Statistics

Dispersion precedes location

Several measures of central tendency can be characterized as solving a variational problem, in the sense of the calculus of variations. However, this center may not be unique.

The dispersion of a vector \(\mathbf{x} = (x_1, \dots, x_n)\) abount a point \(\mathbf{c} = (c,c,\dots, c)\) is the distance in the sense of the (normalized) p-norm between them: \[ \|\mathbf{x} - \mathbf{c}\|_p := \left(\frac{1}{n}\sum_{i=1}^n |x_i - c |^p\right)^{1/p}. \]

• A value that is in the middle.

\[ \operatorname*{arg\,min}_{y\in \mathbb{R}^n}\sum_{i=1}^m\| x_i - y\|_2 \]

• It is equal to the arithmetic-harmonic mean defined by the limit of the sequnces \(a_i\) and \(g_i\):
  • \[ a_0 = x, h_0 = y\\ a_{n+1} = \mathrm{AM}(a_n, h_n), h_{n+1} = \mathrm{HM}(a_n,h_n) \]

• For a continuous function \(u\colon U \to \mathbb{F}\), with \(U\) being the open subset of the Euclidean space \(\mathbb{R}^n\) and \(\mathbb{F}\) being either the real or complex number,
• The spherical mean over the sphere of radius \(r\) centered at \(x\) is defined by:
  • \[ \frac{1}{\omega_{n-1}(r)}\int_{\partial B(x,r)} u(y)\mathrm{d}^{\wedge n-1}y \]
    • where \(B(x,r)\subset U\), \(\mathrm{d}^{n-1}y\) is the spherical measure, and \(\omega_{n-1}(r)\) is the size of the hypersurface of \((n-1)\)-sphere.
• The spherical mean is often denoted as:
  • \[ \raisebox{.4em}{\underline{\smash{\raisebox{-.4em}{\displaystyle\int}}}}\raisebox{-1em}{\scriptstyle\partial B(x,r)} u(y)\,\mathrm{d}S(y) \]
  • This notation is also used for the Cauchy principal value sometimes.

• It is the limit of the sequnces \(a_i\) and \(g_i\):
  • \[ a_0 = x, g_0 = y\\ a_{n+1} = \mathrm{AM}(a_n, g_n), g_{n+1} = \mathrm{GM}(a_n,g_n) \]

• \[ S_{k-1}S_{k+1} \le S_k^2 \]
• with equality if and only if all the numbers \(a_i\) are equal.

• \[ S_1 \ge \sqrt{S_2} \ge \sqrt[3]{S_3} \ge \cdots \ge \sqrt[n]{S_n} \]
• with equality if and only if all the \(a_i\) are equal.
• The case \(n=2\) is already known as the inequality of arithmetic and geometric mean.

• \((1+x)^r \ge 1+rx\)
  • for every integer \(r\ge 1\) and real number \(x\ge -1\), with strict inequality if \(x\neq 0\land r\ge 2\). logseq.order-list-type:: number
  • for every integer \(r\ge 0\) and real number \(x\ge -2\). logseq.order-list-type:: number
  • for every even integer \(r\ge 0\) and real number \(x\). logseq.order-list-type:: number
  • for every real number \(r\ge 1\) and \(x\ge -1\), with strict inequality if \(x\neq 0\land r\neq 1\). logseq.order-list-type:: number
  • for every real number \(0\le r\le 1\) and \(x\ge -1\). logseq.order-list-type:: number

\[ m_{-1} = \frac{n}{\dfrac{1}{x_1}+\dfrac{1}{x_2}+\cdots +\dfrac{1}{x_n}} \]

• First-Order Homogeneity: \(\mathrm{M}(bx_1, \dots, bx_n) = b\mathrm{M}(x_1,\dots, x_n)\)
• Total Symmetry: \(\mathrm{M}(\dots, x_i,\dots,x_j,\dots) = \mathrm{M}(\dots, x_j,\dots,x_i,\dots)\)
• Monotonicity (in all variables): \(a\le b \implies \mathrm{M}(a,x_2,\dots,x_n) \le \mathrm{M}(b,x_2,\dots, x_n)\)
• Idempotence: \(\forall x, \mathrm{M}(x,x,\dots,x) = x\)

• For positive real numbers \(x_1, \dots, x_n\),
• \[ \mathrm{C}(x_1, \dots,x_n) = \frac{\frac1n(x_1^2 + \cdots + x_n^2)}{\frac1n(x_1+\cdots + x_n)} \]

• \[ \mathrm{AM}(\mathrm{HM}(a,b), \mathrm{CM}(a,b)) = \mathrm{AM}(a,b) \]

• For a complete \((M, d)\), the Fréchet variance is:
  • \[ \Psi(p) := \sum_{i=1}^N d(p,x_i)^2 \]

• Karcher Means
  • \[ m = \operatorname*{arg\,min}_{p\in M}\sum_{i=1}^Nd(p,x_i)^2 \]
• If there is a unique \(m\) that strictly minimizes \(\Psi\), then it is Fréchet mean.

• For a nonzero real number \(p\), and positive real numbers \(x_1,\dots, x_n\), the generalized mean with exponent \(p\) is:
  • \[ m_p(x_{i\in I}) = \left(\frac{\sum_{i\in I}x_i^p}{|I|}\right)^{\frac{1}{p}} \]

• \(m_1\) is arithmetic mean
• \(m_0\) is geometric mean via limit.
• \(m_{-1}\) is harmonic mean.
• \(m_{-\infty}\) and \(m_\infty\) are minimum and maximum.
• \(m_2\) is quadratic mean, or root mean square.

