2499 lines
186 KiB
Groff
2499 lines
186 KiB
Groff
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Which library should I use? Possibly scipy?" />
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Create free Team
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<h5 class="pt4 fw-bold">Collectives™ on Stack Overflow</h5>
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<h1 itemprop="name" class="fs-headline1 ow-break-word mb8 flex--item fl1"><a href="/questions/20626994/how-to-calculate-the-inverse-of-the-normal-cumulative-distribution-function-in-p" class="question-hyperlink">How to calculate the inverse of the normal cumulative distribution function in python?</a></h1>
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<time itemprop="dateCreated" datetime="2013-12-17T05:57:49">9 years, 11 months ago</time>
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<p>How do I calculate the inverse of the cumulative distribution function (CDF) of the normal distribution in Python?</p>
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<p>Which library should I use? Possibly scipy? </p>
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data-gps-track="post.click({ item: 4, priv: 0, post_type: 1 })">edited <span title='2015-03-31 14:52:45Z' class='relativetime'>Mar 31, 2015 at 14:52</span></a>
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asked <span title='2013-12-17 05:57:49Z' class='relativetime'>Dec 17, 2013 at 5:57</span>
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<span class="comment-copy">Do you mean the inverse Gaussian distribution (<a href="http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution" rel="nofollow noreferrer">en.wikipedia.org/wiki/Inverse_Gaussian_distribution</a>), or the inverse of the cumulative distribution function of the normal distribution (<a href="http://en.wikipedia.org/wiki/Normal_distribution" rel="nofollow noreferrer">en.wikipedia.org/wiki/Normal_distribution</a>), or something else?</span>
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– <a href="/users/1217358/warren-weckesser"
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title="111,515 reputation"
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class="comment-user">Warren Weckesser</a>
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<span class="comment-date" dir="ltr"><span title='2013-12-17 06:30:48Z, License: CC BY-SA 3.0' class='relativetime-clean'>Dec 17, 2013 at 6:30</span></span>
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<span class="comment-copy">@WarrenWeckesser the second one: inverse of the cumulative distribution function of the normal distribution</span>
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– <a href="/users/1624752/yueyoum"
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<span class="comment-date" dir="ltr"><span title='2013-12-17 06:32:40Z, License: CC BY-SA 3.0' class='relativetime-clean'>Dec 17, 2013 at 6:32</span></span>
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<span class="comment-copy">@WarrenWeckesser i mean the python version of "normsinv" function in excel.</span>
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– <a href="/users/1624752/yueyoum"
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title="2,853 reputation"
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class="comment-user owner">Yueyoum</a>
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<span class="comment-date" dir="ltr"><span title='2013-12-17 06:39:22Z, License: CC BY-SA 3.0' class='relativetime-clean'>Dec 17, 2013 at 6:39</span></span>
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3 Answers
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<p><a href="http://support.microsoft.com/kb/826772" rel="noreferrer">NORMSINV</a> (mentioned in a comment) is the inverse of the CDF of the standard normal distribution. Using <code>scipy</code>, you can compute this with the <code>ppf</code> method of the <a href="http://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.norm.html" rel="noreferrer"><code>scipy.stats.norm</code></a> object. The acronym <code>ppf</code> stands for <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda362.htm#PPF" rel="noreferrer"><em>percent point function</em></a>, which is another name for the <a href="https://en.wikipedia.org/wiki/Quantile_function" rel="noreferrer"><em>quantile function</em></a>.</p>
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<pre><code>In [20]: from scipy.stats import norm
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In [21]: norm.ppf(0.95)
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Out[21]: 1.6448536269514722
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</code></pre>
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<p>Check that it is the inverse of the CDF:</p>
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<pre><code>In [34]: norm.cdf(norm.ppf(0.95))
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Out[34]: 0.94999999999999996
