the correlation function for one of the standard functions included with the c++ compiler. + + // This function computes the autocorrelation function for + // the standard c++ random number generator + + #include + #include + #include + #include + using namespace std; + // output file as global variable + ofstream ofile; + + // Main function begins here + int main(int argc, char* argv[]) + { + int n; + char *outfilename; + + cin >> n; + double MCint = 0.; double MCintsqr2=0.; + double invers_period = 1./RAND_MAX; // initialise the random number generator + srand(time(NULL)); // This produces the so-called seed in MC jargon + // Compute the variance and the mean value of the uniform distribution + // Compute also the specific values x for each cycle in order to be able to + // the covariance and the correlation function + // Read in output file, abort if there are too few command-line arguments + if( argc <= 2 ){ + cout << "Bad Usage: " << argv[0] << + " read also output file and number of cycles on same line" << endl; + exit(1); + } + else{ + outfilename=argv[1]; + } + ofile.open(outfilename); + // Get the number of Monte-Carlo samples + n = atoi(argv[2]); + double *X; + X = new double[n]; + for (int i = 0; i < n; i++){ + double x = double(rand())*invers_period; + X[i] = x; + MCint += x; + MCintsqr2 += x*x; + } + double Mean = MCint/((double) n ); + MCintsqr2 = MCintsqr2/((double) n ); + double STDev = sqrt(MCintsqr2-Mean*Mean); + double Variance = MCintsqr2-Mean*Mean; + // Write mean value and standard deviation + cout << " Standard deviation= " << STDev << " Integral = " << Mean << endl; + + // Now we compute the autocorrelation function + double *autocor; autocor = new double[n]; + for (int j = 0; j < n; j++){ + double sum = 0.0; + for (int k = 0; k < (n-j); k++){ + sum += (X[k]-Mean)*(X[k+j]-Mean); + } + autocor[j] = sum/Variance/((double) n ); + ofile << setiosflags(ios::showpoint | ios::uppercase); + ofile << setw(15) << setprecision(8) << j; + ofile << setw(15) << setprecision(8) << autocor[j] << endl; + } + ofile.close(); // close output file + return 0; + } // end of main program + + +# Which RNG should I use? +* C++ has a class called **random**. The [random class](http://www.cplusplus.com/reference/random/) contains a large selection of RNGs and is highly recommended. Some of these RNGs have very large periods making it thereby very safe to use these RNGs in case one is performing large calculations. In particular, the [Mersenne twister random number engine](http://www.cplusplus.com/reference/random/mersenne_twister_engine/) has a period of $2^{19937}$. + +* Add RNGs in Python + + + + +## How to use the Mersenne generator +The following part of a c++ code (from project 4) sets up the uniform distribution for $x\in [0,1]$. + + /* + + // You need this + #include + + // Initialize the seed and call the Mersienne algo + std::random_device rd; + std::mt19937_64 gen(rd()); + // Set up the uniform distribution for x \in [[0, 1] + std::uniform_real_distribution RandomNumberGenerator(0.0,1.0); + + // Now use the RNG + int ix = (int) (RandomNumberGenerator(gen)*NSpins); + + +## Why blocking? +**Statistical analysis.** + + * Monte Carlo simulations can be treated as *computer experiments* + + * The results can be analysed with the same statistical tools as we would use analysing experimental data. + + * As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors. + +A very good article which explains blocking is H. Flyvbjerg and H. G. Petersen, *Error estimates on averages of correlated data*, [Journal of Chemical Physics 91, 461-466 (1989)](http://scitation.aip.org/content/aip/journal/jcp/91/1/10.1063/1.457480). + + + + + +## Why blocking? +**Statistical analysis.** + + * As in other experiments, Monte Carlo experiments have two classes of errors: + + * Statistical errors + + * Systematical errors + + + * Statistical errors can be estimated using standard tools from statistics + + * Systematical errors are method specific and must be treated differently from case to case. (In VMC a common source is the step length or time step in importance sampling) + + + + +## Code to demonstrate the calculation of the autocorrelation function +The following code computes the autocorrelation function, the covariance and the standard deviation +for standard RNG. +The [following file](https://github.com/CompPhysics/ComputationalPhysics2/tree/gh-pages/doc/Programs/LecturePrograms/programs/Blocking/autocorrelation.cpp) gives the code. + + // This function computes the autocorrelation function for + // the Mersenne random number generator with a uniform distribution + #include + #include + #include + #include + #include + #include + #include + #include + using namespace std; + using namespace arma; + // output file + ofstream ofile; + + // Main function begins here + int main(int argc, char* argv[]) + { + int MonteCarloCycles; + string filename; + if (argc > 1) { + filename=argv[1]; + MonteCarloCycles = atoi(argv[2]); + string fileout = filename; + string argument = to_string(MonteCarloCycles); + fileout.append(argument); + ofile.open(fileout); + } + + // Compute the variance and the mean value of the uniform distribution + // Compute also the specific values x for each cycle in order to be able to + // compute the covariance and the correlation function + + vec X = zeros(MonteCarloCycles); + double MCint = 0.; double MCintsqr2=0.; + std::random_device rd; + std::mt19937_64 gen(rd()); + // Set up the uniform distribution for x \in [[0, 1] + std::uniform_real_distribution RandomNumberGenerator(0.0,1.0); + for (int i = 0; i < MonteCarloCycles; i++){ + double x = RandomNumberGenerator(gen); + X(i) = x; + MCint += x; + MCintsqr2 += x*x; + } + double Mean = MCint/((double) MonteCarloCycles ); + MCintsqr2 = MCintsqr2/((double) MonteCarloCycles ); + double STDev = sqrt(MCintsqr2-Mean*Mean); + double Variance = MCintsqr2-Mean*Mean; + // Write mean value and variance + cout << " Sample variance= " << Variance << " Mean value = " << Mean << endl; + // Now we compute the autocorrelation function + vec autocorrelation = zeros(MonteCarloCycles); + for (int j = 0; j < MonteCarloCycles; j++){ + double sum = 0.0; + for (int k = 0; k < (MonteCarloCycles-j); k++){ + sum += (X(k)-Mean)*(X(k+j)-Mean); + } + autocorrelation(j) = sum/Variance/((double) MonteCarloCycles ); + ofile << setiosflags(ios::showpoint | ios::uppercase); + ofile << setw(15) << setprecision(8) << j; + ofile << setw(15) << setprecision(8) << autocorrelation(j) << endl; + } + // Now compute the exact covariance using the autocorrelation function + double Covariance = 0.0; + for (int j = 0; j < MonteCarloCycles; j++){ + Covariance += autocorrelation(j); + } + Covariance *= 2.0/((double) MonteCarloCycles); + // Compute now the total variance, including the covariance, and obtain the standard deviation + double TotalVariance = (Variance/((double) MonteCarloCycles ))+Covariance; + cout << "Covariance =" << Covariance << "Totalvariance= " << TotalVariance << "Sample Variance/n= " << (Variance/((double) MonteCarloCycles )) << endl; + cout << " STD from sample variance= " << sqrt(Variance/((double) MonteCarloCycles )) << " STD with covariance = " << sqrt(TotalVariance) << endl; + + ofile.close(); // close output file + return 0; + } // end of main program + + + + +## What is blocking? +**Blocking.** + + * Say that we have a set of samples from a Monte Carlo experiment + + * Assuming (wrongly) that our samples are uncorrelated our best estimate of the standard deviation of the mean $\langle \mathbf{M}\rangle$ is given by + +$$ +\sigma=\sqrt{\frac{1}{n}\left(\langle \mathbf{M}^2\rangle-\langle \mathbf{M}\rangle^2\right)} +$$ + +* If the samples are correlated we can rewrite our results to show that + +$$ +\sigma=\sqrt{\frac{1+2\tau/\Delta t}{n}\left(\langle \mathbf{M}^2\rangle-\langle \mathbf{M}\rangle^2\right)} +$$ + +where $\tau$ is the correlation time (the time between a sample and the next uncorrelated sample) and $\Delta t$ is time between each sample + + + +## What is blocking? +**Blocking.** + + * If $\Delta t\gg\tau$ our first estimate of $\sigma$ still holds + + * Much more common that $\Delta t<\tau$ + + * In the method of data blocking we divide the sequence of samples into blocks + + * We then take the mean $\langle \mathbf{M}_i\rangle$ of block $i=1\ldots n_{blocks}$ to calculate the total mean and variance + + * The size of each block must be so large that sample $j$ of block $i$ is not correlated with sample $j$ of block $i+1$ + + * The correlation time $\tau$ would be a good choice + + + +## What is blocking? +**Blocking.** + + * Problem: We don't know $\tau$ or it is too expensive to compute + + * Solution: Make a plot of std. dev. as a function of blocksize + + * The estimate of std. dev. of correlated data is too low $\to$ the error will increase with increasing block size until the blocks are uncorrelated, where we reach a plateau + + * When the std. dev. stops increasing the blocks are uncorrelated + + + +## Implementation + * Do a Monte Carlo simulation, storing all samples to file + + * Do the statistical analysis on this file, independently of your Monte Carlo program + + * Read the file into an array + + * Loop over various block sizes + + * For each block size $n_b$, loop over the array in steps of $n_b$ taking the mean of elements $i n_b,\ldots,(i+1) n_b$ + + * Take the mean and variance of the resulting array + + * Write the results for each block size to file for later + analysis + + + + + + + +## Actual implementation with code, main function +When the file gets large, it can be useful to write your data in binary mode instead of ascii characters. +The [following python file](https://github.com/CompPhysics/MachineLearning/blob/master/doc/Programs/Sampling/analysis.py) reads data from file with the output from every Monte Carlo cycle. + +# Blocking + @timeFunction + def blocking(self, blockSizeMax = 500): + blockSizeMin = 1 + + self.blockSizes = [] + self.meanVec = [] + self.varVec = [] + + for i in range(blockSizeMin, blockSizeMax): + if(len(self.data) % i != 0): + pass#continue + blockSize = i + meanTempVec = [] + varTempVec = [] + startPoint = 0 + endPoint = blockSize + + while endPoint <= len(self.data): + meanTempVec.append(np.average(self.data[startPoint:endPoint])) + startPoint = endPoint + endPoint += blockSize + mean, var = np.average(meanTempVec), np.var(meanTempVec)/len(meanTempVec) + self.meanVec.append(mean) + self.varVec.append(var) + self.blockSizes.append(blockSize) + + self.blockingAvg = np.average(self.meanVec[-200:]) + self.blockingVar = (np.average(self.varVec[-200:])) + self.blockingStd = np.sqrt(self.blockingVar) + +## The Bootstrap method + +The Bootstrap resampling method is also very popular. It is very simple: + +1. Start with your sample of measurements and compute the sample variance and the mean values + +2. Then start again but pick in a random way the numbers in the sample and recalculate the mean and the sample variance. + +3. Repeat this $K$ times. + +It can be shown, see the article by [Efron](https://projecteuclid.org/download/pdf_1/euclid.aos/1176344552) +that it produces the correct standard deviation. + +This method is very useful for small ensembles of data points. + + +## Bootstrapping +Given a set of $N$ data, assume that we are interested in some +observable $\theta$ which may be estimated from that set. This observable can also be for example the result of a fit based on all $N$ raw data. +Let us call the value of the observable obtained from the original +data set $\hat{\theta}$. One recreates from the sample repeatedly +other samples by choosing randomly $N$ data out of the original set. +This costs essentially nothing, since we just recycle the original data set for the building of new sets. + + +## Bootstrapping, recipe +Let us assume we have done this $K$ times and thus have $K$ sets of $N$ +data values each. +Of course some values will enter more than once in the new sets. For each of these sets one computes the observable $\theta$ resulting in values $\theta_k$ with $k = 1,...,K$. Then one determines + +$$ +\tilde{\theta} = \frac{1}{K} \sum_{k=1}^K \theta_k, +$$ + +and + +$$ +sigma^2_{\tilde{\theta}} = \frac{1}{K} \sum_{k=1}^K \left(\theta_k-\tilde{\theta}\right)^2. +$$ + +These are estimators for $\angle\theta\rangle$ and its variance. They are not unbiased and therefore +$\tilde{\theta}\neq\hat{\theta}$ for finite K. + +The difference is called bias and gives an idea on how far away the result may be from +the true $\angle\theta\rangle$. As final result for the observable one quotes $\angle\theta\rangle = \tilde{\theta} \pm \sigma_{\tilde{\theta}}$ . + + + +## Bootstrapping, [code](https://github.com/CompPhysics/MachineLearning/blob/master/doc/Programs/Sampling/analysis.py) + + # Bootstrap + @timeFunction + def bootstrap(self, nBoots = 1000): + bootVec = np.zeros(nBoots) + for k in range(0,nBoots): + bootVec[k] = np.average(np.random.choice(self.data, len(self.data))) + self.bootAvg = np.average(bootVec) + self.bootVar = np.var(bootVec) + self.bootStd = np.std(bootVec) + + + +## Jackknife, [code](https://github.com/CompPhysics/MachineLearning/blob/master/doc/Programs/Sampling/analysis.py) + + # Jackknife + @timeFunction + def jackknife(self): + jackknVec = np.zeros(len(self.data)) + for k in range(0,len(self.data)): + jackknVec[k] = np.average(np.delete(self.data, k)) + self.jackknAvg = self.avg - (len(self.data) - 1) * (np.average(jackknVec) - self.avg) + self.jackknVar = float(len(self.data) - 1) * np.var(jackknVec) + self.jackknStd = np.sqrt(self.jackknVar) diff --git a/doc/src/LectureNotes/_build/jupyter_execute/chapter3_178_0.png b/doc/src/LectureNotes/_build/jupyter_execute/chapter3_178_0.png new file mode 100644 index 000000000..e0dacc37f Binary files /dev/null and b/doc/src/LectureNotes/_build/jupyter_execute/chapter3_178_0.png differ diff --git a/doc/src/LectureNotes/_build/jupyter_execute/chapter3_184_1.png b/doc/src/LectureNotes/_build/jupyter_execute/chapter3_184_1.png new file mode 100644 index 000000000..06bf49bb5 Binary files /dev/null and b/doc/src/LectureNotes/_build/jupyter_execute/chapter3_184_1.png differ diff --git a/doc/src/LectureNotes/_build/jupyter_execute/chapter3_29_0.png b/doc/src/LectureNotes/_build/jupyter_execute/chapter3_29_0.png new file mode 100644 index 000000000..aa693ddeb Binary files /dev/null and b/doc/src/LectureNotes/_build/jupyter_execute/chapter3_29_0.png differ diff --git a/doc/src/LectureNotes/_build/jupyter_execute/regression.ipynb b/doc/src/LectureNotes/_build/jupyter_execute/chapter4.ipynb similarity index 96% rename from doc/src/LectureNotes/_build/jupyter_execute/regression.ipynb rename to doc/src/LectureNotes/_build/jupyter_execute/chapter4.ipynb index a45d89f07..88dd21f52 100644 --- a/doc/src/LectureNotes/_build/jupyter_execute/regression.ipynb +++ b/doc/src/LectureNotes/_build/jupyter_execute/chapter4.ipynb @@ -1085,7 +1085,7 @@ }, "metadata": { "filenames": { - "image/png": "/Users/hjensen/Teaching/FYS-STK4150/doc/src/LectureNotes/_build/jupyter_execute/regression_70_0.png" + "image/png": "/Users/hjensen/Teaching/FYS-STK4150/doc/src/LectureNotes/_build/jupyter_execute/chapter4_70_0.png" }, "needs_background": "light" }, @@ -1611,7 +1611,7 @@ }, "metadata": { "filenames": { - "image/png": "/Users/hjensen/Teaching/FYS-STK4150/doc/src/LectureNotes/_build/jupyter_execute/regression_117_1.png" + "image/png": "/Users/hjensen/Teaching/FYS-STK4150/doc/src/LectureNotes/_build/jupyter_execute/chapter4_117_1.png" }, "needs_background": "light" }, @@ -1742,13 +1742,13 @@ "output_type": "stream", "text": [ "Training R2\n", - "0.9999909942655069\n", + "0.9999941778221182\n", "Training MSE\n", - "4.080219047317478\n", + "2.1663059697316176\n", "Test R2\n", - "0.9999405860292051\n", + "0.9999188193958993\n", "Test MSE\n", - "23.995899617131318\n" + "57.32209754560856\n" ] } ], @@ -1991,7 +1991,7 @@ }, "metadata": { "filenames": { - "image/png": "/Users/hjensen/Teaching/FYS-STK4150/doc/src/LectureNotes/_build/jupyter_execute/regression_129_0.png" + "image/png": "/Users/hjensen/Teaching/FYS-STK4150/doc/src/LectureNotes/_build/jupyter_execute/chapter4_129_0.png" } }, "output_type": "display_data" @@ -2021,7 +2021,7 @@ { "data": { "text/plain": [ - "" + "" ] }, "execution_count": 16, @@ -2037,7 +2037,7 @@ }, "metadata": { "filenames": { - "image/png": "/Users/hjensen/Teaching/FYS-STK4150/doc/src/LectureNotes/_build/jupyter_execute/regression_131_1.png" + "image/png": "/Users/hjensen/Teaching/FYS-STK4150/doc/src/LectureNotes/_build/jupyter_execute/chapter4_131_1.png" } }, "output_type": "display_data" @@ -2072,7 +2072,7 @@ }, "metadata": { "filenames": { - "image/png": "/Users/hjensen/Teaching/FYS-STK4150/doc/src/LectureNotes/_build/jupyter_execute/regression_133_0.png" + "image/png": "/Users/hjensen/Teaching/FYS-STK4150/doc/src/LectureNotes/_build/jupyter_execute/chapter4_133_0.png" } }, "output_type": "display_data" @@ -2223,7 +2223,7 @@ }, "metadata": { "filenames": { - "image/png": "/Users/hjensen/Teaching/FYS-STK4150/doc/src/LectureNotes/_build/jupyter_execute/regression_140_0.png" + "image/png": "/Users/hjensen/Teaching/FYS-STK4150/doc/src/LectureNotes/_build/jupyter_execute/chapter4_140_0.png" } }, "output_type": "display_data" @@ -2321,27 +2321,27 @@ "MSE before scaling: 0.00\n", "R2 score before scaling 0.99\n", "Feature min values before scaling:\n", - " [1.00000000e+00 2.70650053e-04 1.70425352e-03 7.32514510e-08\n", - " 4.61256305e-07 2.90448006e-06 1.98255091e-11 1.24839043e-10\n", - " 7.86097682e-10 4.94997037e-09 5.36577508e-15 3.37876937e-14\n", - " 2.12757379e-13 1.33970974e-12 8.43600443e-12 1.45224731e-18\n", - " 9.14464107e-18 5.75827959e-17 3.62592512e-16 2.28320505e-15\n", - " 1.43770903e-14]\n", + " [1.00000000e+00 3.31070195e-03 4.26950921e-04 1.09607474e-05\n", + " 1.41350724e-06 1.82287089e-07 3.62877676e-08 4.67970118e-09\n", + " 6.03498219e-10 7.78276403e-11 1.20137983e-10 1.54930958e-11\n", + " 1.99800273e-12 2.57664120e-13 3.32285826e-14 3.97741054e-13\n", + " 5.12930224e-14 6.61479151e-15 8.53049103e-16 1.10009933e-16\n", + " 1.41869739e-17]\n", "Feature max values before scaling:\n", - " [1. 0.99958956 0.99908436 0.99917929 0.9986743 0.99816955\n", - " 0.99876919 0.9982644 0.99775987 0.99725559 0.99835926 0.99785468\n", - " 0.99735035 0.99684628 0.99634246 0.9979495 0.99744512 0.996941\n", - " 0.99643713 0.99593352 0.99543016]\n", + " [1. 0.99858638 0.99964819 0.99717477 0.99823507 0.99929651\n", + " 0.99576514 0.99682395 0.99788388 0.99894494 0.99435752 0.99541483\n", + " 0.99647326 0.99753282 0.99859351 0.99295188 0.99400769 0.99506463\n", + " 0.99612269 0.99718188 0.99824219]\n", "Feature min values after scaling:\n", - " [ 0. -1.76022296 -1.83404211 -1.14581009 -1.16015297 -1.17531975\n", - " -0.90221667 -0.9105284 -0.91876649 -0.92694434 -0.76295619 -0.76977308\n", - " -0.77649633 -0.78311771 -0.78962976 -0.66997319 -0.67600919 -0.68199208\n", - " -0.68791314 -0.69376357 -0.69953457]\n", + " [ 0. -1.7247216 -1.76366767 -1.11821924 -1.1207329 -1.12330932\n", + " -0.88380004 -0.88090074 -0.87801917 -0.87516765 -0.75358163 -0.74975133\n", + " -0.74594528 -0.74216924 -0.73842878 -0.66775774 -0.66415352 -0.66058072\n", + " -0.65704182 -0.65353917 -0.65007499]\n", "Feature max values after scaling:\n", - " [0. 1.73102381 1.68493089 2.26096593 2.20791419 2.15452546\n", - " 2.69341363 2.64475059 2.59522254 2.54488068 3.06558633 3.02242943\n", - " 2.97825315 2.93307898 2.88693235 3.394687 3.35696162 3.31817404\n", - " 3.27832771 3.23742905 3.19548764]\n", + " [0. 1.71439421 1.7725581 2.19850437 2.24574687 2.29247479\n", + " 2.59145801 2.63154972 2.67112014 2.71016871 2.93008405 2.96590898\n", + " 3.00128975 3.03623229 3.07074298 3.23004415 3.26340549 3.29641768\n", + " 3.32908665 3.36141837 3.39341886]\n", "MSE after scaling: 0.00\n", "R2 score for scaled data: 0.99\n" ] @@ -3578,10 +3578,10 @@ "name": "stdout", "output_type": "stream", "text": [ - "-0.03657945968920499\n", - "4.0843822059793204\n", - "[[ 1.07379261 3.24702637]\n", - " [ 3.24702637 10.84879359]]\n" + "0.04144248019085487\n", + "4.194659964041423\n", + "[[ 1.10772753 3.41408669]\n", + " [ 3.41408669 11.38266616]]\n" ] } ], @@ -3620,10 +3620,10 @@ "name": "stdout", "output_type": "stream", "text": [ - "0.08730525680217946\n", - "1.6300895034092742\n", - "[[1. 0.67148001]\n", - " [0.67148001 1. ]]\n" + "0.09052813127366242\n", + "1.5055457430392476\n", + "[[1. 0.56806976]\n", + " [0.56806976 1. 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0.072168 0.070584 0.069200 0.067997 \n", + "3 0.084723 0.082939 0.080992 0.079031 0.077297 0.075772 0.074439 \n", + "4 0.085401 0.084154 0.081340 0.079900 0.078621 0.077496 0.076518 \n", + "5 0.085786 0.085015 0.081431 0.080455 0.079584 0.078821 0.078164 \n", + "6 0.082993 0.081923 0.079151 0.077870 0.076722 0.075706 0.074816 \n", + "7 0.082776 0.082128 0.078671 0.077808 0.077031 0.076344 0.075749 \n", + "8 0.082517 0.082246 0.078172 0.077682 0.077236 0.076846 0.076519 \n", + "9 0.082246 0.082311 0.077682 0.077525 0.077378 0.077256 0.077172 \n", + "10 0.078172 0.077682 0.074339 0.073633 0.072989 0.072416 0.071916 \n", + "11 0.077682 0.077525 0.073633 0.073260 0.072914 0.072607 0.072349 \n", + "12 0.077236 0.077378 0.072989 0.072914 0.072835 0.072771 0.072733 \n", + "13 0.076846 0.077256 0.072416 0.072607 0.072771 0.072926 0.073089 \n", + "14 0.076519 0.077172 0.071916 0.072349 0.072733 0.073089 0.073437 \n" ] } ], @@ -4352,10 +4352,10 @@ "name": "stdout", "output_type": "stream", "text": [ - "Runtime: 0.430137 sec\n", + "Runtime: 0.389789 sec\n", "Jackknife Statistics :\n", "original bias std. error\n", - " 100.065 100.055 0.149964\n" + " 100.259 100.249 0.149656\n" ] } ], @@ -4504,10 +4504,10 @@ "name": "stdout", "output_type": "stream", "text": [ - "Runtime: 2.02866 sec\n", + "Runtime: 2.00056 sec\n", "Bootstrap Statistics :\n", "original bias std. error\n", - " 100.052 15.0878 100.051 0.150313\n" + " 99.9087 15.0416 99.9098 0.150465\n" ] }, { @@ -4529,14 +4529,14 @@ }, { "data": { - "image/png": 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\n", 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Released under CC Attribution-NonCommercial 4.0 license\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Logistic Regression\n", + "\n", + "In linear regression our main interest was centered on learning the\n", + "coefficients of a functional fit (say a polynomial) in order to be\n", + "able to predict the response of a continuous variable on some unseen\n", + "data. The fit to the continuous variable $y_i$ is based on some\n", + "independent variables $\\hat{x}_i$. Linear regression resulted in\n", + "analytical expressions for standard ordinary Least Squares or Ridge\n", + "regression (in terms of matrices to invert) for several quantities,\n", + "ranging from the variance and thereby the confidence intervals of the\n", + "parameters $\\hat{\\beta}$ to the mean squared error. If we can invert\n", + "the product of the design matrices, linear regression gives then a\n", + "simple recipe for fitting our data.\n", + "\n", + "\n", + "Classification problems, however, are concerned with outcomes taking\n", + "the form of discrete variables (i.e. categories). We may for example,\n", + "on the basis of DNA sequencing for a number of patients, like to find\n", + "out which mutations are important for a certain disease; or based on\n", + "scans of various patients' brains, figure out if there is a tumor or\n", + "not; or given a specific physical system, we'd like to identify its\n", + "state, say whether it is an ordered or disordered system (typical\n", + "situation in solid state physics); or classify the status of a\n", + "patient, whether she/he has a stroke or not and many other similar\n", + "situations.\n", + "\n", + "The most common situation we encounter when we apply logistic\n", + "regression is that of two possible outcomes, normally denoted as a\n", + "binary outcome, true or false, positive or negative, success or\n", + "failure etc.\n", + "\n", + "## Optimization and Deep learning\n", + "\n", + "Logistic regression will also serve as our stepping stone towards\n", + "neural network algorithms and supervised deep learning. For logistic\n", + "learning, the minimization of the cost function leads to a non-linear\n", + "equation in the parameters $\\hat{\\beta}$. The optimization of the\n", + "problem calls therefore for minimization algorithms. This forms the\n", + "bottle neck of all machine learning algorithms, namely how to find\n", + "reliable minima of a multi-variable function. This leads us to the\n", + "family of gradient descent methods. The latter are the working horses\n", + "of basically all modern machine learning algorithms.\n", + "\n", + "We note also that many of the topics discussed here on logistic \n", + "regression are also commonly used in modern supervised Deep Learning\n", + "models, as we will see later.\n", + "\n", + "\n", + "\n", + "## Basics\n", + "\n", + "We consider the case where the dependent variables, also called the\n", + "responses or the outcomes, $y_i$ are discrete and only take values\n", + "from $k=0,\\dots,K-1$ (i.e. $K$ classes).\n", + "\n", + "The goal is to predict the\n", + "output classes from the design matrix $\\hat{X}\\in\\mathbb{R}^{n\\times p}$\n", + "made of $n$ samples, each of which carries $p$ features or predictors. The\n", + "primary goal is to identify the classes to which new unseen samples\n", + "belong.\n", + "\n", + "Let us specialize to the case of two classes only, with outputs\n", + "$y_i=0$ and $y_i=1$. Our outcomes could represent the status of a\n", + "credit card user that could default or not on her/his credit card\n", + "debt. That is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "y_i = \\begin{bmatrix} 0 & \\mathrm{no}\\\\ 1 & \\mathrm{yes} \\end{bmatrix}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Linear classifier\n", + "\n", + "Before moving to the logistic model, let us try to use our linear\n", + "regression model to classify these two outcomes. We could for example\n", + "fit a linear model to the default case if $y_i > 0.5$ and the no\n", + "default case $y_i \\leq 0.5$.\n", + "\n", + "We would then have our \n", + "weighted linear combination, namely" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + "\\hat{y} = \\hat{X}^T\\hat{\\beta} + \\hat{\\epsilon},\n", + "\\label{_auto1} \\tag{1}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $\\hat{y}$ is a vector representing the possible outcomes, $\\hat{X}$ is our\n", + "$n\\times p$ design matrix and $\\hat{\\beta}$ represents our estimators/predictors.\n", + "\n", + "## Some selected properties\n", + "\n", + "The main problem with our function is that it takes values on the\n", + "entire real axis. In the case of logistic regression, however, the\n", + "labels $y_i$ are discrete variables. A typical example is the credit\n", + "card data discussed below here, where we can set the state of\n", + "defaulting the debt to $y_i=1$ and not to $y_i=0$ for one the persons\n", + "in the data set (see the full example below).\n", + "\n", + "One simple way to get a discrete output is to have sign\n", + "functions that map the output of a linear regressor to values $\\{0,1\\}$,\n", + "$f(s_i)=sign(s_i)=1$ if $s_i\\ge 0$ and 0 if otherwise. \n", + "We will encounter this model in our first demonstration of neural networks. Historically it is called the \"perceptron\" model in the machine learning\n", + "literature. This model is extremely simple. However, in many cases it is more\n", + "favorable to use a ``soft\" classifier that outputs\n", + "the probability of a given category. This leads us to the logistic function.\n", + "\n", + "\n", + "## The logistic function\n", + "\n", + "The perceptron is an example of a ``hard classification\" model. We\n", + "will encounter this model when we discuss neural networks as\n", + "well. Each datapoint is deterministically assigned to a category (i.e\n", + "$y_i=0$ or $y_i=1$). In many cases, it is favorable to have a \"soft\"\n", + "classifier that outputs the probability of a given category rather\n", + "than a single value. For example, given $x_i$, the classifier\n", + "outputs the probability of being in a category $k$. Logistic regression\n", + "is the most common example of a so-called soft classifier. In logistic\n", + "regression, the probability that a data point $x_i$\n", + "belongs to a category $y_i=\\{0,1\\}$ is given by the so-called logit function (or Sigmoid) which is meant to represent the likelihood for a given event," + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "p(t) = \\frac{1}{1+\\mathrm \\exp{-t}}=\\frac{\\exp{t}}{1+\\mathrm \\exp{t}}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Note that $1-p(t)= p(-t)$.\n", + "\n", + "## Examples of likelihood functions used in logistic regression and nueral networks\n", + "\n", + "\n", + "The following code plots the logistic function, the step function and other functions we will encounter from here and on." + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", 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\n", + "text/plain": [ + "