• If \(p

For a injective continuous function \(f\colon I\to \mathbb{R}\) with an interval \(I\), \[ M_f(\mathbf{x}) = f^{-1}\left(\frac{1}{n}\sum_{i=1}^nf(x_i)\right). \]

• RealSoftMax(LSE), Multivariable Softplus

Quasi-arithmetic mean with \( f = \exp \) that smoothly approximate the maximum function: \[ \mathrm{LSE}(x_1,\dots,x_n) := \log(\exp(x_1),\dots,\exp(x_n)) \]

• \(f\ \text{identity}\): Arithmetic mean

• If \( f \) is a convex function, then quasi-arithmetic mean satisfies the Jensen's inequality.

• \[ H = \frac{1}{3}(a + \sqrt{ab} + b) \]

• The volume of a frustum is the product of the height and the Heronian mean of areas of the opposing parallel faces.

• A function of \(n\) variables give rises to a Chisini mean \(M\), if for every vector \((x_1,\dots, x_n)\) there exists a unique \(M\) such that:
  • \[ f(M, M,\cdots, M) = f(x_1, x_2, \dots, x_n). \]

• \(f\ \text{summation}\): arithmetic mean
• \(f\ \text{product}\): geometric mean
• \(f\ \text{reciprocal summation}\): harmonic mean
• \(f\ \text{summation after squaring}\): quadratic mean
• \(f\ \text{summation after exponentiation}\): generalized mean
• \(f\ \text{summation after filtering with a function}\): quasi-arithmetic mean
• \(f\ \text{volume of a frustum in terms of the areas of the bases}\): Heronian mean

• \[ L_p(\mathbf{x}) = \frac{\sum_{k=1}^nx_k^p}{\sum_{k=1}^nx_k^{p-1}} \]

• \[ L_{p,w}(\mathbf{x}) = \frac{\sum_{k=1}^nw_kx_k^p}{\sum_{k=1}^nw_kx_k^{p-1}} \]

• \(L_0\) is the harmonic mean
• \(L_{1/2}((x_1, x_2))\) is the geometric mean
• \(L_1\) is the arithmetic mean
• \(L_2\) is the contraharmonic mean
• \(\lim_{p\to -\infty}L_p\) and \(\lim_{p\to \infty}L_p\) are the minimum and maximum

• For two non-negative number \(a, b\),
• \[ \mathrm{H}_x(a,b) = \frac{a^xb^{1-x} + a^{1-x}b^x}{2} \]
  • with \(0\le x\le 1/2\).

• It interpolates between the arithmetic (\(x=0\)) and geometric \((x=1/2)\) mean.

• For any real vector \(a = (a_1,\dots, a_n)\), the \(\mathbf{a}\)-mean \([a]\) of positive real numbers are:
  • \[ [a] := \frac{1}{n!}\sum_{\sigma\in S_n}x_{\sigma(1)}^{a_1}\cdots x_{\sigma(n)}^{a_n} \]

• \(a = (1,0,\dots, 0)\) is the arithmetic mean
• \(a = (1/n,\dots,1/n)\) is the geometric mean
• \(a = (x,1-x)\) is the Heinz mean
• \(a = (-1, 0, \dots, 0)\) is the harmonic mean

\[ [a] \le [b] \iff a=Pb \] where \(P\) is a doubly stochastic matrix. The equality holds if and only if \(a=b\) or all \(x_i\) are equal.

• \[ \lim_{(\xi,\eta)\to (x,y)} \frac{\eta - \xi}{\ln\eta - \ln\xi} \]

• \(\mathrm{GM} \le \mathrm{LM} \le \mathrm{AM}\)

• For \(0

• \[ S_p(a,b)=f'^{-1}\left(\frac{f(b)-f(a)}{b-a}\right) \]
  • where \(f(x)=x^p\). Here, \(S_p(a,b)\) is guaranteed to be in \((a,b)\) by Mean value theorem.
  • Calculate the average rate of change of \(x^p\) on the interval \((a,b)\) and invert it into the value that has the same instantaneous rate of change.
• \[ S_2(a,b)=\frac{a+b}{2}. \]
  • The property of the standard parabola
• \[ S_{-1}(a,b) = \sqrt{ab}\, . \]

• \(S_{-\infty}\) and \(S_\infty\) are the minimum and maximum
• \(S_{-1}\) is the geometric mean
• \(S_0\) is the logarithmic mean
• \(S_{1/2}\) is the power mean with exponent \(1/2\)
• \(S_1\) is the identric mean
• \(S_2\) is the arithmetic mean
• \(S_3 = \mathrm{QM}(x,y,\mathrm{GM}(x,y))\)

\[ \bar{\alpha} = \operatorname{atan2}\left(\frac{1}{n}\sum_{j=1}^n\sin\alpha_j, \frac{1}{n}\sum_{j=1}^n\cos\alpha_j\right) \]

For a sequence \((a_n)_{n=1}^\infty\), and the partial sum \(s_k\) of the first \(k\) terms, The sequence is called Cesáro summable, if the arithmetic mean of its first \(n\) partial sums tends to a finite number: \[ \lim_{n\to \infty} \frac{1}{n}\sum_{k=1}^ns_k = A < \infty \]