First digit law from Laplace transform
aa r X i v : . [ s t a t . O T ] A p r First digit law from Laplace transform
Mingshu Cong a,d , Congqiao Li a , Bo-Qiang Ma a,b,c, ∗ a School of Physics and State Key Laboratory of Nuclear Physics and Technology,Peking University, Beijing 100871, China b Collaborative Innovation Center of Quantum Matter, Beijing, China c Center for High Energy Physics, Peking University, Beijing 100871, China d The FinTech and Blockchain Laboratory, Department of Computer Science, The University of Hong Kong, Hong Kong, China
Abstract
The occurrence of digits 1 through 9 as the leftmost nonzero digit of numbers from real-world sources is distributedunevenly according to an empirical law, known as Benford’s law or the first digit law. It remains obscure why a varietyof data sets generated from quite different dynamics obey this particular law. We perform a study of Benford’s lawfrom the application of the Laplace transform, and find that the logarithmic Laplace spectrum of the digital indicatorfunction can be approximately taken as a constant. This particular constant, being exactly the Benford term, explainsthe prevalence of Benford’s law. The slight variation from the Benford term leads to deviations from Benford’s lawfor distributions which oscillate violently in the inverse Laplace space. We prove that the whole family of completelymonotonic distributions can satisfy Benford’s law within a small bound. Our study suggests that Benford’s law originatesfrom the way that we write numbers, thus should be taken as a basic mathematical knowledge.
Keywords: first digit law, Benford’s law, Laplace transform
1. Introduction
There is an empirical law concerning the occurrence ofthe first digits in real-world data, stating that the firstdigits of natural numbers prefer small ones rather than auniform distribution as might be expected. More accu-rately, the probability that a number begins with digit d ,where d = 1 , , . . . , P d = log (1 + 1 d ) , d = 1 , , . . . , , (1)as shown in Fig. 1. This is known as Benford’s law, whichis also called the first digit law or the significant digitlaw, first noticed by Newcomb [1] in 1881, and then re-discovered independently by Benford [2] in 1938.Empirically, the areas of lakes, the lengths of rivers, theArabic numbers on the front page of a newspaper [2], phys-ical constants [3], the stock market indices [4], file sizes ina personal computer [5], survival distributions [6], etc.,all conform to this peculiar law well. Due to the pow-erful data analyzing tools provided by computer science,Benford’s law has been verified for a vast number of ex-amples in various domains, such as economics [7, 8], socialscience [6], environmental science [9], biology [10], geol-ogy [11], astronomy [12], statistical physics [13, 14], nu-clear physics [15, 16, 17], particle physics [18], and some ∗ Corresponding author
Email address: [email protected] (Bo-Qiang Ma) Figure 1: Benford’s law of the first digit distribution, from which wesee that the probability of finding numbers with leading digit 1 ismore than 6 times larger than that with 9. dynamical systems [19, 20]. There have been also many ex-plorations on the applications of the law in various fields,e.g., in upgrading the description in precipitation regimeshift [21]. Some applications focus on detecting data andjudging their reasonableness, such as distinguishing andascertaining fraud in taxing and accounting [22, 23, 24, 25],fabrication in clinical trials [26], the authenticity of thepollutant concentrations in ambient air [9], electoral cheatsor voting anomalies [5, 27], and falsified data in scientificexperiments [28]. Moreover, the first digit law is applied incomputer science for speeding up calculation [29], minimiz-ing expected storage space [30, 31], analyzing the behaviorof floating-point arithmetic algorithms [31], and also for
Preprint submitted to PLA, and published as: M. Cong, C. Li, B.