A Proof of First Digit Law from Laplace Transform

TL;DR

This paper provides a simple proof of Benford's law using Laplace transform, showing that the first digit distribution arises from fundamental properties of the number system, thus explaining its widespread empirical occurrence.

Contribution

It offers a novel, elegant proof of the first digit law based on Laplace transform, linking it to basic mathematical properties of number systems.

Findings

The first digit law originates from the fundamental properties of number systems.

Laplace transform provides a straightforward proof of Benford's law.

The law is rooted in basic mathematical principles, not mysterious natural mechanisms.

Abstract

The first digit law, also known as Benford's law or the significant digit law, is an empirical phenomenon that the leading digit of numbers from real world sources favors small ones in a form $\log(1+{1}/{d})$, where $d=1, 2, ..., 9$. Such a law keeps elusive for over one hundred years because it was obscure whether this law is due to the logical consequence of the number system or some mysterious mechanism of the nature. We provide a simple and elegant proof of this law from the application of the Laplace transform, which is an important tool of mathematical methods in physics. We reveal that the first digit law is originated from the basic property of the number system, thus it should be attributed as a basic mathematical knowledge for wide applications.

Equations

$$P_d=\log (1+\frac{1}{d}),d=1,2,...,9,$$

$$P_d={\log }_b(1+\frac{1}{d}),d=1,2,...,b-1,\mathrm{for}b\ge 2.$$

$$P_d=\sum _{n=-\infty }^{\infty }{\int }_{d\cdot 10^n}^{(d+1)\cdot 10^n}F(x)\mathrm{d}x,$$

$$P_d={\int }_0^{\infty }F(x)g_d(x)\mathrm{d}x,$$

$$\displaystyle\eta(x)=\left\{\begin{array}[]{ll}1,&\mbox{if $x\geq 0$, }\\ 0,&\mbox{if $x<0$},\end{array}\right.$$

$$g_d(x)=\sum _{n=-\infty }^{\infty }[\eta (x-d\cdot 10^n)-\eta (x-(d+1)\cdot 10^n)].$$

$$F(x)={\int }_0^{\infty }f(t)e^{-tx}\mathrm{d}t,$$

$$G(t)={\int }_0^{\infty }g(x)e^{-tx}\mathrm{d}x.$$

$${\int }_0^{\infty }F(x)g(x)\mathrm{d}x$$

$$G_d(t)$$

$$\frac{\partial G_d(t)}{\partial d}$$

$$G_d(t)\simeq \frac{1}{t}{\log }_{10}(1+\frac{1}{d}).$$

$$P_d$$

$$1$$

$$P(2nd\mathrm{digit}\mathrm{being}d)=\sum _{k=1}^9{\log }_{10}(1+(10k+d{)}^{-1}),d=0,1,...,9.$$

$$P(D_1=d_1,...,D_k=d_k)={\log }_{10}[1+(\sum _{i=1}^k{d}_i\cdot 10^{k-i}{)}^{-1}].$$

$$g_{b,d,l,k}(x)=\sum _{n=-\infty }^{\infty }[\eta (x-d\cdot b^n)-\eta (x-(d+l)\cdot b^n)],$$

$$G_{b,d,l}(t)\simeq \frac{1}{t}{\log }_b(1+\frac{l}{d}).$$

$$P_{b,d,l,k}={\log }_b(1+\frac{l}{d}).$$

$${\Delta }_{\mathrm{total}}={\int }_0^{\infty }F(x)g_{b,d,l}(x)\mathrm{d}x-{\log }_b(1+\frac{l}{d}).$$