" + ] + }, + "metadata": { + "filenames": { + "image/png": "/Users/hjensen/Teaching/FYS-STK4150/doc/src/LectureNotes/_build/jupyter_execute/chapter5_7_2.png" + }, + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "%matplotlib inline\n", + "\n", + "\"\"\"The sigmoid function (or the logistic curve) is a\n", + "function that takes any real number, z, and outputs a number (0,1).\n", + "It is useful in neural networks for assigning weights on a relative scale.\n", + "The value z is the weighted sum of parameters involved in the learning algorithm.\"\"\"\n", + "\n", + "import numpy\n", + "import matplotlib.pyplot as plt\n", + "import math as mt\n", + "\n", + "z = numpy.arange(-5, 5, .1)\n", + "sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))\n", + "sigma = sigma_fn(z)\n", + "\n", + "fig = plt.figure()\n", + "ax = fig.add_subplot(111)\n", + "ax.plot(z, sigma)\n", + "ax.set_ylim([-0.1, 1.1])\n", + "ax.set_xlim([-5,5])\n", + "ax.grid(True)\n", + "ax.set_xlabel('z')\n", + "ax.set_title('sigmoid function')\n", + "\n", + "plt.show()\n", + "\n", + "\"\"\"Step Function\"\"\"\n", + "z = numpy.arange(-5, 5, .02)\n", + "step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0)\n", + "step = step_fn(z)\n", + "\n", + "fig = plt.figure()\n", + "ax = fig.add_subplot(111)\n", + "ax.plot(z, step)\n", + "ax.set_ylim([-0.5, 1.5])\n", + "ax.set_xlim([-5,5])\n", + "ax.grid(True)\n", + "ax.set_xlabel('z')\n", + "ax.set_title('step function')\n", + "\n", + "plt.show()\n", + "\n", + "\"\"\"tanh Function\"\"\"\n", + "z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)\n", + "t = numpy.tanh(z)\n", + "\n", + "fig = plt.figure()\n", + "ax = fig.add_subplot(111)\n", + "ax.plot(z, t)\n", + "ax.set_ylim([-1.0, 1.0])\n", + "ax.set_xlim([-2*mt.pi,2*mt.pi])\n", + "ax.grid(True)\n", + "ax.set_xlabel('z')\n", + "ax.set_title('tanh function')\n", + "\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Two parameters\n", + "\n", + "We assume now that we have two classes with $y_i$ either $0$ or $1$. Furthermore we assume also that we have only two parameters $\\beta$ in our fitting of the Sigmoid function, that is we define probabilities" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{align*}\n", + "p(y_i=1|x_i,\\hat{\\beta}) &= \\frac{\\exp{(\\beta_0+\\beta_1x_i)}}{1+\\exp{(\\beta_0+\\beta_1x_i)}},\\nonumber\\\\\n", + "p(y_i=0|x_i,\\hat{\\beta}) &= 1 - p(y_i=1|x_i,\\hat{\\beta}),\n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $\\hat{\\beta}$ are the weights we wish to extract from data, in our case $\\beta_0$ and $\\beta_1$. \n", + "\n", + "Note that we used" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "p(y_i=0\\vert x_i, \\hat{\\beta}) = 1-p(y_i=1\\vert x_i, \\hat{\\beta}).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "## Maximum likelihood\n", + "\n", + "In order to define the total likelihood for all possible outcomes from a \n", + "dataset $\\mathcal{D}=\\{(y_i,x_i)\\}$, with the binary labels\n", + "$y_i\\in\\{0,1\\}$ and where the data points are drawn independently, we use the so-called [Maximum Likelihood Estimation](https://en.wikipedia.org/wiki/Maximum_likelihood_estimation) (MLE) principle. \n", + "We aim thus at maximizing \n", + "the probability of seeing the observed data. We can then approximate the \n", + "likelihood in terms of the product of the individual probabilities of a specific outcome $y_i$, that is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{align*}\n", + "P(\\mathcal{D}|\\hat{\\beta})& = \\prod_{i=1}^n \\left[p(y_i=1|x_i,\\hat{\\beta})\\right]^{y_i}\\left[1-p(y_i=1|x_i,\\hat{\\beta}))\\right]^{1-y_i}\\nonumber \\\\\n", + "\\end{align*}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "from which we obtain the log-likelihood and our **cost/loss** function" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathcal{C}(\\hat{\\beta}) = \\sum_{i=1}^n \\left( y_i\\log{p(y_i=1|x_i,\\hat{\\beta})} + (1-y_i)\\log\\left[1-p(y_i=1|x_i,\\hat{\\beta}))\\right]\\right).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## The cost function rewritten\n", + "\n", + "Reordering the logarithms, we can rewrite the **cost/loss** function as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathcal{C}(\\hat{\\beta}) = \\sum_{i=1}^n \\left(y_i(\\beta_0+\\beta_1x_i) -\\log{(1+\\exp{(\\beta_0+\\beta_1x_i)})}\\right).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to $\\beta$.\n", + "Since the cost (error) function is just the negative log-likelihood, for logistic regression we have that" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathcal{C}(\\hat{\\beta})=-\\sum_{i=1}^n \\left(y_i(\\beta_0+\\beta_1x_i) -\\log{(1+\\exp{(\\beta_0+\\beta_1x_i)})}\\right).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This equation is known in statistics as the **cross entropy**. Finally, we note that just as in linear regression, \n", + "in practice we often supplement the cross-entropy with additional regularization terms, usually $L_1$ and $L_2$ regularization as we did for Ridge and Lasso regression.\n", + "\n", + "## Minimizing the cross entropy\n", + "\n", + "The cross entropy is a convex function of the weights $\\hat{\\beta}$ and,\n", + "therefore, any local minimizer is a global minimizer. \n", + "\n", + "\n", + "Minimizing this\n", + "cost function with respect to the two parameters $\\beta_0$ and $\\beta_1$ we obtain" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial \\mathcal{C}(\\hat{\\beta})}{\\partial \\beta_0} = -\\sum_{i=1}^n \\left(y_i -\\frac{\\exp{(\\beta_0+\\beta_1x_i)}}{1+\\exp{(\\beta_0+\\beta_1x_i)}}\\right),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial \\mathcal{C}(\\hat{\\beta})}{\\partial \\beta_1} = -\\sum_{i=1}^n \\left(y_ix_i -x_i\\frac{\\exp{(\\beta_0+\\beta_1x_i)}}{1+\\exp{(\\beta_0+\\beta_1x_i)}}\\right).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## A more compact expression\n", + "\n", + "Let us now define a vector $\\hat{y}$ with $n$ elements $y_i$, an\n", + "$n\\times p$ matrix $\\hat{X}$ which contains the $x_i$ values and a\n", + "vector $\\hat{p}$ of fitted probabilities $p(y_i\\vert x_i,\\hat{\\beta})$. We can rewrite in a more compact form the first\n", + "derivative of cost function as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial \\mathcal{C}(\\hat{\\beta})}{\\partial \\hat{\\beta}} = -\\hat{X}^T\\left(\\hat{y}-\\hat{p}\\right).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "If we in addition define a diagonal matrix $\\hat{W}$ with elements \n", + "$p(y_i\\vert x_i,\\hat{\\beta})(1-p(y_i\\vert x_i,\\hat{\\beta})$, we can obtain a compact expression of the second derivative as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\frac{\\partial^2 \\mathcal{C}(\\hat{\\beta})}{\\partial \\hat{\\beta}\\partial \\hat{\\beta}^T} = \\hat{X}^T\\hat{W}\\hat{X}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Extending to more predictors\n", + "\n", + "Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with $p$ predictors" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\log{ \\frac{p(\\hat{\\beta}\\hat{x})}{1-p(\\hat{\\beta}\\hat{x})}} = \\beta_0+\\beta_1x_1+\\beta_2x_2+\\dots+\\beta_px_p.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Here we defined $\\hat{x}=[1,x_1,x_2,\\dots,x_p]$ and $\\hat{\\beta}=[\\beta_0, \\beta_1, \\dots, \\beta_p]$ leading to" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "p(\\hat{\\beta}\\hat{x})=\\frac{ \\exp{(\\beta_0+\\beta_1x_1+\\beta_2x_2+\\dots+\\beta_px_p)}}{1+\\exp{(\\beta_0+\\beta_1x_1+\\beta_2x_2+\\dots+\\beta_px_p)}}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Including more classes\n", + "\n", + "Till now we have mainly focused on two classes, the so-called binary\n", + "system. Suppose we wish to extend to $K$ classes. Let us for the sake\n", + "of simplicity assume we have only two predictors. We have then\n", + "following model" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "1\n", + "5\n", + " \n", + "<\n", + "<\n", + "<\n", + "!\n", + "!\n", + "M\n", + "A\n", + "T\n", + "H\n", + "_\n", + "B\n", + "L\n", + "O\n", + "C\n", + "K" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\log{\\frac{p(C=2\\vert x)}{p(K\\vert x)}} = \\beta_{20}+\\beta_{21}x_1,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and so on till the class $C=K-1$ class" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\log{\\frac{p(C=K-1\\vert x)}{p(K\\vert x)}} = \\beta_{(K-1)0}+\\beta_{(K-1)1}x_1,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and the model is specified in term of $K-1$ so-called log-odds or\n", + "**logit** transformations.\n", + "\n", + "\n", + "## More classes\n", + "\n", + "In our discussion of neural networks we will encounter the above again\n", + "in terms of a slightly modified function, the so-called **Softmax** function.\n", + "\n", + "The softmax function is used in various multiclass classification\n", + "methods, such as multinomial logistic regression (also known as\n", + "softmax regression), multiclass linear discriminant analysis, naive\n", + "Bayes classifiers, and artificial neural networks. Specifically, in\n", + "multinomial logistic regression and linear discriminant analysis, the\n", + "input to the function is the result of $K$ distinct linear functions,\n", + "and the predicted probability for the $k$-th class given a sample\n", + "vector $\\hat{x}$ and a weighting vector $\\hat{\\beta}$ is (with two\n", + "predictors):" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "p(C=k\\vert \\mathbf {x} )=\\frac{\\exp{(\\beta_{k0}+\\beta_{k1}x_1)}}{1+\\sum_{l=1}^{K-1}\\exp{(\\beta_{l0}+\\beta_{l1}x_1)}}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "It is easy to extend to more predictors. The final class is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "p(C=K\\vert \\mathbf {x} )=\\frac{1}{1+\\sum_{l=1}^{K-1}\\exp{(\\beta_{l0}+\\beta_{l1}x_1)}},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and they sum to one. Our earlier discussions were all specialized to\n", + "the case with two classes only. It is easy to see from the above that\n", + "what we derived earlier is compatible with these equations.\n", + "\n", + "To find the optimal parameters we would typically use a gradient\n", + "descent method. Newton's method and gradient descent methods are\n", + "discussed in the material on [optimization\n", + "methods](https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html).\n", + "\n", + "\n", + "\n", + "\n", + "## A simple classification problem" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", 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" + ] + }, + "metadata": { + "filenames": { + "image/png": "/Users/hjensen/Teaching/FYS-STK4150/doc/src/LectureNotes/_build/jupyter_execute/chapter5_42_0.png" + }, + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "import numpy as np\n", + "from sklearn import datasets, linear_model\n", + "import matplotlib.pyplot as plt\n", + "\n", + "\n", + "def generate_data():\n", + " np.random.seed(0)\n", + " X, y = datasets.make_moons(200, noise=0.20)\n", + " return X, y\n", + "\n", + "\n", + "def visualize(X, y, clf):\n", + " plot_decision_boundary(lambda x: clf.predict(x), X, y)\n", + "\n", + "def plot_decision_boundary(pred_func, X, y):\n", + " # Set min and max values and give it some padding\n", + " x_min, x_max = X[:, 0].min() - .5, X[:, 0].max() + .5\n", + " y_min, y_max = X[:, 1].min() - .5, X[:, 1].max() + .5\n", + " h = 0.01\n", + " # Generate a grid of points with distance h between them\n", + " xx, yy = np.meshgrid(np.arange(x_min, x_max, h), np.arange(y_min, y_max, h))\n", + " # Predict the function value for the whole gid\n", + " Z = pred_func(np.c_[xx.ravel(), yy.ravel()])\n", + " Z = Z.reshape(xx.shape)\n", + " # Plot the contour and training examples\n", + " plt.contourf(xx, yy, Z, cmap=plt.cm.Spectral)\n", + " plt.scatter(X[:, 0], X[:, 1], c=y, cmap=plt.cm.Spectral)\n", + " plt.show()\n", + "\n", + "\n", + "def classify(X, y):\n", + " clf = linear_model.LogisticRegressionCV()\n", + " clf.fit(X, y)\n", + " return clf\n", + "\n", + "\n", + "def main():\n", + " X, y = generate_data()\n", + " # visualize(X, y)\n", + " clf = classify(X, y)\n", + " visualize(X, y, clf)\n", + "\n", + "if __name__ == \"__main__\":\n", + " main()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## The Credit Card example\n", + "Here we use the the [credit card data](https://archive.ics.uci.edu/ml/datasets/default+of+credit+card+clients). \n", + "The data are from an extensive database from Taiwan and include more than ten predictors.\n", + "\n", + "For categorical data -Scikit-Learn- provides a so-called **one-hot encoder**.\n", + "This is called one-hot\n", + "encoding, because only one attribute will be equal to 1 (hot), while the others will be 0 (cold).\n", + "**Scikit-Learn** provides a OneHotEncoder encoder to convert integer categorical values into one-hot" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [], + "source": [ + "from sklearn.preprocessing import OneHotEncoder\n", + "encoder = OneHotEncoder()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## How to read the Credit Card data" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "ename": "FileNotFoundError", + "evalue": "[Errno 2] No such file or directory: '/Users/hjensen/Teaching/FYS-STK4150/doc/src/LectureNotes/default of credit card clients.xls'", + "output_type": "error", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mFileNotFoundError\u001b[0m Traceback (most recent call last)", + "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[1;32m 19\u001b[0m \u001b[0mfilename\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcwd\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0;34m'/default of credit card clients.xls'\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 20\u001b[0m \u001b[0mnanDict\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m{\u001b[0m\u001b[0;34m}\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 21\u001b[0;31m \u001b[0mdf\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mpd\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mread_excel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfilename\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mheader\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m 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confusion_matrix, accuracy_score, roc_auc_score\n", + "\n", + "# Trying to set the seed\n", + "np.random.seed(0)\n", + "import random\n", + "random.seed(0)\n", + "\n", + "# Reading file into data frame\n", + "cwd = os.getcwd()\n", + "filename = cwd + '/default of credit card clients.xls'\n", + "nanDict = {}\n", + "df = pd.read_excel(filename, header=1, skiprows=0, index_col=0, na_values=nanDict)\n", + "\n", + "df.rename(index=str, columns={\"default payment next month\": \"defaultPaymentNextMonth\"}, inplace=True)\n", + "\n", + "# Features and targets \n", + "X = df.loc[:, df.columns != 'defaultPaymentNextMonth'].values\n", + "y = df.loc[:, df.columns == 'defaultPaymentNextMonth'].values\n", + "\n", + "# Categorical variables to one-hot's\n", + "onehotencoder = OneHotEncoder(categories=\"auto\")\n", + "\n", + "X = ColumnTransformer(\n", + " [(\"\", onehotencoder, [3]),],\n", + " remainder=\"passthrough\"\n", + ").fit_transform(X)\n", + "\n", + "y.shape\n", + "\n", + "# Train-test split\n", + "trainingShare = 0.5 \n", + "seed = 1\n", + "XTrain, XTest, yTrain, yTest=train_test_split(X, y, train_size=trainingShare, \\\n", + " test_size = 1-trainingShare,\n", + " random_state=seed)\n", + "\n", + "# Input Scaling\n", + "sc = StandardScaler()\n", + "XTrain = sc.fit_transform(XTrain)\n", + "XTest = sc.transform(XTest)\n", + "\n", + "# One-hot's of the target vector\n", + "Y_train_onehot, Y_test_onehot = onehotencoder.fit_transform(yTrain), onehotencoder.fit_transform(yTest)\n", + "\n", + "# Remove instances with zeros only for past bill statements or paid amounts\n", + "'''\n", + "df = df.drop(df[(df.BILL_AMT1 == 0) &\n", + " (df.BILL_AMT2 == 0) &\n", + " (df.BILL_AMT3 == 0) &\n", + " (df.BILL_AMT4 == 0) &\n", + " (df.BILL_AMT5 == 0) &\n", + " (df.BILL_AMT6 == 0) &\n", + " (df.PAY_AMT1 == 0) &\n", + " (df.PAY_AMT2 == 0) &\n", + " (df.PAY_AMT3 == 0) &\n", + " (df.PAY_AMT4 == 0) &\n", + " (df.PAY_AMT5 == 0) &\n", + " (df.PAY_AMT6 == 0)].index)\n", + "'''\n", + "df = df.drop(df[(df.BILL_AMT1 == 0) &\n", + " (df.BILL_AMT2 == 0) &\n", + " (df.BILL_AMT3 == 0) &\n", + " (df.BILL_AMT4 == 0) &\n", + " (df.BILL_AMT5 == 0) &\n", + " (df.BILL_AMT6 == 0)].index)\n", + "\n", + "df = df.drop(df[(df.PAY_AMT1 == 0) &\n", + " (df.PAY_AMT2 == 0) &\n", + " (df.PAY_AMT3 == 0) &\n", + " (df.PAY_AMT4 == 0) &\n", + " (df.PAY_AMT5 == 0) &\n", + " (df.PAY_AMT6 == 0)].index)\n", + "\n", + "from sklearn.linear_model import LogisticRegression\n", + "from sklearn.model_selection import GridSearchCV\n", + "\n", + "lambdas=np.logspace(-5,7,13)\n", + "parameters = [{'C': 1./lambdas, \"solver\":[\"lbfgs\"]}]#*len(parameters)}]\n", + "scoring = ['accuracy', 'roc_auc']\n", + "logReg = LogisticRegression()\n", + "gridSearch = GridSearchCV(logReg, parameters, cv=5, scoring=scoring, refit='roc_auc')" + ] + } + ], + "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": "3.8.3" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} \ No newline at end of file diff --git a/doc/src/LectureNotes/_build/jupyter_execute/chapter5.py b/doc/src/LectureNotes/_build/jupyter_execute/chapter5.py new file mode 100644 index 000000000..39168e60a --- /dev/null +++ b/doc/src/LectureNotes/_build/jupyter_execute/chapter5.py @@ -0,0 +1,544 @@ + +# Data Analysis and Machine Learning: Logistic Regression + + +**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University + +Date: **Oct 17, 2019** + +Copyright 1999-2019, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license + + + + + +## Logistic Regression + +In linear regression our main interest was centered on learning the +coefficients of a functional fit (say a polynomial) in order to be +able to predict the response of a continuous variable on some unseen +data. The fit to the continuous variable $y_i$ is based on some +independent variables $\hat{x}_i$. Linear regression resulted in +analytical expressions for standard ordinary Least Squares or Ridge +regression (in terms of matrices to invert) for several quantities, +ranging from the variance and thereby the confidence intervals of the +parameters $\hat{\beta}$ to the mean squared error. If we can invert +the product of the design matrices, linear regression gives then a +simple recipe for fitting our data. + + +Classification problems, however, are concerned with outcomes taking +the form of discrete variables (i.e. categories). We may for example, +on the basis of DNA sequencing for a number of patients, like to find +out which mutations are important for a certain disease; or based on +scans of various patients' brains, figure out if there is a tumor or +not; or given a specific physical system, we'd like to identify its +state, say whether it is an ordered or disordered system (typical +situation in solid state physics); or classify the status of a +patient, whether she/he has a stroke or not and many other similar +situations. + +The most common situation we encounter when we apply logistic +regression is that of two possible outcomes, normally denoted as a +binary outcome, true or false, positive or negative, success or +failure etc. + +## Optimization and Deep learning + +Logistic regression will also serve as our stepping stone towards +neural network algorithms and supervised deep learning. For logistic +learning, the minimization of the cost function leads to a non-linear +equation in the parameters $\hat{\beta}$. The optimization of the +problem calls therefore for minimization algorithms. This forms the +bottle neck of all machine learning algorithms, namely how to find +reliable minima of a multi-variable function. This leads us to the +family of gradient descent methods. The latter are the working horses +of basically all modern machine learning algorithms. + +We note also that many of the topics discussed here on logistic +regression are also commonly used in modern supervised Deep Learning +models, as we will see later. + + + +## Basics + +We consider the case where the dependent variables, also called the +responses or the outcomes, $y_i$ are discrete and only take values +from $k=0,\dots,K-1$ (i.e. $K$ classes). + +The goal is to predict the +output classes from the design matrix $\hat{X}\in\mathbb{R}^{n\times p}$ +made of $n$ samples, each of which carries $p$ features or predictors. The +primary goal is to identify the classes to which new unseen samples +belong. + +Let us specialize to the case of two classes only, with outputs +$y_i=0$ and $y_i=1$. Our outcomes could represent the status of a +credit card user that could default or not on her/his credit card +debt. That is + +$$ +y_i = \begin{bmatrix} 0 & \mathrm{no}\\ 1 & \mathrm{yes} \end{bmatrix}. +$$ + +## Linear classifier + +Before moving to the logistic model, let us try to use our linear +regression model to classify these two outcomes. We could for example +fit a linear model to the default case if $y_i > 0.5$ and the no +default case $y_i \leq 0.5$. + +We would then have our +weighted linear combination, namely + + +