-Q. Ma, Phys. Lett. A 383 (2019) 1836-1844 May 2, 2019 arious studies in the image domain [32, 33].Theoretically, several elegant properties of Benford’s lawhave been revealed. In mathematics, Benford’s law is theonly digit law that is scale-invariant [34, 35], which meansthat the law does not depend on any particular choice ofunits. This law is also base-invariant [36, 37, 38], whichmeans that it is independent of the base b . In the octalsystem ( b = 8), the hexadecimal system ( b = 16), or otherbase systems, the data, if fit the law in the decimal system( b = 10), all fit the general Benford’s law P d = log b (1 + 1 d ) , d = 1 , , . . . , b − . (2)The law is also found to be power-invariant [18], i.e., anypower ( = 0) on numbers in the data set does not changethe first digit distribution.There have been many studies on Benford’s law withnumerous breakthroughs. For example, Hill provided ameasure-theoretical proof that Benford’s law is equiva-lent to the scale-invariant property and that random sam-ples taken from randomly-selected distributions convergeto Benford’s law [37, 38]. Pietronero et al. explainedwhy some data sets naturally show scale-invariant prop-erties from a dynamics governed by multiplicative fluctu-ations thus conform to Benford’s law [39]. Gottwald andNicol figured out that deterministic quasiperiodic or peri-odic forced multiplicative process and even affine processesalso tend to Benford’s law [40]. Engel and Leuenbergerfocused on exponential distributions and illustrated thatthey approximately obey Benford’s law within a boundof 0.03 [41]. Smith applied digital signal processing andstudied the distributions on the logarithmic scale and theirfrequency domain, revealing that the first digit law holdsfor distributions with no components of nonzero integerfrequencies [42]. Fewster asserted that any distributionmight tend to Benford’s law if it can span several ordersof magnitude and be reasonably smooth [43].However, there are still various data sets that violateBenford’s law, e.g., the telephone numbers, birthday data,and accounts with a fixed minimum or maximum. Ben-ford’s law still remains obscure whether this law is merelya result of our way of writing numbers. If the answer isyes, why not all number sets obey this law; if the answeris no, why is this law so common that it can be a goodapproximation for most data sets. The situation can alsobe reflected by some puzzles about Benford’s law in theliterature, e.g., it is stated by Tao that no one can reallyprove or derive this law because Benford’s law, being anempirically observed phenomenon rather than an abstractmathematical fact, cannot be “proved” the same way amathematical theorem can be proved [44]. Aldous andPhan also suggested that without checking the assump-tions of Benford’s law for the data sets we studied, thislogically correct mathematical theorem is not relevant tothe real world [45].Therefore, most studies on Benford’s law are case stud-ies in literature, restricted to a specific probability density distribution or a group of them. In this work, we providea general derivation of Benford’s law with the applicationof the Laplace transform, which is an important tool ofmathematical methods in physics [46]. From our deriva-tion, we can safely assert that the deviation from Benford’slaw is always less than a small proportion of the L -normof the logarithmic inverse Laplace transform of the proba-bility density function. This bound is universal. Since the L -norm of the logarithmic inverse Laplace transform isusually small but not zero, Benford’s law is commonly wellobeyed but not strictly obeyed. We introduce a guidelineto judge how well a specific distribution obeys Benford’slaw. In this method, the degree of deviation from the lawis associated with the oscillatory behavior of the proba-bility density function in the inverse Laplace space. Wefind that the whole family of completely monotonic distri-butions can all fulfill Benford’s law within a small bound.We also carry out some numerical estimations of the er-ror term, and present several examples which verify ourmethod. We agree with Goudsmit and Furry [47] and re-veal from our own method that the appearance of the firstdigit law is a logical consequence of the digital system, butnot due to some unknown mechanics of the nature.Our work is organized as follows. In Sec. 2 we intro-duce the digital indicator functions for the given digitalsystem and put forward an intuitive explanation of Ben-ford’s law by revealing the heterogeneity of such functionsamong different first digits. In Sec. 3 we apply the Laplacetransform to the digital indicator functions to reveal theirelegant properties. From these, in Sec. 4 we provide ageneral derivation of a strict version of Benford’s law andprove that the strict Benford’s law is composed of a Ben-ford term and an error term. In Sec. 5 we study the errorterm by applying our general result to four categories ofnumber sets, which obey Benford’s law to varying degrees.Especially, we prove that completely monotonic distribu-tions can satisfy Benford’s law well. Numerical studies arealso provided to verify our method. Sec. 6 is reserved forconclusions.
