Think Nuclear

Expressing the rate of increase or decrease of a quantity as a relative value has been recognized as a challenging proposition for a long time because a consistent relative increase results in compounding effects (think compound interest). The quantity that is commonly used to express such changes, the percent difference, has some serious shortcomings when combining relative changes or when comparing changes over different periods. Here I propose an alternative way of expressing such changes, which is similar to the percent difference but which overcomes these shortcomings.

Background

This is an idea that I first encountered over a decade ago on the blog of a Czech theoretical physicist (which is now closed to the public). He proposed the idea of an “exponential percent” that could be used as a more rational way to express changes in absolute quantities. I agree completely that it makes more sense to “count the logarithms” than to count absolute changes in the quantities themselves.

The problem that Dr. Motl brings up is not new. Plenty of people have noticed the shortcomings of using percentages to express changes in a quantity. For example, a 10% increase followed by a 10% decrease does not get one back to the original value. In addition, it is not straightforward to combine percent differences and the result of such an attempt can be deceptive as Darrell Huff pointed out in his classic book, How to Lie with Statistics. He cleverly demonstrated that a 100% increase in the price of one commodity (bread) and a 50% decrease in another (milk) can be made to look like a 20% increase or decrease in their combined price, depending on the price chosen as a reference point (before or after the change). Mr. Huff’s solution for this dilemma was to switch to using the geometric mean when combining two or more percent differences.

Dr. Motl’s solution is not new either, and it already has a name. What he calls an “exponential percent” is already known as a centineper (cNp). This unit, based on the neper, is an established unit—similar to the more familiar decibel (dB), which is an essential unit for anyone working in acoustics or with audio equipment. While the neper is not part of the International System of Units (SI), it is accepted for use alongside the SI. A change expressed in centinepers is also sometimes called the “log change.”

Although the centineper is quite useful, here I propose an alternative notation and way of thinking about this general concept. The new notation that I propose here accomplishes the same thing as Huff’s geometric mean when combining two or more percent differences—mathematically it’s identical—but hopefully this notation will be more intuitive.

Percent Doubling

I call the measure of change that I propose “percent doubling,” and it is defined as follows. If a quantity \(q\) changes from an initial value of \(q_{\text{i}}\) to a final value of \(q_{\text{f}}\), the percent doubling rate \(r_{{\times}2}\) is the value that satisfies\[
q_{\text{f}} = q_{\text{i}} 2^{r_{{\times}2}/100}
\]Thus, a value of \(r_{{\times}2} = 100\) results in\[
q_{\text{f}} = 2 q_{\text{i}}
\]or a doubling of the original amount. In terms of \(r_{{\times}2}\), this can be written\[
r_{{\times}2} = 100 \times \log_2 \frac{q_{\text{f}}}{q_{\text{i}}}
\]To denote quantities expressed this way, either “\(\%_{{\times}2}\)” or “\(\%{\times}2\)” can be used. The former is more compact, the latter is more straightforward and doesn’t require a change in font size. Another possibility is to use the abbreviation “pcd.”

Because of the choice of scale, percent doubling is somewhat similar to the percent difference, which is defined as\[
\Delta\%q = \frac{q_{\text{f}} – q_{\text{i}}}{q_{\text{i}}} \times 100\%
\]In fact, percent doubling and percent difference are very similar for differences that either are very small (near 0%) or are very large (near 100%), as shown below.

The difference between the two values is largest in the middle region, near 50%. Outside of the 0% to 100% range, the similarity ends and the logarithmic nature of percent doubling dominates.

For example, a percent difference of \(-100\%\) indicates that nothing is left in the original quantity, but a difference of \(-100\%_{{\times}2}\) means that only half of the original quantity is gone. In this case, it might make sense to adopt a separate notation for negative percent doubling. This new quantity—which we can call “percent halving” (or perhaps “half percent”) and express as \(\%_{{\div}2}\) or \(\%{\div}2\) to be consistent with the notation introduced above—is defined as\[
r_{{\div}2} = 100\times\log_2\frac{q_{\text{i}}}{q_{\text{f}}}
\]since \(q_{\text{f}} = \frac{1}{2} q_{\text{i}} \Rightarrow r_{{\div}2} = 100\%_{{\div}2}\). It is immediately clear that this new quantity is simply the negative of the percent doubling:\[
r_{{\div}2} = -100 \times\log_2\frac{q_{\text{f}}}{q_{\text{i}}} = -r_{{\times}2}
\]This is the first example of a generalization of this concept. While a factor of two was chosen because \(+100\%\) means twice the original amount, we can also define a “percent \(n\)-times,” denoted “\(\%_{{\times}n}\)” or “\(\%{\times}n\),” which is given by the formula\[
r_{{\times}n} = 100\times\log_n\frac{q_{\text{f}}}{q_{\text{i}}}
\]Thus, \(r_{{\div}2}\) is equivalent to \(r_{{\times}1/2}\).