$${\Delta }_{b,d,l}(t)=t{G}_{b,d,l}(t)-{\log }_b(1+\frac{l}{d}),$$

$${\Delta }_{\mathrm{total}}$$

$${\Delta }_{b,d,l}(bt)={\Delta }_{b,d,l}(t).$$

$$t=e^s,$$

$${\tilde{\Delta }}_{b,d,l}(s)={\Delta }_{b,d,l}(e^s),$$

$$\tilde{f}(s)=f(e^s).$$

$${\int }_{-\infty }^{+\infty }\tilde{f}(s)\mathrm{d}s=1,$$

$${\tilde{\Delta }}_{b,d,l}(s+\ln b)={\tilde{\Delta }}_{b,d,l}(s).$$

$$0.029<\max |{\tilde{\Delta }}_{10,1,1}(s)|<\mathrm{0.03.}$$

$${\Delta }_{\mathrm{total}}={\int }_{-\infty }^{+\infty }{\tilde{\Delta }}_{b,d,l}(s)\tilde{f}(s)\mathrm{d}s,$$

$${\Delta }_{\mathrm{total}}$$

Full text

first digit law, Benford’s law, Laplace transform

The first digit law, which is also called the significant digit law or Benford’s law, was first noticed by Newcomb in 1881 n81 , and then re-discovered independently by Benford in 1938 b38 . It is an empirical observation that the first digits of natural numbers prefer small ones rather than a uniform distribution as might be expected. More accurately, the probability that a number begins with digit $d$, where $d=1,2,...,9$ respectively, can be expressed as

[TABLE]

as shown in Fig. 1.

Empirically, the populations of countries, the areas of lakes, the lengths of rivers, the arabic numbers on the front page of a newspaper b38 , physical constants bk91 , the stock market indices l96 , file sizes in a personal computer tfgs07 , etc., all conform to the peculiar law well. Benford’s law has been verified to hold true for a vast number of examples in various domains, such as economics l96 , social science lse00 , environmental science b05 , biology biology , geology geology , astronomy sm10a , statistical physics sm10b ; sm10c , nuclear physics bmp93ejp ; nr08 ; ren09 ; liu2011benford ; hui2011benford , particle physics sm09a , and some dynamical systems tbl00 ; bbh05 ; b05dcds . Also, there have been many explorations on applications of the law in various fields, mainly to detect data and judge their reasonableness, such as in distinguishing and ascertaining fraud in taxing and accounting n96 ; n99 ; rr03 and falsified data in scientific experiments d07a .

Benford’s law has several elegant properties. It is scale-invariant p61 ; bhm08 , which means that the law does not depend on any particular choice of units. This law is also base-invariant h95a ; h95b ; h95c , which means that it is independent of the base $b$ with a general form

[TABLE]

The law is also found to be power-invariant sm09a , i.e., any power ($\neq 0$) on numbers in the data set does not change the first digit distribution. Though there have been many studies on Benford’s law Benfordsite , the underlying reason for the success of this law remains elusive for more than one hundred years. It was unclear whether Benford’s Law is due to some unknown mechanism of the nature or it is merely a logical consequence of human number system.

However, the situation has been changed due to the appearance of a general derivation of Benford’s law from the application of the Laplace transform Cong19PLA , where a strict version of Benford’s law is derived as composed of a Benford term and an err term. The Benford term explains the prevalence of Benford’s law and the err term leads to derivations from the law with four categories of number sets. It is the purpose of this Letter to provide a more simple and elegant version of the derivation of Benford’s law compared to Ref. Cong19PLA . Through this derivation, it is easier to understand the rationality of Benford’s law. We reveal that the first digit law can be derived as the main term from the Laplace transform. This explains why Benford’s law is so successful for many number sets. We perform similar analysis on the regularities of the second digit and $i$th-significant digit distributions, and extend the law to a more general rule for the first several digit distribution. We also estimate the error term and point out conditions for the validity of this law.

For simplification, we constrain ourselves to the decimal system first. Let $F(x)$ be an arbitrary normalized probability density function defined on positive real number set $\mathbb{R}^{+}$. (Here we use the capital letter F instead of the lowercase one, as opposed to the convention.) Of course, in the real case, the variable $x$ may be negative or bounded, but this is not harmful to our derivation. When $x$ can be negative, we can use the probability density function of its absolute value, keeping results unchanged.

