Merge pull request #319 from JoramSoch/master

added 4 defintions and 5 proofs

Author: JoramSoch

D/mean-circ.md

@@ -0,0 +1,43 @@
+---
+layout: definition
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "OvGU Magdeburg"
+e_mail: "joram.soch@ovgu.de"
+date: 2026-04-20 16:01:24
+
+title: "Circular expected value"
+chapter: "General Theorems"
+section: "Probability theory"
+topic: "Expected value"
+definition: "Circular expected value"
+
+sources:
+  - authors: "Mardia KV, Jupp PE"
+    year: 2000
+    title: "Moments and Measures of Location and Dispersion"
+    in: "Directional Statistics"
+    pages: "ch. 3.4, pp. 28-31"
+    url: "https://onlinelibrary.wiley.com/doi/book/10.1002/9780470316979"
+    doi: "10.1002/9780470316979"
+
+def_id: "D230"
+shortcut: "mean-circ"
+username: "JoramSoch"
+---
+
+
+**Definition:** The expected value (or, mean) of a [circular](/D/rvar-circ) [random variable](/D/rvar) $X$ with domain $\mathcal{X} = [0, 2 \pi)$ is defined as the value of $\mu$ satisfying
+
+$$ \label{eq:mean-circ-eq}
+\mathrm{E}\left( e^{iX} \right) = r \cdot e^{i\mu}
+$$
+
+or, equivalently,
+
+$$ \label{eq:mean-circ}
+\mu = \mathrm{E}(X) = \mathrm{atan2}\left( \mathrm{E}(\sin X), \mathrm{E}(\sin X) \right)
+$$
+
+where $\sin$ and $\cos$ are the sine and cosine function, respectively, and $\mathrm{atan2}$ is the two-argument arctangent function.

D/rvar-circ.md

@@ -0,0 +1,36 @@
+---
+layout: definition
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "OvGU Magdeburg"
+e_mail: "joram.soch@ovgu.de"
+date: 2026-04-20 15:18:30
+
+title: "Circular random variable"
+chapter: "General Theorems"
+section: "Probability theory"
+topic: "Random variables"
+definition: "Circular random variable"
+
+sources:
+  - authors: "Wikipedia"
+    year: 2026
+    title: "Circular distribution"
+    in: "Wikipedia, the free encyclopedia"
+    pages: "retrieved on 2026-04-20"
+    url: "https://en.wikipedia.org/wiki/Circular_distribution#Examples"
+
+def_id: "D229"
+shortcut: "rvar-circ"
+username: "JoramSoch"
+---
+
+
+**Definition:** Let $X$ be a [random variable](/D/rvar) with possible outcomes $\mathcal{X}$. Then, $X$ is called a circular random variable, if $\mathcal{X} = [0, 2 \pi)$ and if the [probability density function](/D/pdf) of $X$ satisfies
+
+$$ \label{eq:pdf-circ}
+f_X(x + 2 \pi k) = f_X(x)
+\quad \text{for all} \quad
+k \in \mathbb{Z} \; .
+$$

D/vm-data.md

@@ -0,0 +1,34 @@
+---
+layout: definition
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "OvGU Magdeburg"
+e_mail: "joram.soch@ovgu.de"
+date: 2026-04-23 15:45:22
+
+title: "Univariate von Mises"
+chapter: "Statistical Models"
+section: "Periodic data"
+topic: "Univariate von Mises"
+definition: "Definition"
+
+sources:
+  - authors: "Bishop CM"
+    year: 2006
+    title: "Periodic variables"
+    in: "Pattern Recognition for Machine Learning"
+    pages: "sect. 2.3.8, p. 108"
+    url: "http://users.isr.ist.utl.pt/~wurmd/Livros/school/Bishop%20-%20Pattern%20Recognition%20And%20Machine%20Learning%20-%20Springer%20%202006.pdf"
+
+def_id: "D232"
+shortcut: "vm-data"
+username: "JoramSoch"
+---
+
+
+**Definition:** A univariate von Mises data set is given by a set of real numbers $y = \left\lbrace y_1, \ldots, y_n \right\rbrace$ with $y_i \in [0, 2 \pi), \; i = 1,\ldots,n$, [independent and identically distributed](/D/iid) according to a [von Mises distribution](/D/vm) with unknown circular mean $\mu \in [0, 2 \pi)$ and unknown reciprocal dispersion $\kappa > 0$:
+
+$$ \label{eq:vm}
+y_i \sim \mathrm{vM}(\mu, \kappa), \quad i = 1, \ldots, n
+$$

