"
- ]
- }
- ],
- "prompt_number": 3
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "Probably this is immediately recognizable to you as a 'bell curve'. This curve is ubiquitious because under real world conditions most observations are distributed in such a manner. In fact, this is the bell curve for IQ (Intelligence Quotient). You've probably seen this before, and understand it. It tells us that the average IQ is 100, and that the number of people that have IQs higher or lower than that drops off as they get further away from 100. It's hard to see the exact number, but we can see that very few people have an IQ over 150 or under 50, but a lot have an IQ of 90 or 110. \n",
- "\n",
- "This curve is not unique to IQ distributions - a vast amount of natural phenomena exhibits this sort of distribution, including the sensors that we use in filtering problems. As we will see, it also has all the attributes that we are looking for - it represents a unimodal belief or value as a probabilitiy, it is continuous, and it is computationally efficient. We will soon discover that it also other desirable qualities that we do not yet recognize we need.\n",
- "\n",
- "
"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "#### Nomenclature\n",
- "\n",
- "A bit of nomenclature before we continue - this chart depicts the probability of of a *random variable* having any value between ($-\\infty..\\infty)$. For example, for this chart the probability of the variable being 100 is roughly 2.7%, whereas the probability of it being 80 is around 1%.\n",
- "> *Random variable* will be precisely defined later. For now just think of it as a variable that can 'freely' and 'randomly' vary. A dog's position in a hallway, air temperature, and a drone's height above the ground are all random variables. The position of the North Pole is not, nor is a sin wave (a sin wave is anything but 'free').\n",
- "\n",
- "You may object that human IQs cannot be less than zero, let alone $-\\infty$. This is true, but this is a common limitation of mathematical modelling. \"The map is not the territory\" is a common expression, and it is true for Bayesian filtering and statistics. The Gaussian distribution above very closely models the distribution of IQ test results, but being a model it is necessarily imperfect. The difference between model and reality will come up again and again in these filters. \n",
- "\n",
- "You will see these distributions called *Gaussian distributions*, *normal distributions*, and *bell curves*. Bell curve is ambiguous because there are other distributions which also look bell shaped but are not Gaussian distributions, so we will not use it further in this book. But *Gaussian* and *normal* both mean the same thing, and are used interchangeably. I will use both throughout this book as different sources will use either term, and so I want you to be used to seeing both. Finally, as in this paragraph, it is typical to shorten the name and just talk about a *Gaussian* or *normal* - these are both typical shortcut names for the *Gaussian distribution*. "
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "### Gaussian Distributions\n",
- "\n",
- "So let us explore how Gaussians work. A Gaussian is a *continuous probability distribution* that is completely described with two parameters, the mean ($\\mu$) and the variance ($\\sigma^2$). It is defined as:\n",
- "$$ \n",
- "f(x, \\mu, \\sigma) = \\frac{1}{\\sigma\\sqrt{2\\pi}} e^{-\\frac{1}{2}{(x-\\mu)^2}/\\sigma^2 }\n",
- "$$"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "
Don't be dissuaded by the equation if you haven't seen it before; you will not need to memorize or manipulate it. The computation of this function is stored in stats.py. \n",
- "\n",
- "> **Optional:** Let's remind ourselves how to look at a function stored in a file by using the *%load* magic. If you type *%load -s gaussian stats.py* into a code cell and then press CTRL-Enter, the notebook will create a new input cell and load the function into it.\n",
- "\n",
- " %load -s gaussian stats.py\n",
- " \n",
- " def gaussian(x, mean, var):\n",
- " \"\"\"returns normal distribution for x given a \n",
- " gaussian with the specified mean and variance. \n",
