Weber's Law and Fibonacci Numbers: An Exploratory Essay
I use Fibonacci numbers rather frequently. In fact, the Fib retracement tool is the first thing I reach for when I start on a new chart. However, explanations for how Fibonacci numbers work have always sound woolly and mystical to me. They work because "man is subject to rhythmical procedure", because there is a Golden Ratio that is hidden behind all things, because Cthulhu says so?
However, when we take a close hard look at reality, and actually whip out a ruler and measure things, we find that the Fibonacci sequence is *not* found as-is throughout reality. What we do find are *approximations*. However, this is to be expected for approximations of subjects where the rate of growth is proportional to the current size. And this is to be expected because of Weber's Law.
Weber’s law is a psychological law quantifying the perception of change in a given stimulus. The law states that the change in a stimulus that will be just noticeable is a constant ratio of the original stimulus. It has been shown not to hold for extremes of stimulation. And since I will be referencing Mike Cohn's excellent essay (1), I might as well quote his explanation of how Weber's Law apply to Fibonacci numbers:
"Imagine instead being handed a 20kg weight and a 21kg weight. They are the same one kg difference as the one and two kg weights. But you would have a much harder time identifying the heavier of the two weights. The difference from one to two kilograms is 100%. You can probably distinguish the weight of items that differ by 100%. The difference between 20 and 21kg, however, is only 5%. You probably can’t tell the difference. (I know I can’t.) And if you could, it would mean you should be able to distinguish between a 1.00 kg weight and a 1.05 kg weight, as that would also be 5%. The values in the Fibonacci sequence work well because they roughly correspond to Weber’s Law. After the two (which is 100% bigger than one), each number is about 60% larger than the preceding value. According to Weber’s Law, if we can distinguish a 60% difference in effort between two estimates, we can distinguish that same percentage difference between other estimates. So, the Fibonacci values work well because they increase by about the same proportion each time."
So, given that how we think is affected by how we perceive (2), if Weber's Law applies, the Fibonacci retracement tool works for some of us because it allows us to focus our *imagination by visualising discernible and distinct possibilities within a limited range*. This is why the common criticisms of TA, including Fibs, are valid: 1) it is an uncertain business; 2) one cannot consistently identify where levels should be placed and forecasts are prone to revision; 3) its narrative story-telling power may be stronger than its forecasting power; and 4) levels cannot be verified till they have been tested (ie passed). Let's be humble and accept the general validity of these criticisms; for if TA can be an exact science, let he produce an algorithm which could make anyone rich!
That being said, if Weber's Law apply thusly, it simply reaffirms what experience traders often exhort: that it is hard for algorithms to replace (3) the imagination and instincts of an experienced trader!
Having said that, if Weber's Law apply thusly, we ought to 1) pay attention to how other industries, eg Mike Cohn's, have adapted Fibonacci numbers to great success and ask ourselves if our approach to Fibs can be adapted accordingly; and 2) maybe more importantly, reconsider our values, assumptions, beliefs and expectations of those tools we use that are based on Fibs.
(1) www.mountaingoatsoft...-well-for-estimating
(2) www.frontiersin.org/...nhum.2014.00542/full
(3) "Replace", not "aid".
However, when we take a close hard look at reality, and actually whip out a ruler and measure things, we find that the Fibonacci sequence is *not* found as-is throughout reality. What we do find are *approximations*. However, this is to be expected for approximations of subjects where the rate of growth is proportional to the current size. And this is to be expected because of Weber's Law.
Weber’s law is a psychological law quantifying the perception of change in a given stimulus. The law states that the change in a stimulus that will be just noticeable is a constant ratio of the original stimulus. It has been shown not to hold for extremes of stimulation. And since I will be referencing Mike Cohn's excellent essay (1), I might as well quote his explanation of how Weber's Law apply to Fibonacci numbers:
"Imagine instead being handed a 20kg weight and a 21kg weight. They are the same one kg difference as the one and two kg weights. But you would have a much harder time identifying the heavier of the two weights. The difference from one to two kilograms is 100%. You can probably distinguish the weight of items that differ by 100%. The difference between 20 and 21kg, however, is only 5%. You probably can’t tell the difference. (I know I can’t.) And if you could, it would mean you should be able to distinguish between a 1.00 kg weight and a 1.05 kg weight, as that would also be 5%. The values in the Fibonacci sequence work well because they roughly correspond to Weber’s Law. After the two (which is 100% bigger than one), each number is about 60% larger than the preceding value. According to Weber’s Law, if we can distinguish a 60% difference in effort between two estimates, we can distinguish that same percentage difference between other estimates. So, the Fibonacci values work well because they increase by about the same proportion each time."
So, given that how we think is affected by how we perceive (2), if Weber's Law applies, the Fibonacci retracement tool works for some of us because it allows us to focus our *imagination by visualising discernible and distinct possibilities within a limited range*. This is why the common criticisms of TA, including Fibs, are valid: 1) it is an uncertain business; 2) one cannot consistently identify where levels should be placed and forecasts are prone to revision; 3) its narrative story-telling power may be stronger than its forecasting power; and 4) levels cannot be verified till they have been tested (ie passed). Let's be humble and accept the general validity of these criticisms; for if TA can be an exact science, let he produce an algorithm which could make anyone rich!
That being said, if Weber's Law apply thusly, it simply reaffirms what experience traders often exhort: that it is hard for algorithms to replace (3) the imagination and instincts of an experienced trader!
Having said that, if Weber's Law apply thusly, we ought to 1) pay attention to how other industries, eg Mike Cohn's, have adapted Fibonacci numbers to great success and ask ourselves if our approach to Fibs can be adapted accordingly; and 2) maybe more importantly, reconsider our values, assumptions, beliefs and expectations of those tools we use that are based on Fibs.
(1) www.mountaingoatsoft...-well-for-estimating
(2) www.frontiersin.org/...nhum.2014.00542/full
(3) "Replace", not "aid".
This is probably too boring for you, but this is dedicated to you!