diff --git a/04_Gaussians.ipynb b/04_Gaussians.ipynb index 7212b7f..1a47eae 100644 --- a/04_Gaussians.ipynb +++ b/04_Gaussians.ipynb @@ -269,11 +269,44 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "The last chapter ended by discussing some of the drawbacks of the Discrete Bayesian filter. For many tracking and filtering problems our desire is to have a filter that is *unimodal* and *continuous*. That is, we want to model our system using floating point math (continuous) and to have only one belief represented (unimodal). For example, we want to say an aircraft is at (12.34381, -95.54321,2389.5) where that is latitude, longitude, and altitude. We do not want our filter to tell us \"it might be at (1,65,78) or at (34,656,98)\" That doesn't match our physical intuition of how the world works, and as we discussed, it is prohibitively expensive to compute.\n", + "The last chapter ended by discussing some of the drawbacks of the Discrete Bayesian filter. For many tracking and filtering problems our desire is to have a filter that is *unimodal* and *continuous*. That is, we want to model our system using floating point math (continuous) and to have only one belief represented (unimodal). For example, we want to say an aircraft is at (12.34, -95.54, 2389.5) where that is latitude, longitude, and altitude. We do not want our filter to tell us \"it *might be* at (1.65, -78.01, 2100.45) or it *might be* at (34.36, -98.23, 2543.79).\" That doesn't match our physical intuition of how the world works, and as we discussed, it is prohibitively expensive to compute. \n", "\n", - ">So we desire a unimodal, continuous way to represent probabilities that models how the real world works, and that is very computationally efficient to calculate. As you might guess from the chapter name, Gaussian distributions provide all of these features.\n", + ">So we desire a unimodal, continuous way to represent probabilities that models how the real world works, and that is very computationally efficient to calculate. As you might guess from the chapter name, Gaussian distributions provide all of these features." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Mean, Variance, and Standard Deviations\n", "\n", - "Before we go into the math, lets just look at a graph of the Gaussian distribution to get a sense of what we are talking about." + "### Random Variables\n", + "\n", + "To understand Gaussians we first need to understand a few simple mathematical definitions and computations\n", + "\n", + "To understand Gaussians we first need to understand a few simple mathematical computations. We start with a *random variable* x. Maybe x is the height of the students in a class in meters. We call $x$ *random* because the height of one student does not depend on the height of another student - each is independent. For a specific classrom perhaps the heights are\n", + "\n", + "$$x= [1.8, 2.0, 1.7, 1.9, 1.6]$$\n", + "\n", + "Another example of a random variable would be the result of a coin toss. Each coin toss is independent from all other tosses. Rolling a die would be another example. \n", + "\n", + "The coin toss and die roll are examples of *discrete random variables*. That is, the outcome of any given event comes from a discrete set of values. The die roll can never produce a value of 7 or 3.24, for