- " \"\"\"\n",
- " return math.exp((-0.5*(x-mean)**2)/var) / \\\n",
- " math.sqrt(_two_pi*var)"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "
We will plot a Gaussian with a mean of 22 $(\\mu=22)$, with a variance of 4 $(\\sigma^2=4)$, and then discuss what this means. "
- ]
- },
- {
- "cell_type": "code",
- "collapsed": false,
- "input": [
- "from stats import gaussian\n",
- "plot_gaussian(22,4,True,xlabel='$^{\\circ}C$',ylabel=\"Percent\")\n",
- "\n",
- "print('Probability of 22 is %.2f' % (gaussian(22,22,4)*100))\n",
- "print('Probability of 24 is %.2f' % (gaussian(24,22,4)*100))"
- ],
- "language": "python",
- "metadata": {},
- "outputs": [
- {
- "metadata": {},
- "output_type": "display_data",
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YizbYiBFRwJxDI9jb3IcK7qQfUpIk4frSVLx/vFt0KUSkETZiBsW5eSVmoi4YuVTWOLCw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\n+XnO/cyZXLq7u3nePceZXJKTkyP+vOtyufCDH/w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- "text": [
- ""
- ]
- },
- {
- "output_type": "stream",
- "stream": "stdout",
- "text": [
- "Probability of 22 is 19.95\n",
- "Probability of 24 is 12.10\n"
- ]
- }
- ],
- "prompt_number": 4
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "So what does this curve *mean*? Assume for a moment that we have a themometer, which reads 22$\\,^{\\circ}C$. No thermometer is perfectly accurate, and so we normally expect that thermometer will read $\\pm$ that temperature by some amount each time we read it. Furthermore, a theorem called **Central Limit Theorem** states that if we make many measurements that the measurements will be normally distributed. If that is true, then this chart can be interpreted as a continuous curve depicting our belief that the temperature is any given temperature. In this curve, we assign a probability of the temperature being exactly $22\\,^{\\circ}C$ is $19.95%$. Looking to the right, we assign the probability that the temperature is $24\\,^{\\circ}C$ is $12.10%$. Because of the curve's symmetry, the probability of $20\\,^{\\circ}C$ is also $12.10%$.\n",
- "\n",
- "So the mean ($\\mu$) is what it sounds like - the average of all possible probabilities. Because of the symmetric shape of the curve it is also the tallest part of the curve. The thermometer reads $22\\,^{\\circ}C$, so that is what we used for the mean. \n",
- "\n",
- "> *Important*: I will repeat what I wrote at the top of this section: \"A Gaussian...is completely described with two parameters\"\n",
- "\n",
- "The standard notation for a normal distribution for a random variable $X$ is $X \\sim\\ \\mathcal{N}(\\mu,\\sigma^2)$. This means I can express the temperature reading of our thermometer as\n",
- "\n",
- "$$temp = \\mathcal{N}(22,4)$$\n",
- "\n",
- "This is an **extremely important** result. Gaussians allow me to capture an infinite number of possible values with only two numbers! With the values $\\mu=22$ and $\\sigma^2=4$ I can compute the probability of the temperature being $22\\,^{\\circ}C$, $20\\,^{\\circ}C$, $87.34\\,^{\\circ}C$, or any other arbitrary value.\n",
- "\n",
- "###### The Variance\n",
- "\n",
- "Since this is a probability distribution it is required that the area under the curve always equals one. This should be intuitively clear - the area under the curve represents all possible occurances, which must sum to one.\n",
- "\n",
- "This leads to an important insight. If the variance is small the curve will be narrow. To keep the area equal to 1, the curve must also be tall. On the other hand if the variance is large the curve will be wide, and thus it will also have to be short to make the area equal to 1.\n",
- "\n",
- "Let's look at that graphically:"
- ]
- },
- {
- "cell_type": "code",
- "collapsed": false,
- "input": [
- "import numpy as np\n",
- "import matplotlib.pyplot as plt\n",
- "\n",
- "xs = np.arange(15,30,0.1)\n",
- "p1, = plt.plot (xs,[gaussian(x, 23, .2) for x in xs],'b')\n",
- "p2, = plt.plot (xs,[gaussian(x, 23, 1) for x in xs],'g')\n",
- "p3, = plt.plot (xs,[gaussian(x, 23, 5) for x in xs],'r')\n",
- "plt.legend([p1,p2,p3], ['var = .2', 'var = 1', 'var = 5'])\n",
- "plt.show()"
- ],
- "language": "python",
- "metadata": {},
- "outputs": [
- {
- "metadata": {},
