Statistical Distribution Laws for Metallic Mineral Deposit Sizes

Abstract

Given that expanding industrial societies rely on the continuing supply of metallic raw materials, it is of interest to be able to generate estimates of the likely magnitude of undiscovered metal resources. Statistical predictions of this kind require that one knows the frequency distribution of the metal content (or tonnage) of mineral deposits. The distribution of metal content among deposits of different sizes may be treated empirically (i.e., without explicit formulation of a density function), or formally, starting from a density function that is known to represent the distribution of metal in known deposits. The empirical approach has been applied to the prediction of undiscovered resources of some metals, most notably copper. The focus here is on the formal approach, which may be simpler and more general than the empirical one. It has long been suspected that the distribution of metal tonnages is heavy-tailed, in the sense that a relatively few very large deposits are likely to contain much of the metal endowment. There is, however, uncertainty about the precise nature of the distribution of metal tonnages. Lognormal and power law models have been proposed. Under some conditions, which are examined here, the choice of one or the other of these two models may result in widely diverging estimates of the likely magnitude and distribution of undiscovered resources. However, a detailed statistical analysis of the available deposit tonnage data for 20 metals reveals that it is not possible to decide between the two models. Using conservative significance levels, both models fit the data equally well. Two different procedures are used to attempt to decide whether either of the two models is more plausible: likelihood ratio tests and power law fits to lognormally distributed synthetic data sets that mimic the natural data sets. Neither of the two procedures can detect any preference for one or the other of the two models, for any of the metals discussed here. This uncertainty may arise from the relatively small sizes of the samples available, which are limited to known and reasonably well-characterized metallic mineral deposits. There is at present no statistical justification to choose between a lognormal and a power law description of metal distribution in mineral deposits. This uncertainty should be kept in mind when preparing, and interpreting, estimates of the magnitude of undiscovered mineral resources.

Appendices

Appendix 1: Metal Endowment for Power-Law Distributed Deposit Sizes

If deposit sizes, z, follow a probability density function $p\left( z \right)$, then metal endowment is distributed according to $m\left( z \right)$:

$$ m\left( z \right) = \frac{z p\left( z \right)}{{z_{a} }} ,$$

(16)

where $z_{a}$ is the arithmetic mean of the deposit sizes (see Patiño Douce 2016b, and note that uppercase and lowercase conventions for probability density functions and cumulative distribution functions are here opposite to those in the previous paper). If z is power-law distributed, then, using Eq. (7) we find

$$ z_{a} = \mathop \int\limits_{{z_{m} }}^{\infty } z p\left( z \right) dz = \alpha \left( {z_{m} } \right)^{\alpha } \mathop \int\limits_{{z_{m} }}^{\infty } z^{ - \alpha } dz = \frac{\alpha }{1 - \alpha } \left( {z_{m} } \right)^{\alpha } \left[ {\left( {z^{1 - \alpha } } \right)_{z \to \infty } - \left( {z^{1 - \alpha } } \right)_{{z = z_{m} }} } \right] .$$

(17)

If $\alpha\, > \, 1$ then the arithmetic mean exists and is equal to:

$$ z_{a} = \frac{\alpha }{\alpha - 1}z_{m} ,$$

(18)

where $z_{m}$ is the lower bound of validity of the power law (see main text). However, if $0 < \alpha \le 1$ then the arithmetic mean does not exist.

Consider now the fraction of the total metal endowment contained in deposits within the size interval $\left( {z_{1} , kz_{1} } \right)$, with k $>$ 1. This fraction is given by (see also Patiño Douce 2016b)

$$ M \left[ {z\left| {\left( {z_{1} , kz_{1} } \right)} \right.} \right] = \frac{1}{{z_{a} }}\mathop \int\limits_{{z_{1} }}^{{kz_{1} }} z p\left( z \right) dz .$$

(19)

