A New Wavelet-Based Privatization Mechanism for Probability Distributions

Abstract

In this paper, 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. Wavelets are employed to extract information from a wide range of types of data, including audio signals and images often related to sensors, as unstructured data. Specifically, the cumulative wavelet integral function is defined to build the perturbation on a probability 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. Thus, we offer a mathematical method for choosing a suitable probability distribution for data by starting from some guessed initial distribution. Examples of the proposed method are discussed. Computational experiments were carried out using a database-sensor and two related algorithms. Several knowledge areas can benefit from the new approach proposed in this investigation. The areas of artificial intelligence, machine learning, and deep learning constantly need techniques for data fitting, whose areas are closely related to sensors. Therefore, we believe that the proposed privatization mechanism is an important contribution to increasing the spectrum of existing techniques.

Keywords: artificial intelligence; data fitting; database-sensor; digital image sensor; machine learning; perturbation theory; signal-to-noise ratio; statistical modeling; wavelets.

Figure 1

Plots of: (a) a wavelet perturbation to be applied to the $\mathcal{U}[0,1]$ distribution; and (b) wavelet perturbation (— blue), uniform (- - red), and perturbed uniform (- · - orange) CDFs.

Figure 2

Plots of the beta wavelet perturbations: (a) ${\psi }_{\mathrm{beta}}(x,4,3)$; and (b) ${\psi }_{\mathrm{beta}}(x,3,7)$.

Figure 3

Plots of: (a) beta wavelet perturbations to be applied to the $\mathcal{U}[0,1]$ distribution; and (b) ${\psi }_{\mathrm{beta}}(x,4,3)$ perturbed uniform (⋯ blue), ${\psi }_{\mathrm{beta}}(x,3,7)$ perturbed uniform (- · - blue), and uniform (— red) CDFs.

Figure 4

Plots of: (a) a DB4 wavelet perturbation to be applied to the $\mathcal{U}[0,1]$ distribution; and (b) DB4 wavelet perturbation (— blue) and uniform (- · - red) CDFs.

Figure 5

Plots of: (a) a Mexican-hat wavelet perturbation to be applied to the $\mathcal{U}[0,1]$ distribution; and (b) Mexican-hat wavelet perturbation (— blue) and uniform (- · - red) CDFs.

Figure 6

Plots of: (a) a level-2 beta wavelet perturbation to be applied to the $\mathcal{U}[0,1]$ distribution; and (b) level-2 beta wavelet perturbation (— blue) and uniform (- · - red) CDFs.

Figure 7

Plots of: (a) PDF and CDF of the triangular distribution; and (b) wavelet perturbation (— blue), triangular (- - red), and perturbed triangular (- · - orange) CDFs.