example. In contrast, the student heights are continuous; they can take on any value (within biological limits, naturally). For example, heights of 1.7, 1.71, 1.711, 1.7111, 1.71111,.... are all possible. Do not confuse the *measurement* of the random variable with the actual value. If we can only measure the height of a person to 0.1 meters we would only record values from 0.1, 0.2, 0.3...2.7, or 27 discrete choices. Nonetheless the heights can vary between any arbitrary real value between those ranges, and so it is continuous. \n", + "\n", + "### The Mean of a Random Variable\n", + "\n", + "We want to know the *average* height of the students. Another word for *average* is the *mean*. We compute the mean by summing the values and dividing by the number of values. In this case we have\n", + "\n", + "$$mean = (1.8 + 2.0 + 1.7 + 1.9 + 1.6)/5 = 1.8$$\n", + "\n", + "In statistics we normally use the symbol $\\mu$ to denote the mean, so we could write \n", + "\n", + "$$\\mu_x = 1.8$$\n", + "\n", + "We can formalize this computation with the equation\n", + "\n", + "$$ \\mu_x = \\frac{1}{n}\\sum^n_{i=1} x_i$$\n", + "\n", + "NumPy has the function `numpy.mean()` which computes the mean for you." ] }, { @@ -282,6 +315,908 @@ "metadata": { "collapsed": false }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "1.8\n" + ] + } + ], + "source": [ + "import numpy as np\n", + "\n", + "x= [1.8, 2.0, 1.7, 1.9, 1.6]\n", + "print(np.mean(x))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Standard Deviation of a Random Variable\n", + "\n", + "The computation above tells us the average height of the students, but it doesn't tell us everything we might want to know. For example, suppose a second class has students with these heights:\n", + "\n", + "$$y = [2.2, 1.5, 2.3, 1.7, 1.3]$$" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "1.8" + ] + }, + "execution_count": 3, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "y = [2.2, 1.5, 2.3, 1.7, 1.3]\n", + "np.mean(y)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "the mean of these heights is also 1.8 m, but notice that there is a much greater amount of variation in the heights in this class. Finally, suppose a third class has heights\n", + "\n", + "$$ z = [1.8, 1.8, 1.8, 1.8, 1.8]$$\n", + "\n", + "In this third class the average height is again 1.8 m, but here there is no variation in the height between students at all." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We want to be able to specify how much *variation* there is between the heights of the students. You can imagine a number of reasons for this. Perhaps a school district needs to order 5,000 desks, and they want to be sure they buy sizes that accommodate the range of heights of the students. \n", + "\n", + "Statistics has formalized this concept of measuring variation into the notion of *standard deviation* and *variance*:\n", + "\n", + "**The standard deviation is defined as the square root of the average of the squared differences from the mean.