+ +$$ +\begin{equation} +\hat{y} = \hat{X}^T\hat{\beta} + \hat{\epsilon}, +\label{_auto1} \tag{1} +\end{equation} +$$ + +where $\hat{y}$ is a vector representing the possible outcomes, $\hat{X}$ is our +$n\times p$ design matrix and $\hat{\beta}$ represents our estimators/predictors. + +## Some selected properties + +The main problem with our function is that it takes values on the +entire real axis. In the case of logistic regression, however, the +labels $y_i$ are discrete variables. A typical example is the credit +card data discussed below here, where we can set the state of +defaulting the debt to $y_i=1$ and not to $y_i=0$ for one the persons +in the data set (see the full example below). + +One simple way to get a discrete output is to have sign +functions that map the output of a linear regressor to values $\{0,1\}$, +$f(s_i)=sign(s_i)=1$ if $s_i\ge 0$ and 0 if otherwise. +We will encounter this model in our first demonstration of neural networks. Historically it is called the "perceptron" model in the machine learning +literature. This model is extremely simple. However, in many cases it is more +favorable to use a ``soft" classifier that outputs +the probability of a given category. This leads us to the logistic function. + + +## The logistic function + +The perceptron is an example of a ``hard classification" model. We +will encounter this model when we discuss neural networks as +well. Each datapoint is deterministically assigned to a category (i.e +$y_i=0$ or $y_i=1$). In many cases, it is favorable to have a "soft" +classifier that outputs the probability of a given category rather +than a single value. For example, given $x_i$, the classifier +outputs the probability of being in a category $k$. Logistic regression +is the most common example of a so-called soft classifier. In logistic +regression, the probability that a data point $x_i$ +belongs to a category $y_i=\{0,1\}$ is given by the so-called logit function (or Sigmoid) which is meant to represent the likelihood for a given event, + +$$ +p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}. +$$ + +Note that $1-p(t)= p(-t)$. + +## Examples of likelihood functions used in logistic regression and nueral networks + + +The following code plots the logistic function, the step function and other functions we will encounter from here and on. + +%matplotlib inline + +"""The sigmoid function (or the logistic curve) is a +function that takes any real number, z, and outputs a number (0,1). +It is useful in neural networks for assigning weights on a relative scale. +The value z is the weighted sum of parameters involved in the learning algorithm.""" + +import numpy +import matplotlib.pyplot as plt +import math as mt + +z = numpy.arange(-5, 5, .1) +sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z))) +sigma = sigma_fn(z) + +fig = plt.figure() +ax = fig.add_subplot(111) +ax.plot(z, sigma) +ax.set_ylim([-0.1, 1.1]) +ax.set_xlim([-5,5]) +ax.grid(True) +ax.set_xlabel('z') +ax.set_title('sigmoid function') + +plt.show() + +"""Step Function""" +z = numpy.arange(-5, 5, .02) +step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0) +step = step_fn(z) + +fig = plt.figure() +ax = fig.add_subplot(111) +ax.plot(z, step) +ax.set_ylim([-0.5, 1.5]) +ax.set_xlim([-5,5]) +ax.grid(True) +ax.set_xlabel('z') +ax.set_title('step function') + +plt.show() + +"""tanh Function""" +z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1) +t = numpy.tanh(z) + +fig = plt.figure() +ax = fig.add_subplot(111) +ax.plot(z, t) +ax.set_ylim([-1.0, 1.0]) +ax.set_xlim([-2*mt.pi,2*mt.pi]) +ax.grid(True) +ax.set_xlabel('z') +ax.set_title('tanh function') + +plt.show() + +## Two parameters + +We assume now that we have two classes with $y_i$ either $0$ or $1$. Furthermore we assume also that we have only two parameters $\beta$ in our fitting of the Sigmoid function, that is we define probabilities + +$$ +\begin{align*} +p(y_i=1|x_i,\hat{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\ +p(y_i=0|x_i,\hat{\beta}) &= 1 - p(y_i=1|x_i,\hat{\beta}), +\end{align*} +$$ + +where $\hat{\beta}$ are the weights we wish to extract from data, in our case $\beta_0$ and $\beta_1$. + +Note that we used + +$$ +p(y_i=0\vert x_i, \hat{\beta}) = 1-p(y_i=1\vert x_i, \hat{\beta}). +$$ + + +## Maximum likelihood + +In order to define the total likelihood for all possible outcomes from a +dataset $\mathcal{D}=\{(y_i,x_i)\}$, with the binary labels +$y_i\in\{0,1\}$ and where the data points are drawn independently, we use the so-called [Maximum Likelihood Estimation](https://en.wikipedia.org/wiki/Maximum_likelihood_estimation) (MLE) principle. +We aim thus at maximizing +the probability of seeing the observed data. We can then approximate the +likelihood in terms of the product of the individual probabilities of a specific outcome $y_i$, that is + +$$ +\begin{align*} +P(\mathcal{D}|\hat{\beta})& = \prod_{i=1}^n \left[p(y_i=1|x_i,\hat{\beta})\right]^{y_i}\left[1-p(y_i=1|x_i,\hat{\beta}))\right]^{1-y_i}\nonumber \\ +\end{align*} +$$ + +from which we obtain the log-likelihood and our **cost/loss** function + +$$ +\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left( y_i\log{p(y_i=1|x_i,\hat{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\hat{\beta}))\right]\right). +$$ + +## The cost function rewritten + +Reordering the logarithms, we can rewrite the **cost/loss** function as + +$$ +\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right). +$$ + +The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to $\beta$. +Since the cost (error) function is just the negative log-likelihood, for logistic regression we have that + +$$ +\mathcal{C}(\hat{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right). +$$ + +This equation is known in statistics as the **cross entropy**. Finally, we note that just as in linear regression, +in practice we often supplement the cross-entropy with additional regularization terms, usually $L_1$ and $L_2$ regularization as we did for Ridge and Lasso regression. + +## Minimizing the cross entropy + +The cross entropy is a convex function of the weights $\hat{\beta}$ and, +therefore, any local minimizer is a global minimizer. + + +Minimizing this +cost function with respect to the two parameters $\beta_0$ and $\beta_1$ we obtain + +$$ +\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_0} = -\sum_{i=1}^n \left(y_i -\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right), +$$ + +and + +$$ +\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_1} = -\sum_{i=1}^n \left(y_ix_i -x_i\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right). +$$ + +## A more compact expression + +Let us now define a vector $\hat{y}$ with $n$ elements $y_i$, an +$n\times p$ matrix $\hat{X}$ which contains the $x_i$ values and a +vector $\hat{p}$ of fitted probabilities $p(y_i\vert x_i,\hat{\beta})$. We can rewrite in a more compact form the first +derivative of cost function as + +$$ +\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}} = -\hat{X}^T\left(\hat{y}-\hat{p}\right). +$$ + +If we in addition define a diagonal matrix $\hat{W}$ with elements +$p(y_i\vert x_i,\hat{\beta})(1-p(y_i\vert x_i,\hat{\beta})$, we can obtain a compact expression of the second derivative as + +$$ +\frac{\partial^2 \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}\partial \hat{\beta}^T} = \hat{X}^T\hat{W}\hat{X}. +$$ + +## Extending to more predictors + +Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with $p$ predictors + +$$ +\log{ \frac{p(\hat{\beta}\hat{x})}{1-p(\hat{\beta}\hat{x})}} = \beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p. +$$ + +Here we defined $\hat{x}=[1,x_1,x_2,\dots,x_p]$ and $\hat{\beta}=[\beta_0, \beta_1, \dots, \beta_p]$ leading to + +$$ +p(\hat{\beta}\hat{x})=\frac{ \exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}{1+\exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}. +$$ + +## Including more classes + +Till now we have mainly focused on two classes, the so-called binary +system. Suppose we wish to extend to $K$ classes. Let us for the sake +of simplicity assume we have only two predictors. We have then +following model + +1 +5 + +< +< +< +! +! +M +A +T +H +_ +B +L +O +C +K + +$$ +\log{\frac{p(C=2\vert x)}{p(K\vert x)}} = \beta_{20}+\beta_{21}x_1, +$$ + +and so on till the class $C=K-1$ class + +$$ +\log{\frac{p(C=K-1\vert x)}{p(K\vert x)}} = \beta_{(K-1)0}+\beta_{(K-1)1}x_1, +$$ + +and the model is specified in term of $K-1$ so-called log-odds or +**logit** transformations. + + +## More classes + +In our discussion of neural networks we will encounter the above again +in terms of a slightly modified function, the so-called **Softmax** function. + +The softmax function is used in various multiclass classification +methods, such as multinomial logistic regression (also known as +softmax regression), multiclass linear discriminant analysis, naive +Bayes classifiers, and artificial neural networks. Specifically, in +multinomial logistic regression and linear discriminant analysis, the +input to the function is the result of $K$ distinct linear functions, +and the predicted probability for the $k$-th class given a sample +vector $\hat{x}$ and a weighting vector $\hat{\beta}$ is (with two +predictors): + +$$ +p(C=k\vert \mathbf {x} )=\frac{\exp{(\beta_{k0}+\beta_{k1}x_1)}}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}. +$$ + +It is easy to extend to more predictors. The final class is + +$$ +p(C=K\vert \mathbf {x} )=\frac{1}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}, +$$ + +and they sum to one. Our earlier discussions were all specialized to +the case with two classes only. It is easy to see from the above that +what we derived earlier is compatible with these equations. + +To find the optimal parameters we would typically use a gradient +descent method. Newton's method and gradient descent methods are +discussed in the material on [optimization +methods](https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html). + + + + +## A simple classification problem + +import numpy as np +from sklearn import datasets, linear_model +import matplotlib.pyplot as plt + + +def generate_data(): + np.random.seed(0) + X, y = datasets.make_moons(200, noise=0.20) + return X, y + + +def visualize(X, y, clf): + plot_decision_boundary(lambda x: clf.predict(x), X, y) + +def plot_decision_boundary(pred_func, X, y): + # Set min and max values and give it some padding + x_min, x_max = X[:, 0].min() - .5, X[:, 0].max() + .5 + y_min, y_max = X[:, 1].min() - .5, X[:, 1].max() + .5 + h = 0.01 + # Generate a grid of points with distance h between them + xx, yy = np.meshgrid(np.arange(x_min, x_max, h), np.arange(y_min, y_max, h)) + # Predict the function value for the whole gid + Z = pred_func(np.c_[xx.ravel(), yy.ravel()]) + Z = Z.reshape(xx.shape) + # Plot the contour and training examples + plt.contourf(xx, yy, Z, cmap=plt.cm.Spectral) + plt.scatter(X[:, 0], X[:, 1], c=y, cmap=plt.cm.Spectral) + plt.show() + + +def classify(X, y): + clf = linear_model.LogisticRegressionCV() + clf.fit(X, y) + return clf + + +def main(): + X, y = generate_data() + # visualize(X, y) + clf = classify(X, y) + visualize(X, y, clf) + +if __name__ == "__main__": + main() + +## The Credit Card example +Here we use the the [credit card data](https://archive.ics.uci.edu/ml/datasets/default+of+credit+card+clients). +The data are from an extensive database from Taiwan and include more than ten predictors. + +For categorical data -Scikit-Learn- provides a so-called **one-hot encoder**. +This is called one-hot +encoding, because only one attribute will be equal to 1 (hot), while the others will be 0 (cold). +**Scikit-Learn** provides a OneHotEncoder encoder to convert integer categorical values into one-hot + +from sklearn.preprocessing import OneHotEncoder +encoder = OneHotEncoder() + +## How to read the Credit Card data + +import pandas as pd +import os +import numpy as np + + +from sklearn.model_selection import train_test_split +from sklearn.preprocessing import OneHotEncoder +from sklearn.compose import ColumnTransformer +from sklearn.preprocessing import StandardScaler, OneHotEncoder +from sklearn.metrics import confusion_matrix, accuracy_score, roc_auc_score + +# Trying to set the seed +np.random.seed(0) +import random +random.seed(0) + +# Reading file into data frame +cwd = os.getcwd() +filename = cwd + '/default of credit card clients.xls' +nanDict = {} +df = pd.read_excel(filename, header=1, skiprows=0, index_col=0, na_values=nanDict) + +df.rename(index=str, columns={"default payment next month": "defaultPaymentNextMonth"}, inplace=True) + +# Features and targets +X = df.loc[:, df.columns != 'defaultPaymentNextMonth'].values +y = df.loc[:, df.columns == 'defaultPaymentNextMonth'].values + +# Categorical variables to one-hot's +onehotencoder = OneHotEncoder(categories="auto") + +X = ColumnTransformer( + [("", onehotencoder, [3]),], + remainder="passthrough" +).fit_transform(X) + +y.shape + +# Train-test split +trainingShare = 0.5 +seed = 1 +XTrain, XTest, yTrain, yTest=train_test_split(X, y, train_size=trainingShare, \ + test_size = 1-trainingShare, + random_state=seed) + +# Input Scaling +sc = StandardScaler() +XTrain = sc.fit_transform(XTrain) +XTest = sc.transform(XTest) + +# One-hot's of the target vector +Y_train_onehot, Y_test_onehot = onehotencoder.fit_transform(yTrain), onehotencoder.fit_transform(yTest) + +# Remove instances with zeros only for past bill statements or paid amounts +''' +df = df.drop(df[(df.BILL_AMT1 == 0) & + (df.BILL_AMT2 == 0) & + (df.BILL_AMT3 == 0) & + (df.BILL_AMT4 == 0) & + (df.BILL_AMT5 == 0) & + (df.BILL_AMT6 == 0) & + (df.PAY_AMT1 == 0) & + (df.PAY_AMT2 == 0) & + (df.PAY_AMT3 == 0) & + (df.PAY_AMT4 == 0) & + (df.PAY_AMT5 == 0) & + (df.PAY_AMT6 == 0)].index) +''' +df = df.drop(df[(df.BILL_AMT1 == 0) & + (df.BILL_AMT2 == 0) & + (df.BILL_AMT3 == 0) & + (df.BILL_AMT4 == 0) & + (df.BILL_AMT5 == 0) & + (df.BILL_AMT6 == 0)].index) + +df = df.drop(df[(df.PAY_AMT1 == 0) & + (df.PAY_AMT2 == 0) & + (df.PAY_AMT3 == 0) & + (df.PAY_AMT4 == 0) & + (df.PAY_AMT5 == 0) & + (df.PAY_AMT6 == 0)].index) + +from sklearn.linear_model import LogisticRegression +from sklearn.model_selection import GridSearchCV + +lambdas=np.logspace(-5,7,13) +parameters = [{'C': 1./lambdas, "solver":["lbfgs"]}]#*len(parameters)}] +scoring = ['accuracy', 'roc_auc'] +logReg = LogisticRegression() +gridSearch = GridSearchCV(logReg, parameters, cv=5, scoring=scoring, refit='roc_auc') \ No newline at end of file diff --git a/doc/src/LectureNotes/_build/jupyter_execute/chapter5_42_0.png b/doc/src/LectureNotes/_build/jupyter_execute/chapter5_42_0.png new file mode 100644 index 000000000..d4c378f37 Binary files /dev/null and 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This means that you can include\n", - "code blocks and their outputs in your book.\n", - "\n", - "## Markdown + notebooks\n", - "\n", - "As it is markdown, you can embed images, HTML, etc into your posts!\n", - "\n", - "![](https://myst-parser.readthedocs.io/en/latest/_static/logo.png)\n", - "\n", - "You an also $add_{math}$ and\n", - "\n", - "$$\n", - "math^{blocks}\n", - "$$\n", - "\n", - "or\n", - "\n", - "$$\n", - "\\begin{aligned}\n", - "\\mbox{mean} la_{tex} \\\\ \\\\\n", - "math blocks\n", - "\\end{aligned}\n", - "$$\n", - "\n", - "But make sure you \\$Escape \\$your \\$dollar signs \\$you want to keep!\n", - "\n", - "## MyST markdown\n", - "\n", - "MyST markdown works in Jupyter Notebooks as well. For more information about MyST markdown, check\n", - "out [the MyST guide in Jupyter Book](https://jupyterbook.org/content/myst.html),\n", - "or see [the MyST markdown documentation](https://myst-parser.readthedocs.io/en/latest/).\n", - "\n", - "## Code blocks and outputs\n", - "\n", - "Jupyter Book will also embed your code blocks and output in your book.\n", - "For example, here's some sample Matplotlib code:" - ] - }, - { - "cell_type": "code", - "execution_count": 1, - "metadata": {}, - "outputs": [], - "source": [ - "from matplotlib import rcParams, cycler\n", - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "plt.ion()" - ] - }, - { - "cell_type": "code", - "execution_count": 2, - "metadata": {}, - "outputs": [ - { - "data": { - "image/png": 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- "text/plain": [ - "
" - ] - }, - "metadata": { - "filenames": { - "image/png": "/Users/MortenImac/Teaching/MachineLearning/doc/src/LectureNotes/_build/jupyter_execute/notebooks_2_0.png" - }, - "needs_background": "light" - }, - "output_type": "display_data" - } - ], - "source": [ - "# Fixing random state for reproducibility\n", - "np.random.seed(19680801)\n", - "\n", - "N = 10\n", - "data = [np.logspace(0, 1, 100) + np.random.randn(100) + ii for ii in range(N)]\n", - "data = np.array(data).T\n", - "cmap = plt.cm.coolwarm\n", - "rcParams['axes.prop_cycle'] = cycler(color=cmap(np.linspace(0, 1, N)))\n", - "\n", - "\n", - "from matplotlib.lines import Line2D\n", - "custom_lines = [Line2D([0], [0], color=cmap(0.), lw=4),\n", - " Line2D([0], [0], color=cmap(.5), lw=4),\n", - " Line2D([0], [0], color=cmap(1.), lw=4)]\n", - "\n", - "fig, ax = plt.subplots(figsize=(10, 5))\n", - "lines = ax.plot(data)\n", - "ax.legend(custom_lines, ['Cold', 'Medium', 'Hot']);" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "There is a lot more that you can do with outputs (such as including interactive outputs)\n", - "with your book. For more information about this, see [the Jupyter Book documentation](https://jupyterbook.org)" - ] - } - ], - "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": "3.6.8" - }, - "widgets": { - "application/vnd.jupyter.widget-state+json": { - "state": {}, - "version_major": 2, - "version_minor": 0 - } - } - }, - "nbformat": 4, - "nbformat_minor": 4 -} \ No newline at end of file diff --git a/doc/src/LectureNotes/_build/jupyter_execute/notebooks.py b/doc/src/LectureNotes/_build/jupyter_execute/notebooks.py deleted file mode 100644 index 945002c60..000000000 --- a/doc/src/LectureNotes/_build/jupyter_execute/notebooks.py +++ /dev/null @@ -1,65 +0,0 @@ -# Content with notebooks - -You can also create content with Jupyter Notebooks. This means that you can include -code blocks and their outputs in your book. - -## Markdown + notebooks - -As it is markdown, you can embed images, HTML, etc into your posts! - -![](https://myst-parser.readthedocs.io/en/latest/_static/logo.png) - -You an also $add_{math}$ and - -$$ -math^{blocks} -$$ - -or - -$$ -\begin{aligned} -\mbox{mean} la_{tex} \\ \\ -math blocks -\end{aligned} -$$ - -But make sure you \$Escape \$your \$dollar signs \$you want to keep! - -## MyST markdown - -MyST markdown works in Jupyter Notebooks as well. For more information about MyST markdown, check -out [the MyST guide in Jupyter Book](https://jupyterbook.org/content/myst.html), -or see [the MyST markdown documentation](https://myst-parser.readthedocs.io/en/latest/). - -## Code blocks and outputs - -Jupyter Book will also embed your code blocks and output in your book. -For example, here's some sample Matplotlib code: - -from matplotlib import rcParams, cycler -import matplotlib.pyplot as plt -import numpy as np -plt.ion() - -# Fixing random state for reproducibility -np.random.seed(19680801) - -N = 10 -data = [np.logspace(0, 1, 100) + np.random.randn(100) + ii for ii in range(N)] -data = np.array(data).T -cmap = plt.cm.coolwarm -rcParams['axes.prop_cycle'] = cycler(color=cmap(np.linspace(0, 1, N))) - - -from matplotlib.lines import Line2D -custom_lines = [Line2D([0], [0], color=cmap(0.), lw=4), - Line2D([0], [0], color=cmap(.5), lw=4), - Line2D([0], [0], color=cmap(1.), lw=4)] - -fig, ax = plt.subplots(figsize=(10, 5)) -lines = ax.plot(data) -ax.legend(custom_lines, ['Cold', 'Medium', 'Hot']); - -There is a lot more that you can do with outputs (such as including interactive outputs) -with your book. For more information about this, see [the Jupyter Book documentation](https://jupyterbook.org) \ No newline at end of file diff --git a/doc/src/LectureNotes/_build/jupyter_execute/notebooks_2_0.png b/doc/src/LectureNotes/_build/jupyter_execute/notebooks_2_0.png deleted file mode 100644 index fbee95cfe..000000000 Binary files a/doc/src/LectureNotes/_build/jupyter_execute/notebooks_2_0.png and /dev/null differ diff --git a/doc/src/LectureNotes/_build/jupyter_execute/regression_278_2.png b/doc/src/LectureNotes/_build/jupyter_execute/regression_278_2.png deleted file mode 100644 index 602fb57d5..000000000 Binary files a/doc/src/LectureNotes/_build/jupyter_execute/regression_278_2.png and /dev/null differ diff --git a/doc/src/LectureNotes/_build/latex/LICRcyr2utf8.xdy b/doc/src/LectureNotes/_build/latex/LICRcyr2utf8.xdy new file mode 100644 index 000000000..a9ca1c82c --- /dev/null +++ b/doc/src/LectureNotes/_build/latex/LICRcyr2utf8.xdy @@ -0,0 +1,101 @@ +;; -*- coding: utf-8; mode: Lisp; -*- +;; style file for xindy +;; filename: LICRcyr2utf8.xdy +;; description: style file for xindy which maps back LaTeX Internal +;; Character Representation of Cyrillic to utf-8 +;; usage: for use with pdflatex produced .idx files. +;; Contributed by the Sphinx team, July 2018. +(merge-rule "\IeC {\'\CYRG }" "Ѓ" :string) +(merge-rule "\IeC {\'\CYRK }" "ÐŒ" :string) +(merge-rule "\IeC {\'\cyrg }" "Ñ“" :string) +(merge-rule "\IeC {\'\cyrk }" "Ñœ" :string) +(merge-rule "\IeC {\CYRA }" "Ð" :string) +(merge-rule "\IeC {\CYRB }" "Б" :string) +(merge-rule "\IeC {\CYRC }" "Ц" :string) +(merge-rule "\IeC {\CYRCH }" "Ч" :string) +(merge-rule "\IeC {\CYRD }" "Д" :string) +(merge-rule "\IeC {\CYRDJE }" "Ђ" :string) +(merge-rule "\IeC {\CYRDZE }" "Ð…" :string) +(merge-rule "\IeC {\CYRDZHE }" "Ð" :string) +(merge-rule "\IeC {\CYRE }" "Е" :string) +(merge-rule "\IeC {\CYREREV }" "Э" :string) +(merge-rule "\IeC {\CYRERY }" "Ы" :string) +(merge-rule "\IeC {\CYRF }" "Ф" :string) +(merge-rule "\IeC {\CYRG }" "Г" :string) +(merge-rule "\IeC {\CYRGUP }" "Ò" :string) +(merge-rule "\IeC {\CYRH }" "Ð¥" :string) +(merge-rule "\IeC {\CYRHRDSN }" "Ъ" :string) +(merge-rule "\IeC {\CYRI }" "И" :string) +(merge-rule "\IeC {\CYRIE }" "Є" :string) +(merge-rule "\IeC {\CYRII }" "І" :string) +(merge-rule "\IeC {\CYRISHRT }" "Й" :string) +(merge-rule "\IeC {\CYRJE }" "Ј" :string) +(merge-rule "\IeC {\CYRK }" "К" :string) +(merge-rule "\IeC {\CYRL }" "Л" :string) +(merge-rule "\IeC {\CYRLJE }" "Љ" :string) +(merge-rule "\IeC {\CYRM }" "Ðœ" :string) +(merge-rule "\IeC {\CYRN }" "Ð" :string) +(merge-rule "\IeC {\CYRNJE }" "Њ" :string) +(merge-rule "\IeC {\CYRO }" "О" :string) +(merge-rule "\IeC {\CYRP }" "П" :string) +(merge-rule "\IeC {\CYRR }" "Р" :string) +(merge-rule "\IeC {\CYRS }" "С" :string) +(merge-rule "\IeC {\CYRSFTSN }" "Ь" :string) +(merge-rule "\IeC {\CYRSH }" "Ш" :string) +(merge-rule "\IeC {\CYRSHCH }" "Щ" :string) +(merge-rule "\IeC {\CYRT }" "Т" :string) +(merge-rule "\IeC {\CYRTSHE }" "Ћ" :string) +(merge-rule "\IeC {\CYRU }" "У" :string) +(merge-rule "\IeC {\CYRUSHRT }" "ÐŽ" :string) +(merge-rule "\IeC {\CYRV }" "Ð’" :string) +(merge-rule "\IeC {\CYRYA }" "Я" :string) +(merge-rule "\IeC {\CYRYI }" "Ї" :string) +(merge-rule "\IeC {\CYRYO }" "Ð" :string) +(merge-rule "\IeC {\CYRYU }" "Ю" :string) +(merge-rule "\IeC {\CYRZ }" "З" :string) +(merge-rule "\IeC {\CYRZH }" "Ж" :string) +(merge-rule "\IeC {\cyra }" "а" :string) +(merge-rule "\IeC {\cyrb }" "б" :string) +(merge-rule "\IeC {\cyrc }" "ц" :string) +(merge-rule "\IeC {\cyrch }" "ч" :string) +(merge-rule "\IeC {\cyrd }" "д" :string) +(merge-rule "\IeC {\cyrdje }" "Ñ’" :string) +(merge-rule "\IeC {\cyrdze }" "Ñ•" :string) +(merge-rule "\IeC {\cyrdzhe }" "ÑŸ" :string) +(merge-rule "\IeC {\cyre }" "е" :string) +(merge-rule "\IeC {\cyrerev }" "Ñ" :string) +(merge-rule "\IeC {\cyrery }" "Ñ‹" :string) +(merge-rule "\IeC {\cyrf }" "Ñ„" :string) +(merge-rule "\IeC {\cyrg }" "г" :string) +(merge-rule "\IeC {\cyrgup }" "Ò‘" :string) +(merge-rule "\IeC {\cyrh }" "Ñ…" :string) +(merge-rule "\IeC {\cyrhrdsn }" "ÑŠ" :string) +(merge-rule "\IeC {\cyri }" "и" :string) +(merge-rule "\IeC {\cyrie }" "Ñ”" :string) +(merge-rule "\IeC {\cyrii }" "Ñ–" :string) +(merge-rule "\IeC {\cyrishrt }" "й" :string) +(merge-rule "\IeC {\cyrje }" "ј" :string) +(merge-rule "\IeC {\cyrk }" "к" :string) +(merge-rule "\IeC {\cyrl }" "л" :string) +(merge-rule "\IeC {\cyrlje }" "Ñ™" :string) +(merge-rule "\IeC {\cyrm }" "м" :string) +(merge-rule "\IeC {\cyrn }" "н" :string) +(merge-rule "\IeC {\cyrnje }" "Ñš" :string) +(merge-rule "\IeC {\cyro }" "о" :string) +(merge-rule "\IeC {\cyrp }" "п" :string) +(merge-rule "\IeC {\cyrr }" "Ñ€" :string) +(merge-rule "\IeC {\cyrs }" "Ñ" :string) +(merge-rule "\IeC {\cyrsftsn }" "ÑŒ" :string) +(merge-rule "\IeC {\cyrsh }" "ш" :string) +(merge-rule "\IeC {\cyrshch }" "щ" :string) +(merge-rule "\IeC {\cyrt }" "Ñ‚" :string) +(merge-rule "\IeC {\cyrtshe }" "Ñ›" :string) +(merge-rule "\IeC {\cyru }" "у" :string) +(merge-rule "\IeC {\cyrushrt }" "Ñž" :string) +(merge-rule "\IeC {\cyrv }" "в" :string) +(merge-rule "\IeC {\cyrya }" "Ñ" :string) +(merge-rule "\IeC {\cyryi }" "Ñ—" :string) +(merge-rule "\IeC {\cyryo }" "Ñ‘" :string) +(merge-rule "\IeC {\cyryu }" "ÑŽ" :string) +(merge-rule "\IeC {\cyrz }" "з" :string) +(merge-rule "\IeC {\cyrzh }" "ж" :string) diff --git a/doc/src/LectureNotes/_build/latex/LICRlatin2utf8.xdy b/doc/src/LectureNotes/_build/latex/LICRlatin2utf8.xdy new file mode 100644 index 000000000..31c80f97c --- /dev/null +++ b/doc/src/LectureNotes/_build/latex/LICRlatin2utf8.xdy @@ -0,0 +1,239 @@ +;; style file for xindy +;; filename: LICRlatin2utf8.xdy +;; description: style file for xindy which maps back LaTeX Internal +;; Character Representation of letters (as arising in .idx index +;; file) to UTF-8 encoding for correct sorting by xindy. +;; usage: for use with the pdflatex engine, +;; *not* for use with xelatex or lualatex. +;; +;; This is based upon xindy's distributed file tex/inputenc/utf8.xdy. +;; The modifications include: +;; +;; - Updates for compatibility with current LaTeX macro encoding. +;; +;; - Systematic usage of the \IeC {...