Draft:Relative change

Template:Short description A relative or percent difference (or change) measures of the distance between two quantities, relative to their size. The comparison is expressed as a ratio and is generally a dimensionless quantity.Template:Sfn

A special case of relative change, called the relative or percent error, occurs in measurement situations where there is an accepted reference value (perhaps theoretically determined) and the value being compared to it is experimentally determined (by measurement). Corresponding values of percent change can be obtained by multiplying these values by 100 (and using the % sign to indicate that the value is a percentage).

Basic (classical)

Given two numbers—v_ref and v—with v_ref being the reference value, their difference (sometimes absolute difference) is:

This difference is not always a good way to compare values, because it depends on the context and units of measurement. For instance, Template:Val is the same as Template:Val, but the absolute difference between Template:Val is 1 m, while the absolute difference between Template:Val is 100 cm, giving the impression of a larger difference.Template:Sfn Using relative change helps judge the importance of a change, even when comparing changes in quantities of different units.

One way to adjust the comparison so it accounts for the "size" of the quantities involved is to use the basic, classical, or approximate relative change, commonly taught in primary schooling. It is defined as:$$\text{relative change (basic): }\frac{\text{change}}{\text{reference value}}=\frac{\mathrm{\Delta }v}{v_{\text{ref}}}=\frac{v}{v_{\text{ref}}}-1$$The relative change is independent of the units. For example, the relative change from Template:Val is Template:Val, the same as for Template:Val. The relative change is not defined if the reference value (v_ref) is zero.

Percent error

The percent error is a special case of the basic relative change. It is equal to the absolute change between the experimental (measured) and theoretical (accepted) values, and dividing by the theoretical (accepted) value.$$\mathrm{\%}\text{ Error}=\frac{\text{Experimental}-\text{Theoretical}}{\text{Theoretical}}$$The terms "Experimental" and "Theoretical" used in the equation above are commonly replaced with similar terms. Other terms used for experimental could be "measured," "calculated," or "actual" and another term used for theoretical could be "accepted." Experimental value is what has been derived by use of calculation and/or measurement and is having its accuracy tested against the theoretical value, a value that is accepted by the scientific community or a value that could be seen as a goal for a successful result.

Issues

Although intuitive and commonly taught, it has several issues that can create counterintuitive behaviors. For example, basic percent changes are not commutative. Therefore, it is vital to preserve the order as above: subtract the theoretical value from the experimental value and not vice versa.

Logarithmic (scientific)

The logarithmic or scientific relative change is another indicator of relative change. It is generally recommended over the basic relative change used in scientific and statistical analysis, as it has better (more intuitive) mathematical properties than the basic relative change.

Of these indicators of relative change, the most natural arguably is the natural logarithm ($\ln$) of the ratio of the two numbers (final and initial), called log change:Template:Sfn

$$\text{relative change (scientific): }\frac{\text{actual change}}{\text{reference value}}=\ln \frac{v_1}{v_0}=\ln v_1-\ln v_0$$

In the same way that relative change can be scaled by 100 to get percentages, $\ln \frac{V_1}{V_0}$ can be scaled by 100 to get what are commonly called centinepers (cNp), log points, log percentages, or (in some scientific or statistical contexts) simply percentages; these are sometimes denoted % or just % (when the meaning is clear from context).Template:Sfn

Log percentages are approximately equal to basic percentages for small relative changes. For example, an increase of 1% is an increase of 0.995%, and a 5% increase is a 4.88% increase.

Advantages and properties

Log points have several desirable mathematical properties, which are not shared with by have led scientists and statisticians to advocate using them to replace over the basic percent change for most scientific and .

Additivity

Using the log change has the advantage of additivity compared to the basic relative change;Template:Sfn in other words, an x% change and a y% change add up to an x+y% change. With basic percentages, summing is only an approximation that breaks down for large changes. For example:

Log change 0 (cNp)	Log change 1 (cNp)	Total log change (cNp)	Relative change 0 (%)	Relative change 1 (%)	Total relative change (%)
10%	5%	15%	10%	5%	15.5%
10%	−5%	5%	10%	−5%	4.5%
10%	10%	20%	10%	10%	21%
10%	−10%	0%	10%	−10%	−1%
50%	50%	100%	50%	50%	125%
50%	−50%	0%	50%	−50%	−25%

Note that in the above table, since relative change 0 (respectively relative change 1) has the same numerical value as log change 0 (respectively log change 1), it does not correspond to the same variation. The conversion between relative and log changes may be computed as $\text{log change}=\ln (1+\text{relative change})$.

Symmetry

The basic (or classical) relative change is not generally useful as a metric of the "difference" or "distance" between two quantities, because it is not symmetric. In other words, whether

However, by additivity, $\ln \frac{V_1}{V_0}=-\ln \frac{V_0}{V_1}$; thus the magnitude of a change expressed in log change is the same whether V₀ or V₁ is chosen as reference. In contrast, for relative change, $\frac{V_1-V_0}{V_0}\ne -\frac{V_0-V_1}{V_1}$, with the difference $\frac{(V_1-V_0{)}^2}{V_0{V}_1}$ becoming larger as V₁ or V₀ approaches 0 while the other remains fixed. For example:

V0	V1	Log change (cNp)	Relative change (%)
10	9	−10.5	−10.0
9	10	+10.5	+11.1
10	1	−230	−90
1	10	+230	+900
10	0+	−∞	−100
0+	10	+∞	+∞

Here 0⁺ means taking the limit from above towards 0.

