However, I have been out of practice for some time, and I was looking for ways to keep my skills polished. I stumbled across Good Judgment Open (www.gjopen.com), an open site that allows anyone to make forecasts on an array of real life questions. Apparently it was started a number of years ago by a Wharton professor to study what kind of person makes a good predictor, and it has become very popular in the community. I thought this would be a perfect way for me to practice some data analysis.
The first challenge I took on is the prediction of whether iPhone sales in Q4 of 2015 (ending in December) will exceed 75 million units. The strategy is to use simple extrapolation to arrive at a number. The 4th quarter is the first full quarter after the release of the iPhone 6S and 6S Plus on Sept 25th 2015, roughly a year after the release of iPhone 6. the first quarter after iphone 6 sales peaked at 74.5 million. However S versions tend to perform less than brand new versions.
I first approached the problem from a pure data fitting approach. Over the past 4 years (2011-2015), iPhones have been released at around the same time each year. Forming a periodic data set with 16 data points, one for each quarter.
From the looks of it, Apple should easily exceed 75 million mark come the first quarter. But the purpose of these exercises for me is to practice my numerical analysis skills, so I should crunch some numbers first.
Next I fitted a quadratic model to the data: it seemed to make sense that the sales rate of change over time would be linearly decreasing. I averaged the quadratic coefficients of the previous four years to find typical sales rate values. In other words, I took the coefficients of the first and second order terms of the model, believing that the higher order terms would be more stable. To find the constant term of the quadratic, I calculated the annual sales averages of the previous years and performed a least square regression to obtain a expected average sales per quarter of 72 million. In other words, the average quarterly sales of the entire year is already approaching the 75 million threshold. From the annual average, I can calculate for the constant term. Applying the predicted model to this past quarter yields a value of 83 million units, well exceeding the threshold. However, when I tried to find the confidence intervals, the variance on my curve fit coefficients were so large that any conclusion is hard to justify with certainty (i.e. pretty much anything from 0 to twice the value would fit into the confidence interval).
I tried to fit to the exponential decay model in hopes of getting better uncertainties. I fitted the previous years and extrapolated into this past quarter using similar methods. For exponential models, I need to assume some equilibrium values for the models when the transients have decayed away completely. For this model I tried two different values as the equilibrium value. First I tried the average sales of the previous year. Second, I tried the minimum quarterly sales of the previous year. Both of these try to exploit the fact that in the previous four years, a new iPhone has never sold slower than the previous version for any quarter. I averaged the decaying time constants and assumed the averaged time constant is close to the model for the new year. After a linear extrapolation of the expected average in the new quarter, all the other coefficients can be computed. However, when I calculated the expected sales numbers for the first quarter, the results were 60 million and 50 million respectively, again with very large uncertainties.
So far the analysis has been inconclusive. 75 million is roughly in the middle of the two models, so naively one might think the probability of hitting the target is around 50%. The data, however, is just too sparse to conclude anything with certainty.
Nevertheless, there are still some interesting properties that can be extracted, particularly with the decay model. Looking at the decay rates, the one with the least sustained popularity is the 2013 products: the iPhone5 released in late 2012 with a decay rate of 1.26 per quarter. This is followed by the iPhone 5C and 5S release the following year with a decay rate of 1.00 per quarter, then the iPhone 6 at 0.65 per quarter. The product with the most sustaining popularity is the iPhone 4S released late 2011 with a decay rate of just 0.25 per quarter. This is only one side of the picture, however, as year over year growth of overall sales has been growing steadily during this period. Considering this effect, the newer iPhone 6 can be considered quite a success. Despite a large spike upon initial release, it nevertheless was able to sustain its popularity into the second quarter. It would be interesting to see the statistics of the 6s.
A final method of analyzing the market share is used. Apple's market share of global mobile phones has increased in the past number of years, even as global mobile phone cells have stabilized around 450 million units per quarter in recent years.
Extrapolating the linear fit (which, compared with my previous models, gives a much smaller uncertainty), the first quarter of 2015 should result in 83 million sales, with a fairly small confidence interval. This is 8 million more than the 75 million unit threshold. Overall, this calculation gives a more optimistic result compared to my previous analyses.
Therefore, the answer is less certain than what I first thought when I saw the initial graph of the promising year over year growth. Which models should I go with? Should I just simply average them out or should I weight them according to their variances? My last analysis tipped the scale for me due to its small uncertainty. In the end, I decided to play safe and go with 60% confidence (within one standard deviation) that iPhone 6S will exceed 75 million unit sales in its first quarter.