**\n", + "\n", + "That's a mouthful; let's just work through that with the data from three classes. We just subtract the mean of x from each value of x, square it, take the average of those, and then take the square root of the result. The mean of $[1.8, 2.0, 1.7, 1.9, 1.6]$ is 1.8, so we compute the standard deviation as\n", + "\n", + "$$ \n", + "\\begin{aligned}\n", + "\\sigma_x &=\\sqrt{\\frac{(1.8-1.8)^2 + (2-1.8)^2 + (1.7-1.8)^2 + (1.9-1.8)^2 + (1.6-1.8)^2} {5}} \\\\\n", + "&= \\sqrt{\\frac{0^2 + 0.2^2 + (-0.1)^2 + 0.1^2 + (-0.2)^2}{5}} \\\\\n", + "&= \\sqrt{\\frac{0 + 0.04 + 0.01 + 0.01 + 0.04}{5}} \\\\\n", + "&= \\sqrt{0.02} \\\\\n", + "\\sigma_x&= 0.1414\n", + "\\end{aligned}$$\n", + "\n", + "We use the symbol $\\sigma$ to denote the standard deviation.\n", + "\n", + "We can verify this computation with the NumPy method `std()` which computes the standard deviation." + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "0.1414\n" + ] + } + ], + "source": [ + "print('{:.4f}'.format(np.std(x)))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "What does this number *mean* (no pun intended)? *How much* is not a mathematical term. We will be able to define it much more precisely once we introduce the concept of a Gaussian in the next section. For now I'll say that for many things in nature, including the height of people, 68% of all values lie within one standard deviation of the mean. From this we can conclude that for a random class 68% of the students will have heights between 1.66 meters and 1.94 meters. \n", + "\n", + "We can look at this in a plot:" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "image/png": [ + "iVBORw0KGgoAAAANSUhEUgAAAvIAAAGHCAYAAADSnLajAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", + "AAALEgAACxIB0t1+/AAAIABJREFUeJzt3X10VPWdx/HPTHhKIgwhbAKEp+BCxBAeQogQQEAkK0sX\n", + "1sozBwgrUD2AEJfasmIBBbFQ6QolETwYU1q34NENtYsVUpCHgm0iBiki6EYiLiaUkISSFVyTu39Y\n", + "psQ8MBMyufld3q9zck7mzvfOfIcvyfnkzm/udVmWZQkAAACAUdx2NwAAAADAfwR5AAAAwEAEeQAA\n", + "AMBABHkAAADAQAR5AAAAwEAEeQAAAMBABHkAAADAQLYF+bVr12rQoEHyeDyKiIjQ+PHjdfLkyZvu\n", + "d+LECY0YMUIhISHq3LmznnnmmUboFgAAAGhabAvyBw4c0MKFC3X06FHt27dPzZo10/3336+SkpJa\n", + "97l8+bLGjBmjjh07Kjc3Vy+88ILWr1+vDRs2NGLnAAAAgP1cTeXKruXl5fJ4PNq1a5fGjRtXY016\n", + "erqWLVumoqIitWzZUpK0Zs0apaen6/PPP2/MdgEAAABbNZk18pcvX1ZlZaXCwsJqrTl69KiGDx/u\n", + 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len(x)+1), m-s, m+s, \n", + " facecolor='yellow', alpha=0.4)\n", + " plt.xlabel('student')\n", + " plt.ylabel('height (m)')\n", + " plt.show()\n", + " \n", + "x= [1.8, 2.0, 1.7, 1.9, 1.6]\n", + "plot_height_std(x)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "For only 5 students we obviously will not get exactly 68% within one standard deviation (normally denoted as $1\\sigma$), but we do see that 3 out of 5 students are within $1\\sigma$, or 60%, which is as close as you can get to 68% with only 5 samples. I haven't yet introduced enough math or Python for you to fully understand the next bit of code, but let's look at the results if have a class with 100 students." + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "image/png": [ + "iVBORw0KGgoAAAANSUhEUgAAAvQAAAGHCAYAAADfgsbkAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", + "AAALEgAACxIB0t1+/AAAIABJREFUeJzt3X9cVPWC//H3DPgDTck0kEQFexjX36lghZpayY21tUfZ\n", + "D8XN8GbeWjTF7dZqllqWm27sqollN5Fc29RHrt7auqlpmle7FzKqxdS6Jtq1wRUFF79pVzjfP7pO\n", + "jjDDDDI/PjOv5+PB48Gc8zkz7xk+HN4czhxslmVZAgAAAGAke7ADAAAAAGg8Cj0AAABgMAo9AAAA\n", + "YDAKPQAAAGAwCj0AAABgMAo9AAAAYDAKPQAAAGCwoBb6BQsWKC0tTbGxsYqLi9Po0aNVWlrqcZvD\n", + "hw/LbrfX+di8eXOAUgMAAAChI6iFfseOHZoyZYr27Nmjbdu2KTo6WrfddptOnTrV4LYffPCBHA6H\n", + "82PEiBEBSAwAAACEFlso/afYM2fOKDY2Vps2bdKoUaPqHXP48GF169ZNRUVFGjhwYIATAgAAAKEl\n", + "pM6hP336tGpra9WuXbsGx959992Kj4/XkCFD9PbbbwcgHQAAABB6QuoI/X333ac///nPKi4uls1m\n", + 