Privatitation of Probability Distributions by use Wavelet Integral approach

—A naive theory of additive perturbations on a continuous probability distribution is presented. We propose a new privatization mechanism based on a naive theory of a perturbation on a probability using wavelets, such as a noise perturbs the signal of a digital image sensor. The cumulative wavelet integral function is deﬁned and builds up the perturbations with the help of this function. We show that an arbitrary distribution function additively perturbed is still a distribution function, which can be seen as a privatized distribution, with the privatization mechanism being a wavelet function. It is shown that an arbitrary cumulative distribution function added to such an additive perturbation is still a cumulative distribution function. Thus, we offer a mathematical method for choosing a suitable probability distribution to data by starting from some guessed initial distribution. The areas of artiﬁcial intelligence and machine learning are constantly in need of data ﬁtting techniques, closely related to sensors. The proposed privatization mechanism is therefore a contribution to increasing the scope of existing techniques.


I. INTRODUCTION
Given an arbitrary random variable with a continuous cumulative probability distribution (CDF) F X (x), let us consider an additive perturbation ε(x) so that However, the choice of the disturbance cannot be arbitrary because it could lead to breaking the requirements to deal only with a probability distribution. The following conditions must be hold by the perturbation, namely: In order to propose a manageable perturbation, let us deal just with compactly supported wavelets, supp ψ(x) ≡ [a, b].
Definition 1: (wavelet cumulative function) The wavelet cumulative function Ψ(x) is defined by Since only continuous compactly supported wavelets are considered, this can be simplified to ant the following properties can easily been verified: To begin with, let us deal with the distribution U[0, 1], which CDF is given by For this particular choice, the new distribution defined in Eqn 1 has the same support as the original distribution with no perturbation added. Furthermore, imposing the condition: it follows that so that we guarantee that for x ∈ [0, 1] Therefore, the condition F new (x) ≥ 0 is assured. Then, we need to see whether the F new (x) is always a non-descending function or not. So we examine the behavior of the corresponding probability density function (pdf) given by implying  It follows that f new (x) ≥ 0, and that ∞ −∞ f new (ζ)dζ = 1, thereby proving that this is indeed a valid PDF to be considered.
Let us first assume a compactly supported wavelet ψ U (x) defined within [0, 1] proposed in [16] formulated as Another compactly supported wavelets family with parameters that can be adjusted is the beta wavelet family [14]. One of the advantages of adopting beta wavelet perturbations consists of the easy replacement of parameters α and β to shape the perturbation ψ beta (x, α, β). Figure 3 shows the local perturbation generated by beta wavelets for a few selected parameters. The advantage of taking this wavelet family is the simple parametrization that drives the asymmetry of the resulting probability distribution. The parametric plot between the uniform and beta wavelet perturbed CDFs for the two cases is shown in Figure 3. This approach can be employed to introduce asymmetries in a chosen probability distribution, controlling through the beta wavelet parameter.
Among the compactly supported wavelets, certainly the most used are the wavelets of Daubechies. Expressions close to approximate Daubechies wavelets of any order have been proposed in [9]. Here these continuous approximation were used to plot the db4 perturbation adapted to a uniform distribution [0,1], using commands in Matlab TM (see Fig. 4a).

II. CHOOSING A PERTURBATION TO AN ARBITRARY PROBABILITY DISTRIBUTION
Now, we offer a valid perturbation for an arbitrary CDF F X . For a given compactly supported wavelet ψ with a wavelet cumulative function (see Definition 1), consider a new chosen distribution according to It is promptly seen that F new (−∞) = 0 and F new (∞) = 1, which is a consequence of This equation results in 0 ≤ |ε(x)| ≤ f X (x) so this is actually a valid perturbation. Clearly, since that Now, since that the added perturbation is constrained to hold the inequality 0 ≤ |ε(x)| ≤ f X (x). Thus, any wavelet of compact support can be used to induce a different perturbation in the vicinity of the probability distribution initially assigned. To sum up, given a random variable X with CDF F X (x), a perturbation is added, which guarantees that the modified function is still a distribution arround the original CDF. This new distribution could be seen as a privatized version of the reference distribution and the privatization mechanism could be called wavelet perturbation.

III. MOMENTS CORRECTION DUE TO THE PERTURBATION
The hypothesized distribution (initial or prior distribution around which the wavelet-perturbation is introduced) has its moments defined by By introducing the perturbation defined in 7, the new (adjusted/privatized) moments are given by Thus, using it follows that and the second term on the right side of the previous equation accounts for a moment correction due to the introduced wavelet-perturbation. Let us consider now the particular case of a perturbation in a (normalized) uniform distribution, X ∼ U(0, 1). In order to evaluate the moments of the new probability distribution F new (x) under the wavelet-perturbation ψ, with a compactly support [0, 1], one has that is to say the moment of the wavelet [5] used to build the additive perturbation also adds to the moment of the starting distribution, because

IV. GENERALIZING THE PERTURBATION APPROACH AT FURTHER LEVELS
In the case that a (beta)-perturbation occurs over a uniform distribution [0,1], it depends on the parameters α and β of the disturbing wavelet, so it is worth rewriting (via Equations 1-7) F new (x) = x + Ψ [0,1] (x; α, β).

approx + detail
This rich interpretation of wavelet theory (approxima-tion+detail) can be generalized into the lines of a wavelet tree.

B. Level-2 LH
(28) The following is a simple example using (α L = 4, β L = 3 and α H = 3, β H = 7) (same parameters as in Fig. 3b, except that a set, index L, for "the low [0, 1/2]" and another, index H, for "the upper [1/2, 1]". But it is worth to note that different wavelets can be selected to fit different segments of the initial distribution support, for example, in a two-level perturbation, L-level can use a beta wavelet whereas the Hlevel use a sombrero wavelet.
One interpretation for this approach is to consider a distinct perturbation in each quartile of the distribution.

V. CONCLUDING REMARKS
This paper provide a new method to build an additive wavelet-based perturbation (privacy mechanism) to modify a given continuous probability distribution. The initial guess can then be perturbed as some sort of "prospecting within the ensemble of possible probability distributions around the starting distribution". A procedure is also offered to fit four different perturbations, one in each quartile of the distribution, which can be quite attractive in data fittings.