- "output_type": "display_data",
- "png": 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zMxP5+fkAgNmzZ8Medk1eHzlyJE6ePAkAWLhwISRJwtKlSwEAFRUVSExMRF5e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\ngegcgciK67neFobeFoHeGove1huDLQK9NZxIWwQtrM2QVD1mXF9iSiK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- "text": [
- ""
- ]
- }
- ],
- "prompt_number": 5
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "So what is this telling us? The blue gaussian is very narrow. It is saying that we believe $x=23$, and that we are very sure about that $(90%)$. In contrast, the red gaussian also believes that $x=23$, but we are much less sure about that $(18%)$. Our believe that $x=23$ is lower, and so our belief about the likely possible values for $x$ is spread out - we think it is quite likely that $x=20$ or $x=26$, for example. The blue gaussian has almost completely eliminated $22$ or $24$ as possible value - their probability is almost $0%$, whereas the red curve considers them nearly as likely as $23$.\n",
- "\n",
- "If we think back to the thermometer, we can consider these three curves as representing the readings from three different thermometers. The blue curve represents a very accurate thermometer, and the red one represents a fairly inaccurate one. Green of course represents one in between the two others. Note the very powerful property the Gaussian distribution affords us - we can entirely represent both the reading and the error of a thermometer with only two numbers - the mean and the variance.\n",
- "\n",
- "The standard notation for a normal distribution for a random variable $X$ is just $X \\sim\\ \\mathcal{N}(\\mu,\\sigma^2)$ where $\\mu$ is the mean and $\\sigma^2$ is the variance. It may seem odd to use $\\sigma$ squared - why not just $\\sigma$? We will not go into great detail about the math at this point, but in statistics $\\sigma$ is the *standard deviation* of a normal distribution. *Variance* is defined as the square of the standard deviation, hence $\\sigma^2$.\n",
- "\n",
- "It is worth spending a few words on standard deviation now. The standard deviation is a measure of how much variation from the mean exists. For Gaussian distributions, 68% of all the data falls within one standard deviation($1\\sigma$) of the mean, 95% falls within two standard deviations ($2\\sigma$), and 99.7% within three ($3\\sigma$). This is often called the 68-95-99.7 rule. So if you were told that the average test score in a class was 71 with a standard deviation of 9.4, you could conclude that 95% of the students received a score between 52.2 and 89.8 if the distribution is normal (that is calculated with $71 \\pm (2 * 9.4)$). "
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "The following graph depicts the relationship between the standard deviation and the normal distribution. "
- ]
- },
- {
- "cell_type": "code",
- "collapsed": false,
- "input": [
- "from gaussian_internal import display_stddev_plot\n",
- "display_stddev_plot()"
- ],
- "language": "python",
- "metadata": {},
- "outputs": [
- {
- "metadata": {},
- "output_type": "display_data",
- "png": 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rRUWLrpzFJgFANPrcrDy9srdNh9r7TUcBYgqFFRhG96+r0aem5Cgj0WM6CoDj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\n2FgAkY3CiojV0BXQt1/aq5s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\n",
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**Sidebar**: An equivalent formation for a Gaussian is $\\mathcal{N}(\\mu,1/\\tau)$ where $\\mu$ is the *mean* and $tau$ the *precision*. Here $1/\\tau = \\sigma^2$; it is the reciprocal of the variance. While we do not use this formulation in this book, it underscores that the variance is a measure of how precise our data is. A small variance yields large precision - our measurement is very precise. Conversely, a large variance yields low precision - our belief is spread out across a large area. You should become comfortable with thinking about Gaussians in these equivelant forms. Gaussians reflect our *belief* about a measurement, they express the *precision* of the measurement, and they express how much *variance* there is in the measurements. These are all different ways of stating the same fact.
