Fixed errors in probability density function.
For some reason I was calling the PDF the probability, when of course it is the probability density.
This commit is contained in:
Roger Labbe committed
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002de6e8df
commit
cbfd8db15c
4 files changed
+515
-89
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@@ -1,7 +1,7 @@
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{
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"metadata": {
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"name": "",
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"signature": "sha256:f06623f9c65b72eaaa654d00719496d1826c6c5a76a1cb6ee583c4b4844b2cf2"
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"signature": "sha256:00a931decd323bc4f252540ee5dd36a369b885268e890c562b4a210ea2fff34b"
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},
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"nbformat": 3,
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"nbformat_minor": 0,
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@@ -257,13 +257,13 @@
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],
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"metadata": {},
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"output_type": "pyout",
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"prompt_number": 12,
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"prompt_number": 3,
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"text": [
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"<IPython.core.display.HTML at 0x7f8a50209cf8>"
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"<IPython.core.display.HTML at 0x7fdfe851eb00>"
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]
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}
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],
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"prompt_number": 12
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"prompt_number": 3
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},
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{
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"cell_type": "heading",
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@@ -290,7 +290,7 @@
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"input": [
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"from stats import plot_gaussian\n",
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"\n",
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"plot_gaussian(mean=100, variance=15*15, xlabel='IQ', ylabel='percent')"
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"plot_gaussian(mean=100, variance=15*15, xlabel='IQ')"
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],
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"language": "python",
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"metadata": {},
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@@ -298,13 +298,13 @@
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{
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"metadata": {},
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"output_type": "display_data",
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|
||||
"<matplotlib.figure.Figure at 0x7f8a502092b0>"
|
||||
"<matplotlib.figure.Figure at 0x7fdfc797ca90>"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 13
|
||||
"prompt_number": 4
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
@@ -359,7 +359,7 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"<p> 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",
|
||||
"<p> 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` with the function `gaussian(x, mean, var)`.\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",
|
||||
@@ -384,11 +384,53 @@
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [
|
||||
"from stats import gaussian\n",
|
||||
"plot_gaussian(22,4,mean_line=True,xlabel='$^{\\circ}C$',ylabel=\"Percent\")\n",
|
||||
"from stats import gaussian, norm_cdf\n",
|
||||
"plot_gaussian(22, 4, mean_line=True, xlabel='$^{\\circ}C$')"
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"metadata": {},
|
||||
"output_type": "display_data",
|
||||
"png": 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truncated
|
||||
"text": [
|
||||
"<matplotlib.figure.Figure at 0x7fdfc764c940>"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 5
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"So what does this curve *mean*? Assume for a moment that we have a thermometer, 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. So, when we look at this chart we can *sort of* think of it as representing the probability of the thermometer reading a particular value given the actual temperature of 22$^{\\circ}C$. However, that is not quite accurate mathematically. \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))"
|
||||
"Recall that we said that the distribution is *continuous*. Think of an infinitly long straight line - what is the probability that a point you pick randomly is at, say, 2.0. Clearly 0%, as there is an infinite number of choices to choose from. The same is true for normal distributions; in the graph above the probability of being *exactly* 22$^{\\circ}C$ is 0% because there are an infinite number of values the reading can take.\n",
|
||||
"\n",
|
||||
"So what then is this curve? It is something we call the *probability density function.* Later we will delve into this in greater detail; for now just understand that the area under the curve at any region gives you the probability of those values. So, for example, if you compute the area under the curve between 20 and 22 the result will be the probability of the temperature reading being between those two temperatures."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"So how do you compute the probability, or area under the curve? Well, you integrate the equation for the Gaussian \n",
