From cbf70c4ea0ed1b4a5396ed6e33a534c475659a3a Mon Sep 17 00:00:00 2001 From: aleicazatti Date: Fri, 21 Jun 2024 12:15:20 -0300 Subject: [PATCH 1/4] Add Student's T --- docs/examples/normal_distribution.md | 2 +- docs/examples/students_t.png | Bin 30943 -> 196273 bytes docs/examples/students_t_distribution.md | 106 ++++++++++++++++++++++ docs/examples/students_t_distribution.rst | 43 --------- docs/gallery_content.rst | 2 +- 5 files changed, 108 insertions(+), 45 deletions(-) create mode 100644 docs/examples/students_t_distribution.md delete mode 100644 docs/examples/students_t_distribution.rst diff --git a/docs/examples/normal_distribution.md b/docs/examples/normal_distribution.md index 95a6a069..8a1c311b 100644 --- a/docs/examples/normal_distribution.md +++ b/docs/examples/normal_distribution.md @@ -78,7 +78,7 @@ F(x \mid \mu, \sigma) = \frac{1}{2} \left[ 1 + \text{erf} \left( \frac{x - \mu}{\sigma \sqrt{2}} \right) \right] $$ -where [erf](https://en.wikipedia.org/wiki/Error_function) is the error function. +where erf is the [error function](https://en.wikipedia.org/wiki/Error_function). ```{seealso} diff --git a/docs/examples/students_t.png b/docs/examples/students_t.png index f2b55b6f26f5275a95713ef5fe3cbd517f6862ad..d8f8e827243395e7633a2cc4c7e783430bd43448 100644 GIT binary patch literal 196273 zcmeFZXH=6}+cwO7d(SvycMwn!u!FP+NRjG@ib_)f0VzScv`DWBIu0sbK}x71f>I-d zUJ?rgfj~q`2rWv7Pz)tN0^!|JufW{j^M1d+_5FIX)+`(r+1K7@InLuO`wG3Lr?qq2 z!EHP|JUg#kzHozwXE%n2=NG=;eg(gwZ?W47ew_8XXyRpnu=n!4?P15Gd)w=-3&P9g zj`d+5I}gu02sb%tC21MS!;W5FcRiJ1Fjwv$NFzKPV4w4ORly>^-@R<=$-^UkjPu{W z{IWEBc>cx1bLGOHhJGni{l527$1FsdRCHmR*P^s`vE))x-3j-n&QoWS|1Q~iW#;^s z^HxW;H;)tx@K^Zy-apY1B6#@8o*0+kCJKwBDDElNg1*h-WdUTzu$Snz63-bC$?XZM z80(3eF&$PAd!D6+LPZbL;SRW2%r)`2tKiB1_}INhk8l6SZ+Li4FP_3|`RDJhDU5jk z^EW(A*)_ZV`Kxo3XXO7{Uo3e}`JaEpV`pH=5$&eWbC)LCrvK&q^O>%{&_925ulRpA z#qQjnEUvrjP*s*c7Dmn4<9~sg8!G|?;SF#twDjmS?+F7XW-c@Ln7;b zl4fQ3J>-Athv%@CJTLOU)X$Tx{=atfzsAh5l>c>m9-jZ-0kW(r6!+6jR{;1IxjKra 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b/docs/examples/students_t_distribution.md @@ -0,0 +1,106 @@ +--- +jupytext: + text_representation: + extension: .md + format_name: myst +kernelspec: + display_name: Python 3 + language: python + name: python3 +--- + +# Student's t Distribution + +The Student's t distribution, also known as the t-distribution, is a continuous probability distribution that resembles the normal distribution but with heavier tails. It is characterized by its bell-shaped curve, symmetric around the mean, and can defined by three parameters: the degrees of freedom ($\nu$), the location parameter ($\mu$), and the scale parameter ($\sigma$). The smaller the value of ($\nu$), the heavier the tails of the distribution. + +The t-distribution is widely used in statistical analysis such as in Student's t-test, confidence interval estimation and linear regression analysis. It is often used in Bayesian analysis particularly when the sample size is small or the population variance is unknown. The t-distribution is more robust to outliers and deviations from normality compared to the normal distribution, making it a suitable choice for modeling data with heavier tails. + + +## Probability Density Function (PDF): + +```{code-cell} +--- +tags: [remove-input] +mystnb: + image: + alt: Student's t Distribution PDF +--- + +import matplotlib.pyplot as plt +import arviz as az +from preliz import StudentT +az.style.use('arviz-doc') +nus = [2., 5., 5.] +mus = [0., 0., -4.] +sigmas = [1., 1., 2.] +for nu, mu, sigma in zip(nus, mus, sigmas): + StudentT(nu, mu, sigma).plot_pdf(support=(-10,6)) + +plt.savefig('students_t.png') +``` + +## Cumulative Distribution Function (CDF): + +```{code-cell} +--- +tags: [remove-input] +mystnb: + image: + alt: Student's t Distribution CDF +--- + +for nu in nus: + StudentT(nu, mu, sigma).plot_cdf(support=(-10,6)) +``` + +## Key properties and parameters: + +```{eval-rst} +======== ========================================== +Support :math:`x \in \mathbb{R}` +Mean :math:`\mu` for :math:`\nu > 1`, otherwise undefined +Variance :math:`\frac{\nu}{\nu-2}` for :math:`\nu > 2`, + :math:`\infty` for :math:`1 < \nu \le 2`, otherwise undefined +======== ========================================== +``` + +**Probability Density Function (PDF):** + +$$ +f(x \mid \nu, \mu, \sigma) = \frac{\Gamma \left(\frac{\nu+1}{2} \right)} {\sqrt{\nu\pi}\Gamma \left(\frac{\nu}{2} \right)} \left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}} +$$ + +where $\Gamma$ is the [gamma function](https://en.wikipedia.org/wiki/Gamma_function). + +**Cumulative Distribution Function (CDF):** + +$$ +F(y \mid \nu, \mu, \sigma) = +\begin{cases} +1 - \frac{1}{2} I_{\frac{\nu}{x^2 + \nu}} \left( \frac{\nu}{2}, \frac{1}{2} \right) & \text{for } x = \frac{y - \mu}{\sigma} \leq 0, \\[0.5em] +\frac{1}{2} I_{\frac{\nu}{x^2 + \nu}} \left( \frac{\nu}{2}, \frac{1}{2} \right) & \text{for } x = \frac{y - \mu}{\sigma} > 0, +\end{cases} +$$ + +where $I_x(a, b)$ denotes the [regularized incomplete beta function](https://en.wikipedia.org/wiki/Regularized_incomplete_beta_function). + + + +```{seealso} +:class: seealso + +**Common Alternatives:** + +- [Normal Distribution](normal_distribution.md) - When $\nu \to \infty$, the t-distribution converges to the normal distribution. +- [Cauchy Distribution](cauchy_distribution.md) - The Cauchy distribution is a special case of the Student's t-distribution with $\nu=1$. + +**Related Distributions:** + +- [Chi-Squared Distribution](chi_squared_distribution.md) - Student's t-distribution is a transformation of chi-squared distribution and can be obtained from chi-squared distribution and normal distribution. + +``` +## References + +- Wikipedia. [Student's t-distribution](https://en.wikipedia.org/wiki/Student%27s_t-distribution) + + diff --git a/docs/examples/students_t_distribution.rst b/docs/examples/students_t_distribution.rst deleted file mode 100644 index b8abc35f..00000000 --- a/docs/examples/students_t_distribution.rst +++ /dev/null @@ -1,43 +0,0 @@ -Student's T Distribution -========================= - -Story ... - -Probability Density Function (PDF): -____________________________________________________ - -.. image:: students_t_pdf.png - :alt: Student's T Distribution PDF - -Cumulative Distribution Function (CDF): -____________________________________________________ - -.. image:: students_t_cdf.png - :alt: Student's T Distribution CDF - -Key properties and parameters: -____________________________________________________ - -======== ======================== - -======== ======================== - -Relevant mathematical formulas: -____________________________________________________ - -**Probability Density Function (PDF):** - - - -**Cumulative Distribution Function (CDF):** - - -See also: -____________________________________________________ - -- :doc:`Normal Distribution ` - -References: -____________________________________________________ - -- `Wikipedia - Student's t-distribution `_ \ No newline at end of file diff --git a/docs/gallery_content.rst b/docs/gallery_content.rst index c46bfe54..7f5e2e8d 100644 --- a/docs/gallery_content.rst +++ b/docs/gallery_content.rst @@ -295,7 +295,7 @@ Continuous Distributions Skew-Normal .. grid-item-card:: - :link: ./examples/student_t_distribution.html + :link: ./examples/students_t_distribution.html :text-align: center :shadow: none :class-card: example-gallery From c6b113c75431fbcc81ac282fc912deded25248d5 Mon Sep 17 00:00:00 2001 From: Alejandro Icazatti Date: Fri, 21 Jun 2024 20:45:31 -0300 Subject: [PATCH 2/4] Apply suggestion Co-authored-by: Osvaldo A Martin --- docs/examples/students_t_distribution.