In the decimal system, the probability $P_{d}$ of finding a number whose first digit is $d$ is the sum of the probability that it is contained in the interval $[d\cdot 10^{n},(d+1)\cdot 10^{n})$ for all integer $n$, therefore $P_{d}$ can be expressed as

[TABLE]

which can also be rewritten as

[TABLE]

with $g_{d}(x)$ being a new density function whose significance will be clear in the following. (Here the lowercase letter is used, due to conventions for Laplace transform in the following sections.) Adopting the Heaviside step function,

[TABLE]

we can write $g_{d}(x)$ as

[TABLE]

Based on the above discussion, we can understand to some extent why numbers prefer smaller first digits. Naively one might think that the 9 digits in the decimal system play the same roles, but they define different density $g_{d}(x)$ as shown above, thus behave differently in the decimal system. For better illustration, we draw the images of $g_{1}(x)$ and $g_{2}(x)$ in the interval $[1,30)$, as shown in Fig. 2, from which we notice that the two density functions have totally different shapes. Neither of them can simply be a translation or an expansion of the other.

All the above derivations are rigorous. In fact, using Eq. (3) or Eq. (4), we can calculate $P_{d}$ for any given $F(x)$ numerically. Usually, it does not strictly fit in with Eq. (1). In this sense, Benford’s law is not a rigorous “law” with strong predictive power. However, by using the technique of Laplace transform, we show in the following that Benford’s law is a rather good approximation for those well-behaved probability density functions.

We now prove that if a probability density function has an inverse Laplace transform, it satisfies Benford’s law well. Recalling the complex inversion formula bca , if F(x), extended to the complex plane, satisfies:

F(x) is analytic on $\mathbb{C}$ except for a finite number of isolated singularities; 2. 2.

F(x) is analytic on the half plane $\{x|\mathrm{Re}z>0\}$; 3. 3.

There are positive constants $M$, $R$, and $\beta$ such that $|F(x)|\leq M/|x|^{\beta}$ whenever $|z|\geq R$,

F(x) has an inverse Laplace transform.

We call a probability density function “well-behaved” if it satisfies these three conditions and its inverse Laplace transform is smooth enough, i.e., without violent oscillation. Exponential functions, some fractional functions, and a handful of other common functions are all well-behaved. Thus, the derivation in the following has wide application. In what follows, we assume that $F(x)$ is well-behaved.

Let $f(t)$ be the inverse Laplace transform of $F(x)$, and $G(t)$ be the Laplace transform of $g(x)$, i.e.,

[TABLE]

[TABLE]

Laplace transforms have the following property

[TABLE]

which means that Laplace transform can act on either the function $f$ or $g$ with the above integral result keeping unchanged.

To derive the left-hand side of the above equation, we would like to calculate the right-hand side instead. Because it is comparably convenient to calculate the Laplace transform of $g_{d}(x)$,

[TABLE]

which can be treated as a function of two variables $d$ and $t$. Although $d$ is defined on the decimal digit set ${1,2,...,9}$, it can be extended to the whole real axis. Therefore, $G_{d}(t)$ is a continuous function of $d$ as well as $t$. A technique to evaluate $G_{d}(t)$ is to calculate its partial derivative with respect to $d$ approximately, and then integrate the partial derivative to derive the result

[TABLE]

There is one and the only one approximation, i.e., we adopt an integration to replace a summation. Because $G_{d}(t)\to 0$ when $d\to\infty$, Eq. (13) can be integrated to yield

[TABLE]

Then using Eq. (11), we obtain

[TABLE]

where we have used the following normalization condition of $f(t)$,

[TABLE]

Eq. (15) is exactly the first digit law for the decimal system. Thus we show that well-behaved functions satisfy Benford’s law approximately. A more rigorous derivation without the approximately equal signs in Eqs. (13), (14), (15) can be found in Ref. Cong19PLA .

Compared to Ref. Cong19PLA , the method provided above accords with our intuition better. In fact, unnecessary complicated treatments are introduced to guarantee the strictness of the proof in Ref. Cong19PLA . For example, a logarithmic scale is adopted after Laplace transform, merely to derive Eq. (12) of Ref. Cong19PLA , which corresponds to Eq. (15) in this paper. Eq. (15), though approximately holds, is set up on the original linear scale, thus manifests itself as a property of the direct Laplace transform, instead of the logarithmic Laplace transform which bears less intuitive physical meanings. In this paper, no logarithmic transform is required to derive Benford s law.