If $p\left( z \right)$ is a power law (Eq. (7)) and $\alpha \le 1$, then $M \left[ {z\left| {\left( {z_{1} , kz_{1} } \right)} \right.} \right] = 0$ for any finite k. In other words, no matter how large a value of k we consider, the endowment contained in deposits within the size interval $\left( {z_{1} , kz_{1} } \right)$ is always an infinitesimally small fraction of the total endowment. This situation, in a finite planet, obviously lacks physical meaning, but it can be thought of as an extreme case of the same type of behavior that arises from a lognormal distribution of deposit sizes, in the sense that metal endowment tends to be concentrated in a few very large deposits. But this is not the only possible behavior that arises from a power law. Consider the case $\alpha > 1$, and suppose that we wish to find the deposit size $z_{q}$ such that a fraction q of the metal endowment, $\left( {0 < q < 1} \right)$, is contained in deposits larger than $z_{q}$. We make $z_{1} = z_{q}$ and $k \to \infty$, substitute Eq. (7) in Eq. (19), and solve for $z_{q}$:

$$ z_{q} = z_{m} q^{{\frac{1}{1 - \alpha }}} .$$

(20)

Using Eq. (9) we can also find the fraction of deposits, $C\left( {z_{q} } \right)$, that carry the fraction q of the metal endowment:

$$ C\left( {z_{q} } \right) = q^{{\frac{\alpha }{\alpha - 1}}} .$$

(21)

From these two equations, we see that for large α the endowment tends to be evenly distributed across all deposits sizes, but as α approaches 1 from above the endowment becomes concentrated in progressively larger and rarer deposits, until at the limit α = 1 we encounter the unbound behavior described above. The unbound case can also be interpreted to mean that there always exists a single deposit that contains more metal than all smaller deposits combined.

In practice, these results mean that, if metal deposit sizes are power-law distributed with an exponent close to 1, then there is a finite probability that a very few, perhaps only one, very large deposits contain an overwhelming fraction of the metal endowment. But if the exponent is significantly larger than 1 then a lognormal distribution of deposits sizes may in fact result in a stronger concentration of the endowment in large deposits than a power law. Determining whether size distribution is lognormal or power law, and the values of the distribution parameters (see below), are therefore tasks that have important practical applications.

Appendix 2: Maximum Likelihood Estimates for the Parameters of a Truncated Lognormal Distribution

It is necessary to distinguish between the parameters of a statistical distribution and the estimated values of the parameters. The parameters are labeled with Greek letters with no modifiers, e.g., $\theta$, whereas the estimated values of the parameters are labeled with the same letter followed by a superscript asterisk, e.g., $\theta^{*} .$ In this and all subsequent Appendices the random variable in a distribution is labeled $x$, rather than $z$ as in the main text. The reason for this is that the equations derived in these Appendices apply to any arbitrary random variable, whereas the variable $z$ in the main text refers to a random variable normalized to the geometric mean of the data. Thus, $x$ can be thought of as a dummy variable, and substitution with $z$ when necessary should present no problems.

Let $f\left( x \right)$ be a probability density function (PDF), and $F\left( x \right)$ the corresponding cumulative distribution function (CDF). Consider now the same PDF, but truncated on the interval $\left( {a,b} \right)$, i.e., $a \le x \le b.$ Call the truncated PDF $g\left( x \right)$. This function is given by

$$ g\left( x \right) = \frac{f\left( x \right)}{F\left( b \right) - F\left( a \right)} $$

(22)

where the denominator is simply a normalization factor that ensures that $g\left( x \right)$ integrates to unity over its domain.

Let $f\left( x \right)$ and $g\left( x \right)$ be functions of a set of $m$ parameters $\theta_{k} , k = 1 \ldots m$. Given $n$ observations of a random variable $x_{i}$, the log likelihood function of the set of observations under the distribution $g\left( x \right)$ is

$$ l\left( {\theta_{k} } \right) = \mathop \sum \limits_{i = 1}^{n} \ln \left( {g\left( {x_{i} } \right)} \right) .$$

(23)

The maximum likelihood estimators (MLE) of the parameters of a distribution are the values of the parameters that maximize the likelihood function $\mathop \prod \nolimits_{i = 1}^{n} g\left( {x_{i} } \right)$, or equivalently the log likelihood function (i.e., Eq. (23)). Thus, the MLE of the $\theta_{k}$ parameters are given by the simultaneous solution of the $m$ equations:

$$ \frac{\partial }{{\partial \theta_{k}^{*} }}l\left( {\theta_{k}^{*} } \right) = 0 $$