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs/examples/students_t_distribution.md b/docs/examples/students_t_distribution.md index e17affe7..1ac65e7f 100644 --- a/docs/examples/students_t_distribution.md +++ b/docs/examples/students_t_distribution.md @@ -13,7 +13,7 @@ kernelspec: The Student's t distribution, also known as the t-distribution, is a continuous probability distribution that resembles the normal distribution but with heavier tails. It is characterized by its bell-shaped curve, symmetric around the mean, and can defined by three parameters: the degrees of freedom ($\nu$), the location parameter ($\mu$), and the scale parameter ($\sigma$). The smaller the value of ($\nu$), the heavier the tails of the distribution. -The t-distribution is widely used in statistical analysis such as in Student's t-test, confidence interval estimation and linear regression analysis. It is often used in Bayesian analysis particularly when the sample size is small or the population variance is unknown. The t-distribution is more robust to outliers and deviations from normality compared to the normal distribution, making it a suitable choice for modeling data with heavier tails. +It is often used in Bayesian analysis particulary as a robust alternative to the Normal due to the possibility of having heavier tails. ## Probability Density Function (PDF): From 43df6e42635f66d0c7b9f321b498f2cb2dc9dd6f Mon Sep 17 00:00:00 2001 From: Alejandro Icazatti Date: Fri, 21 Jun 2024 21:10:42 -0300 Subject: [PATCH 3/4] Apply suggestions Add HalfStudent and SkewedStudent --- docs/examples/students_t_distribution.md | 7 ++----- 1 file changed, 2 insertions(+), 5 deletions(-) diff --git a/docs/examples/students_t_distribution.md b/docs/examples/students_t_distribution.md index 1ac65e7f..6ed71107 100644 --- a/docs/examples/students_t_distribution.md +++ b/docs/examples/students_t_distribution.md @@ -91,13 +91,10 @@ where $I_x(a, b)$ denotes the [regularized incomplete beta function](https://en. **Common Alternatives:** +- [Skewed Student's t Distribution](skew_studentt_distribution.md) - Extends the Student's t-distribution by introducing a skewness parameter, allowing for the modeling of data that is not symmetrically distributed. +- [Half-Student's t Distribution](halfstudentt_distribution.md) - Considers only the positive values of the Student's t-distribution. - [Normal Distribution](normal_distribution.md) - When $\nu \to \infty$, the t-distribution converges to the normal distribution. - [Cauchy Distribution](cauchy_distribution.md) - The Cauchy distribution is a special case of the Student's t-distribution with $\nu=1$. - -**Related Distributions:** - -- [Chi-Squared Distribution](chi_squared_distribution.md) - Student's t-distribution is a transformation of chi-squared distribution and can be obtained from chi-squared distribution and normal distribution. - ``` ## References From c74c64b1098b76ab44dbede3a539739cfcf3b3f6 Mon Sep 17 00:00:00 2001 From: Alejandro Icazatti Date: Fri, 21 Jun 2024 21:41:11 -0300 Subject: [PATCH 4/4] Apply sugestion Co-authored-by: Osvaldo A Martin --- docs/examples/students_t_distribution.md | 1 - 1 file changed, 1 deletion(-) diff --git a/docs/examples/students_t_distribution.md b/docs/examples/students_t_distribution.md index 6ed71107..39d75792 100644 --- a/docs/examples/students_t_distribution.md +++ b/docs/examples/students_t_distribution.md @@ -36,7 +36,6 @@ sigmas = [1., 1., 2.] for nu, mu, sigma in zip(nus, mus, sigmas): StudentT(nu, mu, sigma).plot_pdf(support=(-10,6)) -plt.savefig('students_t.png') ``` ## Cumulative Distribution Function (CDF):