Nonzero Digit Research Articles

Some new invariant sum tests and MAD tests for the assessment of Benford’s law

AbstractThe Benford law is used world-wide for detecting non-conformance or data fraud of numerical data. It says that the significand of a data set from the universe is not uniformly, but logarithmically distributed. Especially, the first non-zero digit is One with an approximate probability of 0.3. There are several tests available for testing Benford, the best known are Pearson’s $$\chi ^2$$ χ 2 -test, the Kolmogorov–Smirnov test and a modified version of the MAD-test. In the present paper we propose some tests, three of the four invariant sum tests are new and they are motivated by the sum invariance property of the Benford law. Two distance measures are investigated, Euclidean and Mahalanobis distance of the standardized sums to the orign. We use the significands corresponding to the first significant digit as well as the second significant digit, respectively. Moreover, we suggest inproved versions of the MAD-test and obtain critical values that are independent of the sample sizes. For illustration the tests are applied to specifically selected data sets where prior knowledge is available about being or not being Benford. Furthermore we discuss the role of truncation of distributions.

The first digit of the discriminant of Eisenstein polynomials as an invariant of totally ramified extensions of p-adic fields

Let K be an extension of the p-adic numbers with uniformizer π. Let φ and ψ be Eisenstein polynomials over K of degree n that generate isomorphic extensions. We show that if the cardinality of the residue class field of K divides n(n−1), then v(disc(φ)− disc(ψ))>v(disc(φ)). This makes the first (nonzero) digit of the π-adic expansion of disc(φ) an invariant of the extension generated by φ. Furthermore we find that noncyclic extensions of degree p of the field of p-adic numbers are uniquely determined by this invariant.

A new transcendental number from the digits of NN

We first give a summary of the history of transcendental numbers, and then we use a nice technique by G. Dresden to find a new transcendental number. In particular, while previous work looked at the last non-zero digit of nn , we consider the digit immediately before its last non-zero digit and show that the infinite decimal built from these digits is transcendental.

A density version of Cobham’s theorem

Cobham's theorem asserts that if a sequence is automatic with respect to two multiplicatively independent bases, then it is ultimately periodic. We prove a stronger density version of the result: if two sequences which are automatic with respect to two multiplicatively independent bases coincide on a set of density one, then they also coincide on a set of density one with a periodic sequence. We apply the result to a problem of Deshouillers and Ruzsa concerning the least nonzero digit of $n!$ in base $12$.

Usability of the Benford’s law for the results of least square estimation

Benford’s law (BL), also known as the first-digit or significant-digit law, is an intriguing pattern in data sets, considers the frequency of occurrence of the first digits, which are not uniformly distributed as might be expected, conversely follow a specified theoretical distribution. According to BL, the occurrence of first non-zero digit in a numerical data, which is generated or found in nature, depends on a logarithmic distribution. Least square estimation (LSE) method is mostly preferred for the estimation of the unknown parameters from different types of geodetic data. The residuals and the normalized residuals of the LSE method, which follow normal distribution and expected values of them are zero are used in outlier detection problem. In this study, BL is investigated for residuals and the normalized residuals estimated from LSE method. Three types of geodetic data are used: (1) simulated regression models, (2) global positioning system (GPS) data, (3) leveling network. The first group data sets are simulated based on linear regression and univariate models and each simulated group is generated for a number of 100, 1000, and 10,000 samples. To generate second group, an international global navigation satellite system (GNSS) service (IGS) station data (ISTA) is processed by kinematic PPP approach using GIPSY OASIS II v6.4 software. Here, the observation duration of GPS data is 4 days. For the last data, a leveling network with 55 points involving 110 observations of height differences is simulated. BL has been applied to the residuals (v) and normalized residuals (w) estimated from LSE method. Goodness-of-fit test has been implemented to determine whether a population has a specified BL distribution or not. This test is based on how good a fit we have between the frequency of occurrence of residuals and normalized residuals in an observed sample and the expected frequencies obtained from the hypothesized distribution. The results depending on the statistical test show that each data set (residuals and normalized residuals) used in this study follows BL.