With this more general notation, Motl’s “exponential percent” (or “E-percent”), which he denoted as “E%,” can be written in my notation as “\(\%{\times}e\).” In fact, the only difference between our ideas is a choice of notation. A short table below shows equivalent notation in the two systems.

Relation to Percent Difference

As Motl points out, one advantage of the E-percent (and the centineper) is that it approximates the percent difference for very small differences. Expressed in terms of the percent difference, the E-percent is\[
\mathrm{E\%} = 100 \times \ln \left(\frac{\Delta\%q}{100} + 1\right)
\] Since\[
\ln (x+1) = x – \frac{1}{2} x^2 + \frac{1}{3} x^3 – \cdots
\]this quantity can be written\[
\mathrm{E\%} = \Delta\%q + O\left((\Delta\%q)^2\right)
\]or if \(\Delta\%q\) is small,\[
\mathrm{E\%} \approx \Delta\%q
\]

Percent doubling does not have this property. It is the choice of scaling that is exactly equal to the percent difference at values of 0 and 100 so as to preserve the meanings of 0% and 100%.

Relation to Other Units

As mentioned above, Motl’s E-percent is actually a centineper, a unit that is already defined:\[
1\,\mathrm{E\%} = 1\,\mathrm{cNp}
\] Furthermore, the neper is similar to the decibel in that they both measure the same thing: logarithmic-scale increases and decreases. The difference between the two units is that the decibel uses the decadic (base-10) logarithm, whereas the neper uses the natural logarithm. Therefore, the two units are related by a scaling factor:\[
1\,\mathrm{Np} = 20 \log_{10} ( e )\,\mathrm{dB}
\approx 8.69\,\mathrm{dB}
\]

The percent doubling also measures logarithmic-scale increases decreases, but the measurement is according to factors of two. That is, it uses base-2 logarithms. Therefore, it is related to these other two units by scaling factors:\[
1\%_{{\times}2} = \frac{1}{\ln 2}\,\mathrm{cNp}
\approx 1.44\,\mathrm{cNp}
\]and\[
1\%_{{\times}2} = \frac{0.2 \log_{10} ( e )}{\ln 2}\,\mathrm{dB}
\approx 0.125\,\mathrm{dB}
\]

It is important to emphasize that there is nothing conceptually new with the introduction of this unit of measure. It’s much like replacing miles with kilometers or inches with centimeters. (And in fact, the modern inch is defined as exactly 2.54 cm.) The scale is chosen for convenience. For example, feet and inches lend themselves well to factions, since 12 can be divided by 2, 3, 4, and 6 (whereas 10 is divisible only by 2 and 5). Centimeters and their associated metric units, on the other hand, are more amenable to decimal arithmetic.

In this case, a scale is proposed that lends itself well to a two-digit representation of a change that is similar to the familiar percentage convention that is often used for changes in quantities. The end points of this scale are the same—i.e., 0 means no change and 100 means twice the amount. Therefore, this new scale should be intuitive to anyone accustomed to working with percentages.

Outside of the range from 0 to 100, percent doubling loses its intuitive similarity with percent differences. For example, a percent difference of 200% means an addition that is twice the original amount. That is, the original amount has increased by a factor of three. A percent doubling of 200%\(_{{\times}2}\), however, means that the original amount has been doubled twice. The original amount has increased by a factor of four, and the new addition is three times the original amount.

Time Rate of Change

A key feature of percent doubling is that it allows successive relative changes to be combined into a total relative change that can be determined by combining the parts by addition. This lends itself well to time-dependent processes, such as growth or decay, and in fact percent doubling is closely related to these processes.

Constant exponential growth and decay are governed by the following equation:\[
q ( t ) = q_0\, e^{\lambda t}
\]where \(q_0\) is the initial quantity, \(q(t)\) is the quantity at time \(t\), and \(\lambda\) is a constant that determines the rate of growth or decay (\(\lambda > 0\) for growth, \(\lambda < 0\) for decay). Such dynamical systems (especially when \(\lambda > 0\)) are sometimes described by the e-folding time\[
t_{\text{e}} = \frac{1}{\lambda}
\]which is the time required for the quantity to increase by a factor of \(e \approx 2.1416\). Closely related to this quantity is the doubling time, which is the time required for the quantity to increase by a factor of two. The converse of the doubling time for decreasing quantities is the half-life, which is commonly used to quantify the rate of radioactive decay in nuclear physics. In terms of the relevant constant \(\lambda\) (growth constant if \(\lambda > 0\), decay constant if \(\lambda < 0\)), these two times are\[
\text{doubling time}\qquad
t_{\text{d}} = \frac{\ln 2}{\lambda}
\] \[
\text{half-life}\qquad
t_{1/2} = – \frac{\ln 2}{\lambda}
\]

