My Experiments with ‘Natural’ Data!
By: Rishi Bhargava
Tags: Golden Ratio, Time Series Forecasting, Design, Data Science, Fibonacci

Well, that was that. So after scratching my head as well for years, I have developed a knack for using the sequence anywhere/ everywhere (yes, the inquisitor in me, wakes up intermittently).

Now, coming back to the title (the experiment). So last night I was going through the time series forecasting models & stumbled upon a weird coincidence, if it was!?! As I started reading about exponential smoothing model & how magnitude of weights decrease exponentially as the observations get older. Following equation explains the method for the forecasting:

t = no. of time period (t = 0 for the most recent one, t = 1 for the period before)

Y(t) = actual value in period t

Weight(t) = α(1-α)^t

n = total no. of time periods

Here, the value of alpha (α) varies between 0 & 1 such that sum of all the Weights approximate to 1.

The aforementioned inquisitor in me thought why not take the numbers in the fibonacci series & replace them with the weights. So i did precisely that & took first 15 numbers (excluding 0) in the sequence & normalised them.

So now, my weights from the fibonacci sequence are:

0.382, 0.236, 0.146, 0.09, 0.0558, 0.0345, 0.0213, 0.013, 0.0081, 0.005 ..

The first weight at t=0, from these normalised values, comes to 0.3822. Now i replace this value with first weight in the above exponential smoothing weight equation so that α = 0.3822 at t=0. Now lets substitute this value of alpha in the weights equation from exponential smoothing:

at t = 0, W(0) = 0.3822 * (1–0.3822)⁰ = 0.382

at t = 1, W(1) = 0.3822 * (1–0.3822)¹ = 0.236

at t = 2, W(2) = 0.3822 * (1–0.3822)² = 0.1458

at t = 3, W(3) = 0.3822 * (1–0.3822)³ = 0.0901

at t = 4, W(4) = 0.3822 * (1–0.3822)⁴ = 0.055

at t = 5, W(5) = 0.3822 * (1–0.3822)⁵ = 0.0344

See where i am going with this? Yep, the weights are exactly same as the normalised version of fibonacci sequence which means at a particular value of alpha (α) the exponential smoothing weights mimic the nature !?! And that value of alpha after taking 50 first numbers in the sequence is:

α = 0.3819966011

Here’s a sample forecasted exponential smoothing model at an alpha of 0.38199

Food for thought ..

Now, i just wonder if this could just be a coincidence or if this could be by design from the person who first suggested this model, Mr. Robert Goodell Brown?

With this alpha do we get the best forecasted model? Nope, not all the time.

Could this equation (approximated) have further applications with respect to time or a pattern with subsequent layers?

~ 0.38199 * [(0.618)^t ]

Now between 0 & 1, these are the only combination of values (0.38199 & 0.618) which satisfy the following equation:

α = (1-α)² which becomes α²-3α+1=0 (other root of this quadratic equation is 2.618) Also, α²-3α+1=0 is a mirror image of α²-α-1=0 (golden ratio equation) at y=1!!

So, the above equation can be written as

~ (0.618)² * (0.618)^t

~ (0.618)^(t+2)

Now we also know that the multiplicative inverse of 0.618 is 1.618 which is the golden ratio - phi(φ), so now the equation boils down to this:

~ (1/φ)^(t+2)

Could this pattern possibly be an alter ego of the fibonacci one for the micro world?

1, 0.6180, 0.3819, 0.236, 0.145, 0.09, 0.055, 0.0344, 0.0213, 0.0131 ..