} mark-up, because mark-up in +;; tex/inputenc/utf8.xdy was using it on seemingly random basis, and +;; Sphinx coercing of xindy usability for both Latin and Cyrillic scripts +;; with pdflatex requires its systematic presence here. +;; +;; - Support for some extra letters: Ÿ, ÅŠ, Å‹, Å’, Å“, IJ, ij, È· and ẞ. +;; +;; Indeed Sphinx needs to support for pdflatex engine all Unicode letters +;; available in TeX T1 font encoding. The above letters are found in +;; that encoding but not in the Latin1, 2, 3 charsets which are those +;; covered by original tex/inputenc/utf8.xdy. +;; +;; - There is a problem that È· is not supported out-of-the box by LaTeX +;; with inputenc, one must add explicitely +;; \DeclareUnicodeCharacter{0237}{\j} +;; to preamble of LaTeX document. However this character is not supported +;; by the TeX "times" font used by default by Sphinx for pdflatex engine. +;; +;; **Update**: since LaTeX 2018/12/01, the \j as well as \SS, \k{} and +;; \.{} need no extra user declaration anymore. +;; +;; - ẞ needs \DeclareUnicodeCharacter{1E9E}{\SS} (but ß needs no extra set-up). +;; +;; - U+02DB (Ë›) and U+02D9 (Ë™) are also not supported by inputenc +;; out of the box and require +;; \DeclareUnicodeCharacter{02DB}{\k{}} +;; \DeclareUnicodeCharacter{02D9}{\.{}} +;; to be added to preamble. +;; +;; - U+0127 ħ and U+0126 Ħ are absent from TeX T1+TS1 font encodings. +;; +;; - Characters ÅŠ and Å‹ are not supported by TeX font "times" used by +;; default by Sphinx for pdflatex engine but they are supported by +;; some TeX fonts, in particular by the default LaTeX font for T1 +;; encoding. +;; +;; - " and ~ must be escaped as ~" and resp. ~~ in xindy merge rules. +;; +;; Contributed by the Sphinx team, July 2018. +;; +;; See sphinx.xdy for superior figures, as they are escaped by LaTeX writer. +(merge-rule "\IeC {\textonesuperior }" "¹" :string) +(merge-rule "\IeC {\texttwosuperior }" "²" :string) +(merge-rule "\IeC {\textthreesuperior }" "³" :string) +(merge-rule "\IeC {\'a}" "á" :string) +(merge-rule "\IeC {\'A}" "Ã" :string) +(merge-rule "\IeC {\`a}" "à" :string) +(merge-rule "\IeC {\`A}" "À" :string) +(merge-rule "\IeC {\^a}" "â" :string) +(merge-rule "\IeC {\^A}" "Â" :string) +(merge-rule "\IeC {\~"a}" "ä" :string) +(merge-rule "\IeC {\~"A}" "Ä" :string) +(merge-rule "\IeC {\~~a}" "ã" :string) +(merge-rule "\IeC {\~~A}" "Ã" :string) +(merge-rule "\IeC {\c c}" "ç" :string) +(merge-rule "\IeC {\c C}" "Ç" :string) +(merge-rule "\IeC {\'c}" "ć" :string) +(merge-rule "\IeC {\'C}" "Ć" :string) +(merge-rule "\IeC {\^c}" "ĉ" :string) +(merge-rule "\IeC {\^C}" "Ĉ" :string) +(merge-rule "\IeC {\.c}" "Ä‹" :string) +(merge-rule "\IeC {\.C}" "ÄŠ" :string) +(merge-rule "\IeC {\c s}" "ÅŸ" :string) +(merge-rule "\IeC {\c S}" "Åž" :string) +(merge-rule "\IeC {\c t}" "Å£" :string) +(merge-rule "\IeC {\c T}" "Å¢" :string) +(merge-rule "\IeC {\-}" "­" :string); soft hyphen +(merge-rule "\IeC {\textdiv }" "÷" :string) +(merge-rule "\IeC {\'e}" "é" :string) +(merge-rule "\IeC {\'E}" "É" :string) +(merge-rule "\IeC {\`e}" "è" :string) +(merge-rule "\IeC {\`E}" "È" :string) +(merge-rule "\IeC {\^e}" "ê" :string) +(merge-rule "\IeC {\^E}" "Ê" :string) +(merge-rule "\IeC {\~"e}" "ë" :string) +(merge-rule "\IeC {\~"E}" "Ë" :string) +(merge-rule "\IeC {\^g}" "Ä" :string) +(merge-rule "\IeC {\^G}" "Äœ" :string) +(merge-rule "\IeC {\.g}" "Ä¡" :string) +(merge-rule "\IeC {\.G}" "Ä " :string) +(merge-rule "\IeC {\^h}" "Ä¥" :string) +(merge-rule "\IeC {\^H}" "Ĥ" :string) +(merge-rule "\IeC {\H o}" "Å‘" :string) +(merge-rule "\IeC {\H O}" "Å" :string) +(merge-rule "\IeC {\textacutedbl }" "Ë" :string) +(merge-rule "\IeC {\H u}" "ű" :string) +(merge-rule "\IeC {\H U}" "Å°" :string) +(merge-rule "\IeC {\ae }" "æ" :string) +(merge-rule "\IeC {\AE }" "Æ" :string) +(merge-rule "\IeC {\textcopyright }" "©" :string) +(merge-rule "\IeC {\c \ }" "¸" :string) +(merge-rule "\IeC {\dh }" "ð" :string) +(merge-rule "\IeC {\DH }" "Ã" :string) +(merge-rule "\IeC {\dj }" "Ä‘" :string) +(merge-rule "\IeC {\DJ }" "Ä" :string) +(merge-rule "\IeC {\guillemotleft }" "«" :string) +(merge-rule "\IeC {\guillemotright }" "»" :string) +(merge-rule "\IeC {\'\i }" "í" :string) +(merge-rule "\IeC {\`\i }" "ì" :string) +(merge-rule "\IeC {\^\i }" "î" :string) +(merge-rule "\IeC {\~"\i }" "ï" :string) +(merge-rule "\IeC {\i }" "ı" :string) +(merge-rule "\IeC {\^\j }" "ĵ" :string) +(merge-rule "\IeC {\k {}}" "Ë›" :string) +(merge-rule "\IeC {\l }" "Å‚" :string) +(merge-rule "\IeC {\L }" "Å" :string) +(merge-rule "\IeC {\nobreakspace }" " " :string) +(merge-rule "\IeC {\o }" "ø" :string) +(merge-rule "\IeC {\O }" "Ø" :string) +(merge-rule "\IeC {\textsterling }" "£" :string) +(merge-rule "\IeC {\textparagraph }" "¶" :string) +(merge-rule "\IeC {\ss }" "ß" :string) +(merge-rule "\IeC {\textsection }" "§" :string) +(merge-rule "\IeC {\textbrokenbar }" "¦" :string) +(merge-rule "\IeC {\textcent }" "¢" :string) +(merge-rule "\IeC {\textcurrency }" "¤" :string) +(merge-rule "\IeC {\textdegree }" "°" :string) +(merge-rule "\IeC {\textexclamdown }" "¡" :string) +(merge-rule "\IeC {\texthbar }" "ħ" :string) +(merge-rule "\IeC {\textHbar }" "Ħ" :string) +(merge-rule "\IeC {\textonehalf }" "½" :string) +(merge-rule "\IeC {\textonequarter }" "¼" :string) +(merge-rule "\IeC {\textordfeminine }" "ª" :string) +(merge-rule "\IeC {\textordmasculine }" "º" :string) +(merge-rule "\IeC {\textperiodcentered }" "·" :string) +(merge-rule "\IeC {\textquestiondown }" "¿" :string) +(merge-rule "\IeC {\textregistered }" "®" :string) +(merge-rule "\IeC {\textthreequarters }" "¾" :string) +(merge-rule "\IeC {\textyen }" "Â¥" :string) +(merge-rule "\IeC {\th }" "þ" :string) +(merge-rule "\IeC {\TH }" "Þ" :string) +(merge-rule "\IeC {\'I}" "Ã" :string) +(merge-rule "\IeC {\`I}" "ÃŒ" :string) +(merge-rule "\IeC {\^I}" "ÃŽ" :string) +(merge-rule "\IeC {\~"I}" "Ã" :string) +(merge-rule "\IeC {\.I}" "Ä°" :string) +(merge-rule "\IeC {\^J}" "Ä´" :string) +(merge-rule "\IeC {\k a}" "Ä…" :string) +(merge-rule "\IeC {\k A}" "Ä„" :string) +(merge-rule "\IeC {\k e}" "Ä™" :string) +(merge-rule "\IeC {\k E}" "Ę" :string) +(merge-rule "\IeC {\'l}" "ĺ" :string) +(merge-rule "\IeC {\'L}" "Ĺ" :string) +(merge-rule "\IeC {\textlnot }" "¬" :string) +(merge-rule "\IeC {\textmu }" "µ" :string) +(merge-rule "\IeC {\'n}" "Å„" :string) +(merge-rule "\IeC {\'N}" "Ń" :string) +(merge-rule "\IeC {\~~n}" "ñ" :string) +(merge-rule "\IeC {\~~N}" "Ñ" :string) +(merge-rule "\IeC {\'o}" "ó" :string) +(merge-rule "\IeC {\'O}" "Ó" :string) +(merge-rule "\IeC {\`o}" "ò" :string) +(merge-rule "\IeC {\`O}" "Ã’" :string) +(merge-rule "\IeC {\^o}" "ô" :string) +(merge-rule "\IeC {\^O}" "Ô" :string) +(merge-rule "\IeC {\~"o}" "ö" :string) +(merge-rule "\IeC {\~"O}" "Ö" :string) +(merge-rule "\IeC {\~~o}" "õ" :string) +(merge-rule "\IeC {\~~O}" "Õ" :string) +(merge-rule "\IeC {\textpm }" "±" :string) +(merge-rule "\IeC {\r a}" "Ã¥" :string) +(merge-rule "\IeC {\r A}" "Ã…" :string) +(merge-rule "\IeC {\'r}" "Å•" :string) +(merge-rule "\IeC {\'R}" "Å”" :string) +(merge-rule "\IeC {\r u}" "ů" :string) +(merge-rule "\IeC {\r U}" "Å®" :string) +(merge-rule "\IeC {\'s}" "Å›" :string) +(merge-rule "\IeC {\'S}" "Åš" :string) +(merge-rule "\IeC {\^s}" "Å" :string) +(merge-rule "\IeC {\^S}" "Åœ" :string) +(merge-rule "\IeC {\textasciidieresis }" "¨" :string) +(merge-rule "\IeC {\textasciimacron }" "¯" :string) +(merge-rule "\IeC {\.{}}" "Ë™" :string) +(merge-rule "\IeC {\textasciiacute }" "´" :string) +(merge-rule "\IeC {\texttimes }" "×" :string) +(merge-rule "\IeC {\u a}" "ă" :string) +(merge-rule "\IeC {\u A}" "Ä‚" :string) +(merge-rule "\IeC {\u g}" "ÄŸ" :string) +(merge-rule "\IeC {\u G}" "Äž" :string) +(merge-rule "\IeC {\textasciibreve }" "˘" :string) +(merge-rule "\IeC {\'u}" "ú" :string) +(merge-rule "\IeC {\'U}" "Ú" :string) +(merge-rule "\IeC {\`u}" "ù" :string) +(merge-rule "\IeC {\`U}" "Ù" :string) +(merge-rule "\IeC {\^u}" "û" :string) +(merge-rule "\IeC {\^U}" "Û" :string) +(merge-rule "\IeC {\~"u}" "ü" :string) +(merge-rule "\IeC {\~"U}" "Ãœ" :string) +(merge-rule "\IeC {\u u}" "Å­" :string) +(merge-rule "\IeC {\u U}" "Ŭ" :string) +(merge-rule "\IeC {\v c}" "Ä" :string) +(merge-rule "\IeC {\v C}" "ÄŒ" :string) +(merge-rule "\IeC {\v d}" "Ä" :string) +(merge-rule "\IeC {\v D}" "ÄŽ" :string) +(merge-rule "\IeC {\v e}" "Ä›" :string) +(merge-rule "\IeC {\v E}" "Äš" :string) +(merge-rule "\IeC {\v l}" "ľ" :string) +(merge-rule "\IeC {\v L}" "Ľ" :string) +(merge-rule "\IeC {\v n}" "ň" :string) +(merge-rule "\IeC {\v N}" "Ň" :string) +(merge-rule "\IeC {\v r}" "Å™" :string) +(merge-rule "\IeC {\v R}" "Ř" :string) +(merge-rule "\IeC {\v s}" "Å¡" :string) +(merge-rule "\IeC {\v S}" "Å " :string) +(merge-rule "\IeC {\textasciicaron }" "ˇ" :string) +(merge-rule "\IeC {\v t}" "Å¥" :string) +(merge-rule "\IeC {\v T}" "Ť" :string) +(merge-rule "\IeC {\v z}" "ž" :string) +(merge-rule "\IeC {\v Z}" "Ž" :string) +(merge-rule "\IeC {\'y}" "ý" :string) +(merge-rule "\IeC {\'Y}" "Ã" :string) +(merge-rule "\IeC {\~"y}" "ÿ" :string) +(merge-rule "\IeC {\'z}" "ź" :string) +(merge-rule "\IeC {\'Z}" "Ź" :string) +(merge-rule "\IeC {\.z}" "ż" :string) +(merge-rule "\IeC {\.Z}" "Å»" :string) +;; letters not in Latin1, 2, 3 but available in TeX T1 font encoding +(merge-rule "\IeC {\~"Y}" "Ÿ" :string) +(merge-rule "\IeC {\NG }" "ÅŠ" :string) +(merge-rule "\IeC {\ng }" "Å‹" :string) +(merge-rule "\IeC {\OE }" "Å’" :string) +(merge-rule "\IeC {\oe }" "Å“" :string) +(merge-rule "\IeC {\IJ }" "IJ" :string) +(merge-rule "\IeC {\ij }" "ij" :string) +(merge-rule "\IeC {\j }" "È·" :string) +(merge-rule "\IeC {\SS }" "ẞ" :string) diff --git a/doc/src/LectureNotes/_build/latex/LatinRules.xdy b/doc/src/LectureNotes/_build/latex/LatinRules.xdy new file mode 100644 index 000000000..99f14a2ee --- /dev/null +++ b/doc/src/LectureNotes/_build/latex/LatinRules.xdy @@ -0,0 +1,607 @@ +;; style file for xindy +;; filename: LatinRules.xdy +;; +;; It is based upon xindy's files lang/general/utf8.xdy and +;; lang/general/utf8-lang.xdy which implement +;; "a general sorting order for Western European languages" +;; +;; The aim for Sphinx is to be able to index in a Cyrillic document +;; also terms using the Latin alphabets, inclusive of letters +;; with diacritics. To this effect the xindy rules from lang/general +;; got manually re-coded to avoid collisions with the encoding +;; done by xindy for sorting words in Cyrillic languages, which was +;; observed not to use bytes with octal encoding 0o266 or higher. +;; +;; So here we use only 0o266 or higher bytes. +;; (ÅŠ, Å‹, IJ, and ij are absent from +;; lang/general/utf8.xdy and not included here) +;; Contributed by the Sphinx team, 2018. + +(define-letter-group "A" :prefixes ("¶")) +(define-letter-group "B" :after "A" :prefixes ("·")) +(define-letter-group "C" :after "B" :prefixes ("¸")) +(define-letter-group "D" :after "C" :prefixes ("¹")) +(define-letter-group "E" :after "D" :prefixes ("º")) +(define-letter-group "F" :after "E" :prefixes ("»")) +(define-letter-group "G" :after "F" :prefixes ("¼")) +(define-letter-group "H" :after "G" :prefixes ("½")) +(define-letter-group "I" :after "H" :prefixes ("¾")) +(define-letter-group "J" :after "I" :prefixes ("¿")) +(define-letter-group "K" :after "J" :prefixes ("À")) +(define-letter-group "L" :after "K" :prefixes ("Á")) +(define-letter-group "M" :after "L" :prefixes ("Â")) +(define-letter-group "N" :after "M" :prefixes ("Ã")) +(define-letter-group "O" :after "N" :prefixes ("Ä")) +(define-letter-group "P" :after "O" :prefixes ("È")) +(define-letter-group "Q" :after "P" :prefixes ("Ê")) +(define-letter-group "R" :after "Q" :prefixes ("Ë")) +(define-letter-group "S" :after "R" :prefixes ("Ð")) +(define-letter-group "T" :after "S" :prefixes ("Ú")) +(define-letter-group "U" :after "T" :prefixes ("à")) +(define-letter-group "V" :after "U" :prefixes ("å")) +(define-letter-group "W" :after "V" :prefixes ("æ")) +(define-letter-group "X" :after "W" :prefixes ("ë")) +(define-letter-group "Y" :after "X" :prefixes ("í")) +(define-letter-group "Z" :after "Y" :prefixes ("ð")) + +(define-rule-set "sphinx-xy-alphabetize" + + :rules (("À" "¶" :string) + ("Ä‚" "¶" :string) + ("â" "¶" :string) + ("Ä" "¶" :string) + ("à" "¶" :string) + ("Ã…" "¶" :string) + ("Ã" "¶" :string) + ("Ã" "¶" :string) + ("á" "¶" :string) + ("ã" "¶" :string) + ("Â" "¶" :string) + ("ă" "¶" :string) + ("Ã¥" "¶" :string) + ("Ä…" "¶" :string) + ("ä" "¶" :string) + ("Ä„" "¶" :string) + ("æ" "¶º" :string) + ("Æ" "¶º" :string) + ("ć" "¸" :string) + ("ĉ" "¸" :string) + ("ç" "¸" :string) + ("ÄŒ" "¸" :string) + ("Ä" "¸" :string) + ("Ĉ" "¸" :string) + ("Ç" "¸" :string) + ("Ć" "¸" :string) + ("Ä" "¹" :string) + ("Ä" "¹" :string) + ("ÄŽ" "¹" :string) + ("Ä‘" "¹" :string) + ("ê" "º" :string) + ("Ę" "º" :string) + ("Äš" "º" :string) + ("ë" "º" :string) + ("Ä›" "º" :string) + ("é" "º" :string) + ("È" "º" :string) + ("Ë" "º" :string) + ("É" "º" :string) + ("è" "º" :string) + ("Ê" "º" :string) + ("Ä™" "º" :string) + ("Ä" "¼" :string) + ("ÄŸ" "¼" :string) + ("Äž" "¼" :string) + ("Äœ" "¼" :string) + ("Ä¥" "½" :string) + ("Ĥ" "½" :string) + ("Ã" "¾" :string) + ("Ã" "¾" :string) + ("ï" "¾" :string) + ("ÃŽ" "¾" :string) + ("î" "¾" :string) + ("ı" "¾" :string) + ("Ä°" "¾" :string) + ("í" "¾" :string) + ("ÃŒ" "¾" :string) + ("ì" "¾" :string) + ("Ä´" "¿" :string) + ("ĵ" "¿" :string) + ("Å‚" "Á" :string) + ("Å" "Á" :string) + ("ľ" "Á" :string) + ("Ľ" "Á" :string) + ("Å„" "Ã" :string) + ("Ń" "Ã" :string) + ("ñ" "Ã" :string) + ("ň" "Ã" :string) + ("Ñ" "Ã" :string) + ("Ň" "Ã" :string) + ("Õ" "Ä" :string) + ("Å" "Ä" :string) + ("ó" "Ä" :string) + ("ö" "Ä" :string) + ("ô" "Ä" :string) + ("Å‘" "Ä" :string) + ("Ø" "Ä" :string) + ("Ö" "Ä" :string) + ("õ" "Ä" :string) + ("Ô" "Ä" :string) + ("ø" "Ä" :string) + ("Ó" "Ä" :string) + ("Ã’" "Ä" :string) + ("ò" "Ä" :string) + ("Å“" "ĺ" :string) + ("Å’" "ĺ" :string) + ("Ř" "Ë" :string) + ("Å™" "Ë" :string) + ("Å”" "Ë" :string) + ("Å•" "Ë" :string) + ("Å" "Ð" :string) + ("Åš" "Ð" :string) + ("È™" "Ð" :string) + ("ÅŸ" "Ð" :string) + ("Åœ" "Ð" :string) + ("Å›" "Ð" :string) + ("Ș" "Ð" :string) + ("Å¡" "Ð" :string) + ("Åž" "Ð" :string) + ("Å " "Ð" :string) + ("ß" "ÐÐ" :string) + ("Èš" "Ú" :string) + ("Ť" "Ú" :string) + ("È›" "Ú" :string) + ("Å¥" "Ú" :string) + ("û" "à" :string) + ("Å­" "à" :string) + ("ů" "à" :string) + ("ű" "à" :string) + ("ù" "à" :string) + ("Ŭ" "à" :string) + ("Ù" "à" :string) + ("Å°" "à" :string) + ("Ãœ" "à" :string) + ("Å®" "à" :string) + ("ú" "à" :string) + ("Ú" "à" :string) + ("Û" "à" :string) + ("ü" "à" :string) + ("ÿ" "í" :string) + ("Ã" "í" :string) + ("Ÿ" "í" :string) + ("ý" "í" :string) + ("Å»" "ð" :string) + ("Ž" "ð" :string) + ("Ź" "ð" :string) + ("ž" "ð" :string) + ("ż" "ð" :string) + ("ź" "ð" :string) + ("a" "¶" :string) + ("A" "¶" :string) + ("b" "·" :string) + ("B" "·" :string) + ("c" "¸" :string) + ("C" "¸" :string) + ("d" "¹" :string) + ("D" "¹" :string) + ("e" "º" :string) + ("E" "º" :string) + ("F" "»" :string) + ("f" "»" :string) + ("G" "¼" :string) + ("g" "¼" :string) + ("H" "½" :string) + ("h" "½" :string) + ("i" "¾" :string) + ("I" "¾" :string) + ("J" "¿" :string) + ("j" "¿" :string) + ("K" "À" :string) + ("k" "À" :string) + ("L" "Á" :string) + ("l" "Á" :string) + ("M" "Â" :string) + ("m" "Â" :string) + ("n" "Ã" :string) + ("N" "Ã" :string) + ("O" "Ä" :string) + ("o" "Ä" :string) + ("p" "È" :string) + ("P" "È" :string) + ("Q" "Ê" :string) + ("q" "Ê" :string) + ("r" "Ë" :string) + ("R" "Ë" :string) + ("S" "Ð" :string) + ("s" "Ð" :string) + ("t" "Ú" :string) + ("T" "Ú" :string) + ("u" "à" :string) + ("U" "à" :string) + ("v" "å" :string) + ("V" "å" :string) + ("W" "æ" :string) + ("w" "æ" :string) + ("x" "ë" :string) + ("X" "ë" :string) + ("Y" "í" :string) + ("y" "í" :string) + ("z" "ð" :string) + ("Z" "ð" :string) + )) + +(define-rule-set "sphinx-xy-resolve-diacritics" + + :rules (("Ĥ" "£" :string) + ("ó" "£" :string) + ("ľ" "£" :string) + ("Ř" "£" :string) + ("Ä" "£" :string) + ("Ä" "£" :string) + ("Äš" "£" :string) + ("Ä¥" "£" :string) + ("ÄŒ" "£" :string) + ("Ä´" "£" :string) + ("Ä›" "£" :string) + ("ž" "£" :string) + ("ÄŽ" "£" :string) + ("Å™" "£" :string) + ("Ž" "£" :string) + ("ı" "£" :string) + ("Ť" "£" :string) + ("á" "£" :string) + ("Ä" "£" :string) + ("Ã" "£" :string) + ("ň" "£" :string) + ("Å " "£" :string) + ("Ň" "£" :string) + ("ĵ" "£" :string) + ("Å¥" "£" :string) + ("Ó" "£" :string) + ("ý" "£" :string) + ("Äœ" "£" :string) + ("Ú" "£" :string) + ("Ľ" "£" :string) + ("Å¡" "£" :string) + ("Ã" "£" :string) + ("ú" "£" :string) + ("Åš" "¤" :string) + ("ć" "¤" :string) + ("Å" "¤" :string) + ("Å‚" "¤" :string) + ("Å„" "¤" :string) + ("À" "¤" :string) + ("Ź" "¤" :string) + ("à" "¤" :string) + ("Ń" "¤" :string) + ("Ä" "¤" :string) + ("ÿ" "¤" :string) + ("Å›" "¤" :string) + ("Äž" "¤" :string) + ("ÄŸ" "¤" :string) + ("Ù" "¤" :string) + ("Ä°" "¤" :string) + ("Ä‘" "¤" :string) + ("ù" "¤" :string) + ("Èš" "¤" :string) + ("é" "¤" :string) + ("Å•" "¤" :string) + ("Ć" "¤" :string) + ("È›" "¤" :string) + ("ò" "¤" :string) + ("ź" "¤" :string) + ("Ã’" "¤" :string) + ("Ÿ" "¤" :string) + ("Å”" "¤" :string) + ("É" "¤" :string) + ("ĉ" "¥" :string) + ("ô" "¥" :string) + ("Ã" "¥" :string) + ("Å" "¥" :string) + ("Å»" "¥" :string) + ("Ä‚" "¥" :string) + ("Åœ" "¥" :string) + ("ñ" "¥" :string) + ("Å­" "¥" :string) + ("í" "¥" :string) + ("È" "¥" :string) + ("Ô" "¥" :string) + ("Ŭ" "¥" :string) + ("ż" "¥" :string) + ("Ñ" "¥" :string) + ("è" "¥" :string) + ("Ĉ" "¥" :string) + ("ă" "¥" :string) + ("â" "¦" :string) + ("û" "¦" :string) + ("ê" "¦" :string) + ("Õ" "¦" :string) + ("õ" "¦" :string) + ("È™" "¦" :string) + ("ç" "¦" :string) + ("Â" "¦" :string) + ("Ê" "¦" :string) + ("Û" "¦" :string) + ("Ç" "¦" :string) + ("ì" "¦" :string) + ("ÃŒ" "¦" :string) + ("Ș" "¦" :string) + ("ö" "§" :string) + ("Ö" "§" :string) + ("ÅŸ" "§" :string) + ("ů" "§" :string) + ("ë" "§" :string) + ("ã" "§" :string) + ("î" "§" :string) + ("ÃŽ" "§" :string) + ("Ã" "§" :string) + ("Åž" "§" :string) + ("Å®" "§" :string) + ("Ë" "§" :string) + ("ï" "¨" :string) + ("Å" "¨" :string) + ("Ã" "¨" :string) + ("Ę" "¨" :string) + ("Å‘" "¨" :string) + ("Ãœ" "¨" :string) + ("Ã…" "¨" :string) + ("ü" "¨" :string) + ("Ä™" "¨" :string) + ("Ã¥" "¨" :string) + ("Ä" "©" :string) + ("ű" "©" :string) + ("Ø" "©" :string) + ("ø" "©" :string) + ("Å°" "©" :string) + ("ä" "©" :string) + ("Ä„" "ª" :string) + ("Ä…" "ª" :string) + ("Å“" "ÿ" :string) + ("ß" "ÿ" :string) + ("Æ" "ÿ" :string) + ("Å’" "ÿ" :string) + ("æ" "ÿ" :string) + ("e" "¢" :string) + ("t" "¢" :string) + ("L" "¢" :string) + ("Y" "¢" :string) + ("J" "¢" :string) + ("a" "¢" :string) + ("p" "¢" :string) + ("u" "¢" :string) + ("j" "¢" :string) + ("b" "¢" :string) + ("G" "¢" :string) + ("U" "¢" :string) + ("F" "¢" :string) + ("H" "¢" :string) + ("i" "¢" :string) + ("z" "¢" :string) + ("c" "¢" :string) + ("l" "¢" :string) + ("A" "¢" :string) + ("Q" "¢" :string) + ("w" "¢" :string) + ("D" "¢" :string) + ("R" "¢" :string) + ("d" "¢" :string) + ("s" "¢" :string) + ("r" "¢" :string) + ("k" "¢" :string) + ("v" "¢" :string) + ("m" "¢" :string) + ("P" "¢" :string) + ("y" "¢" :string) + ("K" "¢" :string) + ("q" "¢" :string) + ("S" "¢" :string) + ("I" "¢" :string) + ("C" "¢" :string) + ("M" "¢" :string) + ("Z" "¢" :string) + ("T" "¢" :string) + ("W" "¢" :string) + ("B" "¢" :string) + ("h" "¢" :string) + ("x" "¢" :string) + ("X" "¢" :string) + ("f" "¢" :string) + ("E" "¢" :string) + ("V" "¢" :string) + ("N" "¢" :string) + ("O" "¢" :string) + ("o" "¢" :string) + ("g" "¢" :string) + ("n" "¢" :string) + )) + +(define-rule-set "sphinx-xy-resolve-case" + + :rules (("Ú" "8" :string) + ("Ÿ" "8" :string) + ("Ç" "8" :string) + ("Ĉ" "8" :string) + ("Å”" "8" :string) + ("Ľ" "8" :string) + ("Å®" "8" :string) + ("Ã" "8" :string) + ("É" "8" :string) + ("Ë" "8" :string) + ("Ș" "8" :string) + ("ÃŒ" "8" :string) + ("Ê" "8" :string) + ("Ň" "8" :string) + ("Ä„" "8" :string) + ("Å " "8" :string) + ("Û" "8" :string) + ("Åž" "8" :string) + ("Ć" "8" :string) + ("Ã’" "8" :string) + ("Äœ" "8" :string) + ("Ñ" "8" :string) + ("Ó" "8" :string) + ("ÃŽ" "8" :string) + ("Ã" "8" :string) + ("Ã" "8" :string) + ("Èš" "8" :string) + ("Ã…" "8" :string) + ("Äž" "8" :string) + ("Ãœ" "8" :string) + ("È" "8" :string) + ("Ô" "8" :string) + ("Ä°" "8" :string) + ("Å°" "8" :string) + ("Ù" "8" :string) + ("Ŭ" "8" :string) + ("Â" "8" :string) + ("Ť" "8" :string) + ("Ń" "8" :string) + ("ÄŽ" "8" :string) + ("Ź" "8" :string) + ("Ž" "8" :string) + ("Ä" "8" :string) + ("Åœ" "8" :string) + ("ÄŒ" "8" :string) + ("Ä´" "8" :string) + ("Ö" "8" :string) + ("Ø" "8" :string) + ("Å»" "8" :string) + ("Å" "8" :string) + ("Ä‚" "8" :string) + ("Äš" "8" :string) + ("Å" "8" :string) + ("Õ" "8" :string) + ("Ę" "8" :string) + ("Ã" "8" :string) + ("À" "8" :string) + ("Ĥ" "8" :string) + ("Ä" "8" :string) + ("Åš" "8" :string) + ("Ř" "8" :string) + ("Ã" "8" :string) + ("Å’" "89" :string) + ("Æ" "89" :string) + ("ì" "9" :string) + ("è" "9" :string) + ("Ä…" "9" :string) + ("Å¡" "9" :string) + ("ú" "9" :string) + ("Ã¥" "9" :string) + ("ă" "9" :string) + ("Ä™" "9" :string) + ("ü" "9" :string) + ("ź" "9" :string) + ("ò" "9" :string) + ("Å¥" "9" :string) + ("È›" "9" :string) + ("ĵ" "9" :string) + ("Å•" "9" :string) + ("ż" "9" :string) + ("ä" "9" :string) + ("ý" "9" :string) + ("ù" "9" :string) + ("á" "9" :string) + ("é" "9" :string) + ("Ä" "9" :string) + ("ň" "9" :string) + ("Å›" "9" :string) + ("ø" "9" :string) + ("í" "9" :string) + ("Ä‘" "9" :string) + ("ı" "9" :string) + ("ÄŸ" "9" :string) + ("î" "9" :string) + ("ã" "9" :string) + ("à" "9" :string) + ("Å™" "9" :string) + ("Å‘" "9" :string) + ("ů" "9" :string) + ("È™" "9" :string) + ("ÿ" "9" :string) + ("ë" "9" :string) + ("Å­" "9" :string) + ("ç" "9" :string) + ("ű" "9" :string) + ("ñ" "9" :string) + ("õ" "9" :string) + ("Ä›" "9" :string) + ("ÅŸ" "9" :string) + ("ž" "9" :string) + ("Ä" "9" :string) + ("Å" "9" :string) + ("Å„" "9" :string) + ("û" "9" :string) + ("Å‚" "9" :string) + ("Ä" "9" :string) + ("Ä¥" "9" :string) + ("ê" "9" :string) + ("ô" "9" :string) + ("ĉ" "9" :string) + ("â" "9" :string) + ("ć" "9" :string) + ("ï" "9" :string) + ("ö" "9" :string) + ("ľ" "9" :string) + ("ó" "9" :string) + ("æ" "99" :string) + ("ß" "99" :string) + ("Å“" "99" :string) + ("N" "8" :string) + ("V" "8" :string) + ("O" "8" :string) + ("X" "8" :string) + ("E" "8" :string) + ("P" "8" :string) + ("K" "8" :string) + ("T" "8" :string) + ("Z" "8" :string) + ("M" "8" :string) + ("C" "8" :string) + ("I" "8" :string) + ("S" "8" :string) + ("B" "8" :string) + ("W" "8" :string) + ("D" "8" :string) + ("R" "8" :string) + ("H" "8" :string) + ("F" "8" :string) + ("Q" "8" :string) + ("A" "8" :string) + ("G" "8" :string) + ("U" "8" :string) + ("J" "8" :string) + ("Y" "8" :string) + ("L" "8" :string) + ("o" "9" :string) + ("n" "9" :string) + ("g" "9" :string) + ("x" "9" :string) + ("f" "9" :string) + ("y" "9" :string) + ("q" "9" :string) + ("h" "9" :string) + ("w" "9" :string) + ("s" "9" :string) + ("d" "9" :string) + ("v" "9" :string) + ("k" "9" :string) + ("r" "9" :string) + ("m" "9" :string) + ("z" "9" :string) + ("c" "9" :string) + ("i" "9" :string) + ("l" "9" :string) + ("b" "9" :string) + ("j" "9" :string) + ("a" "9" :string) + ("p" "9" :string) + ("u" "9" :string) + ("t" "9" :string) + ("e" "9" :string) + )) + +(use-rule-set :run 0 + :rule-set ("sphinx-xy-alphabetize")) +(use-rule-set :run 1 + :rule-set ("sphinx-xy-resolve-diacritics")) +(use-rule-set :run 2 + :rule-set ("sphinx-xy-resolve-case")) diff --git a/doc/src/LectureNotes/_build/latex/Makefile b/doc/src/LectureNotes/_build/latex/Makefile new file mode 100644 index 000000000..e4653f236 --- /dev/null +++ b/doc/src/LectureNotes/_build/latex/Makefile @@ -0,0 +1,64 @@ +# Makefile for Sphinx LaTeX output + +ALLDOCS = $(basename $(wildcard *.tex)) +ALLPDF = $(addsuffix .pdf,$(ALLDOCS)) +ALLDVI = $(addsuffix .dvi,$(ALLDOCS)) +ALLXDV = +ALLPS = $(addsuffix .ps,$(ALLDOCS)) + +# Prefix for archive names +ARCHIVEPREFIX = +# Additional LaTeX options (passed via variables in latexmkrc/latexmkjarc file) +export LATEXOPTS ?= +# Additional latexmk options +LATEXMKOPTS ?