2. The intuition
Let F ( x ) be an arbitrary normalized probability densityfunction (PDF) defined on the positive real number set R + (here we use the capital letter F instead of the lowercaseone, due to conventions for the Laplace transform intro-duced in Sec. 4). It does not matter if negative data areallowed, for we can instead use the PDFs of their absolutevalues.In the decimal system, the probability P d of finding anumber with first digit d is the sum of the probability thatit is within the interval [ d · n , ( d + 1) · n ) for an integer n , therefore P d can be expressed as P d = ∞ X n = −∞ Z ( d +1) · n d · n F ( x ) d x , (3)2hich can also be rewritten as P d = Z ∞ F ( x ) g d ( x ) d x , (4)where g d ( x ) is the digital indicator function (DIF), indicat-ing numbers with first digit d in the decimal system (herethe lowercase letter is used, also due to conventions of theLaplace transform). Using the notation of the Heavisidestep function, η ( x ) = (cid:26) , if x ≥ , if x < , (5)we can write g d ( x ) as g d ( x ) = ∞ X n = −∞ [ η ( x − d · n ) − η ( x − ( d + 1) · n )] . (6)Different first digits define different g d ( x ) functions, thusbehave differently in the digital system. For a better il-lustration, we draw the images of g ( x ) and g ( x ) in theinterval [1 , g ( x )can be neither a translation nor an expansion of g ( x ), andthat the gap between the shaded areas in g ( x ) is widerthan that in g ( x ). This fact intuitively explains the in-equality among the 9 digits, where smaller leading digitsare more likely to appear. x Figure 2: Images of digital indicator functions g ( x ) and g ( x ). Nei-ther of them can be a translation or an expansion of the other. Furthermore, if drawn on the logarithmic scale, g d ( x )becomes a periodic function with a mean value of log (1+ d ). This gives us the intuition why g d ( x ) has a strongconnection with Benford’s law. In the following sections,through strict mathematical derivations, we verify our in-tuition and show that the Benford term comes exactlyfrom g d ( x ).
3. The Laplace transform of the digital indicatorfunction
In this section, we study the Laplace transform of thedigital indicator function (DIF) and show that the trans-formed DIF is also a log-periodic function which frequently appears in various systems [48], and exhibits some elegantproperties that indicate Benford’s law. For general cases,we can define the DIF under base- b as g b,d,l ( x ), whosevalue is 1 for numbers within the interval [ d · b n , ( d + l ) · b n )for some integer n and 0 otherwise, i.e., g b,d,l ( x ) = ∞ X n = −∞ [ η ( x − d · b n ) − η ( x − ( d + l ) · b n )] . (7)The Laplace transform of this general DIF is defined as G b,d,l ( t ) = Z ∞ g b,d,l ( x ) e − tx d x . (8)We turn to the logarithmic scale again and further define H b,d,l ( t ) = tG b,d,l ( t ) , e H b,d,l ( s ) = H b,d,l ( e s ) . (9)The properties of e H b,d,l ( s ) are given as follows:1. e H b,d,l ( s ) is periodic with period ln b ;2. the mean value of e H b,d,l ( s ) within any single periodis log b (1 + ld ).The first property is obvious by expanding e H b,d,l ( s ) fromEqs. 8 and 9, i.e., e H b,d,l ( s ) = ∞ X n = −∞ (cid:18) exp (cid:2) − d · b ( n + s ln b ) (cid:3) − exp (cid:2) − ( d + l ) · b ( n + s ln b ) (cid:3)(cid:19) . (10)For the second property, we have the mean value of e H b,d,l ( s ) within [0 , ln b ) as (cid:10) e H b,d,l ( s ) (cid:11) = 1ln b Z ln b e H b,d,l ( s ) d s = 1ln b Z ∞−∞ (cid:18) exp[ − d · e s ] − exp[ − ( d + l ) · e s ] (cid:19) d s = 1ln b Z ∞ t (cid:16) e − dt − e − ( d + l ) t (cid:17) d t = log b (1 + ld ) . (11)With these two properties, it is straightforward torewrite e H b,d,l ( s ) as e H b,d,l ( s ) = log b (1 + ld ) + e ∆ b,d,l ( s ) , (12)where e ∆ b,d,l ( s ) represents the periodic fluctuation of e H b,d,l ( s ) around its mean value. It is noted here that e H b,d,l ( s ) is the logarithmic Laplace spectrum of the DIF.Therefore, it is independent of any particular distributionsof number sets.The first term log b (1 + ld ) is here called the Benfordterm and we will show in Sec. 4 that it is the origin of the3lassical Benford’s law, while e ∆ b,d,l ( s ) is responsible forthe possible deviation from the law. We will show in Sec.5that this deviation is small for a big family of distributions.It is worth noting that the Benford term is derivedmerely from the DIF of a certain digital system withoutassuming the exact form of the PDF. Therefore, we assertthat the origin of Benford’s law comes from the way thatthe digital system is constructed, instead of the way thatsome specific number set is formed.