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<p>By default, <code>norm.ppf</code> uses mean=0 and stddev=1, which is the "standard" normal distribution. You can use a different mean and standard deviation by specifying the <code>loc</code> and <code>scale</code> arguments, respectively.</p>
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<pre><code>In [35]: norm.ppf(0.95, loc=10, scale=2)
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Out[35]: 13.289707253902945
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</code></pre>
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<p>If you look at the source code for <code>scipy.stats.norm</code>, you'll find that the <code>ppf</code> method ultimately calls <a href="http://docs.scipy.org/doc/scipy/reference/generated/scipy.special.ndtri.html" rel="noreferrer"><code>scipy.special.ndtri</code></a>. So to compute the inverse of the CDF of the standard normal distribution, you could use that function directly:</p>
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<pre><code>In [43]: from scipy.special import ndtri
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In [44]: ndtri(0.95)
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Out[44]: 1.6448536269514722
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</code></pre>
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<p><code>ndtri</code> is <em>much</em> faster than <code>norm.ppf</code>:</p>
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<pre><code>In [46]: %timeit norm.ppf(0.95)
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240 µs ± 1.75 µs per loop (mean ± std. dev. of 7 runs, 1,000 loops each)
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In [47]: %timeit ndtri(0.95)
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1.47 µs ± 1.3 ns per loop (mean ± std. dev. of 7 runs, 1,000,000 loops each)
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data-gps-track="post.click({ item: 4, priv: 0, post_type: 2 })">edited <span title='2022-08-15 14:05:56Z' class='relativetime'>Aug 15, 2022 at 14:05</span></a>
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answered <span title='2013-12-17 06:44:02Z' class='relativetime'>Dec 17, 2013 at 6:44</span>
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<span class="comment-copy">I always think "percent point function" (ppf) is a terrible name. Most people in statistics just use "quantile function".</span>
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– <a href="/users/1670078/william-zhang"
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title="699 reputation"
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class="comment-user">William Zhang</a>
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<span class="comment-date" dir="ltr"><span title='2014-10-04 00:44:33Z, License: CC BY-SA 3.0' class='relativetime-clean'>Oct 4, 2014 at 0:44</span></span>
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<span class="comment-copy">Don't you need to specify the mean and the std on both ppf and cdf?</span>
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– <a href="/users/1391660/bones-felipe"
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<span class="comment-date" dir="ltr"><span title='2021-01-29 19:23:44Z, License: CC BY-SA 4.0' class='relativetime-clean'>Jan 29, 2021 at 19:23</span></span>
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<span class="comment-copy">@bones.felipe, the "standard" normal distribution has mean 0 and standard deviation 1. These are the default values for the location and scale of the <code>scipy.stats.norm</code> methods.</span>
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<span class="comment-date" dir="ltr"><span title='2021-01-29 19:55:51Z, License: CC BY-SA 4.0' class='relativetime-clean'>Jan 29, 2021 at 19:55</span></span>
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<span class="comment-copy">Right, I thought I saw this <code>norm.cdf(norm.ppf(0.95, loc=10, scale=2))</code> and I thought it was weird <code>norm.cdf</code> did not have <code>loc=10</code> and <code>scale=2</code> too, I guess it should.</span>
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<span class="comment-date" dir="ltr"><span title='2021-01-30 05:33:43Z, License: CC BY-SA 4.0' class='relativetime-clean'>Jan 30, 2021 at 5:33</span></span>
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<p>Starting <code>Python 3.8</code>, the standard library provides the <a href="https://docs.python.org/3.8/library/statistics.html?highlight=normaldist#statistics.NormalDist" rel="noreferrer"><code>NormalDist</code></a> object as part of the <a href="https://docs.python.org/3.8/library/statistics.html" rel="noreferrer"><code>statistics</code></a> module.</p>
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<p>It can be used to get the <strong><em>inverse cumulative distribution function</em></strong> (<strong><a href="https://docs.python.org/3.8/library/statistics.html#statistics.NormalDist.inv_cdf" rel="noreferrer"><code>inv_cdf</code></a></strong> - inverse of the <a href="https://docs.python.org/3.8/library/statistics.html#statistics.NormalDist.cdf" rel="noreferrer"><code>cdf</code></a>), also known as the <strong>quantile function</strong> or the <strong>percent-point function</strong> for a given <em>mean</em> (<code>mu</code>) and <em>standard deviation</em> (<code>sigma</code>):</p>