= +# format: pdf or dvi (used only by archive targets) +FMT = pdf + +LATEX = latexmk -dvi +PDFLATEX = latexmk -pdf -dvi- -ps- + + +%.dvi: %.tex FORCE_MAKE + $(LATEX) $(LATEXMKOPTS) '$<' + +%.ps: %.dvi + dvips '$<' + +%.pdf: %.tex FORCE_MAKE + $(PDFLATEX) $(LATEXMKOPTS) '$<' + +all: $(ALLPDF) + +all-dvi: $(ALLDVI) + +all-ps: $(ALLPS) + +all-pdf: $(ALLPDF) + +zip: all-$(FMT) + mkdir $(ARCHIVEPREFIX)docs-$(FMT) + cp $(ALLPDF) $(ARCHIVEPREFIX)docs-$(FMT) + zip -q -r -9 $(ARCHIVEPREFIX)docs-$(FMT).zip $(ARCHIVEPREFIX)docs-$(FMT) + rm -r $(ARCHIVEPREFIX)docs-$(FMT) + +tar: all-$(FMT) + mkdir $(ARCHIVEPREFIX)docs-$(FMT) + cp $(ALLPDF) $(ARCHIVEPREFIX)docs-$(FMT) + tar cf $(ARCHIVEPREFIX)docs-$(FMT).tar $(ARCHIVEPREFIX)docs-$(FMT) + rm -r $(ARCHIVEPREFIX)docs-$(FMT) + +gz: tar + gzip -9 < $(ARCHIVEPREFIX)docs-$(FMT).tar > $(ARCHIVEPREFIX)docs-$(FMT).tar.gz + +bz2: tar + bzip2 -9 -k $(ARCHIVEPREFIX)docs-$(FMT).tar + +xz: tar + xz -9 -k $(ARCHIVEPREFIX)docs-$(FMT).tar + +clean: + rm -f *.log *.ind *.aux *.toc *.syn *.idx *.out *.ilg *.pla *.ps *.tar *.tar.gz *.tar.bz2 *.tar.xz $(ALLPDF) $(ALLDVI) $(ALLXDV) *.fls *.fdb_latexmk + +.PHONY: 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hyperref aware footnote.sty for sphinx (JFB)] +%% +%% Package: footnotehyper-sphinx +%% Version: based on footnotehyper.sty 2017/03/07 v1.0 +%% as available at https://www.ctan.org/pkg/footnotehyper +%% License: the one applying to Sphinx +%% +%% Refer to the PDF documentation at https://www.ctan.org/pkg/footnotehyper for +%% the code comments. +%% +%% Differences: +%% 1. a partial tabulary compatibility layer added (enough for Sphinx mark-up), +%% 2. use of \spx@opt@BeforeFootnote from sphinx.sty, +%% 3. use of \sphinxunactivateextrasandspace from sphinx.sty, +%% 4. macro definition \sphinxfootnotemark, +%% 5. macro definition \sphinxlongtablepatch +%% 6. replaced an \undefined by \@undefined +\DeclareOption*{\PackageWarning{footnotehyper-sphinx}{Option `\CurrentOption' is unknown}}% +\ProcessOptions\relax +\newbox\FNH@notes +\newdimen\FNH@width +\let\FNH@colwidth\columnwidth +\newif\ifFNH@savingnotes +\AtBeginDocument {% + \let\FNH@latex@footnote \footnote + \let\FNH@latex@footnotetext\footnotetext + \let\FNH@H@@footnotetext \@footnotetext + \newenvironment{savenotes} + {\FNH@savenotes\ignorespaces}{\FNH@spewnotes\ignorespacesafterend}% + \let\spewnotes \FNH@spewnotes + \let\footnote \FNH@footnote + \let\footnotetext \FNH@footnotetext + \let\endfootnote \FNH@endfntext + \let\endfootnotetext\FNH@endfntext + \@ifpackageloaded{hyperref} + {\ifHy@hyperfootnotes + \let\FNH@H@@footnotetext\H@@footnotetext + \else + \let\FNH@hyper@fntext\FNH@nohyp@fntext + \fi}% + {\let\FNH@hyper@fntext\FNH@nohyp@fntext}% +}% +\def\FNH@hyper@fntext{\FNH@fntext\FNH@hyper@fntext@i}% +\def\FNH@nohyp@fntext{\FNH@fntext\FNH@nohyp@fntext@i}% +\def\FNH@fntext #1{% + \ifx\ifmeasuring@\@undefined + \expandafter\@secondoftwo\else\expandafter\@firstofone\fi +% these two lines modified for Sphinx (tabulary compatibility): + {\ifmeasuring@\expandafter\@gobbletwo\else\expandafter\@firstofone\fi}% + {\ifx\equation$\expandafter\@gobbletwo\fi #1}%$ +}% +\long\def\FNH@hyper@fntext@i#1{% + \global\setbox\FNH@notes\vbox + {\unvbox\FNH@notes + \FNH@startnote + \@makefntext + {\rule\z@\footnotesep\ignorespaces + \ifHy@nesting\expandafter\ltx@firstoftwo + \else\expandafter\ltx@secondoftwo + \fi + {\expandafter\hyper@@anchor\expandafter{\Hy@footnote@currentHref}{#1}}% + {\Hy@raisedlink + {\expandafter\hyper@@anchor\expandafter{\Hy@footnote@currentHref}% + {\relax}}% + \let\@currentHref\Hy@footnote@currentHref + \let\@currentlabelname\@empty + #1}% + \@finalstrut\strutbox + }% + \FNH@endnote + }% +}% +\long\def\FNH@nohyp@fntext@i#1{% + \global\setbox\FNH@notes\vbox + {\unvbox\FNH@notes + \FNH@startnote + \@makefntext{\rule\z@\footnotesep\ignorespaces#1\@finalstrut\strutbox}% + \FNH@endnote + }% +}% +\def\FNH@startnote{% + \hsize\FNH@colwidth + \interlinepenalty\interfootnotelinepenalty + \reset@font\footnotesize + \floatingpenalty\@MM + \@parboxrestore + \protected@edef\@currentlabel{\csname p@\@mpfn\endcsname\@thefnmark}% + \color@begingroup +}% +\def\FNH@endnote{\color@endgroup}% +\def\FNH@savenotes{% + \begingroup + \ifFNH@savingnotes\else + \FNH@savingnotestrue + \let\@footnotetext \FNH@hyper@fntext + \let\@mpfootnotetext \FNH@hyper@fntext + \let\H@@mpfootnotetext\FNH@nohyp@fntext + \FNH@width\columnwidth + \let\FNH@colwidth\FNH@width + \global\setbox\FNH@notes\box\voidb@x + \let\FNH@thempfn\thempfn + \let\FNH@mpfn\@mpfn + \ifx\@minipagerestore\relax\let\@minipagerestore\@empty\fi + \expandafter\def\expandafter\@minipagerestore\expandafter{% + \@minipagerestore + \let\thempfn\FNH@thempfn + \let\@mpfn\FNH@mpfn + }% + \fi +}% +\def\FNH@spewnotes {% + \endgroup + \ifFNH@savingnotes\else + \ifvoid\FNH@notes\else + \begingroup + \let\@makefntext\@empty + \let\@finalstrut\@gobble + \let\rule\@gobbletwo + \FNH@H@@footnotetext{\unvbox\FNH@notes}% + \endgroup + \fi + \fi +}% +\def\FNH@footnote@envname {footnote}% +\def\FNH@footnotetext@envname{footnotetext}% +\def\FNH@footnote{% +% this line added for Sphinx: + \spx@opt@BeforeFootnote + \ifx\@currenvir\FNH@footnote@envname + \expandafter\FNH@footnoteenv + \else + \expandafter\FNH@latex@footnote + \fi +}% +\def\FNH@footnoteenv{% +% this line added for Sphinx (footnotes in parsed literal blocks): + \catcode13=5 \sphinxunactivateextrasandspace + \@ifnextchar[% + \FNH@footnoteenv@i %] + {\stepcounter\@mpfn + \protected@xdef\@thefnmark{\thempfn}% + \@footnotemark + \def\FNH@endfntext@fntext{\@footnotetext}% + \FNH@startfntext}% +}% +\def\FNH@footnoteenv@i[#1]{% + \begingroup + \csname c@\@mpfn\endcsname #1\relax + \unrestored@protected@xdef\@thefnmark{\thempfn}% + \endgroup + \@footnotemark + \def\FNH@endfntext@fntext{\@footnotetext}% + \FNH@startfntext +}% +\def\FNH@footnotetext{% + \ifx\@currenvir\FNH@footnotetext@envname + \expandafter\FNH@footnotetextenv + \else + \expandafter\FNH@latex@footnotetext + \fi +}% +\def\FNH@footnotetextenv{% + \@ifnextchar[% + \FNH@footnotetextenv@i %] + {\protected@xdef\@thefnmark{\thempfn}% + \def\FNH@endfntext@fntext{\@footnotetext}% + \FNH@startfntext}% +}% +\def\FNH@footnotetextenv@i[#1]{% + \begingroup + \csname c@\@mpfn\endcsname #1\relax + \unrestored@protected@xdef\@thefnmark{\thempfn}% + \endgroup + \ifFNH@savingnotes + \def\FNH@endfntext@fntext{\FNH@nohyp@fntext}% + \else + \def\FNH@endfntext@fntext{\FNH@H@@footnotetext}% + \fi + \FNH@startfntext +}% +\def\FNH@startfntext{% + \setbox\z@\vbox\bgroup + \FNH@startnote + \FNH@prefntext + \rule\z@\footnotesep\ignorespaces +}% +\def\FNH@endfntext {% + \@finalstrut\strutbox + \FNH@postfntext + \FNH@endnote + \egroup + \begingroup + \let\@makefntext\@empty\let\@finalstrut\@gobble\let\rule\@gobbletwo + \FNH@endfntext@fntext {\unvbox\z@}% + \endgroup +}% +\AtBeginDocument{% + \let\FNH@@makefntext\@makefntext + \ifx\@makefntextFB\@undefined + \expandafter\@gobble\else\expandafter\@firstofone\fi + {\ifFBFrenchFootnotes \let\FNH@@makefntext\@makefntextFB \else + \let\FNH@@makefntext\@makefntextORI\fi}% + \expandafter\FNH@check@a\FNH@@makefntext{1.2!3?4,}% + \FNH@@@1.2!3?4,\FNH@@@\relax +}% +\long\def\FNH@check@a #11.2!3?4,#2\FNH@@@#3{% + \ifx\relax#3\expandafter\@firstoftwo\else\expandafter\@secondoftwo\fi + \FNH@bad@makefntext@alert + {\def\FNH@prefntext{#1}\def\FNH@postfntext{#2}\FNH@check@b}% +}% +\def\FNH@check@b #1\relax{% + \expandafter\expandafter\expandafter\FNH@check@c + \expandafter\meaning\expandafter\FNH@prefntext + \meaning\FNH@postfntext1.2!3?4,\FNH@check@c\relax +}% +\def\FNH@check@c #11.2!3?4,#2#3\relax{% + \ifx\FNH@check@c#2\expandafter\@gobble\fi\FNH@bad@makefntext@alert +}% +% slight reformulation for Sphinx +\def\FNH@bad@makefntext@alert{% + \PackageWarningNoLine{footnotehyper-sphinx}% + {Footnotes will be sub-optimal, sorry. This is due to the document class or^^J + some package modifying macro \string\@makefntext.^^J + You can try to report this incompatibility at^^J + https://github.com/sphinx-doc/sphinx with this info:}% + \typeout{\meaning\@makefntext}% + \let\FNH@prefntext\@empty\let\FNH@postfntext\@empty +}% +% this macro from original footnote.sty is not used anymore by Sphinx +% but for simplicity sake let's just keep it as is +\def\makesavenoteenv{\@ifnextchar[\FNH@msne@ii\FNH@msne@i}%] +\def\FNH@msne@i #1{% + \expandafter\let\csname FNH$#1\expandafter\endcsname %$ + \csname #1\endcsname + \expandafter\let\csname endFNH$#1\expandafter\endcsname %$ + \csname end#1\endcsname + \FNH@msne@ii[#1]{FNH$#1}%$ +}% +\def\FNH@msne@ii[#1]#2{% + \expandafter\edef\csname#1\endcsname{% + \noexpand\savenotes + \expandafter\noexpand\csname#2\endcsname + }% + \expandafter\edef\csname end#1\endcsname{% + \expandafter\noexpand\csname end#2\endcsname + \noexpand\expandafter + \noexpand\spewnotes + \noexpand\if@endpe\noexpand\@endpetrue\noexpand\fi + }% +}% +% end of footnotehyper 2017/02/16 v0.99 +% some extras for Sphinx : +% \sphinxfootnotemark: usable in section titles and silently removed from TOCs. +\def\sphinxfootnotemark [#1]% + {\ifx\thepage\relax\else\protect\spx@opt@BeforeFootnote + \protect\footnotemark[#1]\fi}% +\AtBeginDocument{% + % let hyperref less complain + \pdfstringdefDisableCommands{\def\sphinxfootnotemark [#1]{}}% + % to obtain hyperlinked footnotes in longtable environment we must replace + % hyperref's patch of longtable's patch of \@footnotetext by our own + \let\LT@p@ftntext\FNH@hyper@fntext + % this *requires* longtable to be used always wrapped in savenotes environment +}% +\endinput +%% +%% End of file `footnotehyper-sphinx.sty'. diff --git a/doc/src/LectureNotes/_build/latex/index.html b/doc/src/LectureNotes/_build/latex/index.html new file mode 100644 index 000000000..14448b092 --- /dev/null +++ b/doc/src/LectureNotes/_build/latex/index.html @@ -0,0 +1,2 @@ + + diff --git a/doc/src/LectureNotes/_build/latex/latexmkjarc b/doc/src/LectureNotes/_build/latex/latexmkjarc new file mode 100644 index 000000000..6e36b195b --- /dev/null +++ b/doc/src/LectureNotes/_build/latex/latexmkjarc @@ -0,0 +1,22 @@ +$latex = 'pdflatex ' . $ENV{'LATEXOPTS'} . ' -kanji=utf8 %O %S'; +$dvipdf = 'dvipdfmx %O -o %D %S'; +$makeindex = 'internal mendex %S %B %D'; +sub mendex { + my ($source, $basename, $destination) = @_; + my $dictfile = $basename . ".dic"; + unlink($destination); + system("mendex", "-U", "-f", "-d", $dictfile, "-s", "python.ist", $source); + if ($? > 0) { + print("mendex exited with error code $? (ignored)\n"); + } + if (!-e $destination) { + # create an empty .ind file if nothing + open(FH, ">" . $destination); + close(FH); + } + return 0; +} +add_cus_dep( "glo", "gls", 0, "makeglo" ); +sub makeglo { + return system( "mendex -J -f -s gglo.ist -o '$_[0].gls' '$_[0].glo'" ); +} \ No newline at end of file diff --git a/doc/src/LectureNotes/_build/latex/latexmkrc b/doc/src/LectureNotes/_build/latex/latexmkrc new file mode 100644 index 000000000..bba17fa6b --- /dev/null +++ b/doc/src/LectureNotes/_build/latex/latexmkrc @@ -0,0 +1,9 @@ +$latex = 'latex ' . $ENV{'LATEXOPTS'} . ' %O %S'; +$pdflatex = 'pdflatex ' . $ENV{'LATEXOPTS'} . ' %O %S'; +$lualatex = 'lualatex ' . $ENV{'LATEXOPTS'} . ' %O %S'; +$xelatex = 'xelatex --no-pdf ' . $ENV{'LATEXOPTS'} . ' %O %S'; +$makeindex = 'makeindex -s python.ist %O -o %D %S'; +add_cus_dep( "glo", "gls", 0, "makeglo" ); +sub makeglo { + return system( "makeindex -s gglo.ist -o '$_[0].gls' '$_[0].glo'" ); +} \ No newline at end of file diff --git a/doc/src/LectureNotes/_build/latex/make.bat b/doc/src/LectureNotes/_build/latex/make.bat new file mode 100644 index 000000000..94bda2139 --- /dev/null +++ b/doc/src/LectureNotes/_build/latex/make.bat @@ -0,0 +1,31 @@ +@ECHO OFF + +REM Command file for Sphinx documentation + +pushd %~dp0 + +set PDFLATEX=latexmk -pdf -dvi- -ps- + +set "LATEXOPTS= " + +if "%1" == "" goto all-pdf + +if "%1" == "all-pdf" ( + :all-pdf + for %%i in (*.tex) do ( + %PDFLATEX% %LATEXMKOPTS% %%i + ) + goto end +) + +if "%1" == "all-pdf-ja" ( + goto all-pdf +) + +if "%1" == "clean" ( + del /q /s *.dvi *.log *.ind *.aux *.toc *.syn *.idx *.out *.ilg *.pla *.ps *.tar *.tar.gz *.tar.bz2 *.tar.xz *.fls *.fdb_latexmk + goto end +) + +:end +popd \ No newline at end of file diff --git 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symbols B +) +LaTeX Font Info: Font shape `T1/phv/bx/n' in size <12> not available +(Font) Font shape `T1/phv/b/n' tried instead on input line 85. + [1 + +{/usr/local/texlive/2015/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] [2 + + +] +LaTeX Font Info: Font shape `T1/phv/bx/n' in size <14.4> not available +(Font) Font shape `T1/phv/b/n' tried instead on input line 87. +\tf@toc=\write6 +\openout6 = `python.toc'. + + [1 + +] [2 + +] +LaTeX Font Info: Font shape `T1/ptm/bx/n' in size <10> not available +(Font) Font shape `T1/ptm/b/n' tried instead on input line 98. + [1] +[2 + +] +Chapter 1. +[3] [4] +Chapter 2. +[5 + +] [6 + +] +Chapter 3. +[7] [8 + +] +Chapter 4. +[9] [10 + +] +Chapter 5. +[11] [12 + +] +Chapter 6. +LaTeX Font Info: Try loading font information for TS1+ptm on input line 303. + +(/usr/local/texlive/2015/texmf-dist/tex/latex/psnfss/ts1ptm.fd +File: ts1ptm.fd 2001/06/04 font definitions for TS1/ptm. +) [13] [14 + +] +Chapter 7. +[15] [16 + +] +Chapter 8. +[17] [18 + +] +Chapter 9. +[19] [20 + +] +Chapter 10. +[21] [22 + +] +Chapter 11. +[23] [24 + +] +Chapter 12. +[25] [26 + +] +Chapter 13. +[27] [28] [29] [30] +LaTeX Font Info: Try loading font information for T1+pcr on input line 827. + +(/usr/local/texlive/2015/texmf-dist/tex/latex/psnfss/t1pcr.fd +File: t1pcr.fd 2001/06/04 font definitions for T1/pcr. +) [31] [32] +LaTeX Font Info: Font shape `T1/pcr/bx/n' in size <9> not available +(Font) Font shape `T1/pcr/b/n' tried instead on input line 949. + +[33] [34] +LaTeX Font Info: Font shape `T1/pcr/m/it' in size <9> not available +(Font) Font shape `T1/pcr/m/sl' tried instead on input line 1075. + [35] [36] +LaTeX Font Info: Try loading font information for TS1+pcr on input line 1251 +. + +(/usr/local/texlive/2015/texmf-dist/tex/latex/psnfss/ts1pcr.fd +File: ts1pcr.fd 2001/06/04 font definitions for TS1/pcr. +) + +File: chapter2_42_1.png Graphic file (type png) + +Package pdftex.def Info: chapter2_42_1.png used on input line 1266. +(pdftex.def) Requested size: 387.44655pt x 248.92938pt. + +[37] [38 <./chapter2_42_1.png>] [39] [40] + +File: chapter2_54_2.png Graphic file (type png) + +Package pdftex.def Info: chapter2_54_2.png used on input line 1501. +(pdftex.def) Requested size: 469.75908pt x 281.55002pt. + +[41] +File: chapter2_54_3.png Graphic file (type png) + + +Package pdftex.def Info: chapter2_54_3.png used on input line 1503. +(pdftex.def) Requested size: 469.75603pt x 278.13068pt. + [42 <./chapter2_54_2.png> <./chapter2_54_3.png>] +[43] +File: chapter2_60_0.png Graphic file (type png) + + +Package pdftex.def Info: chapter2_60_0.png used on input line 1637. +(pdftex.def) Requested size: 469.76872pt x 338.63754pt. + [44] [45 <./chapter2_60_0.png>] + +File: chapter2_68_0.png Graphic file (type png) + +Package pdftex.def Info: chapter2_68_0.png used on input line 1725. +(pdftex.def) Requested size: 469.77116pt x 332.62776pt. + +[46] [47 <./chapter2_68_0.png>] + +File: chapter2_70_1.png Graphic file (type png) + +Package pdftex.def Info: chapter2_70_1.png used on input line 1781. +(pdftex.def) Requested size: 469.76872pt x 338.63754pt. + +[48 <./chapter2_70_1.png>] +! Argument of \split has an extra }. + + \par +l.1844 ...elta ),&{\text{otherwise.}}\end{cases}}} + . +? s +OK, entering \scrollmode... +Runaway argument? + H_{\delta }(a)={\begin {cases}{\frac {1}{2}}{a^{2}}&{\text {for }}|a\ETC. +! Paragraph ended before \split was complete. + + \par +l.1844 ...elta ),&{\text{otherwise.}}\end{cases}}} + . +I suspect you've forgotten a `}', causing me to apply this +control sequence to too much text. How can we recover? +My plan is to forget the whole thing and hope for the best. + +! Missing $ inserted. + + $ +l.1844 ...elta ),&{\text{otherwise.}}\end{cases}}} + . +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Missing \endgroup inserted. + + \endgroup +l.1844 ...elta ),&{\text{otherwise.}}\end{cases}}} + . +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing \endgroup inserted. + + \endgroup +l.1844 ...elta ),&{\text{otherwise.}}\end{cases}}} + . +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \par +l.1844 ...elta ),&{\text{otherwise.}}\end{cases}}} + . +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + +! Extra }, or forgotten \endgroup. + } + +l.1844 ...elta ),&{\text{otherwise.}}\end{cases}}} + . +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Misplaced \crcr. +\endsplit ->\crcr + \egroup \egroup \iftagsleft@ \@xp \lendsplit@ \else \@xp \... +l.1845 \end{split} + +I can't figure out why you would want to use a tab mark +or \cr or \span just now. If something like a right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Extra }, or forgotten \endgroup. +\endsplit ->\crcr \egroup + \egroup \iftagsleft@ \@xp \lendsplit@ \else \@xp \... +l.1845 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Extra }, or forgotten \endgroup. +\endsplit ->\crcr \egroup \egroup + \iftagsleft@ \@xp \lendsplit@ \else \@xp \... +l.1845 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + + +! LaTeX Error: \begin{equation*} on input line 1842 ended by \end{split}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1845 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.1845 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +! LaTeX Error: \begin{document} ended by \end{equation*}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1846 \end{equation*} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.1846 \end{equation*} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Extra \endgroup. + \endgroup + +l.1846 \end{equation*} + +Things are pretty mixed up, but I think the worst is over. + +[49] +File: chapter2_84_0.png Graphic file (type png) + + +Package pdftex.def Info: chapter2_84_0.png used on input line 1885. +(pdftex.def) Requested size: 469.76321pt x 318.01346pt. + [50 <./chapter2_84_0.png>] [51] [52] +Underfull \vbox (badness 10000) detected at line 2059 + [] + + +Overfull \vbox (15.7258pt too high) detected at line 2059 + [] + +[53] +LaTeX Font Info: Font shape `TS1/pcr/m/it' in size <9> not available +(Font) Font shape `TS1/pcr/m/sl' tried instead on input line 2074. + + [54] +File: chapter2_114_1.png Graphic file (type png) + + +Package pdftex.def Info: chapter2_114_1.png used on input line 2178. +(pdftex.def) Requested size: 469.77574pt x 330.08154pt. + [55] +Overfull \vbox (0.55579pt too high) detected at line 2215 + [] + +[56 <./chapter2_114_1.png>] + +File: chapter2_116_0.png Graphic file (type png) + + +Package pdftex.def Info: chapter2_116_0.png used on input line 2217. +(pdftex.def) Requested size: 469.77574pt x 345.38416pt. + [57 <./chapter2_116_0.png>] +Underfull \vbox (badness 2443) detected at line 2295 + [] + + +Overfull \vbox (2.18254pt too high) detected at line 2295 + [] + +[58] +Underfull \vbox (badness 10000) detected at line 2337 + [] + +[59] +Underfull \vbox (badness 10000) detected at line 2379 + [] + +[60] +File: chapter2_118_16.png Graphic file (type png) + + +Package pdftex.def Info: chapter2_118_16.png used on input line 2388. +(pdftex.def) Requested size: 469.77148pt x 478.36523pt. + [61] [62 <./chapter2_118_16.png>] [63] +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.2551 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.2551 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +[64] + +! LaTeX Error: Bad math environment delimiter. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.2568 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! Package amsmath Error: \tag not allowed here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.2568 \end{split} + +\tag cannot be used at this point. If you don't understand why +you should consult the documentation. +But don't worry: just continue, and I'll forget what happened. + +! You can't use `\eqno' in math mode. +\endmathdisplay@a ...\df@tag \@empty \else \veqno + \alt@tag \df@tag \fi \ifx ... +l.2568 \end{split} + +Sorry, but I'm not programmed to handle this case; +I'll just pretend that you didn't ask for it. +If you're in the wrong mode, you might be able to +return to the right one by typing `I}' or `I$' or `I\par'. + +! Missing \endgroup inserted. + + \endgroup +l.2568 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.2568 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{split} on input line 2568 ended by \end{equation}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.2568 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.2568 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.2568 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing \cr inserted. + + \cr +l.2568 \end{split} + +I'm guessing that you meant to end an alignment here. + +! Missing { inserted. + + { +l.2568 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing $ inserted. + + $ +l.2568 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Missing } inserted. + + } +l.2568 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing { inserted. + + { +l.2568 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing } inserted. + + } +l.2568 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{equation*} on input line 2562 ended by \end{split}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.2568 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.2568 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +\math@cr@@@ ->\ifst@rred \nonumber \fi & + \relax \make@display@tag \ifst@rred ... +l.2568 \end{split} + +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced \cr. +\math@cr@@@ ...fi \global \advance \row@ \@ne \cr + +l.2568 \end{split} + +I can't figure out why you would want to use a tab mark +or \cr or \span just now. If something like a right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Extra }, or forgotten $. +\gmeasure@ ...savetaglength@ \crcr #1\math@cr@@@ } + }\restorecounters@ \if@fle... +l.2568 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Extra }, or forgotten $. +\gmeasure@ ...avetaglength@ \crcr #1\math@cr@@@ }} + \restorecounters@ \if@fleq... +l.2568 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Missing $ inserted. + + $ +l.2568 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.2568 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing $ inserted. + + $ +l.2568 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.2568 \end{split} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + + +! Package amsmath Error: \begin{split} won't work here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.2568 \end{split} + +\Did you forget a preceding \begin{equation}? +If not, perhaps the `aligned' environment is what you want. + +! Missing number, treated as zero. + + \relax +l.2569 \end{equation*} + +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + \relax +l.2569 \end{equation*} + +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +! Too many }'s. +\endmathdisplay@a ...@ \totwidth@ \egroup \egroup + +l.2569 \end{equation*} + +You've closed more groups than you opened. +Such booboos are generally harmless, so keep going. + + +! LaTeX Error: \begin{document} ended by \end{equation*}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.2569 \end{equation*} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.2569 \end{equation*} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.2569 \end{equation*} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + +! Extra \endgroup. + \endgroup + +l.2569 \end{equation*} + +Things are pretty mixed up, but I think the worst is over. + +[65] +File: chapter3_29_0.png Graphic file (type png) + + +Package pdftex.def Info: chapter3_29_0.png used on input line 2639. +(pdftex.def) Requested size: 388.45029pt x 290.08304pt. + [66 <./chapter3_29_0.png>] + +! LaTeX Error: Bad math environment delimiter. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.2666 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! Package amsmath Error: \tag not allowed here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.2666 \end{split} + +\tag cannot be used at this point. If you don't understand why +you should consult the documentation. +But don't worry: just continue, and I'll forget what happened. + +! You can't use `\eqno' in math mode. +\endmathdisplay@a ...\df@tag \@empty \else \veqno + \alt@tag \df@tag \fi \ifx ... +l.2666 \end{split} + +Sorry, but I'm not programmed to handle this case; +I'll just pretend that you didn't ask for it. +If you're in the wrong mode, you might be able to +return to the right one by typing `I}' or `I$' or `I\par'. + +! Missing \endgroup inserted. + + \endgroup +l.2666 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.2666 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{split} on input line 2666 ended by \end{equation}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.2666 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.2666 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.2666 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing \cr inserted. + + \cr +l.2666 \end{split} + +I'm guessing that you meant to end an alignment here. + +! Missing { inserted. + + { +l.2666 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing $ inserted. + + $ +l.2666 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Missing } inserted. + + } +l.2666 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing { inserted. + + { +l.2666 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing } inserted. + + } +l.2666 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{equation*} on input line 2660 ended by \end{split}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.2666 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.2666 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +\math@cr@@@ ->\ifst@rred \nonumber \fi & + \relax \make@display@tag \ifst@rred ... +l.2666 \end{split} + +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced \cr. +\math@cr@@@ ...fi \global \advance \row@ \@ne \cr + +l.2666 \end{split} + +I can't figure out why you would want to use a tab mark +or \cr or \span just now. If something like a right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Extra }, or forgotten $. +\gmeasure@ ...savetaglength@ \crcr #1\math@cr@@@ } + }\restorecounters@ \if@fle... +l.2666 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Extra }, or forgotten $. +\gmeasure@ ...avetaglength@ \crcr #1\math@cr@@@ }} + \restorecounters@ \if@fleq... +l.2666 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Missing $ inserted. + + $ +l.2666 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.2666 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing $ inserted. + + $ +l.2666 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.2666 \end{split} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + + +! Package amsmath Error: \begin{split} won't work here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.2666 \end{split} + +\Did you forget a preceding \begin{equation}? +If not, perhaps the `aligned' environment is what you want. + +! Missing number, treated as zero. + + \relax +l.2667 \end{equation*} + +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + \relax +l.2667 \end{equation*} + +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +! Too many }'s. +\endmathdisplay@a ...