4. The derivation of the general digit law
We see that the logarithmic Laplace spectrum of thedigital indicator function fluctuates around the Benfordterm. Another reason why we choose the Laplace trans-form is that the inverse Laplace transform can be served asa method to judge how well a specific PDF obeys the law,as well as to derive the general digit law. For an arbitraryPDF F ( x ), we can assume that it has an inverse Laplacetransform f ( t ) which belongs to L ( R + ), satisfying F ( x ) = Z ∞ f ( t ) e − tx d t . (13)The probability that a number drawn from a data setwith a PDF F ( x ) is within the set S ∞ n = −∞ [ d, d + l ) × b n can be expressed as P b,d,l = Z ∞ F ( x ) g b,d,l ( x ) d x . (14)We turn to the logarithmic scale again and define e f ( s ) = f ( e s ) , (15)then e f ( s ) also satisfies the normalization condition, i.e., Z ∞−∞ e f ( s ) d s = Z ∞ f ( t ) t d t = Z ∞ F ( x ) d x = 1 . (16)According to the property of the Laplace transform, Eq. 14can be rewritten in the inverse Laplace space of the PDFas Z ∞ F ( x ) g b,d,l ( x ) d x = Z ∞ f ( t ) G b,d,l ( t ) d t = Z ∞−∞ e f ( s ) e H b,d,l ( s ) d s . (17)Combining the expression of e H b,d,l ( s ) in Eq. 12 and thenormalization condition of e f ( s ) in Eq. 16, we derive thestrict form of Benford’s law, which is composed of a Ben-ford term and an error term, as follows, P b,d,l = log b (1 + ld ) + Z ∞−∞ e f ( s ) e ∆ b,d,l ( s ) d s . (18)Since e ∆ b,d,l ( s ) is slightly fluctuating, the error term issmall for most circumstances, as we intend to illustrate further in Sec. 5. If we ignore the error term in Eq. 18, thestrict Benford’s law turns into the general digit law, i.e., P b,d,l ≈ log b (1 + ld ) . (19)Lots of variations of the classical Benford’s law can beseen as corollaries of the general digit law. For example,the base b can be set to 100 to derive the second significantdigit law given by Newcomb [1]. A number ( x ) in thedecimal system can be equally treated as a number ( x ) in the base-100 system, so that the second digit of ( x ) being d is equivalent to that either the first “digit” of ( x ) belongs to the set S d = {
10 + d,
20 + d, · · · ,
90 + d } , orthat the first “digit” of (10 x ) belongs to the same set S d . Therefore, we have P (cid:0) x ) = d (cid:1) = P (cid:0) x ) ∈ S d (cid:1) + P (cid:0) x ) ∈ S d (cid:1) ≈ X k =1 log (cid:18) k + d (cid:19) + X k =1 log (cid:18) k + d (cid:19) = X k =1 log (cid:18) k + d (cid:19) . (20)Similar reasoning can also be applied to the i th-significant digit law of Hill [38]: letting D i ( D , D , ... )denotes the i th-significant digit (with base 10) of a number(e.g., D (0 . D (0 . D (0 . k and all d j ∈ , , · · · , j = 1 , , · · · , k , one has P ( D = d , · · · , D k = d k ) ≈ log k X i =1 d i · k − i ! − . (21)
5. The error term
In this section, we introduce a method to judge how wella certain PDF obeys the classical Benford’s law by analyz-ing the total error term in Eq. 18, which is the interrelationof e ∆ b,d,l ( s ) and e f ( s ), i.e.,∆ total ,b,d,l = Z ∞−∞ e f ( s ) e ∆ b,d,l ( s ) d s . (22)We know in Sec. 4 that e ∆ b,d,l ( s ) is a ln b -periodic functionwith a mean value of 0. For instance, a graph of e ∆ , , ( s )is shown in Fig. 3. The amplitude of this periodic functionis small compared with the Benford term, e.g., the ampli-tude of e ∆ , , ( s ) is less than 0.03 while the Benford termis 