Beta wavelet

Continuous wavelets of compact support alpha can be built, which are related to the beta distribution. The process is derived from probability distributions using blur derivative. These new wavelets have just one cycle, so they are termed unicycle wavelets. They can be viewed as a soft variety of Haar wavelets whose shape is fine-tuned by two parameters \alpha and \beta . Closed-form expressions for beta wavelets and scale functions as well as their spectra are derived. Their importance is due to the Central Limit Theorem by Gnedenko and Kolmogorov applied for compactly supported signals.

Continuous wavelets of compact support alpha can be built, which are related to the beta distribution. The process is derived from probability distributions using blur derivative. These new wavelets have just one cycle, so they are termed unicycle wavelets. They can be viewed as a soft variety of Haar wavelets whose shape is fine-tuned by two parameters \alpha and \beta . Closed-form expressions for beta wavelets and scale functions as well as their spectra are derived. Their importance is due to the Central Limit Theorem by Gnedenko and Kolmogorov applied for compactly supported signals. P(t)=\frac{1}{B(\alpha ,\beta )}t^{\alpha -1}\cdot (1-t)^{\beta -1},\quad 1\leq \alpha ,\beta \leq +\infty . The normalising factor is B(\alpha ,\beta )=\frac{\Gamma (\alpha )\cdot \Gamma (\beta )}{\Gamma (\alpha +\beta )} , where \Gamma (\cdot ) is the generalised factorial function of Euler and B(\cdot ,\cdot ) is the Beta function. Let p_{i}(t) be a probability density of the random variable t_{i} , i=1,2,3..N i.e. p_{i}(t)\ge 0 , (\forall t) and \int_{-\infty }^{+\infty }p_{i}(t)dt=1 . Suppose that all variables are independent. The mean and the variance of a given random variable t_{i} are, respectively m_{i}=\int_{-\infty }^{+\infty }\tau \cdot p_{i}(\tau )d\tau , \sigma _{i}^{2}=\int_{-\infty }^{+\infty }(\tau -m_{i})^{2}\cdot p_{i}(\tau )d\tau . The mean and variance of t are therefore m=\sum_{i=1}^{N}m_{i} and \sigma^2 =\sum_{i=1}^{N}\sigma _{i}^{2} . The density p(t) of the random variable corresponding to the sum t=\sum_{i=1}^{N}t_{i} is given by the Central Limit Theorem for distributions of compact support (Gnedenko and Kolmogorov). Let \{p_{i}(t)\} be distributions such that Supp\{(p_{i}(t))\}=(a_{i},b_{i})(\forall i) . Let a=\sum_{i=1}^{N}a_{i} , and b=\sum_{i=1}^{N}b_{i} . Without loss of generality assume that a=0 and b=1 . The random variable t holds, as N\rightarrow \infty , p(t)\approx \begin{cases} {k \cdot t^{\alpha }(1-t)^{\beta}}, \\otherwise \end{cases} where \alpha =\frac{m(m-m^{2}-\sigma ^{2})}{\sigma ^{2}}, and \beta =\frac{(1-m)(\alpha +1)}{m}. \psi _{beta}(t|\alpha ,\beta )=(-1)\frac{dP(t|\alpha ,\beta )}{dt} has only one-cycle (a negative half-cycle and a positive half-cycle). The main features of beta wavelets of parameters \alpha and \beta are: Supp(\psi )=[ -\sqrt{\frac{\alpha}{\beta}}\sqrt{\alpha + \beta +1},\sqrt{ \frac{\beta }{\alpha }} \sqrt{\alpha +\beta +1}]=[a,b]. lengthSupp(\psi )=T(\alpha ,\beta )=(\alpha +\beta )\sqrt{\frac{\alpha +\beta +1}{\alpha \beta }}. The parameter R=b/|a| =\beta / \alpha is referred to as “cyclic balance”, and is defined as the ratio between the lengths of the causal and non-causal piece of the wavelet. The instant of transition t_{zerocross} from the first to the second half cycle is given by t_{zerocross}=\frac{(\alpha -\beta )}{(\alpha +\beta -2)}\sqrt{\frac{\alpha +\beta +1}{\alpha \beta }}. The (unimodal) scale function associated with the wavelets is given by \phi _{beta}(t|\alpha ,\beta )=\frac{1}{B(\alpha ,\beta )T^{\alpha +\beta -1}}\cdot (t-a)^{\alpha -1}\cdot (b-t)^{\beta -1}, a\leq t\leq b . A closed-form expression for first-order beta wavelets can easily be derived. Within their support, \psi_{beta}(t|\alpha ,\beta ) =\frac{-1}{B(\alpha ,\beta )T^{\alpha +\beta -1}} \cdot [\frac{\alpha -1}{t-a}-\frac{\beta -1}{b-t}] \cdot(t-a)^{\alpha -1} \cdot(b-t)^{\beta -1} \Psi _{BETA}(\omega ) =-j\omega \cdot M(\alpha ,\alpha +\beta ,-j\omega (\alpha +\beta )\sqrt{\frac{\alpha +\beta +1}{\alpha \beta}})\cdot exp\{(j\omega \sqrt{\frac{\alpha (\alpha +\beta +1)}{\beta }})\} where M(\alpha ,\alpha +\beta ,j\nu )=\frac{\Gamma (\alpha +\beta )}{\Gamma (\alpha )\cdot \Gamma (\beta )}\cdot \int_{0}^{1}e^{j\nu t}t^{\alpha -1}(1-t)^{\beta -1}dt . Only symmetrical (\alpha =\beta ) cases have zeroes in the spectrum. A few asymmetric (\alpha \neq \beta ) beta wavelets are shown in Fig. Inquisitively, they are parameter-symmetrical in the sense that they hold |\Psi _{BETA}(\omega |\alpha ,\beta )|=|\Psi _{BETA}(\omega |\beta ,\alpha )|. Higher derivatives may also generate further beta wavelets. Higher order beta wavelets are defined by \psi _{beta}(t|\alpha ,\beta )=(-1)^{N}\frac{d^{N}P(t|\alpha ,\beta )}{dt^{N}}. This is henceforth referred to as an N -order beta wavelet. They exist for order N\leq Min(\alpha ,\beta )-1 . After some algebraic handling, their closed-form expression can be found: \Psi _{beta}(t|\alpha ,\beta ) =\frac{(-1)^{N}}{B(\alpha ,\beta ) \cdot T^{\alpha +\beta -1}} \sum_{n=0}^{N}sgn(2n-N)\cdot \frac{\Gamma (\alpha )}{\Gamma (\alpha -(N-n))}(t-a)^{\alpha -1-(N-n)} \cdot \frac{\Gamma (\beta )}{\Gamma (\beta -n)}(b-t)^{\beta -1-n}. An N^{th} -order beta wavelet has N vanishing moments, and is therefore insensitive to polynomial trends of degree up to N-1 . ==Further reading== W.B. Davenport, Probability and Random Processes, McGraw-Hill, Kogakusha, Tokyo, 1970. ==External links== https://jcis.sbrt.org.br/jcis/issue/view/27 http://www.de.ufpe.br/~hmo/WEBLET.html http://www.de.ufpe.br/~hmo/beta.html

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