First digit law from Laplace transform

The occurrence of digits 1 through 9 as the leftmost nonzero digit of numbers from real-world sources is distributed unevenly according to an empirical law, known as Benford's law or the first digit law. It remains obscure why a variety of data sets generated from quite different dynamics obey this particular law. We perform a study of Benford's law from the application of the Laplace transform, and find that the logarithmic Laplace spectrum of the digital indicator function can be approximately taken as a constant. This particular constant, being exactly the Benford term, explains the prevalence of Benford's law. The slight variation from the Benford term leads to deviations from Benford's law for distributions which oscillate violently in the inverse Laplace space. We prove that the whole family of completely monotonic distributions can satisfy Benford's law within a small bound. Our study suggests that the origin of Benford's law is from the way that we write numbers, thus should be taken as a basic mathematical knowledge.

Automaticity of the sequence of the last nonzero digits of n! in a fixed base

In 2011 Deshouillers and Ruzsa tried to argument that the sequence of the last nonzero digit of $n!$ in base 12 is not automatic. This statement was proved few years later by Deshoulliers. In this paper we provide alternate proof that lets us generalize the problem and give an exact characterization in which bases the sequence of the last nonzero digits of $n!$ is automatic.

Round prices and price rigidity: Evidence from outlawing odd prices

This paper exploits a legal change in Israel that banned the use of non-zero-digit price endings (e.g., 6.99) to study the relationship between digit price endings and price rigidity. We compare the propensity of product prices to change before and after the ban, while distinguishing between products whose prices ended with a zero and products whose prices did not end with a zero digit before the ban. We find that before the ban, zero-digit price endings were more likely to change, typically upward, compared with products with non-zero digit price endings. After the legal change these differences disappeared. Overall, these findings support the Price Point Theory (Blinder, 1991).

Generalized Benford's Law as a Lie Detector.

The first significant (leftmost nonzero) digit of seemingly random numbers often appears to conform to a logarithmic distribution, with more 1s than 2s, more 2s than 3s, and so forth, a phenomenon known as Benford’s law. When humans try to produce random numbers, they often fail to conform to this distribution. This feature grounds the so-called Benford analysis, aiming at detecting fabricated data. A generalized Benford’s law (GBL), extending the classical Benford’s law, has been defined recently. In two studies, we provide some empirical support for the generalized Benford analysis, broadening the classical Benford analysis. We also conclude that familiarity with the numerical domain involved as well as cognitive effort only have a mild effect on the method’s accuracy and can hardly explain the positive results provided here.

Is place-value processing in four-digit numbers fully automatic? Yes, but not always.

Knowing the place-value of digits in multi-digit numbers allows us to identify, understand and distinguish between numbers with the same digits (e.g., 1492 vs. 1942). Research using the size congruency task has shown that the place-value in a string of three zeros and a non-zero digit (e.g., 0090) is processed automatically. In the present study, we explored whether place-value is also automatically activated when more complex numbers (e.g., 2795) are presented. Twenty-five participants were exposed to pairs of four-digit numbers that differed regarding the position of some digits and their physical size. Participants had to decide which of the two numbers was presented in a larger font size. In the congruent condition, the number shown in a bigger font size was numerically larger. In the incongruent condition, the number shown in a smaller font size was numerically larger. Two types of numbers were employed: numbers composed of three zeros and one non-zero digit (e.g., 0040-0400) and numbers composed of four non-zero digits (e.g., 2795-2759). Results showed larger congruency effects in more distant pairs in both type of numbers. Interestingly, this effect was considerably stronger in the strings composed of zeros. These results indicate that place-value coding is partially automatic, as it depends on the perceptual and numerical properties of the numbers to be processed.

English

English

Natural taxonomic categories of angiosperms obey Benford's law, but artificial ones do not

Benford's phenomenological law gives the expected frequencies of the first significant digit (i.e., the leftmost non-zero digit) of any given series of numbers. According to this law, the frequency of 1 is higher than that of 2; this in turn appears more often than 3, and so on decreasing until 9. Similarly, Benford's law can also be applied to the first two significant digits (i.e., from 10 to 99), and so on. We applied Benford's law to sets of taxonomic data sets consisting of the number of taxa included in taxa of higher rank. We chose the angiosperms (Magnoliophyta) as a model case, because they are very diverse, are monophyletic, and a consensus on taxonomy of orders and families has been achieved (classification APG III), and we used as sets of data the number of species, genera, families, and orders. Only the number of species per family and per order are Benford's sets, but the remaining data sets do not obey Benford. Furthermore, in the case of the analysis of the first two significant digits of species per genus, the deviation from Benford was very large, but they fit to a power law. Given that the conformity to Benford's law is fulfilled for ‘natural' taxonomic categories of angiosperms (i.e., species and family), but not for those with more artificiality (genus), we speculate, ‘the more natural, the more Benford'.