0.30. Therefore, intuitively speaking, if e f ( s ) is smoothenough and changes slowly, its interrelation with e ∆ b,d,l ( s )tends to be averaged out; thus the total error tends to be4 - - - - s (cid:1) ˜ , , Figure 3: The image of the ln 10-periodic function e ∆ , , ( s ) for thefirst digit 1 in the decimal system. small. On the other hand, if e f ( s ) oscillates violently, theinterrelation is highly sensitive to the exact form of e f ( s ),and the total error is likely to be large.Rigorously, we can classify real-world number sets intothe following four categories, with different degrees of de-viation from the classical Benford’s law.1. The error term equals to the constant 0 for scale-invariant distributions.2. Since the error term is bounded by a proportion ofthe L -norm of e f ( s ) (noted by k e f k ), i.e.,∆ total ,b,d,l ∈ h k e f k min { e ∆ b,d,l } , k e f k max { e ∆ b,d,l } i , (23)when e f ( s ) oscillates mildly between positive and neg-ative values, k e f k is close to 1, thus the error term issmall.3. Specifically, if e f ( s ) ≥ ∀ s ∈ R , k e f k reachesits minimum value 1, so the bound is the tightest, i.e.,∆ total ,b,d,l ∈ h min { e ∆ b,d,l } , max { e ∆ b,d,l } i . (24)Such distributions are called completely monotonicdistributions.4. When e f ( s ) oscillates dramatically, its L -norm be-comes large, so the error term becomes uncertain andthe classical Benford’s law is generally violated.We prove and explain the above assertions one by one inthe following sections. Some numerical examples are pro-vided for better illustration. We first turn to scale-invariant distributions which areinitially discussed by Hill on the prospect of the probabil-ity measure theory [37]. Pietronero et al. [39] has shownthat scale invariant distributions arise naturally from anymultiplicative stochastic process such as the dynamics ofstock prices. With Laplace transform, we can also showthat scale invariance leads to Benford’s law.Following the definition of Hill [37], a scale-invariantprobability measure P is a measure defined on the fol- lowing σ -algebra M = (cid:26) S (cid:12)(cid:12)(cid:12)(cid:12) S = ∞ [ n = −∞ B × b n (cid:27) , for some Borel B ⊆ [1 , b ) , (25)satisfying P ( S ) = P ( λS ) for all λ > S ∈ M . Hillproved that such a scale-invariant measure strictly satisfiesBenford’s law. We can easily prove this result again withthe language of PDF, if we set S to be S ∞ n = −∞ [ d, d + l ) × b n and notice that the scale-invariance property implies thefollowing statement, i.e, for all ǫ ∈ R , Z ∞−∞ e f ( s ) e ∆ b,d,l ( s ) d s = Z ∞−∞ e f ( s ) e ∆ b,d,l ( s + ǫ ) d s = C (26)holds, where C is independent of ǫ . According to the pe-riodic and zero-mean properties of e ∆ b,d,l ( s ), we have Z ln b (cid:18) Z ∞−∞ e f ( s ) e ∆ b,d,l ( s + ǫ ) d s (cid:19) d ǫ = Z ∞−∞ e f ( s ) Z ln b e ∆ b,d,l ( s + ǫ ) d ǫ ! d s = Z ∞−∞ e f ( s ) 0 · d s = 0 . (27)Thus, we get R ln b C d s = 0, i.e., C = 0. We noticethat C is also the error term in Eq. 22. Therefore, suchscale-invariant distributions conform strictly to the classi-cal Benford’s law. k e f k distribution Although scale invariance is common in nature, not allnatural data sets are scale invariant. Even for those datasets which are not scale invariant, Benford’s law can stillbe a good approximation for most cases. This is becausethe error term in Eq. 22 can be well bounded by the L -norm of e