For systems of constant exponential growth, percent doubling is the percentage of elapsed time \(t\) to the doubling time\[
\%_{{\times}2} = 100 \times \frac{t}{t_{\text{d}}}
\]Similarly, percent halving is the percentage of elapsed time to the half-life\[
\%_{{\div}2} = 100 \times \frac{t}{t_{1/2}}
\]

A common term encountered when considering growth over a time period is the Compound Annual Growth Rate (CAGR), which can be useful for comparing growth rates of various quantities in a common domain and is often employed to compare financial growth. This term is defined as\[
\text{CAGR}(t_0, t_n) = \left [ \frac{q(t_n)}{q(t_0)} \right ]^{\frac{1}{t_n – t_0}} – 1
\] where \(q(t_0)\) is the initial value, \(q(t_n)\) is the final value, and \(t\) is measured in years. CAGR expresses an increase over several years as if it were the result of exponential growth over the time period with a constant relative increase each year, which allows comparisons of the rate of growth for increases over different time periods (e.g., the growth of one quantity over five years versus the growth of another quantity over ten years).

Although not exactly equivalent to CAGR, percent doubling can be used for the same purpose as CAGR for comparing growth over differing time periods. In fact, percent doubling is superior for comparison purposes because CAGR does not provide a completely reliable measure of the rate of growth. For example, a 10% CAGR is not twice the rate of increase of a 5% CAGR. There is more after two years at 5% (10.25% total) than one year at 10%. On the other hand, two years of 5%\(_{{\times}2}\) growth is exactly equivalent to one year of 10%\(_{{\times}2}\) growth. Percent doubling also simplifies the math involved since the total value over the entire period can be divided by the number of time intervals (e.g., years) to arrive at a per-interval value.

To see how the two measures of growth are related, it is useful to note that the definition of CAGR implies\[
\ln (\text{CAGR} + 1) = \frac{\ln q_n – \ln q_0}{n}
\]where \(q_n \equiv q(t_n)\) is the quantity after \(n\) intervals, \(q_0 \equiv q(t_0)\) is the initial quantity, and \(t_n – t_0 = n\). Meanwhile, the same increase expressed as percent doubling is\[
r_{{\times}2} = 100 \times \log_2 \frac{q_n}{q_0}
= 100 \times \frac{\ln q_n – \ln q_0}{\ln 2}
\]so that the average percent doubling over each interval is\[
\bar{r}_{{\times}2} = \frac{100}{\ln 2} \times
\frac{\ln q_n – \ln q_0}{n}
\]Both the expression for CAGR and the expression for percent doubling include the following term on the right hand side of the equations:\[
\frac{\ln q_n – \ln q_0}{n}
\]

This similarity can be explored further. Since\[
\ln (x+1) = x – \frac{1}{2} x^2 + \frac{1}{3} x^3 – \cdots
\]small values of CAGR can be approximated with\[
\text{CAGR} \approx \frac{\ln q_n – \ln q_0}{n}
\]Therefore, if CAGR is expressed as a percentage, it is related to the average percent doubling per year by\[
\text{CAGR(%)} \approx \ln 2 \times \bar{r}_{{\times}2} \approx 0.693\, \bar{r}_{{\times}2}
\]in the limit of small growth.

Summary

Here I have introduced a new terminology and notation for expressing relative changes. It is qualitatively similar to percent change, but it has the advantage that multiple successive changes can be combined through addition, which is not possible when using percent differences.

This new notation uses a logarithmic scale, which is nothing new. Both the neper and the decibel provide the same functionality for comparing quantities. The only novel feature is the choice of scale.

The new terminology has the following features and limitations:

• It is similar to the more familiar percent change: 0% and 100% mean exactly the same thing in both systems.
• Two or more changes can be combined through addition to reach a total change through this system.
• Doubling the pcd means twice the exponential growth.
• It is best applied to fractional changes: changes outside of \(0\%_{{\times}2}\) to \(100\%_{{\times}2}\) quickly loose their intuitive association with percent change.
• Its converse, the percent halving (\(\%_{{\div}2}\)), performs a similar feature for decreasing quantities.
• It is useful for describing exponential growth and decay.
• It can serve a function similar to Compound Annual Growth Rate (CAGR) for growth rates, while simplifying the mathematics required and providing a more genuine measure of geometric growth.