@ \totwidth@ \egroup \egroup + +l.2667 \end{equation*} + +You've closed more groups than you opened. +Such booboos are generally harmless, so keep going. + + +! LaTeX Error: \begin{document} ended by \end{equation*}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.2667 \end{equation*} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.2667 \end{equation*} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.2667 \end{equation*} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + +! Extra \endgroup. + \endgroup + +l.2667 \end{equation*} + +Things are pretty mixed up, but I think the worst is over. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.2720 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.2720 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.2720 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.2720 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.2720 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.2720 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +[67] [68] + +! LaTeX Error: Bad math environment delimiter. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.2838 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! Package amsmath Error: \tag not allowed here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.2838 \end{split} + +\tag cannot be used at this point. If you don't understand why +you should consult the documentation. +But don't worry: just continue, and I'll forget what happened. + +! You can't use `\eqno' in math mode. +\endmathdisplay@a ...\df@tag \@empty \else \veqno + \alt@tag \df@tag \fi \ifx ... +l.2838 \end{split} + +Sorry, but I'm not programmed to handle this case; +I'll just pretend that you didn't ask for it. +If you're in the wrong mode, you might be able to +return to the right one by typing `I}' or `I$' or `I\par'. + +! Missing \endgroup inserted. + + \endgroup +l.2838 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.2838 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{split} on input line 2838 ended by \end{equation}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.2838 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.2838 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.2838 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing \cr inserted. + + \cr +l.2838 \end{split} + +I'm guessing that you meant to end an alignment here. + +! Missing { inserted. + + { +l.2838 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing $ inserted. + + $ +l.2838 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Missing } inserted. + + } +l.2838 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing { inserted. + + { +l.2838 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing } inserted. + + } +l.2838 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{equation*} on input line 2832 ended by \end{split}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.2838 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.2838 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +\math@cr@@@ ->\ifst@rred \nonumber \fi & + \relax \make@display@tag \ifst@rred ... +l.2838 \end{split} + +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced \cr. +\math@cr@@@ ...fi \global \advance \row@ \@ne \cr + +l.2838 \end{split} + +I can't figure out why you would want to use a tab mark +or \cr or \span just now. If something like a right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Extra }, or forgotten $. +\gmeasure@ ...savetaglength@ \crcr #1\math@cr@@@ } + }\restorecounters@ \if@fle... +l.2838 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Extra }, or forgotten $. +\gmeasure@ ...avetaglength@ \crcr #1\math@cr@@@ }} + \restorecounters@ \if@fleq... +l.2838 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Missing $ inserted. + + $ +l.2838 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.2838 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing $ inserted. + + $ +l.2838 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.2838 \end{split} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + + +! Package amsmath Error: \begin{split} won't work here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.2838 \end{split} + +\Did you forget a preceding \begin{equation}? +If not, perhaps the `aligned' environment is what you want. + +! Missing number, treated as zero. + + \relax +l.2839 \end{equation*} + +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + \relax +l.2839 \end{equation*} + +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +! Too many }'s. +\endmathdisplay@a ...@ \totwidth@ \egroup \egroup + +l.2839 \end{equation*} + +You've closed more groups than you opened. +Such booboos are generally harmless, so keep going. + + +! LaTeX Error: \begin{document} ended by \end{equation*}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.2839 \end{equation*} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.2839 \end{equation*} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.2839 \end{equation*} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + +! Extra \endgroup. + \endgroup + +l.2839 \end{equation*} + +Things are pretty mixed up, but I think the worst is over. + +[69] [70] + +! LaTeX Error: Bad math environment delimiter. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.2967 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! Package amsmath Error: \tag not allowed here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.2967 \end{split} + +\tag cannot be used at this point. If you don't understand why +you should consult the documentation. +But don't worry: just continue, and I'll forget what happened. + +! You can't use `\eqno' in math mode. +\endmathdisplay@a ...\df@tag \@empty \else \veqno + \alt@tag \df@tag \fi \ifx ... +l.2967 \end{split} + +Sorry, but I'm not programmed to handle this case; +I'll just pretend that you didn't ask for it. +If you're in the wrong mode, you might be able to +return to the right one by typing `I}' or `I$' or `I\par'. + +! Missing \endgroup inserted. + + \endgroup +l.2967 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.2967 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{split} on input line 2967 ended by \end{equation}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.2967 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.2967 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.2967 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing \cr inserted. + + \cr +l.2967 \end{split} + +I'm guessing that you meant to end an alignment here. + +! Missing { inserted. + + { +l.2967 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing $ inserted. + + $ +l.2967 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Missing } inserted. + + } +l.2967 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing { inserted. + + { +l.2967 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing } inserted. + + } +l.2967 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{equation*} on input line 2961 ended by \end{split}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.2967 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.2967 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +\math@cr@@@ ->\ifst@rred \nonumber \fi & + \relax \make@display@tag \ifst@rred ... +l.2967 \end{split} + +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced \cr. +\math@cr@@@ ...fi \global \advance \row@ \@ne \cr + +l.2967 \end{split} + +I can't figure out why you would want to use a tab mark +or \cr or \span just now. If something like a right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Extra }, or forgotten $. +\gmeasure@ ...savetaglength@ \crcr #1\math@cr@@@ } + }\restorecounters@ \if@fle... +l.2967 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Extra }, or forgotten $. +\gmeasure@ ...avetaglength@ \crcr #1\math@cr@@@ }} + \restorecounters@ \if@fleq... +l.2967 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Missing $ inserted. + + $ +l.2967 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.2967 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing $ inserted. + + $ +l.2967 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.2967 \end{split} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + + +! Package amsmath Error: \begin{split} won't work here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.2967 \end{split} + +\Did you forget a preceding \begin{equation}? +If not, perhaps the `aligned' environment is what you want. + +! Missing number, treated as zero. + + \relax +l.2968 \end{equation*} + +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + \relax +l.2968 \end{equation*} + +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +! Too many }'s. +\endmathdisplay@a ...@ \totwidth@ \egroup \egroup + +l.2968 \end{equation*} + +You've closed more groups than you opened. +Such booboos are generally harmless, so keep going. + + +! LaTeX Error: \begin{document} ended by \end{equation*}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.2968 \end{equation*} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.2968 \end{equation*} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.2968 \end{equation*} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + +! Extra \endgroup. + \endgroup + +l.2968 \end{equation*} + +Things are pretty mixed up, but I think the worst is over. + + +! LaTeX Error: Bad math environment delimiter. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.2978 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! Package amsmath Error: \tag not allowed here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.2978 \end{split} + +\tag cannot be used at this point. If you don't understand why +you should consult the documentation. +But don't worry: just continue, and I'll forget what happened. + +! You can't use `\eqno' in math mode. +\endmathdisplay@a ...\df@tag \@empty \else \veqno + \alt@tag \df@tag \fi \ifx ... +l.2978 \end{split} + +Sorry, but I'm not programmed to handle this case; +I'll just pretend that you didn't ask for it. +If you're in the wrong mode, you might be able to +return to the right one by typing `I}' or `I$' or `I\par'. + +! Missing \endgroup inserted. + + \endgroup +l.2978 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.2978 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{split} on input line 2978 ended by \end{equation}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.2978 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.2978 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.2978 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing \cr inserted. + + \cr +l.2978 \end{split} + +I'm guessing that you meant to end an alignment here. + +! Missing { inserted. + + { +l.2978 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing $ inserted. + + $ +l.2978 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Missing } inserted. + + } +l.2978 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing { inserted. + + { +l.2978 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing } inserted. + + } +l.2978 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{equation*} on input line 2972 ended by \end{split}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.2978 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.2978 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +\math@cr@@@ ->\ifst@rred \nonumber \fi & + \relax \make@display@tag \ifst@rred ... +l.2978 \end{split} + +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced \cr. +\math@cr@@@ ...fi \global \advance \row@ \@ne \cr + +l.2978 \end{split} + +I can't figure out why you would want to use a tab mark +or \cr or \span just now. If something like a right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Extra }, or forgotten $. +\gmeasure@ ...savetaglength@ \crcr #1\math@cr@@@ } + }\restorecounters@ \if@fle... +l.2978 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Extra }, or forgotten $. +\gmeasure@ ...avetaglength@ \crcr #1\math@cr@@@ }} + \restorecounters@ \if@fleq... +l.2978 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Missing $ inserted. + + $ +l.2978 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.2978 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing $ inserted. + + $ +l.2978 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.2978 \end{split} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + + +! Package amsmath Error: \begin{split} won't work here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.2978 \end{split} + +\Did you forget a preceding \begin{equation}? +If not, perhaps the `aligned' environment is what you want. + +! Missing number, treated as zero. + + \relax +l.2979 \end{equation*} + +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + \relax +l.2979 \end{equation*} + +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +! Too many }'s. +\endmathdisplay@a ...@ \totwidth@ \egroup \egroup + +l.2979 \end{equation*} + +You've closed more groups than you opened. +Such booboos are generally harmless, so keep going. + + +! LaTeX Error: \begin{document} ended by \end{equation*}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.2979 \end{equation*} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.2979 \end{equation*} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.2979 \end{equation*} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + +! Extra \endgroup. + \endgroup + +l.2979 \end{equation*} + +Things are pretty mixed up, but I think the worst is over. + +[71] [72] +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.3070 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.3070 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.3070 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.3070 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.3070 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.3070 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.3070 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.3070 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +[73] + +! LaTeX Error: Bad math environment delimiter. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3151 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! Package amsmath Error: \tag not allowed here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.3151 \end{split} + +\tag cannot be used at this point. If you don't understand why +you should consult the documentation. +But don't worry: just continue, and I'll forget what happened. + +! You can't use `\eqno' in math mode. +\endmathdisplay@a ...\df@tag \@empty \else \veqno + \alt@tag \df@tag \fi \ifx ... +l.3151 \end{split} + +Sorry, but I'm not programmed to handle this case; +I'll just pretend that you didn't ask for it. +If you're in the wrong mode, you might be able to +return to the right one by typing `I}' or `I$' or `I\par'. + +! Missing \endgroup inserted. + + \endgroup +l.3151 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3151 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{split} on input line 3151 ended by \end{equation}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3151 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3151 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3151 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing \cr inserted. + + \cr +l.3151 \end{split} + +I'm guessing that you meant to end an alignment here. + +! Missing { inserted. + + { +l.3151 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing $ inserted. + + $ +l.3151 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Missing } inserted. + + } +l.3151 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing { inserted. + + { +l.3151 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing } inserted. + + } +l.3151 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{equation*} on input line 3145 ended by \end{split}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3151 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3151 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +\math@cr@@@ ->\ifst@rred \nonumber \fi & + \relax \make@display@tag \ifst@rred ... +l.3151 \end{split} + +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced \cr. +\math@cr@@@ ...fi \global \advance \row@ \@ne \cr + +l.3151 \end{split} + +I can't figure out why you would want to use a tab mark +or \cr or \span just now. If something like a right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Extra }, or forgotten $. +\gmeasure@ ...savetaglength@ \crcr #1\math@cr@@@ } + }\restorecounters@ \if@fle... +l.3151 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Extra }, or forgotten $. +\gmeasure@ ...avetaglength@ \crcr #1\math@cr@@@ }} + \restorecounters@ \if@fleq... +l.3151 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Missing $ inserted. + + $ +l.3151 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3151 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing $ inserted. + + $ +l.3151 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.3151 \end{split} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + + +! Package amsmath Error: \begin{split} won't work here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.3151 \end{split} + +\Did you forget a preceding \begin{equation}? +If not, perhaps the `aligned' environment is what you want. + +! Missing number, treated as zero. + + \relax +l.3152 \end{equation*} + +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + \relax +l.3152 \end{equation*} + +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +! Too many }'s. +\endmathdisplay@a ...@ \totwidth@ \egroup \egroup + +l.3152 \end{equation*} + +You've closed more groups than you opened. +Such booboos are generally harmless, so keep going. + + +! LaTeX Error: \begin{document} ended by \end{equation*}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3152 \end{equation*} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3152 \end{equation*} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.3152 \end{equation*} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + +! Extra \endgroup. + \endgroup + +l.3152 \end{equation*} + +Things are pretty mixed up, but I think the worst is over. + + +! LaTeX Error: Bad math environment delimiter. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3171 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! Package amsmath Error: \tag not allowed here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.3171 \end{split} + +\tag cannot be used at this point. If you don't understand why +you should consult the documentation. +But don't worry: just continue, and I'll forget what happened. + +! You can't use `\eqno' in math mode. +\endmathdisplay@a ...\df@tag \@empty \else \veqno + \alt@tag \df@tag \fi \ifx ... +l.3171 \end{split} + +Sorry, but I'm not programmed to handle this case; +I'll just pretend that you didn't ask for it. +If you're in the wrong mode, you might be able to +return to the right one by typing `I}' or `I$' or `I\par'. + +! Missing \endgroup inserted. + + \endgroup +l.3171 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3171 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{split} on input line 3171 ended by \end{equation}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3171 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3171 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3171 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing \cr inserted. + + \cr +l.3171 \end{split} + +I'm guessing that you meant to end an alignment here. + +! Missing { inserted. + + { +l.3171 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing $ inserted. + + $ +l.3171 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Missing } inserted. + + } +l.3171 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing { inserted. + + { +l.3171 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing } inserted. + + } +l.3171 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{equation*} on input line 3165 ended by \end{split}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3171 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3171 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +\math@cr@@@ ->\ifst@rred \nonumber \fi & + \relax \make@display@tag \ifst@rred ... +l.3171 \end{split} + +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced \cr. +\math@cr@@@ ...fi \global \advance \row@ \@ne \cr + +l.3171 \end{split} + +I can't figure out why you would want to use a tab mark +or \cr or \span just now. If something like a right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Extra }, or forgotten $. +\gmeasure@ ...savetaglength@ \crcr #1\math@cr@@@ } + }\restorecounters@ \if@fle... +l.3171 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Extra }, or forgotten $. +\gmeasure@ ...avetaglength@ \crcr #1\math@cr@@@ }} + \restorecounters@ \if@fleq... +l.3171 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Missing $ inserted. + + $ +l.3171 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3171 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing $ inserted. + + $ +l.3171 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.3171 \end{split} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + + +! Package amsmath Error: \begin{split} won't work here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.3171 \end{split} + +\Did you forget a preceding \begin{equation}? +If not, perhaps the `aligned' environment is what you want. + +! Missing number, treated as zero. + + \relax +l.3172 \end{equation*} + +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + \relax +l.3172 \end{equation*} + +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +! Too many }'s. +\endmathdisplay@a ...@ \totwidth@ \egroup \egroup + +l.3172 \end{equation*} + +You've closed more groups than you opened. +Such booboos are generally harmless, so keep going. + + +! LaTeX Error: \begin{document} ended by \end{equation*}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3172 \end{equation*} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3172 \end{equation*} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.3172 \end{equation*} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + +! Extra \endgroup. + \endgroup + +l.3172 \end{equation*} + +Things are pretty mixed up, but I think the worst is over. + + +! LaTeX Error: Bad math environment delimiter. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3187 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! Package amsmath Error: \tag not allowed here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.3187 \end{split} + +\tag cannot be used at this point. If you don't understand why +you should consult the documentation. +But don't worry: just continue, and I'll forget what happened. + +! You can't use `\eqno' in math mode. +\endmathdisplay@a ...\df@tag \@empty \else \veqno + \alt@tag \df@tag \fi \ifx ... +l.3187 \end{split} + +Sorry, but I'm not programmed to handle this case; +I'll just pretend that you didn't ask for it. +If you're in the wrong mode, you might be able to +return to the right one by typing `I}' or `I$' or `I\par'. + +! Missing \endgroup inserted. + + \endgroup +l.3187 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3187 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{split} on input line 3187 ended by \end{equation}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3187 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3187 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3187 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing \cr inserted. + + \cr +l.3187 \end{split} + +I'm guessing that you meant to end an alignment here. + +! Missing { inserted. + + { +l.3187 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing $ inserted. + + $ +l.3187 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Missing } inserted. + + } +l.3187 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing { inserted. + + { +l.3187 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing } inserted. + + } +l.3187 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{equation*} on input line 3181 ended by \end{split}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3187 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3187 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +\math@cr@@@ ->\ifst@rred \nonumber \fi & + \relax \make@display@tag \ifst@rred ... +l.3187 \end{split} + +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced \cr. +\math@cr@@@ ...fi \global \advance \row@ \@ne \cr + +l.3187 \end{split} + +I can't figure out why you would want to use a tab mark +or \cr or \span just now. If something like a right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Extra }, or forgotten $. +\gmeasure@ ...savetaglength@ \crcr #1\math@cr@@@ } + }\restorecounters@ \if@fle... +l.3187 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Extra }, or forgotten $. +\gmeasure@ ...avetaglength@ \crcr #1\math@cr@@@ }} + \restorecounters@ \if@fleq... +l.3187 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Missing $ inserted. + + $ +l.3187 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3187 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing $ inserted. + + $ +l.3187 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.3187 \end{split} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + + +! Package amsmath Error: \begin{split} won't work here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.3187 \end{split} + +\Did you forget a preceding \begin{equation}? +If not, perhaps the `aligned' environment is what you want. + +! Missing number, treated as zero. + + \relax +l.3188 \end{equation*} + +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + \relax +l.3188 \end{equation*} + +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +! Too many }'s. +\endmathdisplay@a ...