According to derivations so far, we can already explain the rationality of Benford’s law through a clear chain of logic, as follows:

The integral of the product of $F(x)$ and $g(x)$ equals the integral of the product of the inverse Laplace transform of $F(x)$ and the Laplace transform of $g(x)$, i.e., Eq. (11). 2. 2.

The Laplace transform of $g(x)$ approximately equals the Benford term divided by $t$, i.e., Eq. (14). 3. 3.

The normalization condition of $F(x)$ guarantees that the integral of the inverse Laplace transform of $F(x)$ divided by $t$ equals $1$, i.e., Eq. (16). 4. 4.

Therefore, the integral of the product of $F(x)$ and $g(x)$ approximately equals the Benford term, i.e., Eq. (15).

Such a chain of logic is not apparent in Ref. Cong19PLA .

The second significant digit law was also given by Newcomb n81 . In the decimal system, it is

[TABLE]

Hill derived a general $i$th-significant digit law h95c : letting $D_{i}$ ($D_{1},D_{2},...$) denote the $i$th-significant digit (with base 10) of a number (e.g. $D_{1}(0.0314)=3$, $D_{2}(0.0314)=1$, $D_{3}(0.0314)=4$), then for all positive integers $k$ and all $d_{j}\in{0,1,...,9}$, $j=1,2,...k$, one has

[TABLE]

We propose here a general form of digit law, and show that both the second significant digit law and the general $i$th-significant digit law are only corollaries of this general form.

We calculate $P_{b,d,l,k}$, which is the probability that the integer composed of the first $k$ digits (base $b$) of an arbitrary number [e.g. for the number $0.0314$ and $k=2$, this integer is $31$] is between $d$ and $d+l$ ($b^{k-1}\leq d<d+l<b^{k}$). Correspondingly we introduce the density function $g_{b,d,l,k}(x)$ as

[TABLE]

where the right hand side is independent of $k$ (while $k$ puts restrictions on $d$ and $l$). Thus we can omit the subscript $k$ in the following.

Similar technique gives the Laplace transform of $g_{b,d,l}(x)$

[TABLE]

Thus we arrive at the general significant digit law

[TABLE]

We find that Benford’s law (2) corresponds to a special case of this general form for $k=1$ and $l=1$, whereas Hill’s general $i$th-significant law (Eq. (18)) corresponds to the case for $b=10$, $d=\sum_{i=1}^{k}d_{i}\cdot 10^{k-i}$ and $l=1$. Newcomb’s second significant digit law can be considered as a corollary of Hill’s law according to the addition principle in probability theory.

We now calculate the error brought by our replacement of the summation to the integration in Eq. (13). Since Eq. (4) is always an accurate expression, the total error is

[TABLE]

If we define

[TABLE]

the total error can be written as

[TABLE]

Checking the definitions of the two terms of $\Delta_{b,d,l}$, we find that the variables of both of them can be multiplied by $b$ and the results keep unchanged, i.e., $\Delta_{b,d,l}$ is scale invariant. Hence

[TABLE]

For clarity, we define

[TABLE]

The corresponding normalization condition is

[TABLE]

and the property Eq. (25) becomes

[TABLE]

Clearly, $\widetilde{\Delta}_{b,d,l}$ is a function of period $\ln b$. Furthermore, according to the result for exponential distribution in Ref. el03 (Corollary 2, $\widetilde{f}(s)$ here is exactly $h_{1}(x)$ in Ref. el03 , $|\widetilde{\Delta}_{10,d,1}(s)|$ is $|h_{1}(x)-\log_{10}(1+\frac{1}{d})|$ in the equation of Corollary 2), a rather good estimation can be made when $b=10$ and $d=l=1$

[TABLE]