D/vm.md

@@ -0,0 +1,46 @@
+---
+layout: definition
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "OvGU Magdeburg"
+e_mail: "joram.soch@ovgu.de"
+date: 2026-04-21 15:01:45
+
+title: "von Mises distribution"
+chapter: "Probability Distributions"
+section: "Univariate continuous distributions"
+topic: "von Mises distribution"
+definition: "Definition"
+
+sources:
+  - authors: "Bishop CM"
+    year: 2006
+    title: "Probability Distributions"
+    in: "Pattern Recognition for Machine Learning"
+    pages: "Appendix B, p. 693, eq. B.77"
+    url: "http://users.isr.ist.utl.pt/~wurmd/Livros/school/Bishop%20-%20Pattern%20Recognition%20And%20Machine%20Learning%20-%20Springer%20%202006.pdf"
+
+def_id: "D231"
+shortcut: "vm"
+username: "JoramSoch"
+---
+
+
+**Definition:** Let $X$ be a [circular](/D/rvar-circ) [random variable](/D/rvar). Then, $X$ is said to follow a von Mises distribution with circular mean $\mu$ and reciprocal dispersion $\kappa$
+
+$$ \label{eq:vm}
+X \sim \mathrm{vM}(\mu, \kappa) \; ,
+$$
+
+if and only if its [probability density function](/D/pdf) is given by
+
+$$ \label{eq:vm-pdf}
+\mathrm{vM}(x; \mu, \kappa) = \frac{1}{2 \pi I_0(\kappa)} \cdot \exp \left[ \kappa \cos(x-\mu) \right]
+$$
+
+where $\mu \in \mathbb{R}$, $\kappa > 0$ and $I_0(\kappa)$ is the zeroth-order modified Bessel function of the first kind:
+
+$$ \label{eq:vm-bessel}
+I_0(\kappa) = \frac{1}{2\pi} \int_0^{2\pi} \exp \left[ \kappa \cos(x) \right] \, \mathrm{d}x \; .
+$$

I/LoC.md

@@ -27,6 +27,7 @@ Templates: **[Proof](/P/-temp-)** – *[Definition](/D/-temp-)*
 - *[Constant](/D/const)*
 - *[Discrete vs. continuous](/D/rvar-disc)*
 - *[Univariate vs. multivariate](/D/rvar-uni)*
+- *[Circular random variable](/D/rvar-circ)*
 - *[independent and identically distributed](/D/iid)*
 ### Probability
 - *[Probability](/D/prob)*
@@ -128,6 +129,8 @@ Templates: **[Proof](/P/-temp-)** – *[Definition](/D/-temp-)*
 - **[Chebyshev's inequality](/P/cheb-ineq)**
 - **[Weak law of large numbers](/P/mean-wlln)**
 - **[Expected value minimizes squared error](/P/mean-mse)**
+- *[Circular expected value](/D/mean-circ)*
+- **[Circular sample mean](/D/meancirc-samp)**
 - *[Expected value of a random vector](/D/mean-rvec)*
 - **[Expectation of a quadratic form](/P/mean-qf)**
 - **[Expectation of a bilinear form](/P/mean-blf)**
@@ -520,6 +523,11 @@ Templates: **[Proof](/P/-temp-)** – *[Definition](/D/-temp-)*
 - **[Variance](/P/exg-var)**
 - **[Skewness](/P/exg-skew)**
 - **[Method of moments](/P/exg-mome)**
+### von Mises distribution
+- *[Definition](/D/vm)*
+- **[Relationship to bivariate normal distribution](/P/vm-bvn)**
+- **[Probability density function](/P/vm-pdf)**
+- **[Full width at half maximum](/P/vm-fwhm)**
 ## Multivariate continuous distributions
 ### Multivariate normal distribution
 - *[Definition](/D/mvn)*
@@ -806,6 +814,10 @@ Templates: **[Proof](/P/-temp-)** – *[Definition](/D/-temp-)*
 ### Beta-binomial data
 - *[Definition](/D/betabin-data)*
 - **[Method of moments](/P/betabin-mome)**
+## Periodic data
+### Univariate von Mises
+- *[Definition](/D/vm-data)*
+- **[Maximum likelihood estimation](/P/vm-mle)**
 ## Categorical data
 ### Logistic regression
 - *[Definition](/D/logreg)*