@ \totwidth@ \egroup \egroup + +l.3188 \end{equation*} + +You've closed more groups than you opened. +Such booboos are generally harmless, so keep going. + + +! LaTeX Error: \begin{document} ended by \end{equation*}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3188 \end{equation*} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3188 \end{equation*} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.3188 \end{equation*} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + +! Extra \endgroup. + \endgroup + +l.3188 \end{equation*} + +Things are pretty mixed up, but I think the worst is over. + +[74] + +LaTeX Warning: Hyper reference `chapter3:eq:exptvariance' on page 75 undefined +on input line 3223. + +[75] + +! LaTeX Error: Bad math environment delimiter. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3290 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! Package amsmath Error: \tag not allowed here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.3290 \end{split} + +\tag cannot be used at this point. If you don't understand why +you should consult the documentation. +But don't worry: just continue, and I'll forget what happened. + +! You can't use `\eqno' in math mode. +\endmathdisplay@a ...\df@tag \@empty \else \veqno + \alt@tag \df@tag \fi \ifx ... +l.3290 \end{split} + +Sorry, but I'm not programmed to handle this case; +I'll just pretend that you didn't ask for it. +If you're in the wrong mode, you might be able to +return to the right one by typing `I}' or `I$' or `I\par'. + +! Missing \endgroup inserted. + + \endgroup +l.3290 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3290 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{split} on input line 3290 ended by \end{equation}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3290 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3290 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3290 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing \cr inserted. + + \cr +l.3290 \end{split} + +I'm guessing that you meant to end an alignment here. + +! Missing { inserted. + + { +l.3290 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing $ inserted. + + $ +l.3290 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Missing } inserted. + + } +l.3290 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing { inserted. + + { +l.3290 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing } inserted. + + } +l.3290 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{equation*} on input line 3284 ended by \end{split}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3290 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3290 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +\math@cr@@@ ->\ifst@rred \nonumber \fi & + \relax \make@display@tag \ifst@rred ... +l.3290 \end{split} + +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced \cr. +\math@cr@@@ ...fi \global \advance \row@ \@ne \cr + +l.3290 \end{split} + +I can't figure out why you would want to use a tab mark +or \cr or \span just now. If something like a right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Extra }, or forgotten $. +\gmeasure@ ...savetaglength@ \crcr #1\math@cr@@@ } + }\restorecounters@ \if@fle... +l.3290 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Extra }, or forgotten $. +\gmeasure@ ...avetaglength@ \crcr #1\math@cr@@@ }} + \restorecounters@ \if@fleq... +l.3290 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Missing $ inserted. + + $ +l.3290 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3290 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing $ inserted. + + $ +l.3290 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.3290 \end{split} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + + +! Package amsmath Error: \begin{split} won't work here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.3290 \end{split} + +\Did you forget a preceding \begin{equation}? +If not, perhaps the `aligned' environment is what you want. + +! Missing number, treated as zero. + + \relax +l.3291 \end{equation*} + +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + \relax +l.3291 \end{equation*} + +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +! Too many }'s. +\endmathdisplay@a ...@ \totwidth@ \egroup \egroup + +l.3291 \end{equation*} + +You've closed more groups than you opened. +Such booboos are generally harmless, so keep going. + + +! LaTeX Error: \begin{document} ended by \end{equation*}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3291 \end{equation*} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3291 \end{equation*} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.3291 \end{equation*} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + +! Extra \endgroup. + \endgroup + +l.3291 \end{equation*} + +Things are pretty mixed up, but I think the worst is over. + + +! LaTeX Error: Bad math environment delimiter. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3311 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! Package amsmath Error: \tag not allowed here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.3311 \end{split} + +\tag cannot be used at this point. If you don't understand why +you should consult the documentation. +But don't worry: just continue, and I'll forget what happened. + +! You can't use `\eqno' in math mode. +\endmathdisplay@a ...\df@tag \@empty \else \veqno + \alt@tag \df@tag \fi \ifx ... +l.3311 \end{split} + +Sorry, but I'm not programmed to handle this case; +I'll just pretend that you didn't ask for it. +If you're in the wrong mode, you might be able to +return to the right one by typing `I}' or `I$' or `I\par'. + +! Missing \endgroup inserted. + + \endgroup +l.3311 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3311 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{split} on input line 3311 ended by \end{equation}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3311 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3311 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3311 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing \cr inserted. + + \cr +l.3311 \end{split} + +I'm guessing that you meant to end an alignment here. + +! Missing { inserted. + + { +l.3311 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing $ inserted. + + $ +l.3311 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Missing } inserted. + + } +l.3311 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing { inserted. + + { +l.3311 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing } inserted. + + } +l.3311 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{equation*} on input line 3304 ended by \end{split}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3311 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3311 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +\math@cr@@@ ->\ifst@rred \nonumber \fi & + \relax \make@display@tag \ifst@rred ... +l.3311 \end{split} + +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced \cr. +\math@cr@@@ ...fi \global \advance \row@ \@ne \cr + +l.3311 \end{split} + +I can't figure out why you would want to use a tab mark +or \cr or \span just now. If something like a right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Extra }, or forgotten $. +\gmeasure@ ...savetaglength@ \crcr #1\math@cr@@@ } + }\restorecounters@ \if@fle... +l.3311 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Extra }, or forgotten $. +\gmeasure@ ...avetaglength@ \crcr #1\math@cr@@@ }} + \restorecounters@ \if@fleq... +l.3311 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Missing $ inserted. + + $ +l.3311 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3311 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing $ inserted. + + $ +l.3311 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.3311 \end{split} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + + +! Package amsmath Error: \begin{split} won't work here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.3311 \end{split} + +\Did you forget a preceding \begin{equation}? +If not, perhaps the `aligned' environment is what you want. + +! Missing number, treated as zero. + + \relax +l.3312 \end{equation*} + +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + \relax +l.3312 \end{equation*} + +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +! Too many }'s. +\endmathdisplay@a ...@ \totwidth@ \egroup \egroup + +l.3312 \end{equation*} + +You've closed more groups than you opened. +Such booboos are generally harmless, so keep going. + + +! LaTeX Error: \begin{document} ended by \end{equation*}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3312 \end{equation*} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3312 \end{equation*} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.3312 \end{equation*} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + +! Extra \endgroup. + \endgroup + +l.3312 \end{equation*} + +Things are pretty mixed up, but I think the worst is over. + + +! LaTeX Error: Bad math environment delimiter. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3327 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! Package amsmath Error: \tag not allowed here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.3327 \end{split} + +\tag cannot be used at this point. If you don't understand why +you should consult the documentation. +But don't worry: just continue, and I'll forget what happened. + +! You can't use `\eqno' in math mode. +\endmathdisplay@a ...\df@tag \@empty \else \veqno + \alt@tag \df@tag \fi \ifx ... +l.3327 \end{split} + +Sorry, but I'm not programmed to handle this case; +I'll just pretend that you didn't ask for it. +If you're in the wrong mode, you might be able to +return to the right one by typing `I}' or `I$' or `I\par'. + +! Missing \endgroup inserted. + + \endgroup +l.3327 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3327 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{split} on input line 3327 ended by \end{equation}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3327 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3327 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3327 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing \cr inserted. + + \cr +l.3327 \end{split} + +I'm guessing that you meant to end an alignment here. + +! Missing { inserted. + + { +l.3327 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing $ inserted. + + $ +l.3327 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Missing } inserted. + + } +l.3327 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing { inserted. + + { +l.3327 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing } inserted. + + } +l.3327 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{equation*} on input line 3321 ended by \end{split}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3327 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3327 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +\math@cr@@@ ->\ifst@rred \nonumber \fi & + \relax \make@display@tag \ifst@rred ... +l.3327 \end{split} + +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced \cr. +\math@cr@@@ ...fi \global \advance \row@ \@ne \cr + +l.3327 \end{split} + +I can't figure out why you would want to use a tab mark +or \cr or \span just now. If something like a right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Extra }, or forgotten $. +\gmeasure@ ...savetaglength@ \crcr #1\math@cr@@@ } + }\restorecounters@ \if@fle... +l.3327 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Extra }, or forgotten $. +\gmeasure@ ...avetaglength@ \crcr #1\math@cr@@@ }} + \restorecounters@ \if@fleq... +l.3327 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Missing $ inserted. + + $ +l.3327 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3327 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing $ inserted. + + $ +l.3327 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.3327 \end{split} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + + +! Package amsmath Error: \begin{split} won't work here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.3327 \end{split} + +\Did you forget a preceding \begin{equation}? +If not, perhaps the `aligned' environment is what you want. + +! Missing number, treated as zero. + + \relax +l.3328 \end{equation*} + +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + \relax +l.3328 \end{equation*} + +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +! Too many }'s. +\endmathdisplay@a ...@ \totwidth@ \egroup \egroup + +l.3328 \end{equation*} + +You've closed more groups than you opened. +Such booboos are generally harmless, so keep going. + + +! LaTeX Error: \begin{document} ended by \end{equation*}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3328 \end{equation*} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3328 \end{equation*} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.3328 \end{equation*} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + +! Extra \endgroup. + \endgroup + +l.3328 \end{equation*} + +Things are pretty mixed up, but I think the worst is over. + + +LaTeX Warning: Hyper reference `chapter3:eq:error_estimate_corr_time' on page 7 +6 undefined on input line 3340. + +[76] [77] [78] + +! LaTeX Error: Bad math environment delimiter. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3536 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! Package amsmath Error: \tag not allowed here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.3536 \end{split} + +\tag cannot be used at this point. If you don't understand why +you should consult the documentation. +But don't worry: just continue, and I'll forget what happened. + +! You can't use `\eqno' in math mode. +\endmathdisplay@a ...\df@tag \@empty \else \veqno + \alt@tag \df@tag \fi \ifx ... +l.3536 \end{split} + +Sorry, but I'm not programmed to handle this case; +I'll just pretend that you didn't ask for it. +If you're in the wrong mode, you might be able to +return to the right one by typing `I}' or `I$' or `I\par'. + +! Missing \endgroup inserted. + + \endgroup +l.3536 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3536 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{split} on input line 3536 ended by \end{equation}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3536 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3536 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3536 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing \cr inserted. + + \cr +l.3536 \end{split} + +I'm guessing that you meant to end an alignment here. + +! Missing { inserted. + + { +l.3536 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing $ inserted. + + $ +l.3536 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Missing } inserted. + + } +l.3536 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing { inserted. + + { +l.3536 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing } inserted. + + } +l.3536 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{equation*} on input line 3530 ended by \end{split}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3536 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3536 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +\math@cr@@@ ->\ifst@rred \nonumber \fi & + \relax \make@display@tag \ifst@rred ... +l.3536 \end{split} + +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced \cr. +\math@cr@@@ ...fi \global \advance \row@ \@ne \cr + +l.3536 \end{split} + +I can't figure out why you would want to use a tab mark +or \cr or \span just now. If something like a right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Extra }, or forgotten $. +\gmeasure@ ...savetaglength@ \crcr #1\math@cr@@@ } + }\restorecounters@ \if@fle... +l.3536 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Extra }, or forgotten $. +\gmeasure@ ...avetaglength@ \crcr #1\math@cr@@@ }} + \restorecounters@ \if@fleq... +l.3536 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Missing $ inserted. + + $ +l.3536 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3536 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing $ inserted. + + $ +l.3536 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.3536 \end{split} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + + +! Package amsmath Error: \begin{split} won't work here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.3536 \end{split} + +\Did you forget a preceding \begin{equation}? +If not, perhaps the `aligned' environment is what you want. + +! Missing number, treated as zero. + + \relax +l.3537 \end{equation*} + +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + \relax +l.3537 \end{equation*} + +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +! Too many }'s. +\endmathdisplay@a ...@ \totwidth@ \egroup \egroup + +l.3537 \end{equation*} + +You've closed more groups than you opened. +Such booboos are generally harmless, so keep going. + + +! LaTeX Error: \begin{document} ended by \end{equation*}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3537 \end{equation*} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3537 \end{equation*} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.3537 \end{equation*} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + +! Extra \endgroup. + \endgroup + +l.3537 \end{equation*} + +Things are pretty mixed up, but I think the worst is over. + + +! LaTeX Error: Bad math environment delimiter. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3549 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! Package amsmath Error: \tag not allowed here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.3549 \end{split} + +\tag cannot be used at this point. If you don't understand why +you should consult the documentation. +But don't worry: just continue, and I'll forget what happened. + +! You can't use `\eqno' in math mode. +\endmathdisplay@a ...\df@tag \@empty \else \veqno + \alt@tag \df@tag \fi \ifx ... +l.3549 \end{split} + +Sorry, but I'm not programmed to handle this case; +I'll just pretend that you didn't ask for it. +If you're in the wrong mode, you might be able to +return to the right one by typing `I}' or `I$' or `I\par'. + +! Missing \endgroup inserted. + + \endgroup +l.3549 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3549 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{split} on input line 3549 ended by \end{equation}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3549 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3549 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3549 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing \cr inserted. + + \cr +l.3549 \end{split} + +I'm guessing that you meant to end an alignment here. + +! Missing { inserted. + + { +l.3549 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing $ inserted. + + $ +l.3549 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Missing } inserted. + + } +l.3549 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing { inserted. + + { +l.3549 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing } inserted. + + } +l.3549 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{equation*} on input line 3543 ended by \end{split}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3549 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3549 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +\math@cr@@@ ->\ifst@rred \nonumber \fi & + \relax \make@display@tag \ifst@rred ... +l.3549 \end{split} + +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced \cr. +\math@cr@@@ ...fi \global \advance \row@ \@ne \cr + +l.3549 \end{split} + +I can't figure out why you would want to use a tab mark +or \cr or \span just now. If something like a right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Extra }, or forgotten $. +\gmeasure@ ...savetaglength@ \crcr #1\math@cr@@@ } + }\restorecounters@ \if@fle... +l.3549 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Extra }, or forgotten $. +\gmeasure@ ...avetaglength@ \crcr #1\math@cr@@@ }} + \restorecounters@ \if@fleq... +l.3549 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Missing $ inserted. + + $ +l.3549 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3549 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing $ inserted. + + $ +l.3549 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.3549 \end{split} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + + +! Package amsmath Error: \begin{split} won't work here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.3549 \end{split} + +\Did you forget a preceding \begin{equation}? +If not, perhaps the `aligned' environment is what you want. + +! Missing number, treated as zero. + + \relax +l.3550 \end{equation*} + +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + \relax +l.3550 \end{equation*} + +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +! Too many }'s. +\endmathdisplay@a ...@ \totwidth@ \egroup \egroup + +l.3550 \end{equation*} + +You've closed more groups than you opened. +Such booboos are generally harmless, so keep going. + + +! LaTeX Error: \begin{document} ended by \end{equation*}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3550 \end{equation*} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3550 \end{equation*} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.3550 \end{equation*} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + +! Extra \endgroup. + \endgroup + +l.3550 \end{equation*} + +Things are pretty mixed up, but I think the worst is over. + +[79] + +! LaTeX Error: Bad math environment delimiter. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3621 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! Package amsmath Error: \tag not allowed here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.3621 \end{split} + +\tag cannot be used at this point. If you don't understand why +you should consult the documentation. +But don't worry: just continue, and I'll forget what happened. + +! You can't use `\eqno' in math mode. +\endmathdisplay@a ...\df@tag \@empty \else \veqno + \alt@tag \df@tag \fi \ifx ... +l.3621 \end{split} + +Sorry, but I'm not programmed to handle this case; +I'll just pretend that you didn't ask for it. +If you're in the wrong mode, you might be able to +return to the right one by typing `I}' or `I$' or `I\par'. + +! Missing \endgroup inserted. + + \endgroup +l.3621 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3621 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{split} on input line 3621 ended by \end{equation}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3621 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3621 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3621 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing \cr inserted. + + \cr +l.3621 \end{split} + +I'm guessing that you meant to end an alignment here. + +! Missing { inserted. + + { +l.3621 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing $ inserted. + + $ +l.3621 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Missing } inserted. + + } +l.3621 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing { inserted. + + { +l.3621 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing } inserted. + + } +l.3621 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{equation*} on input line 3615 ended by \end{split}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3621 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3621 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +\math@cr@@@ ->\ifst@rred \nonumber \fi & + \relax \make@display@tag \ifst@rred ... +l.3621 \end{split} + +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced \cr. +\math@cr@@@ ...fi \global \advance \row@ \@ne \cr + +l.3621 \end{split} + +I can't figure out why you would want to use a tab mark +or \cr or \span just now. If something like a right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Extra }, or forgotten $. +\gmeasure@ ...savetaglength@ \crcr #1\math@cr@@@ } + }\restorecounters@ \if@fle... +l.3621 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Extra }, or forgotten $. +\gmeasure@ ...avetaglength@ \crcr #1\math@cr@@@ }} + \restorecounters@ \if@fleq... +l.3621 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Missing $ inserted. + + $ +l.3621 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3621 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing $ inserted. + + $ +l.3621 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.3621 \end{split} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + + +! Package amsmath Error: \begin{split} won't work here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.3621 \end{split} + +\Did you forget a preceding \begin{equation}? +If not, perhaps the `aligned' environment is what you want. + +! Missing number, treated as zero. + + \relax +l.3622 \end{equation*} + +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + \relax +l.3622 \end{equation*} + +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +! Too many }'s. +\endmathdisplay@a ...