Applications of the first digit law to measure correlations

The quasiempirical Benford law predicts that the distribution of the first significant digit of random numbers obtained from mixed probability distributions is surprisingly meaningful and reveals some universal behavior. We generalize this finding to examine the joint first-digit probability of a pair of two random numbers and show that undetectable correlations by means of the usual covariance-based measure can be identified in the statistics of the corresponding first digits. We illustrate this new measure by analyzing the correlations and anticorrelations of the positions of two interacting particles in their quantum mechanical ground state. This suggests that by using this measure, the presence or absence of correlations can be determined even if only the first digit of noisy experimental data can be measured accurately.

Figure 1

The joint first digit (JFD) probability $P(d_x,d_y;W)$ obtained from pairs of random numbers with a density $\rho (x,y;W)=N\mathrm{Exp}[-{(x-y)}^2/\left(2W^2\right)]\theta (x,0,1)\theta (y,0,1)$ as a function of the correlation $W^{-1}$ for the four pairs $(d_x=1,d_y=1),(d_x=9,d_y=9),(d_x=1,d_y=2)$, and $(d_x=1,d_y=9)$.

Figure 2

The correlation $\mathrm{\Xi }$ of the joint first digit (JFD) probability $P(d_x,d_y)$ obtained from pairs of random numbers x and $y={\left|x\right|}^{\alpha }$, where $-1<x<1$ is uniformly distributed as a function of the exponent α. Here the corresponding Pearson correlation coefficient χ is inappropriate to detect a nonlinear correlation as it vanishes for any α. For comparison, the dashed line is $\mathrm{\Xi }\left(\alpha \right)=\mathrm{Exp}(-|\alpha |/4)/2$.

Figure 3

The contour plots spatial probability density for the two particles in the ground state for repulsive (left) and attractive (right) interactions. For both densities, the six contour levels were chosen at $\rho (x,y)={10}^{-n}$ with $n=-6,-5,...-1$).

Figure 4

Three measures of correlations χ, $\mathrm{\Xi }$, and Κ between two particles in the ground state wave function of a harmonic oscillator as a function of the interparticle force strength γ. For better graphical clarity, the parameter $\mathrm{\Xi }$ was multiplied by a factor of 10. In the inset we display the parameter $\mathrm{\Theta }$ (defined in the text) in the short transition range for $-0.49<\gamma <0.5$.

Figure 5

The probabilities $P(d,\alpha )$ that the first digit of the random number y is $d=1,2$ or $d=9$. The random number $y$ is obtained from a uniformly distributed random number $x$ as $y=x^{\alpha }$. For $\alpha =1$ (uniformly distributed random numbers) all digits are equally likely, $P\left(d\right)=1/9$. For comparison, the three dashed lines are the probabilities $\mathrm{Lo}{\mathrm{g}}_{10}(1+1/d)$ according to Benford's law.