Algorithm design and theoretical analysis of a novel CMM modular exponentiation algorithm for large integers

Modular exponentiation is an important operation in public-key cryptography. The Common-Multiplicand-Multiplication (CMM) modular exponentiation is an efficient exponentiation algorithm. This paper presents a novel method for speeding up the CMM modular exponentiation algorithm based on a Modified Montgomery Modular Multiplication (M4) algorithm. The M4 algorithm uses a new multi bit scan-multi bit shift technique by employing a modified encoding algorithm. In the M4 algorithm, three operations (the zero chain multiplication, the required additions and the nonzero digit multiplication) are relaxed to a multi bit shift and one binary addition in only one clock cycle. Our computational complexity analysis shows that the average number of required multiplication steps (clock cycles) is considerably reduced in comparison with other CMM modular exponentiation algorithms.

The Experimenting Society

Non-zero correlation coefficients have non-normal distributions, affecting both means and standard deviations. Previous research suggests that z transformation may effectively correct mean bias for N's less than 30. In this study, simulations with small (20 and 30) and large (50 and 100) N's found that mean bias adjustments for larger N's are seldom needed. However, z transformations improved confidence intervals even for N = 100. The improvement was not in the estimated standard errors so much as in the asymmetrical CI's estimates based upon the z transformation. The resulting observed probabilities were generally accurate to within 1 point in the first non-zero digit. These issues are an order of magnitude less important for accuracy than design issues influencing the accuracy of the results, such as reliability, restriction of range, and N.

Analysis of the Binary Asymmetric Joint Sparse Form

We consider redundant binary joint digital expansions of integer vectors. The redundancy is used to minimize the Hamming weight,i.e., the number of non-zero digit vectors. This leads to efficient linear combination algorithms in abelian groups, which are used in elliptic curve cryptography, for instance.If the digit set is a set of contiguous integers containing zero, a special syntactical condition is known to minimize the weight. We analyse the optimal weight of all non-negative integer vectors with maximum entry less thanN. The expectation and the variance are given with a main term and a periodic fluctuation in the second-order term. Finally, we prove asymptotic normality.

Analysis of the width-[formula omitted] non-adjacent form in conjunction with hyperelliptic curve cryptography and with lattices

In this work the number of occurrences of a fixed non-zero digit in the width-w non-adjacent forms of all elements of a lattice in some region (e.g. a ball) is analysed. As bases, expanding endomorphisms with eigenvalues of the same absolute value are allowed. Applications of the main result are on numeral systems with an algebraic integer as base. Those come from efficient scalar multiplication methods (Frobenius-and-add methods) in hyperelliptic curves cryptography, and the result is needed for analysing the running time of such algorithms.The counting result itself is an asymptotic formula, where its main term coincides with the full block length analysis. In its second order term a periodic fluctuation is exhibited. The proof follows Delange’s method.

Existence and optimality of w-non-adjacent forms with an algebraic integer base

We consider digital expansions in lattices with endomorphisms acting as base. We focus on the $w$-non-adjacent form ($w$-NAF), where each block of $w$ consecutive digits contains at most one non-zero digit. We prove that for sufficiently large $w$ and an expanding endomorphism, there is a suitable digit set such that each lattice element has an expansion as a $w$-NAF. If the eigenvalues of the endomorphism are large enough and $w$ is sufficiently large, then the $w$-NAF is shown to minimise the weight among all possible expansions of the same lattice element using the same digit system.