(24)

which, by substituting Eq. (22) in Eq. (23), can be written as

$$ \frac{1}{n} \mathop \sum \limits_{i = 1}^{n} \frac{\partial }{{\partial \theta_{k}^{*} }} \ln \left( {f\left( {x_{i} } \right)} \right) - \frac{1}{F\left( b \right) - F\left( a \right) } \frac{\partial }{{\partial \theta_{k}^{*} }}\left[ {F\left( b \right) - F\left( a \right)} \right] = 0. $$

(25)

Consider now the lognormal distribution, i.e.,

$$ f\left( x \right) = \frac{1}{{x \sigma \sqrt {2 \pi } }} e^{{ - \frac{{\left( {\ln x - \mu } \right)^{2} }}{{2\sigma^{2} }}}} $$

(26)

and

$$ F\left( x \right) = \frac{1}{2} + \frac{1}{2}{\text{erf}}\left( {\frac{\ln x - \mu }{\sigma \sqrt 2 } } \right). $$

(27)

These are functions of two parameters, $\theta_{1} = \mu$ and $\theta_{2} = \sigma$. The first term in Eq. (25) for each of these parameters is

$$ \frac{1}{n} \mathop \sum \limits_{i = 1}^{n} \frac{\partial }{{\partial \mu^{*} }} \ln \left( {f\left( {x_{i} } \right)} \right) = \frac{1}{{n \left( {\sigma^{*} } \right)^{2} }} S_{1} - \frac{\mu }{{\left( {\sigma^{*} } \right)^{2} }} $$

(28)

and

$$ \frac{1}{n} \mathop \sum \limits_{i = 1}^{n} \frac{\partial }{{\partial \sigma^{*} }} \ln \left( {f\left( {x_{i} } \right)} \right) = - \frac{1}{{\sigma^{*} }} + \frac{1}{{n \left( {\sigma^{*} } \right)^{3} }} S_{2} + \frac{{\left( {\mu^{*} } \right)^{2} }}{{\left( {\sigma^{*} } \right)^{3} }} - \frac{{2 \mu^{*} }}{{n\left( {\sigma^{*} } \right)^{3} }} S_{1}, $$

(29)

where

$$ S_{1} = \mathop \sum \limits_{i = 1}^{n} \ln x_{i} $$

(30)

and

$$ S_{2} = \mathop \sum \limits_{i = 1}^{n} \left( {\ln x_{i} } \right)^{2}. $$

(31)

Also,

$$ F\left( b \right) - F\left( a \right) = \frac{1}{2}\left[ {{\text{erf}}\left( {\frac{{\ln b - \mu^{*} }}{{\sigma^{*} \sqrt 2 }} } \right) - {\text{erf}}\left( {\frac{{\ln a - \mu^{*} }}{{\sigma^{*} \sqrt 2 }} } \right)} \right] $$

(32)

so

$$ \frac{\partial }{{\partial \mu^{*} }}\left[ {F\left( b \right) - F\left( a \right)} \right] = \frac{1}{{\sigma^{*} \sqrt {2\pi } }}\left[ {e^{{ - \frac{{\left( {\ln a - \mu^{*} } \right)^{2} }}{{2\left( {\sigma^{*} } \right)^{2} }}}} - e^{{ - \frac{{\left( {\ln b - \mu^{*} } \right)^{2} }}{{2\left( {\sigma^{*} } \right)^{2} }}}} } \right] $$

(33)

and

$$ \frac{\partial }{{\partial \sigma^{*} }}\left[ {F\left( b \right) - F\left( a \right)} \right] = \frac{1}{{\left( {\sigma^{*} } \right)^{2} \sqrt {2\pi } }}\left[ {\left( {\ln a - \mu^{*} } \right)e^{{ - \frac{{\left( {\ln a - \mu^{*} } \right)^{2} }}{{2\left( {\sigma^{*} } \right)^{2} }}}} - \left( {\ln b - \mu^{*} } \right)e^{{ - \frac{{\left( {\ln b - \mu^{*} } \right)^{2} }}{{2\left( {\sigma^{*} } \right)^{2} }}}} } \right]. $$

(34)