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- "metadata": {},
- "source": [
- "### Interactive Gaussians\n",
- "\n",
- "For those that are reading this in IPython Notebook, here is an interactive version of the Gaussian plots. Use the sliders to modify $\\mu$ and $\\sigma^2$. Adjusting $\\mu$ will move the graph to the left and right because you are adjusting the mean, and adjusting $\\sigma^2$ will make the bell curve thicker and thinner."
- ]
- },
- {
- "cell_type": "code",
- "collapsed": false,
- "input": [
- "import math\n",
- "from IPython.html.widgets import interact, interactive, fixed\n",
- "import IPython.html.widgets as widgets\n",
- "\n",
- "def plt_g (mu,variance):\n",
- " xs = np.arange(2,8,0.1)\n",
- " ys = [gaussian (x, mu,variance) for x in xs]\n",
- " plt.plot (xs, ys)\n",
- " plt.ylim((0,1))\n",
- " plt.show()\n",
- "\n",
- "interact (plt_g, mu=(0,10), variance=widgets.FloatSliderWidget(value=0.6,min=0.2,max=4.5))"
- ],
- "language": "python",
- "metadata": {},
- "outputs": [
- {
- "metadata": {},
- "output_type": "display_data",
- "png": 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- "text": [
- ""
- ]
- },
- {
- "metadata": {},
- "output_type": "pyout",
- "prompt_number": 7,
- "text": [
- ""
- ]
- }
- ],
- "prompt_number": 7
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "#### Computational Properties of the Gaussian\n",
- "\n",
- "Recall how our discrete Bayesian filter worked. We had a vector implemented as a numpy array representing our belief at a certain moment in time. When we performed another measurement using the *sense()* function we had to multiply probabilities together, and when we performed the motion step using the *update()* function we had to shift and add probabilities. I've promised you that the Kalman filter uses essentially the same process, and that it uses Gaussians instead of histograms, so you might reasonable expect that we will be multipling, adding, and shifting Gaussians in the Kalman filter.\n",
- "\n",
- "A typical textbook would directly launch into a multipage proof of the behavior of Gaussians under these operations, but I don't see the value in that right now. I think the math will be much more intuitive and clear if we just start developing a Kalman filter using Gaussians. I will provide the equations for multiplying and shifting Gaussians at the appropriate time. You will then be able to develop a physical intuition for what these operations do, rather than be forced to digest a lot of fairly abstract math.\n",
- "\n",
- "The key point, which I will only assert for now, is that all the operations are very simple, and that they preserve the properties of the Gaussian. This is somewhat remarkable, in that the Gaussian is a nonlinear function, and typically if you multiply a nonlinear equation with itself you end up with a different equation. For example, the shape of $sin(x)sin(x)$ is very different from $sin(x)$. But the result of multiplying two Gaussians is yet another Gaussian. This is a fundamental property, and the key reason why Kalman filters are possible."
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "#### Summary and Key Points\n",
- "\n",
- "The following points **must** be understood by you before we continue:\n",
- "\n",
- "* Normal distributions occur throughout nature\n",
- "* They express a continuous probability distribution\n",
- "* They are completely described by two parameters: the mean ($\\mu$) and variance ($\\sigma^2$)\n",
- "* $\\mu$ is the average of all possible values\n",
- "* $\\sigma^2$ represents how much our measurements vary from the mean\n",
- "\n"
- ]
- },
- {
- "cell_type": "code",
- "collapsed": false,
- "input": [
- "#format the book\n",
- "from IPython.core.display import HTML\n",
- "def css_styling():\n",
- " styles = open(\"./styles/custom2.css\", \"r\").read()\n",
- " return HTML(styles)\n",
- "css_styling()"
- ],
- "language": "python",
- "metadata": {},
- "outputs": [
- {
- "html": [
- "