@ \totwidth@ \egroup \egroup + +l.3622 \end{equation*} + +You've closed more groups than you opened. +Such booboos are generally harmless, so keep going. + + +! LaTeX Error: \begin{document} ended by \end{equation*}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3622 \end{equation*} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3622 \end{equation*} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.3622 \end{equation*} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + +! Extra \endgroup. + \endgroup + +l.3622 \end{equation*} + +Things are pretty mixed up, but I think the worst is over. + + +LaTeX Warning: Hyper reference `chapter3:eq:rntrick1' on page 80 undefined on i +nput line 3630. + + +! LaTeX Error: Bad math environment delimiter. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3641 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! Package amsmath Error: \tag not allowed here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.3641 \end{split} + +\tag cannot be used at this point. If you don't understand why +you should consult the documentation. +But don't worry: just continue, and I'll forget what happened. + +! You can't use `\eqno' in math mode. +\endmathdisplay@a ...\df@tag \@empty \else \veqno + \alt@tag \df@tag \fi \ifx ... +l.3641 \end{split} + +Sorry, but I'm not programmed to handle this case; +I'll just pretend that you didn't ask for it. +If you're in the wrong mode, you might be able to +return to the right one by typing `I}' or `I$' or `I\par'. + +! Missing \endgroup inserted. + + \endgroup +l.3641 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3641 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{split} on input line 3641 ended by \end{equation}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3641 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3641 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3641 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing \cr inserted. + + \cr +l.3641 \end{split} + +I'm guessing that you meant to end an alignment here. + +! Missing { inserted. + + { +l.3641 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing $ inserted. + + $ +l.3641 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Missing } inserted. + + } +l.3641 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing { inserted. + + { +l.3641 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing } inserted. + + } +l.3641 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{equation*} on input line 3635 ended by \end{split}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3641 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3641 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +\math@cr@@@ ->\ifst@rred \nonumber \fi & + \relax \make@display@tag \ifst@rred ... +l.3641 \end{split} + +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced \cr. +\math@cr@@@ ...fi \global \advance \row@ \@ne \cr + +l.3641 \end{split} + +I can't figure out why you would want to use a tab mark +or \cr or \span just now. If something like a right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Extra }, or forgotten $. +\gmeasure@ ...savetaglength@ \crcr #1\math@cr@@@ } + }\restorecounters@ \if@fle... +l.3641 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Extra }, or forgotten $. +\gmeasure@ ...avetaglength@ \crcr #1\math@cr@@@ }} + \restorecounters@ \if@fleq... +l.3641 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Missing $ inserted. + + $ +l.3641 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3641 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing $ inserted. + + $ +l.3641 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.3641 \end{split} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + + +! Package amsmath Error: \begin{split} won't work here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.3641 \end{split} + +\Did you forget a preceding \begin{equation}? +If not, perhaps the `aligned' environment is what you want. + +! Missing number, treated as zero. + + \relax +l.3642 \end{equation*} + +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + \relax +l.3642 \end{equation*} + +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +! Too many }'s. +\endmathdisplay@a ...@ \totwidth@ \egroup \egroup + +l.3642 \end{equation*} + +You've closed more groups than you opened. +Such booboos are generally harmless, so keep going. + + +! LaTeX Error: \begin{document} ended by \end{equation*}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3642 \end{equation*} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3642 \end{equation*} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.3642 \end{equation*} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + +! Extra \endgroup. + \endgroup + +l.3642 \end{equation*} + +Things are pretty mixed up, but I think the worst is over. + + +! LaTeX Error: Bad math environment delimiter. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3655 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! Package amsmath Error: \tag not allowed here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.3655 \end{split} + +\tag cannot be used at this point. If you don't understand why +you should consult the documentation. +But don't worry: just continue, and I'll forget what happened. + +! You can't use `\eqno' in math mode. +\endmathdisplay@a ...\df@tag \@empty \else \veqno + \alt@tag \df@tag \fi \ifx ... +l.3655 \end{split} + +Sorry, but I'm not programmed to handle this case; +I'll just pretend that you didn't ask for it. +If you're in the wrong mode, you might be able to +return to the right one by typing `I}' or `I$' or `I\par'. + +! Missing \endgroup inserted. + + \endgroup +l.3655 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3655 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{split} on input line 3655 ended by \end{equation}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3655 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3655 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3655 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing \cr inserted. + + \cr +l.3655 \end{split} + +I'm guessing that you meant to end an alignment here. + +! Missing { inserted. + + { +l.3655 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing $ inserted. + + $ +l.3655 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Missing } inserted. + + } +l.3655 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing { inserted. + + { +l.3655 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing } inserted. + + } +l.3655 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{equation*} on input line 3649 ended by \end{split}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3655 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3655 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +\math@cr@@@ ->\ifst@rred \nonumber \fi & + \relax \make@display@tag \ifst@rred ... +l.3655 \end{split} + +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced \cr. +\math@cr@@@ ...fi \global \advance \row@ \@ne \cr + +l.3655 \end{split} + +I can't figure out why you would want to use a tab mark +or \cr or \span just now. If something like a right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Extra }, or forgotten $. +\gmeasure@ ...savetaglength@ \crcr #1\math@cr@@@ } + }\restorecounters@ \if@fle... +l.3655 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Extra }, or forgotten $. +\gmeasure@ ...avetaglength@ \crcr #1\math@cr@@@ }} + \restorecounters@ \if@fleq... +l.3655 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Missing $ inserted. + + $ +l.3655 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3655 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing $ inserted. + + $ +l.3655 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.3655 \end{split} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + + +! Package amsmath Error: \begin{split} won't work here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.3655 \end{split} + +\Did you forget a preceding \begin{equation}? +If not, perhaps the `aligned' environment is what you want. + +! Missing number, treated as zero. + + \relax +l.3656 \end{equation*} + +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + \relax +l.3656 \end{equation*} + +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +! Too many }'s. +\endmathdisplay@a ...@ \totwidth@ \egroup \egroup + +l.3656 \end{equation*} + +You've closed more groups than you opened. +Such booboos are generally harmless, so keep going. + + +! LaTeX Error: \begin{document} ended by \end{equation*}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3656 \end{equation*} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3656 \end{equation*} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.3656 \end{equation*} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + +! Extra \endgroup. + \endgroup + +l.3656 \end{equation*} + +Things are pretty mixed up, but I think the worst is over. + +[80] + +! LaTeX Error: Bad math environment delimiter. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3668 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! Package amsmath Error: \tag not allowed here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.3668 \end{split} + +\tag cannot be used at this point. If you don't understand why +you should consult the documentation. +But don't worry: just continue, and I'll forget what happened. + +! You can't use `\eqno' in math mode. +\endmathdisplay@a ...\df@tag \@empty \else \veqno + \alt@tag \df@tag \fi \ifx ... +l.3668 \end{split} + +Sorry, but I'm not programmed to handle this case; +I'll just pretend that you didn't ask for it. +If you're in the wrong mode, you might be able to +return to the right one by typing `I}' or `I$' or `I\par'. + +! Missing \endgroup inserted. + + \endgroup +l.3668 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3668 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{split} on input line 3668 ended by \end{equation}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3668 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3668 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3668 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing \cr inserted. + + \cr +l.3668 \end{split} + +I'm guessing that you meant to end an alignment here. + +! Missing { inserted. + + { +l.3668 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing $ inserted. + + $ +l.3668 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Missing } inserted. + + } +l.3668 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing { inserted. + + { +l.3668 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing } inserted. + + } +l.3668 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{equation*} on input line 3662 ended by \end{split}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3668 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3668 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +\math@cr@@@ ->\ifst@rred \nonumber \fi & + \relax \make@display@tag \ifst@rred ... +l.3668 \end{split} + +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced \cr. +\math@cr@@@ ...fi \global \advance \row@ \@ne \cr + +l.3668 \end{split} + +I can't figure out why you would want to use a tab mark +or \cr or \span just now. If something like a right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Extra }, or forgotten $. +\gmeasure@ ...savetaglength@ \crcr #1\math@cr@@@ } + }\restorecounters@ \if@fle... +l.3668 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Extra }, or forgotten $. +\gmeasure@ ...avetaglength@ \crcr #1\math@cr@@@ }} + \restorecounters@ \if@fleq... +l.3668 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Missing $ inserted. + + $ +l.3668 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.3668 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing $ inserted. + + $ +l.3668 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.3668 \end{split} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + + +! Package amsmath Error: \begin{split} won't work here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.3668 \end{split} + +\Did you forget a preceding \begin{equation}? +If not, perhaps the `aligned' environment is what you want. + +! Missing number, treated as zero. + + \relax +l.3669 \end{equation*} + +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + \relax +l.3669 \end{equation*} + +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +! Too many }'s. +\endmathdisplay@a ...@ \totwidth@ \egroup \egroup + +l.3669 \end{equation*} + +You've closed more groups than you opened. +Such booboos are generally harmless, so keep going. + + +! LaTeX Error: \begin{document} ended by \end{equation*}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.3669 \end{equation*} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.3669 \end{equation*} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.3669 \end{equation*} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + +! Extra \endgroup. + \endgroup + +l.3669 \end{equation*} + +Things are pretty mixed up, but I think the worst is over. + +[81] +File: chapter3_178_0.png Graphic file (type png) + + +Package pdftex.def Info: chapter3_178_0.png used on input line 3788. +(pdftex.def) Requested size: 396.48027pt x 275.02682pt. + [82] + +LaTeX Warning: Hyper reference `chapter3:eq:autocorrelformal' on page 83 undefi +ned on input line 3795. + +[83 <./chapter3_178_0.png>] + +File: chapter3_184_1.png Graphic file (type png) + + +Package pdftex.def Info: chapter3_184_1.png used on input line 3856. +(pdftex.def) Requested size: 392.46527pt x 275.02682pt. + [84 <./chapter3_184_1.png>] [85] [86] [87] [88] +[89] [90] [91] [92] [93] +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4445 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4445 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4445 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4445 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4445 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4445 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4445 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4445 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4445 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4445 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +[94] +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4508 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4508 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4508 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4508 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4508 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4508 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4508 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4508 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4508 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4508 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4508 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4508 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4508 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4508 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4551 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4551 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4551 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4551 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4551 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4551 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4551 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4551 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4551 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4551 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4551 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4551 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4551 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.4551 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +[95] [96] [97] [98] [99] +File: chapter4_70_0.png Graphic file (type png) + + +Package pdftex.def Info: chapter4_70_0.png used on input line 4909. +(pdftex.def) Requested size: 381.42406pt x 264.98935pt. + [100 <./chapter4_70_0.png>] [101] [102] [103] [104] + +File: chapter4_117_1.png Graphic file (type png) + +Package pdftex.def Info: chapter4_117_1.png used on input line 5246. +(pdftex.def) Requested size: 396.48027pt x 269.00433pt. + +[105] [106 <./chapter4_117_1.png>] +Underfull \vbox (badness 10000) detected at line 5345 + [] + +[107] [108] +File: chapter4_129_0.png Graphic file (type png) + + +Package pdftex.def Info: chapter4_129_0.png used on input line 5475. +(pdftex.def) Requested size: 469.76407pt x 334.48817pt. + [109] + +File: chapter4_131_1.png Graphic file (type png) + + +Package pdftex.def Info: chapter4_131_1.png used on input line 5491. +(pdftex.def) Requested size: 469.75784pt x 336.92499pt. + [110 <./chapter4_129_0.png>] + +File: chapter4_133_0.png Graphic file (type png) + + +Package pdftex.def Info: chapter4_133_0.png used on input line 5511. +(pdftex.def) Requested size: 469.74797pt x 134.16124pt. + [111 <./chapter4_131_1.png>] +Overfull \vbox (0.88394pt too high) detected at line 5575 + [] + +[112 <./chapter4_133_0.png>] + +File: chapter4_140_0.png Graphic file (type png) + + +Package pdftex.def Info: chapter4_140_0.png used on input line 5597. +(pdftex.def) Requested size: 469.7642pt x 329.3196pt. + [113] [114 <./chapter4_140_0.png>] [115] [116] +[117] + +! LaTeX Error: Bad math environment delimiter. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.5869 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! Package amsmath Error: \tag not allowed here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.5869 \end{split} + +\tag cannot be used at this point. If you don't understand why +you should consult the documentation. +But don't worry: just continue, and I'll forget what happened. + +! You can't use `\eqno' in math mode. +\endmathdisplay@a ...\df@tag \@empty \else \veqno + \alt@tag \df@tag \fi \ifx ... +l.5869 \end{split} + +Sorry, but I'm not programmed to handle this case; +I'll just pretend that you didn't ask for it. +If you're in the wrong mode, you might be able to +return to the right one by typing `I}' or `I$' or `I\par'. + +! Missing \endgroup inserted. + + \endgroup +l.5869 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.5869 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{split} on input line 5869 ended by \end{equation}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.5869 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.5869 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.5869 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing \cr inserted. + + \cr +l.5869 \end{split} + +I'm guessing that you meant to end an alignment here. + +! Missing { inserted. + + { +l.5869 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing $ inserted. + + $ +l.5869 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Missing } inserted. + + } +l.5869 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing { inserted. + + { +l.5869 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing } inserted. + + } +l.5869 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + + +! LaTeX Error: \begin{equation*} on input line 5863 ended by \end{split}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.5869 \end{split} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.5869 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Misplaced alignment tab character &. +\math@cr@@@ ->\ifst@rred \nonumber \fi & + \relax \make@display@tag \ifst@rred ... +l.5869 \end{split} + +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced \cr. +\math@cr@@@ ...fi \global \advance \row@ \@ne \cr + +l.5869 \end{split} + +I can't figure out why you would want to use a tab mark +or \cr or \span just now. If something like a right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Extra }, or forgotten $. +\gmeasure@ ...savetaglength@ \crcr #1\math@cr@@@ } + }\restorecounters@ \if@fle... +l.5869 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Extra }, or forgotten $. +\gmeasure@ ...avetaglength@ \crcr #1\math@cr@@@ }} + \restorecounters@ \if@fleq... +l.5869 \end{split} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Missing $ inserted. + + $ +l.5869 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.5869 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing $ inserted. + + $ +l.5869 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.5869 \end{split} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + + +! Package amsmath Error: \begin{split} won't work here. + +See the amsmath package documentation for explanation. +Type H for immediate help. + ... + +l.5869 \end{split} + +\Did you forget a preceding \begin{equation}? +If not, perhaps the `aligned' environment is what you want. + +! Missing number, treated as zero. + + \relax +l.5870 \end{equation*} + +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + \relax +l.5870 \end{equation*} + +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +! Too many }'s. +\endmathdisplay@a ...@ \totwidth@ \egroup \egroup + +l.5870 \end{equation*} + +You've closed more groups than you opened. +Such booboos are generally harmless, so keep going. + + +! LaTeX Error: \begin{document} ended by \end{equation*}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.5870 \end{equation*} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.5870 \end{equation*} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.5870 \end{equation*} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + +! Extra \endgroup. + \endgroup + +l.5870 \end{equation*} + +Things are pretty mixed up, but I think the worst is over. + +[118] [119] [120] [121] [122] [123] [124] [125] [126] [127] [128] +Overfull \hbox (138.79266pt too wide) in paragraph at lines 6679--6681 +\T1/ptm/m/n/10 where we wrote $$[]\OT1/cmr/m/n/10 [[][]\OML/cmm/m/it/10 ; [][]\ +OT1/cmr/m/n/10 ] = [][[]]$$\OML/cmm/m/it/10 toindicatethatthisthecovarianceofth +evectors$\T1/ptm/m/n/10 \boldsymbol{x}$\OML/cmm/m/it/10 ofthedesign=featurematr +ix$\T1/ptm/m/n/10 \boldsymbol{X}$. + [] + +[129] [130] [131] +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.6873 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.6873 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.6873 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.6873 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.6873 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.6873 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.6873 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Misplaced \omit. +\math@cr@@@ ...@ \@ne \add@amps \maxfields@ \omit + \kern -\alignsep@ \iftag@ ... +l.6873 \end{split} + +I expect to see \omit only after tab marks or the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Missing \endgroup inserted. + + \endgroup +l.6914 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.6914 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! You can't use `\halign' in math mode. +\H@eqnarray ...let \\\@eqncr $$\everycr {}\halign + to\displaywidth \bgroup \h... +l.6914 \end{split} + +Sorry, but I'm not programmed to handle this case; +I'll just pretend that you didn't ask for it. +If you're in the wrong mode, you might be able to +return to the right one by typing `I}' or `I$' or `I\par'. + +! Missing number, treated as zero. + + \bgroup +l.6914 \end{split} + +A number should have been here; I inserted `0'. +(If you can't figure out why I needed to see a number, +look up `weird error' in the index to The TeXbook.) + +! Illegal unit of measure (pt inserted). + + \bgroup +l.6914 \end{split} + +Dimensions can be in units of em, ex, in, pt, pc, +cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! +I'll assume that you meant to say pt, for printer's points. +To recover gracefully from this error, it's best to +delete the erroneous units; e.g., type `2' to delete +two letters. (See Chapter 27 of The TeXbook.) + +! Missing } inserted. + + } +l.6914 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing $ inserted. + + $ +l.6914 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! You can't use `macro parameter character #' in math mode. +\H@eqnarray ...$\displaystyle \tabskip \z@skip {## + }$\@eqnsel &\global \@eqcn... +l.6914 \end{split} + +Sorry, but I'm not programmed to handle this case; +I'll just pretend that you didn't ask for it. +If you're in the wrong mode, you might be able to +return to the right one by typing `I}' or `I$' or `I\par'. + +! Missing { inserted. + + { +l.6914 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing { inserted. + + { +l.6914 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Missing $ inserted. + + $ +l.6914 \end{split} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Missing } inserted. + + } +l.6914 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.6914 \end{split} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! You can't use `macro parameter character #' in restricted horizontal mode. +\H@eqnarray ...\hskip \tw@ \arraycolsep \hfil ${## + }$\hfil &\global \@eqcnt \... +l.6914 \end{split} + +Sorry, but I'm not programmed to handle this case; +I'll just pretend that you didn't ask for it. +If you're in the wrong mode, you might be able to +return to the right one by typing `I}' or `I$' or `I\par'. + +! Missing { inserted. + + { +l.6914 \end{split} + +I've put in what seems to be necessary to fix +the current column of the current alignment. +Try to go on, since this might almost work. + +! Extra alignment tab has been changed to \cr. +