arXiv subjects

Alex Ely Kossovsky

Publications and source records attributed to Alex Ely Kossovsky.

8 recordsLinked to original sources

arXiv2020

Quantitative Partition Models and Benford's Law

Benford's Law predicts that the first significant digit on the leftmost side of numbers in real-life data is proportioned between all possible 1 to 9 digits approximately as in LOG(1 + 1/digit), so that low digits occur much more frequently than high digits in the first place. The two essential prerequisites for data configuration with regards to compliance with Benford's Law are high order of magnitude and positive skewness with a tail falling to the right of the histogram, so that quantitative configuration is such that the small is numerous and the big is rare. In this article various quantitative partition models are examined in terms of the quantitative and digital behavior of the resultant set of parts. The universal feature found across all partition models is having many small parts but only very few big parts, while Benford's Law is valid only in some particular partition cases and under certain constraints. Hence another suggested vista of Benford's Law is viewing it as a particular subset of the broader positive skewness phenomenon in quantitative partitioning. Significantly, such a vista is true in all other causes and explanations of Benford's Law where the small consistently outnumbers the big also in partial structures of the model or well before full convergence to Benford is achieved - endowing the principle universality in a sense. In conclusion, either the active act of partitioning or the passive consideration of a large quantity as the composition of smaller parts can be considered as another independent explanation for the widespread empirical observation of Benford's Law in the physical sciences.

physics.soc-ph↗

arXiv2019

Exponential Growth Series and Benford's Law

Exponential growth occurs when the growth rate of a given quantity is proportional to the quantity's current value. Surprisingly, when exponential growth data is plotted as a simple histogram disregarding the time dimension, a remarkable fit to the positively skewed k/x distribution is found, where the small is numerous and the big is rare. Such quantitative preference for the small has a corresponding digital preference known as Benford's Law which predicts that the first significant digit on the left-most side of numbers in typical real-life data is proportioned between all possible 1 to 9 digits approximately as in LOG(1 + 1/digit), so that low digits occur much more frequently than high digits in the first place. Exponential growth series with high growth rate are nearly perfectly Benford given that plenty of elements are considered. An additional constraint is that the logarithm of the growth factor must be an irrational number. Since the irrationals vastly outnumber the rationals, on the face of it, this constraint seems to constitute the explanation of why almost all growth series are Benford, yet, in reality this is all too simplistic, and the real and more complex explanation is provided in this article. Empirical examinations of close to a half a million growth series via computerized programs almost perfectly match the prediction of the theoretical study on rational versus irrational occurrences, thus in a sense confirming both, the empirical work as well as the theoretical study. In addition, a rigorous mathematical proof is provided in the continuous growth case showing that it exactly obeys Benford's Law. A non-rigorous proof is given in the discrete case via uniformity of mantissa argument. Finally cases of discrete series embedded within continuous series are studied, detailing the degree of deviation from the ideal Benford configuration.

math.ST↗

arXiv2019

Arithmetical Tugs of War and Benford's Law

Benford's Law predicts that the first significant digit on the leftmost side of numbers in real-life data is proportioned between all possible 1 to 9 digits approximately as in LOG(1 + 1/digit), so that low digits occur much more frequently than high digits in the first place. The two essential prerequisites for data configuration with regards to compliance with Benford's Law are high order of magnitude and positive skewness with a tail falling to the right of the histogram, so that quantitative configuration is such that the small is numerous and the big is rare. A related topic in the study of Benford's Law is the stark contrast between multiplications and additions of random variables and their distinct resultant quantitative and digital configurations. Random multiplication processes induce substantial increase in order of magnitude and they tend to the skewed Lognormal Distribution, favoring the small over the big. Random addition processes on the other hand do not induce any increase in order of magnitude and they tend to the symmetrical Normal Distribution as predicated by the Central Limit Theorem, favoring the medium over the small and the big. Thus, while multiplication processes are highly conducive to Benford behavior, addition processes are highly detrimental to Benford behavior. In this article it is shown that often in real-life data, multiplication and addition processes mix together within one measurement or expression, and consequently they fiercely compete for dominance, each attempting to exert the greatest influence upon sizes and digits. Such tugs of war between additions and multiplications are won or lost depending on the orders of magnitude of the generating random variables, as well as on the relative strength of the two warring sides, measured in terms of the comparative arithmetical involvement in the algebraic expression of the process.