P/meancirc-samp.md

@@ -0,0 +1,77 @@
+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "OvGU Magdeburg"
+e_mail: "joram.soch@ovgu.de"
+date: 2026-04-20 16:38:24
+
+title: "Circular sample mean"
+chapter: "General Theorems"
+section: "Probability theory"
+topic: "Expected value"
+theorem: "Circular sample mean"
+
+sources:
+  - authors: "Bishop CM"
+    year: 2006
+    title: "Periodic variables"
+    in: "Pattern Recognition for Machine Learning"
+    pages: "pp. 105-106, eq. 2.169"
+    url: "http://users.isr.ist.utl.pt/~wurmd/Livros/school/Bishop%20-%20Pattern%20Recognition%20And%20Machine%20Learning%20-%20Springer%20%202006.pdf"
+
+proof_id: "P533"
+shortcut: "meancirc-samp"
+username: "JoramSoch"
+---
+
+
+**Theorem:** Let $x = \left\lbrace x_1, \ldots, x_n \right\rbrace$ be a [sample](/D/samp) from a [circular](/D/rvar-circ) [random variable](/D/rvar) $X$. Then, the [sample mean](/D/mean-samp) of $x$ is given by
+
+$$ \label{eq:meancirc-samp}
+\bar{x} = \arctan \left( \frac{\sum_{i=1}^n \sin x_i}{\sum_{i=1}^n \cos x_i} \right)
+$$
+
+where $\sin$ and $\cos$ are the sine and cosine function, respectively, and $\arctan$ is the arctangent function.
+
+
+**Proof:** For observations of a [circular random variable](/D/rvar-circ) satisfying
+
+$$ \label{eq:rvar-circ-samp-x}
+0 \leq x_i < 2 \pi, \; i = 1,\ldots,n
+$$
+
+calculating the [sample mean as the arithmetic mean](/D/mean-samp) is not appropriate as this does not preserve the periodic nature of the random variable.
+
+Instead and [in accordance with the definition of the circular expected value](/D/mean-circ), observation values are transformed to points on the unit circle ($r = 1$):
+
+$$ \label{eq:rvar-circ-samp-y}
+y_i = \left[ \begin{matrix} \cos x_i \\ \sin x_i \end{matrix} \right], \; i = 1,\ldots,n \; .
+$$
+
+Then, we calculate the [sample mean](/D/mean-samp) of transformed data points
+
+$$ \label{eq:y-mean}
+  \bar{y}
+= \frac{1}{n} \sum_{i=1}^n y_i
+= \left[ \begin{matrix} \frac{1}{n} \sum_{i=1}^n \cos x_i \\ \frac{1}{n} \sum_{i=1}^n \sin x_i \end{matrix} \right]
+$$
+
+and equating this sample mean with its polar-coordinate representation
+
+$$ \label{eq:y-mean-pc}
+  \left[ \begin{matrix} \frac{1}{n} \sum_{i=1}^n \cos x_i \\ \frac{1}{n} \sum_{i=1}^n \sin x_i \end{matrix} \right]
+= \left[ \begin{matrix} \bar{r} \cos \bar{x} \\ \bar{r} \sin \bar{x} \end{matrix} \right] \; ,
+$$
+
+we obtain the solution for $\bar{x}$ by solving the equation system:
+
+$$ \label{eq:x-mean}
+\begin{split}
+\frac{\bar{r} \sin \bar{x}}{\bar{r} \cos \bar{x}} &= \frac{\frac{1}{n} \sum_{i=1}^n \sin x_i}{\frac{1}{n} \sum_{i=1}^n \cos x_i} \\
+\frac{\sin \bar{x}}{\cos \bar{x}} &= \frac{\sum_{i=1}^n \sin x_i}{\sum_{i=1}^n \cos x_i} \\
+\tan \bar{x} &= \frac{\sum_{i=1}^n \sin x_i}{\sum_{i=1}^n \cos x_i} \\
+\bar{x} &= \arctan \left( \frac{\sum_{i=1}^n \sin x_i}{\sum_{i=1}^n \cos x_i} \right) \; .
+\end{split}
+$$