{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# The beauty of computational efficiency and the FFT\n", "\n", "When we discuss \"computational efficiency\", you often hear people throw around phrases like $O(n^2)$ or $O(nlogn)$. We talk about them in the abstract, and it can be hard to appreciate what these distinctions mean and how important they are. So let's take a quick look at what computational efficiency looks like in the context of a very famous algorithm: The Fourier Transform.\n", "\n", "## A short primer on the Fourier Transform\n", "Briefly, A Fourier Transform is used for uncovering the spectral information that is present in a signal. AKA, it tells us about oscillatory components in the signal, and has [a wide range](http://dsp.stackexchange.com/questions/69/why-is-the-fourier-transform-so-important) of uses in communications, signal processing, and even neuroscience analysis.\n", "\n", "Here's a [Quora post](https://www.quora.com/What-is-an-intuitive-way-of-explaining-how-the-Fourier-transform-works) that discusses Fourier Transforms more generally. The first explanation is fantastic and full of history and detail.\n", "\n", "The challenge with the Fourier Transform is that it can take a really long time to compute. If you h ave a signal of length $n$, then you're calculating $n$ Fourier components for each point in the (length $n$) signal. This means that the number of operations required to calculate a fourier transform is $n * n$ or $O(n^2)$.\n", "\n", "For a quick intuition into what a difference this makes. Consider two signals, one of length 10, and the other of length 100. Since the Fourier Transform is $O(n^2)$, the length 100 signal will take *2 orders of magnitude* longer to compute, even though it is only *1 order of magnitude longer in length*.\n", "\n", "Think this isn't a big deal? Let's see what happens when the signal gets longer. First off, a very short signal:" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [], "source": [ "# We can use the `time` and the `numpy` module to time how long it takes to do an FFT\n", "from time import time\n", "import numpy as np\n", "import seaborn as sns\n", "sns.set_style('white')\n", "\n", "# For a signal of length ~1000. Say, 100ms of a 10KHz audio sample.\n", "signal = np.random.randn(1009)\n", "start = time()\n", "_ = np.fft.fft(signal)\n", "stop = time()\n", "print('It takes {} seconds to do the FFT'.format(stop-start))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "That's not too bad - ~.003 seconds is pretty fast. But here's where the $O(n^2)$ thing really gets us..." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true, "jupyter": { "outputs_hidden": true } }, "outputs": [], "source": [ "# We'll test out how long the FFT takes for a few lengths\n", "test_primes = [11, 101, 1009, 10009, 100019]" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [], "source": [ "# Let's try a few slightly longer signals\n", "for i_length in test_primes:\n", " # Calculate the number of factors for this length (we'll see why later)\n", " factors = [ii for ii in range(1, 1000) if i_length % ii == 0]\n", " # Generate a random signal w/ this length\n", " signal = np.random.randn(i_length)\n", " # Now time the FFT\n", " start = time()\n", " _ = np.fft.fft(signal)\n", " stop = time()\n", " print('With data of length {} ({} factors), it takes {} seconds to do the FFT'.format(\n", " i_length, len(factors), stop-start))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Whoah wait a sec, that last one took way longer than everything else. We increased the length of the data by a factor of 10, but the time it took went up by a factor of 100. Not good. That means that if we want to perform an FFT on a signal that was 10 times longer, it'd take us about 42 minutes. 100 times longer? That'd take *~3 days.*\n", "\n", "Given how important the Fourier Transform is, it'd be great if we could speed it up somehow. \n", "\n", "> *You'll notice that I chose a very particular set of numbers above. Specifically, I chose numbers that were primes (or nearly primes) meaning that they couldn't be broken down into products of smaller numbers. That turns out to be really important in allowing the FFT to do its magic. When your signal length is a prime number, then you don't gain any speedup from the FFT, as I'll show below.*\n", "\n", "## Enter the Fast Fourier Transform\n", "The Fast Fourier Transform (FFT) is one of the most important algorithms to come out of the last century because it drastically speeds up the performance of the Fourier Transform. It accomplishes this by breaking down all those $n^2$ computations into a smaller number of computations, and then putting them together at the end to get the same result. This is called **factorizing**.\n", "\n", "You can think of factorizing like trying to move a bunch of groceries from your car to your fridge. Say you have 20 items in your car. One way to do this is to individually take each item, pull it from the car, walk to the house, place it in the fridge. It'd take you 20 trips to do this. Factorizing is like putting your 20 items into 2 grocery bags. Now you only need to make 2 trips to the house - one for each grocery bag. The first approach requires 20 trips to the house, and the second requires 2 trips. You've just reduced the number of trips by an order of magnitude!\n", "\n", "The FFT accomplishes its factorization by recognizing that signals of a certain length can be broken down (factorized) into smaller signals. How many smaller signals? Well, that depends on the length of the original signal. If a number has many *factors*, it means that it can be broken down into a product of many different, smaller, signals.\n", "\n", "In practice, this means that if the input to an FFT has a lot of factors, then you gain a bigger speedup from the FFT algorithm. On one end, a signal with a length == a power of two will have a ton of factors, and yield the greatest speedups. A signal with length == a prime number will be the slowest because it has no factors. Below is a quick simulation to see how much of a difference this makes.