math.ST↗

arXiv2019

Random Consolidations and Fragmentations Cycles Lead to Benford' Law

Benford's Law predicts that the first significant digit on the leftmost side of numbers in real-life data is proportioned between all possible 1 to 9 digits approximately as in LOG(1 + 1/digit), so that low digits occur much more frequently than high digits in the first place. For example, digit 1 occurs approximately 30.1% in the first place in random numbers, while digit 9 occurs only approximately 4.6%. In this article it is shown that a process where a large enough set of identical quantities constantly alternates between minuscule random consolidations (summing two randomly chosen values into a singular value) and tiny random fragmentations (division of one randomly chosen value into two new values) converges digit-wise to the Benford proportions after sufficiently many such cycles. The statistical tendency of the system after numerous cycles is to have approximately 2/3 multiplicative expressions which are conducive to Benford behavior as they tend to the Lognormal Distribution, and 1/3 additive expressions which are detrimental to Benford behavior as they tend to the Normal Distribution, hence the process represents in essence a tug of war between addition and multiplication. Since the process encounters the so-called Achilles' heel of the Central Limit Theorem, namely additions of skewed distributions with high order of magnitude, additions are not very effective, and the war is decisively won by multiplication, leading to Benford behavior. Randomness in selecting the particular quantity to be fragmented, as well as randomness in selecting the two particular quantities to be consolidated, is essential for convergence. Not surprisingly then, fragmentation itself could be performed either randomly say via a realization from the continuous Uniform on (0, 1), or deterministically via any fixed split ratio such as say 25% - 75%, and Benford's Law emerges in either case.

math.ST↗

arXiv2016

Prime Numbers, Dirichlet Density, and Benford's Law

The Prime Numbers are well-known for their paradoxical stand regarding Benford's Law. On one hand they adamantly refuse to obey the law of Benford in the usual sense, namely that of a normal density of the proportion of primes with d as the leading digit, yet on the other hand, the Dirichlet density for the subset of all primes with d as the leading digit is indeed LOG(1 + 1/d). In this article the superficiality of the Dirichlet density result is demonstrated and explained in terms of other well-known and established results in the discipline of Benford's Law, conceptually concluding that prime numbers cannot be considered Benford at all, in spite of the Dirichlet density result. In addition, a detailed examination of the digital behavior of prime numbers is outlined, showing a distinct digital development pattern, from a slight preference for low digits at the start for small primes, to a complete digital equality for large primes in the limit as the prime number sequence goes to infinity. Finally an exact analytical expression for the density of the logarithms of primes is derived and shown to be always on the rise at the macro level, an observation that is also confirmed empirically.

math.GM↗

arXiv2014

On the Relative Quantities Occurring within Physical Data Sets

A statistical measure is given expressing relative occurrences of quantities within a given data set. Application of this measure on several real life physical data sets and some abstract distributions are shown to yield consistent results. These empirical results also correspond almost exactly to the theoretical converging limit of such a measure mathematically constructed for k over x distribution defined over an infinite range.

math.ST↗

arXiv2012

Towards A Better Understanding Of The Leading Digits Phenomena

That the logarithmic distribution manifests itself in the random as well as in the deterministic (multiplication processes) has long intrigued researchers in Benford's Law. In this article it is argued that it springs from one common intrinsic feature of their density curves. On the other hand, the profound dichotomy between the random and the deterministic in the context of Benford's Law is noted here, acknowledging the need to distinguish between them. From its very inception, the field has been suffering from a profound confusion and mixing of these two very different logarithmic flavors, causing mistaken conclusions. One example is Allaart's proof of equality of sums along digital lines, which can only be applied to deterministic processes. Random data lack this equality and consistently show significantly larger sums for lower digits, thus rendering any attempt at test of summation equality irrelevant and futile in the context of forensic analysis regarding accounting and financial fraud detection. Another digital regularity is suggested here, one that is found in logarithmic as well as non-logarithmic random data sets. In addition, chains of distributions that are linked via parameter selection are found to be logarithmic, either in the limit where the number of the sequences in the chain approaches infinity, or where the distributions generating the parameters are themselves logarithmic. A new forensic data analysis method in the context of fraud detection is suggested here even for data types that do not obey Benford's Law, and in particularly regarding tax evasion applications. This can also serve as a robust forensic tool to investigate fraudulent fake data provided by the sophisticated cheater already aware of Benford's Law, a challenge that would become increasing problematic to tax authorities in the future as Benford's Law becomes almost common knowledge.

math.ST↗

arXiv2011

Scale invariance versus translation variance in Nash bargaining problem

Nash's solution in his celebrated article on the bargaining problem calling for maximization of product of marginal utilities is revisited; a different line of argument supporting such a solution is suggested by straightforward or more direct reasoning, and a conjecture is raised which purports uniqueness of algorithm, namely his solution. Other alternative inferior algorithms are also suggested. It is argued in this article that the scale invariance principle for utility functions should and could be applied here, namely that utility rescaling u'=a*u is allowed, while translations, adding a constant to utility functions u'=u+b could not be applied here, since it is not invariant and leads to contradictory behavior. Finally, special situations of ownership and utilities, where trading is predicted not to take place at all because none is profitable are examined, and then shown to be consistent with the scale invariance principle.

math.ST↗