|
||||
"\n",
|
||||
"$$ \\int^{x_1}_{x_0} \\frac{1}{\\sigma\\sqrt{2\\pi}} e^{-\\frac{1}{2}{(x-\\mu)^2}/\\sigma^2 } dx$$\n",
|
||||
"\n",
|
||||
"I wrote `stats.norm_cdf` which computes the integral for you. So, for example, we can compute"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [
|
||||
"print('Probability of value in range 21.5 to 22.5 is {:.2f}%'.format(\n",
|
||||
" norm_cdf((21.5, 22.5), 22,4)*100))\n",
|
||||
"print('Probability of value in range 23.5 to 24.5 is {:.2f}%'.format(\n",
|
||||
" norm_cdf((23.5, 24.5), 22,4)*100))"
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
@@ -397,27 +439,17 @@
|
||||
"output_type": "stream",
|
||||
"stream": "stdout",
|
||||
"text": [
|
||||
"Probability of 22 is 19.95\n",
|
||||
"Probability of 24 is 12.10\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"metadata": {},
|
||||
"output_type": "display_data",
|
||||
"png": 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truncated
|
||||
"text": [
|
||||
"<matplotlib.figure.Figure at 0x7f8a50276630>"
|
||||
"Probability of value in range 21.5 to 22.5 is 19.74%\n",
|
||||
"Probability of value in range 23.5 to 24.5 is 12.10%\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 14
|
||||
"prompt_number": 6
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"So what does this curve *mean*? Assume for a moment that we have a thermometer, 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",
|
||||
@@ -426,13 +458,42 @@
|
||||
"\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",
|
||||
"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 distribution of measurements for over any range."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"###### 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 occurences, 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",
|
||||
"Since this is a probability density 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 occurences, which must sum to one. We can prove this ourselves with a bit of code. (If you are mathematically inclined, integrate the Gaussian equation from $-\\infty$ to $\\infty$)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [
|
||||
"print(norm_cdf((-1e8, 1e8), mu=0, var=4))"
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"output_type": "stream",
|
||||
"stream": "stdout",
|
||||
"text": [
|
||||
"1.0\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 7
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"This leads to an important insight. If the variance is small the curve will be narrow. this is because the variance is a measure of *how much* the samples vary from the mean. 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:"
|
||||
]
|
||||
@@ -457,19 +518,19 @@
|
||||
{
|
||||
"metadata": {},
|
||||
"output_type": "display_data",
|
||||
"png": 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truncated
|
||||
"text": [
|
||||
"<matplotlib.figure.Figure at 0x7f8a501220b8>"
|
||||
"<matplotlib.figure.Figure at 0x7fdfc759fa58>"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 15
|
||||
"prompt_number": 8
|
||||
},
|
||||
{
|
||||
"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",
|
||||
"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. In contrast, the red gaussian also believes that $x=23$, but we are much less sure about that. 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",
|
||||
@@ -499,13 +560,13 @@
|
||||
{
|
||||
"metadata": {},
|
||||
"output_type": "display_data",
|
||||
"png": 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||||
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truncated
|
||||
"text": [
|
||||
"<matplotlib.figure.Figure at 0x7f8a50039be0>"
|
||||
"<matplotlib.figure.Figure at 0x7fdfc759f390>"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 16
|
||||
"prompt_number": 9
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
@@ -554,21 +615,30 @@
|
||||
{
|
||||
"metadata": {},
|
||||
"output_type": "display_data",
|
||||
"png": 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truncated
|
||||
"text": [
|
||||
"<matplotlib.figure.Figure at 0x7f8a50078a90>"
|
||||
"<matplotlib.figure.Figure at 0x7fdfc7551f28>"
|
||||
]
|
||||
},
|
||||
{
|
||||
"metadata": {},
|
||||
"output_type": "pyout",
|
||||
"prompt_number": 17,
|
||||
"prompt_number": 10,
|
||||
"text": [
|
||||
"<function __main__.plt_g>"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 17
|
||||
"prompt_number": 10
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Finally, if you are reading this in an IPython Notebook, here is an animation of a Gaussian. First, the mean is being shifted to the right. Then the mean is centered at $\\mu=5$ and the variance is modified.\n",
|
||||
"\n",
|
||||
"<img src='gaussian_animate.gif'>"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "heading",
|
||||
@@ -582,11 +652,11 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"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 multiplying, adding, and shifting Gaussians in the Kalman filter.\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 `update()` function we had to multiply probabilities together, and when we performed the motion step using the `predict()` 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 multiplying, 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."
|
||||
"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 a key reason why Kalman filters are possible."
