# Log differencing Top Chef

July 9, 2009 in Math

On the most recent Top Chef, one of the chefs received just half a star (out of five) for his initial dish - impressive because the dish hadn't finished cooking and wasn't even entirely served. More interesting perhaps was his response to getting half a star:

"I was sure I'd get nothing... but that's 50% more than I thought I would get."

Naturally, I found the statement confusing. What's 50% more than zero? Not a half - it's still zero! Of course, the error is semantic rather than mathematic - he clearly meant "that's 50% of a star [stars being the relevant unit of measurement] more than the zero stars I expected." Still, the phrasing was awkward and highlights the dangers of measuring differences in values by percent change.

Unless there is a clear base measure and transitive measure, percent changes can be misleading. When there is a known quantity which then transforms into another quantity, then the change has a clear direction and a percent change is ok: today a stock is worth \$100, tomorrow it is worth \$110; its percent change is unambiguously 10% because we know which value came first (the base) and which came second (the transitive).

But what if the question were unclear, and instead of "how much higher is the stock today than yesterday" it was "how much lower was the stock yesterday than today?"

Consider an extreme case: first the stock is at \$100, then it moves to \$150. It ends up 50% higher than it started; conversely it started just 33% lower than it finished. These two measurements are incompatible, and unless the direction of change is very clearly established, this is a prime area for distortion via statistics. For example, a company reducing inventory can inflate their activities by reporting how much higher the level was previously as opposed to how much lower it is now. In some cases, it will be obvious which percent change should be used; it others it may not.

Recently I dealt with an issue of comparing prices discovered in a liquidity exercise. The questions of "how much higher is price Y than price X" and "how much lower is price X than price Y" were equally valid and I could not determine ahead of time which would be of interest. To eliminate any ambiguity, I disregarded percent changes altogether and used log-differences instead. Log-differences have the nice property of roughly approximating percent changes for small numbers, without the asymmetry of having to choose a base and transitive value. Back to my extreme example, 100 to 150 is either a 50% or 33% change, depending on the question of interest; using logs it is ln(150)-ln(100) = 40.5%. Note that 40.5% is quite close to the geometric mean of 50% and 33%, or 40.8%. Also, it doesn't matter if 100 or 150 is my base; the change is 40.5% either way. Thus, the ambiguity of using percentages to describe relative changes is eliminated.

Unfortunately, after all that, the chef's statement still can't be saved - the log difference of any number and 0 is undefined (or negative infinity, if you prefer). So yes, this was all just an excuse to discuss logs as an alternative to percents. C'est la vie.

It's important to note that for large changes, as in my example, logs will not approximate the "true" difference, but rather something close to the average of the assymetry. For small numbers, they will approximate it well. The math which determines this property is related to that of stationarity in brownian motion - for example, stock prices are modeled in finance as log-differences, not percent changes. But that's another story.

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