Analysis of width- w non-adjacent forms to imaginary quadratic bases

We consider digital expansions to the base of $\tau$, where $\tau$ is an algebraic integer. For a $w \geq 2$, the set of admissible digits consists of 0 and one representative of every residue class modulo $\tau^w$ which is not divisible by $\tau$. The resulting redundancy is avoided by imposing the width $w$-NAF condition, i.e., in an expansion every block of $w$ consecutive digits contains at most one non-zero digit. Such constructs can be efficiently used in elliptic curve cryptography in conjunction with Koblitz curves. The present work deals with analysing the number of occurrences of a fixed non-zero digit. In the general setting, we study all $w$-NAFs of given length of the expansion. We give an explicit expression for the expectation and the variance of the occurrence of such a digit in all expansions. Further a central limit theorem is proved. In the case of an imaginary quadratic $\tau$ and the digit set of minimal norm representatives, the analysis is much more refined: We give an asymptotic formula for the number of occurrence of a digit in the $w$-NAFs of all elements of $\Z[\tau]$ in some region (e.g. a disc). The main term coincides with the full block length analysis, but a periodic fluctuation in the second order term is also exhibited. The proof follows Delange's method. We also show that in the case of imaginary quadratic $\tau$ and $w \geq 2$, the digit set of minimal norm representatives leads to $w$-NAFs for \emph{all} elements of $\Z[\tau]$. Additionally some properties of the fundamental domain are stated.

Optimality of the Width-w Non-adjacent Form: General Characterisation and the Case of Imaginary Quadratic Bases

Efficient scalar multiplication in Abelian groups (which is an important operation in public key cryptography) can be performed using digital expansions. Apart from rational integer bases (double-and-add algorithm), imaginary quadratic integer bases are of interest for elliptic curve cryptography, because the Frobenius endomorphism fulfils a quadratic equation. One strategy for improving the efficiency is to increase the digit set (at the prize of additional precomputations). A common choice is the width\nbd-$w$ non-adjacent form (\wNAF): each block of $w$ consecutive digits contains at most one non-zero digit. Heuristically, this ensures a low weight, i.e.\ number of non-zero digits, which translates in few costly curve operations. This paper investigates the following question: Is the \wNAF{}-expansion optimal, where optimality means minimising the weight over all possible expansions with the same digit set? The main characterisation of optimality of \wNAF{}s can be formulated in the following more general setting: We consider an Abelian group together with an endomorphism (e.g., multiplication by a base element in a ring) and a finite digit set. We show that each group element has an optimal \wNAF{}-expansion if and only if this is the case for each sum of two expansions of weight 1. This leads both to an algorithmic criterion and to generic answers for various cases. Imaginary quadratic integers of trace at least 3 (in absolute value) have optimal \wNAF{}s for $w\ge 4$. The same holds for the special case of base $(\pm 3\pm\sqrt{-3})/2$ and $w\ge 2$, which corresponds to Koblitz curves in characteristic three. In the case of $\tau=\pm1\pm i$, optimality depends on the parity of $w$. Computational results for small trace are given.

A SPT Treatment to the Realization of the Sign-LMS Based Adaptive Filters

The “sum of power of two (SPT)” is an effective format to represent filter coefficients in a digital filter which reduces the complexity of multiplications in the filtering process to just a few shift and add operations. The canonic SPT is a special sparse SPT representation that guarantees presence of at least one zero between every two non-zero SPT digits. In the case of adaptive filters, as the coefficients are updated with time continuously, conversion to such canonic SPT forms is, however, required at each time index, which is quite impractical and requires additional circuitry. Also, as the position of the non-zero SPT terms in each coefficient word changes with time, it is not possible to carry out multiplications involving the coefficients via a few fixed “shift and add” operations. This paper addresses these problems, in the context of a SPT based realization of adaptive filters belonging to the sign-LMS family. Firstly, it proposes a bit serial adder that takes as input two numbers in canonic SPT and produces an output also in canonic SPT, which is then extended to the case where one of the inputs is given in 2's complement form. This allows weight updating purely in the canonic SPT domain. It is also shown how the canonic SPT property of the input can be used to reduce the complexity of the proposed adder. For multiplication, the canonic SPT word for each coefficient is partitioned into non-overlapping digit pairs and the data word is multiplied by each pair separately. The fact that each pair can have at the most one non-zero digit is exploited further to reduce the complexity of the multiplication.