\n", "\n", "> Here are some useful links explaining Fourier Transforms, as well as the FFT:\n", "> * [A Quora post](https://www.quora.com/What-is-an-intuitive-explanation-of-the-FFT-algorithm) with some great answers on the intuition behind the Fast Fourier Transform.\n", "> * [The wikipedia entry](https://en.wikipedia.org/wiki/Fast_Fourier_transform) for FFTs also has some nice links.\n", "> * [A post on the FFT](https://jakevdp.github.io/blog/2013/08/28/understanding-the-fft/) from Jake Vanderplas is also a great explanation of how it works." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [], "source": [ "import pandas as pd\n", "from sklearn import linear_model\n", "from matplotlib import pyplot as plt\n", "%matplotlib inline" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## The beautiful efficiency of the FFT\n", "To see what the FFT's efficiency looks like, we'll simulate data of different lengths and see how long it takes to compute the FFT at each length. We'll create a random vector of gaussian noise ranging from length 1 to 10,000. For each vector, we'll compute the FFT, and time how long it took to compute. I've already taken the liberty of doing this (repeated 3 times, and then averaged together). Those times are stored in fft_times.csv." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [], "source": [ "# Let's read in the data and see how it looks\n", "df = pd.read_csv('../data/fft_times.csv', index_col=0)\n", "df = df.apply(pd.to_numeric)\n", "df = df.mean(0).to_frame('time')\n", "df.index = df.index.astype(int)\n", "df['length'] = df.index.values" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": {}, "execution_count": 27, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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}, "metadata": {}, "output_type": "display_data" } ], "source": [ "# First off, it's clear that computation time grows nonlinearly with signal length\n", "df.plot('length', 'time', figsize=(10, 5))" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": {}, "execution_count": 28, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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1QnPKJZqyL1ueY6kYtUD5EcC2PgUErtJuKrbPeDLBcuU53GEpbmVtZGjzSVsY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\nWs2kGLPPufOkFcktUKvvU867c//T+D9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mUEVFc8xY1LRA4tCcWGyZvMlFsz5HxqgtyirQ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\nh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7+/C39/txGpFLB0zmgsX1CXVrnb09KP/378I7T3hHDpGWPTlnknEklI+3rgsFlRW+Hj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\niRoHBwcHBwcHx1GK/we7AqeKX26m6gAAAABJRU5ErkJggg==\n" }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# However, upon closer inspection, it's clear that there's much variability\n", "winsize = 500\n", "i = 0\n", "j = i + winsize\n", "df.iloc[i:j]['time'].plot(figsize=(10, 5))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As you can see, there appear to be multiple trends in the data. There seems to be a \"most inefficient\" line of growth in the data, as well as a \"more efficient\" and a \"most efficient\" trend. These correspond to lengths that are particularly good for an FFT.\n", "\n", "We can use regression to find the \"linear\" relationship between length of signal and time of FFT. However, if there are any trends in the data that are **nonlinear**, then they should show up as errors in the regression model. Let's see if that happens..." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [], "source": [ "# We'll use a regression model to try and fit how length predicts time\n", "mod = linear_model.LinearRegression()\n", "xfit = df['length']\n", "xfit = np.vstack([xfit, xfit**2, xfit**3, xfit**4]).T\n", "yfit = df['time'].reshape([-1, 1])" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [], "source": [ "# Now fit to our data, and calculate the error for each datapoint\n", "mod.fit(xfit, yfit)\n", "df['ypred'] = mod.predict(xfit)\n", "df['diff'] = df['time'] - df.ypred" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": {}, "execution_count": 31, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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NZwG8tPMljpwcIe36iJb76Tr6MsFQgKTPprXl3XmOVCwW//Pfv09/phbbTOLJ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}, "metadata": {}, "output_type": "display_data" } ], "source": [ "# As the length grows, the trends in the data begin to diverge more and more\n", "ax = df.plot('length', 'diff', kind='scatter',\n", " style='.', alpha=.5, figsize=(10, 5))\n", "ax.set_ylim([0, .05])\n", "ax.set_title('Error of linear fit for varying signal lengths')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It looks like there are some clear components of the data that *don't* follow a linear relationship. Moreover, this seems to be systematic. We clearly see several separate traces in the error plot, which means that there are patterns in the data that follow different non-linear trends.\n", "\n", "But we already know that the FFT efficiency will differ depending on the number of factors of the signal's length. Let's see if that's related to the plot above..." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "
\n", " | time | \n", "length | \n", "ypred | \n", "diff | \n", "n_factors | \n", "
---|---|---|---|---|---|
9995 | \n", "0.031189 | \n", "9995 | \n", "0.035864 | \n", "-0.004675 | \n", "2 | \n", "
9996 | \n", "0.000974 | \n", "9996 | \n", "0.035873 | \n", "-0.034899 | \n", "18 | \n", "
9997 | \n", "0.012076 | \n", "9997 | \n", "0.035881 | \n", "-0.023805 | \n", "2 | \n", "
9998 | \n", "0.080163 | \n", "9998 | \n", "0.035889 | \n", "0.044274 | \n", "2 | \n", "
9999 | \n", "0.002135 | \n", "9999 | \n", "0.035897 | \n", "-0.033763 | \n", "6 | \n", "