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"stdout", + "output_type": "stream", + "text": [ + "mean of x = 1.803\n", + "std of x = 0.146\n" + ] + } + ], + "source": [ + "from numpy.random import randn\n", + "x = [1.8 + .1414*randn() for i in range(100)]\n", + "plot_height_std(x, lw=2)\n", + "\n", + "print('mean of x = {:.3f}'.format(np.mean(x)))\n", + "print('std of x = {:.3f}'.format(np.std(x)))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can see by eye that roughly 68% of the heights lie within $1\\sigma$ of the mean 1.8." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We'll discuss this in greater depth soon. For now let's compute the standard deviation for \n", + "\n", + "$$y = [2.2, 1.5, 2.3, 1.7, 1.3]$$\n", + "\n", + "The mean of y is $\\mu=1.8$ m, so \n", + "$$ \n", + "\\begin{aligned}\n", + "\\sigma_y &=\\sqrt{\\frac{(2.2-1.8)^2 + (1.5-1.8)^2 + (2.3-1.8)^2 + (1.7-1.8)^2 + (1.3-1.8)^2} {5}} \\\\\n", + "&= \\sqrt{\\frac{0.4^2 + (-0.3)^2 + 0.5^2 + (-0.1)^2 + (-0.5)^2}{5}} \\\\\n", + "&= \\sqrt{\\frac{0 + 0.04 + 0.01 + 0.01 + 0.04}{5}} \\\\\n", + "&= \\sqrt{0.152} \\\\\n", + "\\sigma_y&= 0.39 \\ m\n", + "\\end{aligned}$$\n", + "\n", + "We will verify that with NumPy with" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "0.3899 m\n" + ] + } + ], + "source": [ + "print('{:.4f} m'.format(np.std(y)))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This corresponds with what we would expect. There is more variation in the heights for y, and the standard deviation is larger.\n", + "\n", + "Finally, let's compute the standard deviation for \n", + "\n", + "$$ z = [1.8, 1.8, 1.8, 1.8, 1.8]$$\n", + "\n", + "There is no variation in the values, so we would expect the standard deviation to be zero. We show this to be true with\n", + "\n", + "$$ \n", + "\\begin{aligned}\n", + "\\sigma_z &=\\sqrt{\\frac{(1.8-1.8)^2 + (1.8-1.8)^2 + (1.8-1.8)^2 + (1.8-1.8)^2 + (183-1.8)^2} {5}} \\\\\n", + "&= \\sqrt{\\frac{0+0+0+0+0}{5}} \\\\\n", + "\\sigma_Z&= 0.0 \\ m\n", + "\\end{aligned}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Variance of a Random Variable\n", + "Finally, the *variance* is defined as the square of the standard deviation. Some texts define this in the opposite way, which gives the definitions\n", + "\n", + "* **The variance is the average of the squared differences from the mean.**\n", + "\n", + "* **The standard deviation is the square root of the variance.**\n", + "\n", + "Both ways of thinking about it are equivalent. We use the notation $\\sigma^2$ for the variance, and the equation for the variance is\n", + "\n", + "\n", + "$$\\sigma = \\frac{1}{N}\\sum_{i=1}^N(x_i - \\mu)^2$$\n", + "\n", + "\n", + "To make sure we understand this let's compute the variance for $x$:\n", + "\n", + "$$ \n", + "\\begin{aligned}\n", + "\\sigma_x^2 &=\\frac{(1.8-1.8)^2 + (2-1.8)^2 + (1.7-1.8)^2 + (1.9-1.8)^2 + (1.6-1.8)^2} {5} \\\\\n", + "&= \\frac{0^2 + 0.2^2 + (-0.1)^2 + 0.1^2 + (-0.2)^2}{5} \\\\\n", + "&= \\frac{0 + 0.04 + 0.01 + 0.01 + 0.04}{5} \\\\\n", + "&=0.02\n", + "\\end{aligned}$$\n", + "\n", + "We previously computed $\\sigma_x=0.1414$, and indeed the square of 0.1414 is 0.02. We can verify this computation with the NumPy function `numpy.var`:" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "VAR(x) = 0.0213 m\n" + ] + } + ], + "source": [ + "print('VAR(x) = {:.4f} m'.format(np.var(x)))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Many texts alternatively use *VAR(x)* to denote the variance of x, and I opted to use that here just to expose you to it." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Why the Square of the Differences\n", + "\n", + "As an aside, why are we taking the *square* of the difference? I could go into a lot of math, but let's just look at this in a simple way. Here is a chart of the values of x plotted against the mean for $x=[3,-3,3,-3]$\n" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "image/png": [ + "iVBORw0KGgoAAAANSUhEUgAAAhkAAAELCAYAAAB07JSGAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", + "AAALEgAACxIB0t1+/AAADZJJREFUeJzt3VuIVXX7wPFnT6aNJTuZMgrH6qVJIsrSUdALKdJoSiSI\n", + "oqAjUWQHTC+iC6OiqOgEFWp0U5EEaYGUyVSQKWGEUwQdKaKDYSM11XaULHPv96rhNW365//3NHvb\n", + "5wPSPs3vWbBazHfWXnum0mg0GgEAUFjbSG8AAHBgEhkAQAqRAQCkEBkAQAqRAQCkEBkAQAqRAQCk\n", + "KBoZS5cujSlTpkS1Wo1qtRqzZs2KtWvXlhwBALSISslfxvXiiy/GmDFjoqurK+r1ejz11FNx//33\n", + "x6ZNm2LKlCmlxgAALaBoZOxLR0dH3HfffXHNNddkjgEAmsyorIV3794dq1atip07d8bs2bOzxgAA\n", + "Tap4ZLz//vsxc+bM+OWXX6K9vT1WrlwZkydPLj0GAGhyxd8u2bVrV2zevDlqtVqsWrUqHnvssVi3\n", + "bl10d3dHREStVis5DgBoAtVqda/H0q/JmDt3bkycODGefPLJiBAZAHAg2ldkpP+ejN27d0e9Xs8e\n", + "AwA0maLXZNx6660xb968mDhxYgwODsazzz4b69evj97e3n2+fl/Vwz+nUqnscT/5pBb/R319fRER\n", + 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np.average(x)\n", + " for i in range(len(x)):\n", + " plt.plot([i,i], [m,x[i]], color='k')\n", + " plt.axhline(m)\n", + " plt.xlim(-1, len(x))\n", + " plt.tick_params(axis='x', labelbottom='off')\n", + " plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "If we didn't take the square of the differences the signs would cancel everything out:\n", + "\n", + "$$\\frac{(3-0) + (-3-0) + (3-0) + (-3-0)}{4} = 0$$\n", + "\n", + "This is clearly incorrect. \n", + "\n", + "Maybe we can use the absolute value? We can see by inspection that the result is $12/4=3$ which is certainly correct. But what if we change $x=[6, -2, -3, 1]$? In this case we get $12/4=3$. X is clearly more spread out, but we get the same variance, so this cannot be correct. If we use the correct formula we get a variance of 3.5, which correctly reflects the larger variatin in $x$.\n", + "\n", + "This is clearly not a proof of correctness; indeed, Gauss recognized that is is somewhat arbitrary. If the variable is distributed normally we can prove that the variance is computed by this function. If there are outliers, meaning values that are not normally distributed, than squaring the difference gives disproportinate weight to that term. For example, let's see what happens if we have $x = [1,-1,1,-2,3,2,100]$. I'll use NumPy's `var()` method to compute the variance for me." + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Variance of x = 1210.69\n" + ] + } + ], + "source": [ + "x=[1,-1,1,-2,3,2,100]\n", + "print('Variance of x = {:.2f}'.format(np.var(x)))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Is this *correct*? You tell me. Without the outlier of 100 we get $\\sigma^2=2.89$, which accurately reflects how $x$ absent the outlier is varying. The one outlier swamps the computation. I will not continue down this path; if you are interested you might want to look at the work that James Berger. For Kalman filters, which assume Gaussian distributions, we can prove that squaring the terms produces optimal results. More about that soon." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Gaussians" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We are now ready to learn about Gaussians. Let's remind ourselves of the motivation for this chapter.\n", + "\n", + ">We desire a unimodal, continuous way to represent probabilities that models how the real world works, and that is very computationally efficient to calculate. As you might guess from the chapter name, Gaussian distributions provide all of these features." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Before we go into the math, lets just look at a graph of the Gaussian distribution to get a sense of what we are talking about." + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": { + "collapsed": false + }, "outputs": [ { "data": { @@ -751,7 +1686,7 @@ "sEknIiIiIjIybNKJiIiIiIwMm3QiIiIiIiPDJp2IiIiIyMj8P8WIzB3z7ixBAAAAAElFTkSuQmCC\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -770,9 +1705,7 @@ "source": [ "Probably this is immediately recognizable to you as a 'bell curve'. This curve is ubiquitous 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 probability, 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", - "

" + "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 probability, 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." ] }, { @@ -791,7 +1724,7 @@ }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 12, "metadata": { "collapsed": false }, @@ -1282,7 +2215,7 @@ "90ccRAAAAABJRU5ErkJggg==\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -1350,9 +2283,10 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 13, "metadata": { - "collapsed": false + "collapsed": false, + "scrolled": true }, "outputs": [ { @@ -1785,7 +2719,7 @@ "RK5CYII=\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -1821,7 +2755,7 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 14, "metadata": { "collapsed": false }, @@ -1873,7 +2807,7 @@ }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 15, "metadata": { "collapsed": false }, @@ -1901,9 +2835,10 @@ }, { "cell_type": "code", - "execution_count": 7, + "execution_count": 16, "metadata": { - "collapsed": false + "collapsed": false, + "scrolled": true }, "outputs": [ { @@ -2493,7 +3428,7 @@ "4AAAAABJRU5ErkJggg==\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -2505,9 +3440,9 @@ "import matplotlib.pyplot as plt\n", " \n", "xs = np.arange(15, 30, 0.05)\n", - "plt.plot(xs, [gaussian(x, 23, .2) for x in xs], label='var=0.2')\n", - "plt.plot(xs, [gaussian(x, 23, 1) for x in xs], label='var=1')\n", - "plt.plot(xs, [gaussian(x, 23, 5) for x in xs], label='var=5')\n", + "plt.plot(xs, gaussian(xs, 23, 0.2), label='var=0.2')\n", + "plt.plot(xs, gaussian(xs, 23, 1), label='var=1')\n", + "plt.plot(xs, gaussian(xs, 23, 5), label='var=5')\n", "plt.legend()\n", "plt.show()" ] @@ -2534,7 +3469,7 @@ }, { "cell_type": "code", - "execution_count": 8, + "execution_count": 17, "metadata": { "collapsed": false }, @@ -2897,7 +3832,7 @@ "fyIiIiKiMsDgT0RERERUBhj8iYiIiIjKAIM/EREREVEZ+B/H1tUZSdQuawAAAABJRU5ErkJggg==\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -2933,7 +3868,7 @@ }, 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Recall the equation for the Gaussian:\n", + "\n", + "$$ f(x, \\mu, \\sigma) = \\frac{1}{\\sigma\\sqrt{2\\pi}} e^{-\\frac{1}{2}{(x-\\mu)^2}/\\sigma^2 }$$\n", + "\n", + "$\\mu$ and $\\sigma$ are constants, so a simplified form of this equation is\n", + "\n", + "$$ \n", + "f(x, \\mu, \\sigma) = c_1 * e^{-c_2{(x-\\mu)^2} }\n", + "$$\n", + "\n", + "Let's just simplify it further by assuming all the constants are one (not true for Gaussians, but the general shape will be the same).\n", + "\n", + "$$ \n", + "f(x, \\mu, \\sigma) = e^{-x^2}\n", + "$$\n", + "\n", + "Raising a value to a negative power is the same as the reciprocal of the value raised to a positive power. I.e.\n", + "\n", + "$$e^{-x^2} = \\frac{1}{e^{x^2}}$$\n", + "\n", + "\n", + "Let's plot this in stages. Here is $e^{x^2}$." + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "image/png": [ + "iVBORw0KGgoAAAANSUhEUgAAAy4AAAGaCAYAAADpdxfhAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", + "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3Xt0VfWd///XPrmSCwkJOQkkCqEGiahUQSFxhCLhYgUj\n", + "TqdoNKjFwrQ0I2XNSsX6FaqIxXEwUtQFtAiFXwb1W8Y6rQMpxSqQWLmqlZtyEQPkJIQQCIaQy+f3\n", + "B1+27gRIIgfOSc7zsRZr+fnszz7nvfNZS/Lisz97W8YYIwAAAADwYy5fFwAAAAAArSG4AAAAAPB7\n", + "BBcAAAAAfo/gAgAAAMDvEVwAAAAA+D2CCwAAAAC/R3ABAFxRI0eOVN++fdXU1OTrUtqlqalJ/fv3\n", + "17Bhw3xdCgAEJIILAOCKeeedd/TXv/5VM2fOlMvl3b+CFi9eLJfLJZfLpV/96letjt+4caMef/xx\n", + "DR48WImJiQoLC1OvXr2Um5urjz/+uMV4l8ulWbNmaf369Xrrrbe8WjsAoHUWL6AEAFwpN910k6qq\n", + "qrR//35ZluW1z12+fLkefvhhdevWTVFRUTp48KCee+45/eIXv7jgOUlJSTp69KiGDBmiQYMGKTw8\n", + "XFu2bNFf//pXBQcH680331R2dnaL89LS0hQWFqZ//OMfXqsfANA6VlwAAFdEcXGxPvroI+Xk5Hg1\n", + 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If we take the reciprocal of a rapidly increasing number the result should approach, but never reach 0." + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "image/png": [ + "iVBORw0KGgoAAAANSUhEUgAAAyAAAAGaCAYAAAD3vidSAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", + "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3Xd0FGXfxvFrN40khF5MKIHQQYFAKEakKB0pAiKIIiAi\n", + "qI8I6AMojwiiqChiA0URBEWKIB0FpBMEYghKR0KPiZQQatruvH/4Eh0TmiSZbPb7OYdz9HfP7F7J\n", + "0C529x6bYRiGAAAAACAH2K0OAAAAAMB9UEAAAAAA5BgKCAAAAIAcQwEBAAAAkGMoIAAAAAByDAUE\n", + "AAAAQI6hgAAAMmjRooUqV64sp9NpdRRLzZkzR3a7XWvXrrU6CgDkGRQQAIDJ8uXL9eOPP2rUqFGy\n", + "2//6YyItLU3vv/+++vTpo9q1a8vb21t2u12ffvrpTT3u0KFDZbfbdfz48SzJefbsWdWqVUt2u13l\n", + "ypXTsWPHbnj81KlT1aVLF1WuXFn+/v4qWLCgGjZsqA8++EBpaWkZzunWrZtq1KihF154Qdw2CwCy\n", + "ho0bEQIA/i40NFQJCQk6fPiwbDZb+vzcuXMqUqSIbDabSpYsKS8vLx0/flyffPKJ+vfvf8PHDQkJ\n", + 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The Gaussian equation has different constants, and hence a somewhat different shape, but both are 'bell curves', as seen below." + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "image/png": [ + "iVBORw0KGgoAAAANSUhEUgAAAxkAAAGNCAYAAAB9i7yDAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", + "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3XtUVOe9P/73nuF+F0EEFRRRkKtcROUiahQ1iYnNvZe0\n", + "SZPTNqenbZpfz8rJ75fvatOmJ6enXTlpTk9sv21O0iRNT9K0NfEu3kFQQEQugiJ4VxBEucPAzP79\n", + "4WGGZw8gl5nZc3m/1upafT7M3vuDQ3Q+PM/neSRZlmUQERERERFZiEbtBIiIiIiIyLmwyCAiIiIi\n", + "IotikUFERERERBbFIoOIiIiIiCyKRQYREREREVkUiwwiIiIiIrIoFhlERERERGRR9ywyjh49ioce\n", + "eghz586FRqPBH//4x3vetLq6Gnl5efDx8cHcuXPxs5/9zCLJEhERERGR/btnkdHT04Pk5GT8+te/\n", + "hre3NyRJGvf1nZ2dWL9+PcLDw1FeXo5f//rX+OUvf4k333zTYkkTEREREZH9kiZz4re/vz/+67/+\n", + "C1//+tfHfM3WrVvxyiuvoKWlBZ6engCAn//859i6dSuuXr06/YyJiIiIiMiuWbwno6SkBLm5ucYC\n", + 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"text": [ "0.131146572034\n", - "0.13114657203397997\n" + "0.131146572034\n" ] } ], @@ -3378,7 +6642,7 @@ }, { "cell_type": "code", - "execution_count": 11, + "execution_count": 25, "metadata": { "collapsed": false }, @@ -3411,7 +6675,7 @@ }, { "cell_type": "code", - "execution_count": 12, + "execution_count": 26, "metadata": { "collapsed": false }, @@ -3420,9 +6684,9 @@ "name": "stdout", "output_type": "stream", "text": [ - "[-0.14310837 0.84075413 1.17083607 3.94112376 1.88644565 3.06415797\n", - " 3.95553435 -1.66300556 1.18171991 -0.93707154 0.30555009 3.46944566\n", - " 1.73282306 2.8570935 -2.56755922]\n" + "[ 3.77999795 -0.18956982 2.69579025 2.56248639 4.57891564 3.00244391\n", + " 3.0723681 7.75724739 -0.25014797 0.85577169 3.94158436 5.14455528\n", + " 6.62881127 1.93136557 1.61332113]\n" ] } ], @@ -3439,7 +6703,7 @@ }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 27, "metadata": { "collapsed": false }, @@ -3466,7 +6730,7 @@ }, { "cell_type": "code", - "execution_count": 14, + "execution_count": 28, "metadata": { "collapsed": false }, @@ -3514,7 +6778,7 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 29, "metadata": { "collapsed": false }, @@ -3884,7 +7148,7 @@ "IiIiIiKfxMkKERERERH5JE5WiIiIiIjIJ/0XkMJRB/+WilkAAAAASUVORK5CYII=\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -3913,7 +7177,7 @@ }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 30, "metadata": { "collapsed": false }, @@ -3934,7 +7198,7 @@ }, { "cell_type": "code", - "execution_count": 17, + "execution_count": 31, "metadata": { "collapsed": false }, @@ -3943,956 +7207,953 @@ "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAyIAAAGNCAYAAADtmMVMAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", - "AAALEgAACxIB0t1+/AAAIABJREFUeJzsvXeUXNWV8LurlZAAk5MsEYxlMEFA0+N5682a9/CbN2/A\n", - "sxbDmLH9+QODv5nx+BsHTBoDA1jGiGjASEIggQjKKIAioJxQK3Wrg9RBrW51UOecu6u7q+q+P6pu\n", - 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variance. All must be scalars. + specified mean and variance. x can either be a scalar or an + array-like. gaussian (1,2,3) is equivalent to scipy.stats.norm(2,math.sqrt(3)).pdf(1) It is quite a bit faster albeit much less flexible than the latter. @@ -39,7 +40,7 @@ def gaussian(x, mean, var): Parameters ---------- - x : scalar + x : scalar or array-like The value for which we compute the probability mean : scalar @@ -50,12 +51,18 @@ def gaussian(x, mean, var): Returns ------- - probability : float + probability : float, or array-like probability of x for the Gaussian (mean, var). E.g. 0.101 denotes 10.1%. - + + Examples + -------- + gaussian(3, 1, 2) + gaussian([3,4,3,2,1], 1, 2) """ - return math.exp((-0.5*(x-mean)**2)/var) / math.sqrt(_two_pi*var) + + return (np.exp((-0.5*(np.asarray(x)-mean)**2)/var) / + np.sqrt(_two_pi*var)) def mul (mean1, var1, mean2, var2):