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -626,13 +696,13 @@
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 18
|
||||
"prompt_number": 11
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"The call `norm(2,3)` creates what scipy calls a 'frozen' distribution - it creates and returns an object with a mean of 2 and a standard deviation of 3. You can then use this object multiple times to get the probability of various values, like so:"
|
||||
"The call `norm(2,3)` creates what scipy calls a 'frozen' distribution - it creates and returns an object with a mean of 2 and a standard deviation of 3. You can then use this object multiple times to get the probability density of various values, like so:"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -640,9 +710,9 @@
|
||||
"collapsed": false,
|
||||
"input": [
|
||||
"n23 = norm(2,3)\n",
|
||||
"print ('probability of 1.5 is %.4f' % n23.pdf(1.5))\n",
|
||||
"print ('probability of 2.5 is also %.4f' % n23.pdf(2.5))\n",
|
||||
"print ('whereas probability of 2 is %.4f' % n23.pdf(2))"
|
||||
"print ('probability density of 1.5 is %.4f' % n23.pdf(1.5))\n",
|
||||
"print ('probability density of 2.5 is also %.4f' % n23.pdf(2.5))\n",
|
||||
"print ('whereas probability density of 2 is %.4f' % n23.pdf(2))"
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
@@ -651,13 +721,13 @@
|
||||
"output_type": "stream",
|
||||
"stream": "stdout",
|
||||
"text": [
|
||||
"probability of 1.5 is 0.1311\n",
|
||||
"probability of 2.5 is also 0.1311\n",
|
||||
"whereas probability of 2 is 0.1330\n"
|
||||
"probability density of 1.5 is 0.1311\n",
|
||||
"probability density of 2.5 is also 0.1311\n",
|
||||
"whereas probability density of 2 is 0.1330\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 19
|
||||
"prompt_number": 12
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
@@ -676,18 +746,7 @@
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"output_type": "stream",
|
||||
"stream": "stdout",
|
||||
"text": [
|
||||
"[ 0.38741053 -4.91803911 -2.00879426 -1.15175952 2.63354369 -0.96097707\n",
|
||||
" 2.45698146 4.22587531 3.69924957 -0.27811435 -0.94132557 4.96510082\n",
|
||||
" 3.43324112 4.09462176 0.6446632 ]\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 20
|
||||
"outputs": []
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
@@ -705,16 +764,7 @@
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"output_type": "stream",
|
||||
"stream": "stdout",
|
||||
"text": [
|
||||
"0.5\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 21
|
||||
"outputs": []
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
@@ -733,18 +783,7 @@
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"output_type": "stream",
|
||||
"stream": "stdout",
|
||||
"text": [
|
||||
"variance is 9.0\n",
|
||||
"standard deviation is 3.0\n",
|
||||
"mean is 2.0\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 23
|
||||
"outputs": []
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
@@ -771,7 +810,8 @@
|
||||
"* 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"
|
||||
"* The variance $\\sigma^2$ represents how much our measurements vary from the mean\n",
|
||||
"* The standard deviation ($\\sigma$) is the square root of the variance ($\\sigma^2$)"
|
||||
]
|
||||
},
|
||||
{
|
||||
|
||||
@@ -0,0 +1,322 @@
|
||||
{
|
||||
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"name": "",
|
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"signature": "sha256:c37d0ec0133c79aa5807b6f8cb1fc339f0365aa42cb11bfe3a91cba980759b23"
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"nbformat_minor": 0,
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{
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"cells": [
|
||||
{
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [
|
||||
"from __future__ import division, print_function\n",
|
||||
"%matplotlib inline\n",
|
||||
"import sys\n",
|
||||
"sys.path.insert(0,'../code') # allow us to format the book\n",
|
||||
"\n",
|
||||
"# use same formatting as rest of book so that the plots are\n",
|
||||
"# consistant with that look and feel.\n",
|
||||
"import book_format\n",
|
||||
"book_format.load_style()"
|
||||
],
|
||||
"language": "python",
|
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"metadata": {},
|
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"<style>\n",
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|
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"\n",
|
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" div.cell{\n",
|
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" width: 850px;\n",
|
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|
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|
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"\n",
|
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" font-weight: 200;\n",
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" display: block;\n",
|
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|