We notice that the total error can be expressed as

[TABLE]

where $\widetilde{f}(s)$ is dependent on $F(x)$ ultimately. In most cases, the correlation between $\widetilde{f}(s)$ and $\widetilde{\Delta}_{b,d,l}(s)$ is small, so is the total error. Therefore, Benford’s law can be a rather good approximation. However, if $\widetilde{f}(s)$ is close to a periodic function with the exact period $\ln b$, or $\widetilde{f}(s)$ changes signs very fast between positive and negative numbers (this may happen when $F(x)$ is artificially chosen, as the case of telephone numbers in a given region), the small $\widetilde{\Delta}_{b,d,l}(s)$ is counted and accumulated for many times, therefore the correlation becomes large. Similar problems also exist for some special types of probability density functions, whose inverse Laplace transforms oscillate violently between positive and negative numbers, e.g., uniform distributions or normal distributions with small variances. Number sets drawn from such distributions, e.g., heights or ages of people, though being natural, still violate Benford’s law. By arguing this, we point out that although the above derivation seems quite general, it cannot be universally true. More rigorous discussions about the err term with general applications to four types of number sets can be found in Ref. Cong19PLA .

A special case is when the integral of $\widetilde{f}(s)$ is not only convergent to 1, but also absolutely convergent to a positive real number $M$, then

[TABLE]

If $f(s)$ is a positive or negative definite function, it is absolutely integrable. Such an $F(x)$ is called the completely monotonic function in mathematics. This means that $M$ is 1, thus $\Delta_{\mathrm{total}}$ is not greater than 0.03. Consequently Benford’s law is a good estimation. For example, when $F(x)=\frac{2}{\sqrt{x}}e^{-\sqrt{x}}$, $f(t)=\frac{2}{\sqrt{\pi t}}e^{-\frac{1}{4t}}>0$, and when $F(x)=\frac{4}{(x+1)^{5}}$, $f(x)=\frac{e^{-t}}{6}>0$. We can assert that in these cases, the total errors are less than 0.03. In fact, numerical results are 0.0005 and 0.009. This verifies our estimation.

As a rule of thumb, distributions with monotonic decreasing and relatively smooth probability density functions often conform to Benford’s law well Cong19PLA . Inverse Laplace transforms of such probability density functions generally change signs only for finite times, thus being absolutely convergent. To understand this, one can view inverse Laplace transform as decomposing the original probability density function into a series of exponential functions, among which some are positive and others negative. If a monotonic decreasing probability density function is relatively smooth, i.e., without a sharp change of probability density, it can be approached mainly by positive exponential distributions, therefore its inverse Laplace transform does not oscillate between positive and negative numbers very much. As an application of this rule of thumb, for non-monotonic decreasing distributions, we can transform them into monotonic decreasing distributions, resulting in better performance of Benford’s law, e.g., for normal distributions, we can subtract the mean value from the original data set and obtain a monotonic decreasing distribution.

The above calculations and derivations tell us that the significant digit behaviors demonstrate that although our nature has no preference to any specific number, it does have discrimination to digits in numbers as a logical consequence of human’s counting system. Therefore our results justify the conventional wisdom that the violation of Benford’s law is a sign that a table of numbers is artificial or anomalous. The underlying reason for the uneven distribution of first digits is due to the basic property of digital system, but not some dynamic source behind the nature as people suspected. This also explains why we can use Benford’s law to distinguish anomalies or unnaturalness in artificial numbers.

The mathematical expressions and derivations provided in this paper are simple, elegant, and all with clear intuitive pictures. They are easily comprehensible. Therefore, this version of proof of Benford’s law can also serve as an example for the application of the Laplace transform.

The first digit law reveals an astonishing regularity in realistic numbers. We provide in this Letter a proof of this law from the Laplace transform, and point out the condition for the validity of the law. Compared to Ref. Cong19PLA , the derivation in this Letter is simple and elegant, and it directly reveals the rationality of the first digit law. From our work, the first digit law is due to the basic structure of the number system. Thus the first digit law is a general rule that applies to vast data sets in natural world as well as in human social activities. It is not strange anymore why Benford’s law is so successful in various domains. Such a law should be regarded as a basic mathematical knowledge with great potential for vast applications.