f , as is shown in Eq. 23.The proof is from the fact that k e f k is an upper boundof the integral of a function. According to Eq. 22, we have k e f k min { e ∆ b,d,l } = min { e ∆ b,d,l } Z ∞−∞ (cid:12)(cid:12) e f ( s ) (cid:12)(cid:12) d s ≤ ∆ total ,b,d,l ≤ max { e ∆ b,d,l } Z ∞−∞ (cid:12)(cid:12) e f ( s ) (cid:12)(cid:12) d s = k e f k max { e ∆ b,d,l } , (28)where k e f k = R ∞−∞ (cid:12)(cid:12) e f ( s ) (cid:12)(cid:12) d s is the L -norm of e f ( s ).In the decimal system, we numerically calculate the Ben-ford terms (noted by P B10 ,d, ) and the maximum value of (cid:12)(cid:12) e ∆ b,d,l (cid:12)(cid:12) (noted by ∆ max10 ,d, ) in Table 1. The relative errors δ max10 ,d, = ∆ max10 ,d, /P B10 ,d, are also listed. From Table 1, we5 able 1: Numerical results of the Benford term P B10 ,d, and the maximum of the absolute error term max { (cid:12)(cid:12) e ∆ b,d,l (cid:12)(cid:12) } in the decimal system ,together with the relative errors δ max10 ,d, . d P B10 ,d, / % 30.10 17.61 12.49 9.69 7.92 6.69 5.80 5.12 4.58∆ max10 ,d, / % 2.97 1.94 1.41 1.11 0.91 0.76 0.77 0.59 0.53 δ max10 ,d, / % 9.9 11.0 11.3 11.4 11.5 11.5 11.6 11.6 11.7notice that when k e f k is small, Benford’s law holds well,with a maximum relative error of 12 ∗ k e f k % for all digits.The exponential distribution is a good example of small- k e f k distributions. F ( x ) = λe − λx ( λ > . (29)Engel and Leuenberger showed that the exponential distri-bution obeys Benford’s law approximately within boundsof 0.03 (for b = 10 and d = l = 1) [41]. This fact canbe explained by Eq. 24 if we notice that the logarithmicinverse Laplace transform of the exponential distributionis e f ( s ) = δ ( s − ln λ ). Therefore, k e f k = 1, so∆ total , , , ∈ h min { e ∆ , , } , max { e ∆ , , } i ⊂ ( − . , . . (30)The log-normal distribution with a big variance is an-other example. The PDF is F ( x ) = 1 xσ √ π e − (ln x − µ )22 σ . (31)As long as σ is not too small relative to the base b , e f ( s )oscillates mildly between positive and negative values, so k e f k is also considerably small. For example, when µ = 5and σ = 1, k e f k = Z ∞−∞ (cid:12)(cid:12) e f ( s ) (cid:12)(cid:12) d s = 1 . . (32)Then, from Eq. 23 we have∆ total , , , ∈ h k e f k min { e ∆ , , } , k e f k max { e ∆ , , } i ⊂ ( − . , . , (33)which is also acceptable compared to the Benford term0.301. The family of completely monotonic (c.m.) distributionsis a special case of small- k e f k distributions. Completelymonotonic distributions are probability distributions withc.m. PDFs. This is equivalent to say that e f ( s ) is non-negative for all s ∈ R . In this case, k e f k reaches its mini-mum value 1 due to the normalization condition in Eq. 16, thus we get the tightest bound in Eq. 24. In fact, the expo-nential distributions and the scale invariant distributionsthat we have discussed above are both c.m., but the familyof c.m. functions are much more prosperous. Miller andSamko [49] surveyed a series of good properties of com-pletely monotonic functions. Herein we summarize theseproperties again for the convenience of the readers.1. A function F with domain (0 , ∞ ) is said to be c.m. ifit possesses derivatives F ( n ) ( x ) for all n = 0 , , , , ... and if ( − n F ( n ) ( x ) ≥ x > F ( x ) is c.m. if and onlyif F ( x ) is the Laplace transform of a non-negativemeasurable function f ( t ) : t → [0 , + ∞ ) and F ( x ) < + ∞ for 0 < x < + ∞ .3. The following elementary functions are all c.m. func-tions: e − ax , a ≥ , a + cx ) α , a, c, α ≥ , ln( a + cx ) , a ≥ , c > . (34)4. If F ( x ) is c.m., then e F ( x ) is also c.m.5. If F ( x ) and F ( x ) are c.m., then aF ( x )+ cF ( x ) , a ≥ , c ≥ F ( x ) and F ( x ) are c.m., then F ( x ) F ( x ) is alsoc.m.7. Let F ( x ) be c.m. and let τ ( x ) be nonnegative with ac.m. derivative, then F ( τ ( x )) is also c.m.From properties 4, 5, 6, 7, we can generate a large fam-ily of c.m. functions from elementary c.m. functions inProperty 3. For such a c.m. function F ( x ) to be a validPDF, it should also satisfy the normalization condition inEq. 16. When Z ∞ F ( x ) d x < ∞ , (35)the normalization condition can be guaranteed by intro-ducing a normalization factor.Several examples of c.m. PDFs are listed below. The pa-rameters are thus chosen so that the integral in Eq. 35 con-verges. The normalization factors are omitted in Eq. 36.Distributions generated from these PDFs all satisfy Ben-6ord’s law within a very small bound: e − a ( x + c ) α , a, c ≥ , ≤ α ≤ ,e a ( x + c ) α − , a, c > , α < , x + c ) α , c > , α > , x ν e − ax α , a ≥ , ≤ α ≤ , ≤ ν < ,e − a (ln x + c ) α , a, c ≥ , ≤ α ≤ . (36)Some non-c.m. distributions can be converted to c.m.distributions through a non-linear transformation of thedata. Literature has shown that non-linear transforma-tions on some data sets yield more robust results whenBenford’s law is used to detect fraud [50]. To explainthis, suppose x is a random variable with PDF F ( x ), ifwe transform x into y = τ ( x ), then the PDF of y becomes n ( y ) X k =1 F ( τ − k ( y )) (cid:12)(cid:12) τ ′ k ( τ − k ( y )) (cid:12)(cid:12) , (37)where n ( y ) is the number of solutions in x for the equation τ ( x ) = y and τ − k ( y ) is the k th solution.One example of this case is the normal distribution withthe PDF F ( x ) = 1 √ πσ e − ( x − µ )22 σ . (38)Through the transformation y = ( x − µ ) , the PDF be-comes F ( y ) = 1 p πσ y e − y σ , y > . (39)Eq. 39 is c.m. Therefore, the transformed data set of y fulfills Benford’s law with an error bound of 0.03, same tothat of exponential distributions. e f ( s ) distribution The only case in which Benford’s law loses its power iswhen e f ( s ) oscillates violently between positive and nega-tive values. The fast oscillation of e f ( s ) makes the smallterm of e ∆ b,d,l be counted and accumulated again andagain. Hence, k e f k becomes large, and e f ( s ) is highly sen-sitive to some tiny perturbation on F ( x ), reflecting theinstability of the inverse Laplace transform [51]. There-fore, the total error is also highly sensitive to the exactform of F ( x ) and Benford’s law is generally violated inthis case.Examples of such violently-oscillating- e f ( s ) distributionsare log-normal distributions with small σ and uniform dis-tributions. We have shown that the log-normal distribu-tion with parameters µ = ln 5 and σ = 1 . σ becomes smaller, the distribution is concentrated on somespecific first digits, as shown in Fig. 4 for σ = 1 .
0, 0.5,0.3. In the logarithmic inverse Laplace space, we numer-ically calculate e f ( s ) by the Stehfest method [52, 53, 54] Figure 4: The images of log-normal PDF F ( x ) with σ = 1 .
0, 0 . .
3, with µ = ln 5 fixed.Figure 5: The images of e ∆ , , ( s ) together with e f ( s ) for the log-normal distribution with σ = 1 .
0, 0 . .