If we consider a lognormal distribution truncated below at $x_{\text{min}}$ but untruncated above, then $a = x_{\text{min}}$ and $b \to \infty$. Substituting these values in Eqs. (32), (33) and (34), and then substituting Eqs. (28), (32) and (33), and Eqs. (29), (32) and (34), in (25), we get two equations in the two unknown MLE of the parameters, $\mu^{*}$ and $\sigma^{*}$:

$$ \frac{{\sqrt {2\pi } }}{2}\left[ {1 - {\text{erf}}\left( {\frac{{\ln x_{\text{min}} - \mu^{*} }}{{\sigma^{*} \sqrt 2 }} } \right)} \right]\left( {S_{1} - n\mu^{*} } \right) - n\sigma^{*} e^{{ - \frac{{\left( {\ln x_{\text{min}} - \mu^{*} } \right)^{2} }}{{2\left( {\sigma^{*} } \right)^{2} }}}} = 0 $$

(35)

and

$$ \frac{{\sqrt {2\pi } }}{2}\left[ {1 - {\text{erf}}\left( {\frac{{\ln x_{\text{min}} - \mu^{*} }}{{\sigma^{*} \sqrt 2 }} } \right)} \right]\left( {n\left( {\mu^{*} } \right)^{2} - n\left( {\sigma^{*} } \right)^{2} + S_{2} - 2\mu^{*} S_{1} } \right) - n\sigma^{*} \left( {\ln x_{\text{min}} - \mu^{*} } \right)e^{{ - \frac{{\left( {\ln x_{\text{min}} - \mu^{*} } \right)^{2} }}{{2\left( {\sigma^{*} } \right)^{2} }}}} = 0. $$

(36)

Equation (35) is Eq. (4). Equation (5) is obtained after dividing Eq. (35) by Eq. (36), collecting terms and simplifying.

Appendix 3: Confidence Interval of Model Parameters by Bootstrapping

Confidence intervals for the lognormal parameters, $\mu^{*}$ and $\sigma^{*}$, and for the power law parameters α* and z ^* _m (see below), were generated by a bootstrap procedure. Say that we wish to estimate the confidence limits of the MLE for a parameter $\theta$, and, as usual, we call the MLE value of the parameter $\theta^{*}$. The true value of the parameter $\theta$ remains unknown, but we seek two values, $c_{\theta ,L}^{*}$ and $c_{\theta ,H}^{*}$, such that there is a $\left( {1 - \gamma } \right)$ probability that the value of $\theta$ is trapped inside the interval: $c_{\theta ,L}^{*} \le \theta \le c_{\theta ,H}^{*}$. For each metal a large number, n, of independent samples are drawn from the natural data set. The samples are drawn with replacement, and have the same size as the original data set. We estimate the MLE $\theta^{*}$ for each of the n samples, and call these values $\theta_{b}^{*}$, $b = 1 \ldots n$. The resulting set of n values is sorted in order of increasing value of the parameter $\theta_{b}^{*}$, and the $\left( {\gamma / 2} \right)$ and $\left( {1 - {\raise0.7ex\hbox{$\gamma $} \!\mathord{\left/ {\vphantom {\gamma 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)$ percentiles are located in the ordered sets. Let us call the values of $\theta_{b}^{*}$ at these percentiles: $\theta_{{\left( {{\raise0.7ex\hbox{$\gamma $} \!\mathord{\left/ {\vphantom {\gamma 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)}}^{*}$ and $\theta_{{\left( {1 - {\raise0.7ex\hbox{$\gamma $} \!\mathord{\left/ {\vphantom {\gamma 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)}}^{*}$. Estimates of the confidence intervals based on the pivot function, which is the difference between $\theta$ and $\theta^{*}$, are then given by (see Wasserman 2005, for full details)

$$ c_{\theta ,L}^{*} = 2\theta^{*} - \theta_{{\left( {1 - {\raise0.7ex\hbox{$\gamma $} \!\mathord{\left/ {\vphantom {\gamma 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)}}^{*} $$

$$ c_{\theta ,H}^{*} = 2\theta^{*} - \theta_{{\left( {{\raise0.7ex\hbox{$\gamma $} \!\mathord{\left/ {\vphantom {\gamma 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)}}^{*}. $$