Uniqueness and extensions

The log change is the unique two-variable function that is additive, and whose line

Generalized indicators of relative change

The relative one of the possible measures/indicators of relative change. An indicator of relative change from x (initial or reference value) to y (new value) $R(x,y)$ is a binary real-valued function defined for the domain of interest which satisfies the following properties:Template:Sfn

• Appropriate sign: $\{\begin{array}{cc}R(x,y)>0& \text{iff }y>x\\ R(x,y)=0& \text{iff }y=x\\ R(x,y)<0& \text{iff }y<x\end{array}.$
• Template:Mvar is an increasing function of Template:Mvar when Template:Mvar is fixed.
• Template:Mvar is continuous.
• Independent of the unit of measurement: for all $a>0$, $R(ax,ay)=R(x,y)$.
• Normalized: ${\frac{d}{dy}R(1,y)|}_{y=1}=1$

The normalization condition is motivated by the observation that Template:Mvar scaled by a constant $c>0$ still satisfies the other conditions besides normalization. Furthermore, due to the independence condition, every Template:Mvar can be written as a single argument function Template:Mvar of the ratio $y/x$.Template:Sfn The normalization condition is then that $H^{\prime }(1)=1$. This implies all indicators behave like the classical one when $y/x$ is close to Template:Val.

Usually the indicator of relative change is presented as the actual change Δ scaled by some function of the values x and y, say Template:Math.Template:Sfn

$$\text{Relative change}(x,y)=\frac{\text{Actual change}\mathrm{\Delta }}{f(x,y)}=\frac{y-x}{f(x,y)}.$$

As with classical relative change, the general relative change is undefined if Template:Math is zero. Various choices for the function Template:Math have been proposed:Template:Sfn

Indicators of relative changeTemplate:Sfn

Name	$f(x,y)$where the indicator's value is $\frac{y-x}{f(x,y)}$	$H(y/x)$
Classical relative change	Template:Mvar	$\frac{y}{x}-1$
Arithmetic mean change	$\frac{1}{2}(x+y)$	$\frac{\frac{y}{x}-1}{\frac{1}{2}(1+\frac{y}{x})}$
Geometric mean change	$\sqrt{xy}$	$\frac{\frac{y}{x}-1}{\sqrt{\frac{y}{x}}}$
Harmonic mean change	$\frac{2}{\frac{1}{x}+\frac{1}{y}}$	$\frac{(\frac{y}{x}-1)(1+\frac{x}{y})}{2}$
Moment mean change of order Template:Mvar	${[\frac{1}{2}(x^k+y^k)]}^{\frac{1}{k}}$	$\frac{\frac{y}{x}-1}{{\left[\frac{1}{2}(1+{\left(\frac{y}{x}\right)}^k)\right]}^{\frac{1}{k}}}$
Maximum mean change	$\max (x,y)$	$\frac{\frac{y}{x}-1}{\max (1,\frac{y}{x})}$
Minimum mean change	$\min (x,y)$	$\frac{\frac{y}{x}-1}{\min (1,\frac{y}{x})}$
Logarithmic (mean) change	$\{\begin{array}{cc}\frac{y-x}{\ln \frac{y}{x}}& x\ne y\\ x& x=y\end{array}$	$\ln \frac{y}{x}$

As can be seen in the table, all but the first two indicators have, as denominator a mean. One of the properties of a mean function $m(x,y)$ is:Template:Sfn $m(x,y)=m(y,x)$, which means that all such indicators have a "symmetry" property that the classical relative change lacks: $R(x,y)=-R(y,x)$. This agrees with intuition that a relative change from x to y should have the same magnitude as a relative change in the opposite direction, y to x, just like the relation $\frac{y}{x}=\frac{1}{\frac{x}{y}}$ suggests.

Maximum mean change has been recommended when comparing floating point values in programming languages for equality with a certain tolerance. Another application is in the computation of approximation errors when the relative error of a measurement is required. Minimum mean change has been recommended for use in econometrics.Template:Sfn Logarithmic change has been recommended as a general-purpose replacement.

Tenhunen defines a general relative difference function from L (reference value) to K:Template:Sfn$$H(K,L)=\{\begin{array}{cc}\underset{1}{\overset{K/L}{\int }}{t}^{c-1}dt& \text{when }K>L\\ -\underset{K/L}{\overset{1}{\int }}{t}^{c-1}dt& \text{when }K<L\end{array}$$which leads to$$H(K,L)=\{\begin{array}{cc}\frac{1}{c}\cdot ((K/L{)}^c-1)& c\ne 0\\ \ln (K/L)& c=0,K>0,L>0\end{array}$$In particular for the special cases $c=\pm 1$,$$H(K,L)=\{\begin{array}{cc}(K-L)/K& c=-1\\ (K-L)/L& c=1\end{array}$$

See also

• Approximation error
• Errors and residuals in statistics
• Relative standard deviation
• Logarithmic scale

References

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