3. Noted that e f ( s )displays stronger oscillatory behavior when σ decreases.Figure 6: P , , of the log-normal distribution compared with theBenford term P B , , under 0 < σ ≤ . and plot them together with e ∆ , , ( s ) in Fig. 5. We no-tice that e f ( s ) displays stronger oscillatory behavior as σ decreases. Thus the interrelation between e ∆ , , ( s ) and e f ( s ) is highly sensitive to the exact form of e f ( s ), or sometiny perturbation on F ( x ).In fact, we can numerically calculate P , , directlyfrom Eq. 4 for 0 < σ ≤ . σ becomessmaller, e f ( s ) oscillates stronger and P , , deviates fur-7 igure 7: The images of e ∆ , , ( s ) together with e f ( s ) for the uniformdistribution with a = 10, 20 and 30. Be noted that e f ( s ) functionsoscillate much more fiercely than those in Fig. 5.Figure 8: P , , of the uniform distribution compared with the Ben-ford term P B , , under 0 < a ≤ ther from the Benford term.For the uniform distribution on the interval [1 , a ] ( a > e f ( s ) oscillates even stronger. The PDF is given by F ( x ) = η ( x − − η ( x − a ) a − . (40)We use an analytic function to approach F ( x ), and calcu-late the numerical values of the inverse Laplace transformfor a = 10, 20 and 30, as shown in Fig. 7. Since e f ( s ) isextremely unstable for such distributions, we can expectthe total error is unstable as well, generally large.Also for verification, we draw the numerical values of P , , under 1 < a ≤
50 in Fig. 8. P , , in this case de-pends greatly on the endpoints of the PDFs, occasionallycoincides with the Benford term but generally violates theclassical Benford’s law. This is again expected.At last, we want to rectify two typical misunderstand-ings about Benford’s law, i.e.,1. random data sets without human manipulation aresupposed to fulfill Benford’s law;2. smooth distributions that span many orders of mag-nitude should satisfy Benford’s law. Unfortunately, however, neither of these two assertions arecorrect.As we have shown, the first statement is only approx-imately true for random variables generated from PDFswith small k e f k . Even natural random data sets could vi-olate Benford’s law if their PDFs oscillate violently in theinverse Laplace space. For example, lots of natural num-ber sets are distributed normally or lognormally near theirmean values with very small variances, such as heights ofall trees on the earth. Such data sets, although natural,do not obey Benford’s law.As for the second statement, whether or not a distri-bution satisfies Benford’s law is determined by the shape,instead of the scale, of the PDF. Therefore, even an ex-tremely flat PDF which spans many orders of magnitudemay still violate Benford’s law. To understand this, wenote that one can change the scale of any PDF F ( x ) bymultiplying the original data with an arbitrary number a .The PDF of the new data set is F ( x ) = 1 a F ( xa ) . (41)When a turns bigger, F ( x ) is flattened out, and it canspan as many orders of magnitude as we desire. However,the logarithmic inverse Laplace transform of F ( x ) and F ( x ) differ only by a horizontal shift, i.e., e f ( s ) = e f ( s + ln( a )) . (42)Such a horizontal shift does not change the unstable natureof the error term in Eq. 22.If we desire to reduce the error term, we need to flattenout e f ( s ) directly, e.g., into e f ( s ) = α e f ( sα ). When α becomes bigger, the total error of Benford’s law in Eq. 22,as the interrelation between a periodic function and anextremely flat e f ( s ), tends to vanish. In this case, theshape of the PDF F ( x ) has been changed. In fact, when α is big enough and f (1) = 0 (this can be guaranteedup to a scaling factor in Eq. 41), F ( x ) approaches to thescale invariant distribution, i.e., F ( x ) F ( cx ) = R ∞ α f ( t α ) e − tx d t R ∞ α f ( t α ) e − ctx d t = R ∞ f ( t α ) e − tx d t c R ∞ f (( tc ) α ) e − tx d t α → + ∞ −−−−−−→ pointwise c, for ∀ c > . (43)Such an operation brings Benford’s law back to poweragain because the shape of the original PDF, not only thescale, has been changed.
6. Summary
The first digit law has revealed an astonishing regularityof natural number sets. We introduce a method of the8aplace transform to study the law in depth. Our methodcan explain the long-standing puzzle about Benford’s law,i.e., whether or not Benford’s law is merely a result of theway of writing numbers. Our answer is yes in the sensethat the Benford term can be derived independently of anyspecific probability distributions. This does not conflictwith the fact that when the L -norm of the logarithmicinverse Laplace transform of the PDF is large, Benford’slaw is always violated.Besides, the method sets a bound on the error term,allowing us to predict the validity of Benford’s law bythe logarithmic inverse Laplace transform of an arbitraryPDF. Real-world distributions can be categorized into fourtypes, corresponding to their oscillatory behavior in theinverse Laplace space. A milder oscillation guaranteeshigher conformity to the law, and vice versa. Especially,the whole family of completely monotonic distributions allobey Benford’s law within a small bound. Numerical ex-amples are shown to verify our method. It is not strangeanymore why Benford’s law is so successful in various do-mains of human knowledge. Such a law should receiveattention as a basic mathematical knowledge, with greatpotential for vast application. Acknowledgments
This work is supported by National Natural ScienceFoundation of China (Grant No. 11475006). It is alsosupported by National Innovation Training Program forUndergraduates.
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