(37)

For each metal, 10,000 independent samples were drawn from the natural data set, i.e., n = 10,000. This number of samples is in every case a vanishingly small fraction of the total number of possible samples, so that the likelihood of repetitions is negligible. The confidence interval was chosen at 95%, so $\gamma = 0.05$. The values of $c_{\mu ,L}^{*}$, $c_{\mu ,H}^{*}$, $c_{\sigma ,L}^{*}$, $c_{\sigma ,H}^{*}$ are listed in Table 2, and those of $c_{\alpha ,L}^{*}$, $c_{\alpha ,H}^{*}$, $c_{{z_{m} ,L}}^{*} , c_{{z_{m} ,H}}^{*}$ are given in Table 3.

Appendix 4: Maximum Likelihood Estimates for the Parameters of a Power Law Distribution

Call the lower bound of power law behavior $x_{m} = a,$ and take $b \to \infty$. We see from Eq. (8) that, for an untruncated power law, $F\left( b \right) - F\left( a \right) = 1$, so Eq. (22) becomes $g\left( x \right) = f\left( x \right)$. Substituting the power law density function, Eq. (7), in Eq. (23), and differentiating as in Eq. (24), we find that the MLE for the exponent of the power law is the solution to the equation:

$$ \frac{\partial }{{\partial \alpha^{*} }}l\left( {\alpha^{*} } \right) = \frac{n}{{\alpha^{*} }} + n\ln x_{m} - \mathop \sum \limits_{i = 1}^{n} \ln x_{i} = 0 $$

(38)

or

$$ \alpha^{*} = \frac{n}{{\mathop \sum \nolimits_{i = 1}^{n} \ln \left( {\frac{{x_{i} }}{{x_{m} }}} \right)}} $$

(39)

which is Eq. (10).

Appendix 5: The Likelihood Ratio Test

Consider $n$ independent random values, and the likelihood of each of these values under two different distributions, $p_{1,i}$ and $p_{2,i}$, $i = 1 \ldots n.$ Each of the ratios $p_{1,i} /p_{2,i}$ is then also an independent random value, and so is each of the differences $\left( {\ln p_{1,i } - \ln p_{2,i } } \right)$. If $n$ is large enough then by the central limit theorem, the sum of the independent random variables $\mathop \sum \nolimits_{i = 1}^{n} \left( {\ln p_{1,i } - \ln p_{2,i } } \right)$ is a normally distributed random variable, with variance $\sigma^{2}$ equal to the sum of the variance of each of the terms, i.e.,

$$ \sigma^{2} = \mathop \sum \limits_{i = 1}^{n} \left[ {\left( {\ln p_{1,i} - \ln p_{2,i} } \right) - \left( {\ln p_{1,i}^{m} - \ln p_{2,i}^{m} } \right)} \right]^{2} $$

(40)

where $\ln p_{1,i}^{m} , \ln p_{2,i}^{m}$ are the means of each of the variables (the mean is taken on the values of $\ln p_{j,i }$). We formulate the hypothesis that the fluctuations in these differences are random statistical fluctuations, so that the sum $\mathop \sum \nolimits_{i = 1}^{n} \left( {\ln p_{1,i } - \ln p_{2,i } } \right)$ is zero, and so is the mean of the sum, $R$ (Eq. 15). The standard deviation of $R$ is

$$ \sigma_{R} = \left( { \frac{{\sigma^{2} }}{n}} \right)^{{\frac{1}{2}}} = \left\{ { \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \left[ {\left( {\ln p_{1,i} - \ln p_{2,i} } \right) - \left( {\ln p_{1,i}^{m} - \ln p_{2,i}^{m} } \right)} \right]^{2} } \right\}^{{\frac{1}{2}}}. $$

(41)

We construct a normal distribution with mean = 0 and standard deviation = $\sigma_{R}$, and use it to determine the probability with which we are likely to observe a value of $R$ different from zero if the true value of the mean is zero. This is the p value of that observation. If this measured p value is less than our chosen significance level, then we conclude that the measured value of $R$ is more different from zero than can be accounted for by random fluctuations, and hence that the sign of R, indicating which of the two distributions is preferred, is significant.