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" }\n",
|
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" .text_cell_render h2 {\n",
|
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" font-weight: 200;\n",
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" font-size: 20pt;\n",
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" font-style: italic;\n",
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" font-style: normal;\n",
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" margin-top: 1.5em;\n",
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" display: block;\n",
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" white-space: nowrap;\n",
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" }\n",
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" font-family: 'Open sans',verdana,arial,sans-serif;\n",
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" line-height: 135%;\n",
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" font-size: 110%;\n",
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" width:750px;\n",
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" }\n",
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" div.output_subarea.output_text.output_pyout {\n",
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" overflow-x: auto;\n",
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" overflow-y: scroll;\n",
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" max-height: 300px;\n",
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" div.output_subarea.output_stream.output_stdout.output_text {\n",
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" overflow-x: auto;\n",
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" overflow-y: scroll;\n",
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" max-height: 300px;\n",
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" font-size: 70%;\n",
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" }\n",
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" .rendered_html code{\n",
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" background-color: transparent;\n",
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" }\n",
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" ul{\n",
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" }\n",
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" ol{\n",
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" margin: 2em;\n",
|
||||
" }\n",
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|
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" margin-top: 0.5em; \n",
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|
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|
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|
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|
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|
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|
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|
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|
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" hr {\n",
|
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" color: #f3f3f3;\n",
|
||||
" background-color: #f3f3f3;\n",
|
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" height: 1px;\n",
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
||||
{
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [
|
||||
"import numpy as np\n",
|
||||
"import matplotlib.pyplot as plt\n",
|
||||
"from gif_animate import animate\n",
|
||||
"from stats import gaussian\n",
|
||||
"\n",
|
||||
"def plt_g (mu, variance):\n",
|
||||
" xs = np.arange(2,8,0.05)\n",
|
||||
" ys = [gaussian (x, mu, variance) for x in xs]\n",
|
||||
" plt.plot (xs, ys)\n",
|
||||
" plt.ylim((0,1))\n",
|
||||
"\n",
|
||||
" \n",
|
||||
"mu = 2\n",
|
||||
"sigma = 0.6\n",
|
||||
"def ganimate(frame):\n",
|
||||
" global mu, sigma\n",
|
||||
" if frame < 25:\n",
|
||||
" mu += .2\n",
|
||||
" elif frame == 25:\n",
|
||||
" mu = 5\n",
|
||||
" elif frame < 37:\n",
|
||||
" sigma -= 0.05\n",
|
||||
" else:\n",
|
||||
" sigma += 0.05\n",
|
||||
" \n",
|
||||
" plt.cla()\n",
|
||||
" plt_g(mu,sigma)\n",
|
||||
"\n",
|
||||
"animate('gaussian_animate.gif', ganimate, frames=80, interval=50)\n",
|
||||
"animate??"
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"metadata": {},
|
||||
"output_type": "display_data",
|
||||
"png": 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truncated
|
||||
"text": [
|
||||
"<matplotlib.figure.Figure at 0x7f9c26dfab38>"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 46
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": []
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"<img src='gaussian_animate.gif'>"
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {}
|
||||
}
|
||||
]
|
||||
}
|
||||
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After Width: | Height: | Size: 754 KiB |
Reference in new issue
Block a user