q-Calculus Formalism for Non-extensive Particle Filter

Abstract

A class of sequential Monte Carlo estimation is frequently called particle filter. This filter belongs to the Bayesian strategy for estimation, where a non-linear and non-Gaussian assumptions can be applied. Here, the Tsallis’ distribution, from the non-extensive thermo-statistics, is used to design the best likelihood operator. Therefore, no previous likelihood operator is assumed. The new filter formulation will be named as non-extensive particle filter (NEx-PF). The distribution estimated by the NEx-PF can compute the standard form of the central limit theorem, as well as the Levy-Gnedenko central limit theorem. The q-calculus formalism is employed to generalize some definitions and properties.

Appendix: Non-extensive Tsallis’ Thermostatistics

A non-extensive form to the entropy can be expressed as [Ts88]:

$$\displaystyle \begin{aligned} S_q(p) \equiv \frac{k}{q-1} \left( 1-\sum_{i=1}^{N_p} p_i^q \right)~. {} \end{aligned} $$

(3.26)

and the q-expectation of an observable is given by

$$\displaystyle \begin{aligned} O_q \equiv \left\langle O \right\rangle_q = \sum_{i=1}^{N_p}p_i^q O_i~. {} \end{aligned} $$

(3.27)

Some properties are derived to the non-extensive entropy S _q.

• 1. If q → 1:

  $$\displaystyle \begin{aligned} S_1=k \sum_{i=1}^{N_p}p_i \ln p_i~, \end{aligned} $$

  (3.28)
  $$\displaystyle \begin{aligned} O_1=\sum_{i=1}^{N_p}p_i O_i~. \end{aligned} $$

  (3.29)

• 2. The non-extensive entropy has a positive value: S _q ≥ 0.

• 3. Non-extensivity:

  $$\displaystyle \begin{aligned} S_q(A+B)=S_q(A)+S_q(B)+(1-q)S_q(A)S_q(B) \end{aligned} $$

  (3.30)
  $$\displaystyle \begin{aligned} O_q(A+B) = O_q(A)+O_q(B) + (1-q) [ O_q(A)S_q(B)+O_qS_q(A) ]~. \end{aligned} $$

  (3.31)

• 4. Maximum for the S _q under the constraint $O_q = \sum _i p_i^q \epsilon _i$ (canonical ensemble):

  $$\displaystyle \begin{aligned} p_i=\frac{1}{Z_q}[1-\beta(1-q)\epsilon_i]^{1/(1-q)} \end{aligned} $$

  (3.32)

  where 𝜖 _i is the energy for the state-i, O _q = U _q is the non-extensive form to the internal energy and the normalization factor Z _q (partition function), for 1 < q < 3, is given by

  $$\displaystyle \begin{aligned} Z_q=\left[ \frac {\pi} {\beta(1-q)} \right]^{1/2} \frac{\varGamma [(3-q)/2(q-1)]}{\varGamma[1/(q-1)]}~. \end{aligned} $$

  (3.33)

  For q = 1, the normalization is written as

  $$\displaystyle \begin{aligned} p_i=e^{\beta \epsilon_i}/Z_1~. \end{aligned} $$

  (3.34)