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<pre><code>from statistics import NormalDist
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NormalDist(mu=10, sigma=2).inv_cdf(0.95)
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# 13.289707253902943
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</code></pre>
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<p>Which can be simplified for the <em>standard normal distribution</em> (<code>mu = 0</code> and <code>sigma = 1</code>):</p>
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<pre><code>NormalDist().inv_cdf(0.95)
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# 1.6448536269514715
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answered <span title='2019-03-19 21:58:57Z' class='relativetime'>Mar 19, 2019 at 21:58</span>
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<span class="comment-copy">Great tip! This allows me to drop the dependency on scipy, which I needed just for the single stats.norm.ppf method</span>
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– <a href="/users/5437918/jethro-cao"
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class="comment-user">Jethro Cao</a>
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<span class="comment-date" dir="ltr"><span title='2020-02-21 16:56:27Z, License: CC BY-SA 4.0' class='relativetime-clean'>Feb 21, 2020 at 16:56</span></span>
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<span class="comment-copy">can you use that to transform data with uniform distribution to normal ?</span>
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– <a href="/users/10824677/vanetoj"
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<span class="comment-date" dir="ltr"><span title='2022-03-31 20:51:01Z, License: CC BY-SA 4.0' class='relativetime-clean'>Mar 31, 2022 at 20:51</span></span>
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<pre><code># given random variable X (house price) with population muy = 60, sigma = 40
|
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import scipy as sc
|
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import scipy.stats as sct
|
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sc.version.full_version # 0.15.1
|
|||
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|
|||
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#a. Find P(X<50)
|
|||
|
|
sct.norm.cdf(x=50,loc=60,scale=40) # 0.4012936743170763
|
|||
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|||
|
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#b. Find P(X>=50)
|
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|
sct.norm.sf(x=50,loc=60,scale=40) # 0.5987063256829237
|
|||
|
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|
|||
|
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#c. Find P(60<=X<=80)
|
|||
|
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sct.norm.cdf(x=80,loc=60,scale=40) - sct.norm.cdf(x=60,loc=60,scale=40)
|
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|
|||
|
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#d. how much top most 5% expensive house cost at least? or find x where P(X>=x) = 0.05
|
|||
|
|
sct.norm.isf(q=0.05,loc=60,scale=40)
|
|||
|
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|
|||
|
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#e. how much top most 5% cheapest house cost at least? or find x where P(X<=x) = 0.05
|
|||
|
|
sct.norm.ppf(q=0.05,loc=60,scale=40)
|
|||
|
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</code></pre>
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answered <span title='2015-09-26 15:04:38Z' class='relativetime'>Sep 26, 2015 at 15:04</span>
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<span class="reputation-score" title="reputation score " dir="ltr">3,376</span><span title="4 gold badges" aria-hidden="true"><span class="badge1"></span><span class="badgecount">4</span></span><span class="v-visible-sr">4 gold badges</span><span title="32 silver badges" aria-hidden="true"><span class="badge2"></span><span class="badgecount">32</span></span><span class="v-visible-sr">32 silver badges</span><span title="45 bronze badges" aria-hidden="true"><span class="badge3"></span><span class="badgecount">45</span></span><span class="v-visible-sr">45 bronze badges</span>
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<span class="comment-copy">PS: You can assume 'loc' as 'mean' and 'scale' as 'standard deviation'</span>
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– <a href="/users/4815313/suresh2692"
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title="3,873 reputation"
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class="comment-user">Suresh2692</a>
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<span class="comment-date" dir="ltr"><span title='2017-07-05 11:11:41Z, License: CC BY-SA 3.0' class='relativetime-clean'>Jul 5, 2017 at 11:11</span></span>
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