## Updated US Smartphone Saturation Forecast

The latest comScore US smartphone survey showing three months’ ending October data has been released and there were no surprises. Smartphone penetration grew to 62.5% representing 149.2 million users. I made a slight adjustment to the predictive logistic function parameters (p1 = 93, p2 = 22.5).

The correlation between predictive and actual logistic function (P/(1-P)) is shown below.

90% penetration (P/(1-P) = 9.23) is now expected by December 2016 when about 230 million users will be using smartphones in the US.

Individual platforms adoption curves are shown as colored areas above and as lines on a logistic plot below:

The parallel behavior of iPhone and the overall market remains unchanged.

• donsleeter

Thank you, Horace. Great work. The formula is a bit small to read, though.

• To avoid the increasingly clumped data toward the origin on the first chart, both axes should be logarithmic just like the vertical axis on the last chart. Or to simplify the interpretation, just plot measured penetration against predicted penetration instead of p/(1-p), but then you’ll get clumping at both ends.

• Ian Ollmann

Unfortunately log log plots start to engender skepticism from both ends of the readership scale. The innumerate don’t understand them. The scientists know that log log plots look linear regardless of what the trend actually is, so are often a last refuge of the scoundrel.

• “log log plots look linear regardless of what the trend actually is”

Sorry but that’s just innacurate. The only functions that look linear in log-log plots are power functions of the form y=bx^a for any real constants a, b.

An exponential trend wouldn’t show up as a line, nor a linear trend, or a parabola unless its vertex is at the origin, or any polynomial except very few special cases, nor a logarithmic curve ir any curve that doesn’t pass through the origin.

Your comment about the data analyst scoundrel using log-log plots as a ploy to counterfit linear relationships at will is disingenuous and overdramatic.

• Ian Ollmann

Log log plots, not log plots.

• Yes, log-log as in both axes are logarithmic. As I noted, power functions (y=b*x^a) become linear in such plots, and modeling power laws is important for lots of legitimate science (including economics and business).

Instead, exponential functions become linear in log (aka semi-log) plots. These are very different from power functions, but also very useful in tons of legitimate research. Exponential relationships are of the form y=b*a^x (linear when the y-axis is logarithmic aka log-lin) or the flipped y=a*log(b*x) (linear when instead the x-axis is logarithmic aka lin-log).

• JaneDoe12

To avoid the increasingly clumped data toward the origin on the first chart, both axes should be logarithmic…

I’m probably missing something, but if you do a log-log plot wouldn’t you get clumped data at the other end?

• No, the data is increasingly sparser as plotted. The sole purpose of the p/(1-p) transformation is to make it linear when using a log scale, which is the same as spreading it uniformly.

• JaneDoe12

If I use the data points from first chart, which are fairly evenly spread (although they get sparser), and plot them on log-log paper with 2×2 decades, wouldn’t there be clumping when x is between .75 – 1 (the end of the 1st decade)? Then when x > 1 the points would spread out more.

• The text before the first chart introduces it as “the correlation between predictive and actual logistic function (P/(1-p))” and the x-axis is labeled “Predictive Logistic”. Now look at the last chart, the thin blue line is precisely this measure, the y-values of the thin blue line are the theoretical P/(1-P) values of the predictive logistic function with P given by the formula and parameters provided, and these P/(P-1) becoming linear when using the logged vertical scale, as shown. This means the y-values are spread evenly on this log scale, because they’re the values on that line, and in fact, the predictive formula gives log(P/(1-P)) = (x-93)/22.5, a line, and for evenly spaced x time periods you must get evenly spaced y-values on the last chart as shown. Since these are exactly the same values now plotted along the x-axis on the first chart, they must once again spread evenly when plotted on a log x-scale, as in the third chart. There’s no need to plot them to know this, as the points come from a formula that is linear.

As for the y-values on the first chart labeled “Measured Logistic”, these correspond to the y-values of the green line in the last chart, the logistic-linear transform P/(1-P) with P now given by the actual penetration data from comScore instead of the predictive formula, and this green line falls tightly along the thin blue line, so these y-values (the same as the y-values on the first chart) also spread roughly evenly along the vertical log scale for each evenly distributed time period. Since both series on the first chart spread evenly on log scales, the corresponding pairs of points will also spread evenly roughly along the y=x line.

If I had the data I’d show you a quick log-log chart done in Excel instead of log paper, and I would start both axes values at 0.2 and use only one decade up to 2.0, with each small tick mark increasing by 0.2.

• JaneDoe12

If I had the data I’d show you a quick log-log chart done in Excel instead of log paper, and I would start both axes values at 0.2 and use only one decade up to 2.0, with each small tick mark increasing by 0.2.

I made a semi-log chart of the x/(1-x) transformation using Excel 2003. I started both axes at 0.1, and used 2 decades up to 10.0 with each tick increasing by 0.1. As you said, x/(1-x) became linear with the exception that when x = 0.1 the point was a little below the line and at x = 0.9, a little above. The points were evenly spaced.

My Excel can’t do log-log charts so I have to use my imagination. Here it is: the line would become parabolic. The points between 0.2 – 2 would be evenly spaced, but at x < 0.2 or > 2, they would be further apart.

• No, the points would spread evenly over the whole range from 0 to 100% penetration (0 to infinity for P/(1-P)). Since I don’t have the actual data (I know it’s public from comScore but gathering it all seems quite tedious) I went ahead and simulated it trying to preserve the features shown in the chart, so I can make my own charts and show what I mean visually.

Putting aside the doubtful notion of regressing a variable against its own predictor (instead of just showing the residuals), and the presence of serial correlation which is a whole other issue, my argument is that the clumping and tightening together of points not only makes reading the plot difficult, but more critically renders any regression inferences invalid since the plot as done features very obvious heteroscedasticity (http://en.wikipedia.org/wiki/Heteroscedasticity).

There a several options to fix this through extension of the ordinary least squares algorithm—some quite complicated—but in this case the simplest solution is to log the regression variables. Plotting it with log-log axes only makes it visually sound (points spread evenly), but to really achieve a valid regression model one must regress log(P/(1-P) instead of P/(1-P), an obvious choice since we already know, from the theoretical logistic model, that P/(1-P) is exponential.

I’ve attached my plots showing this with the simulated data. The key insight is in the pattern of residuals below the charts. Hope it helps.

• Walt French

Your point about autocorrelation goes unaddressedas you note, even your improved models have “trends” that are not captured by the trend, making us wonder what the trends really are.

First differencing (modeling the changes month-to-month) is the usual solution, with the obvious problem of making the models even LESS intuitive.

In the longer-term histories of competing technologies, seasonality and brand introductions eventually get washed out. In the ultra-rapid change of smartphones, a single product — most obviously, the original iPhone — can so redefine the industry that the chart becomes about the company’s ability to go from 0 to 150 in a couple of years, about the timing of its contracting with carriers, etc. These may account for some of the bumps in penetration, in fact, as the China Mobile anticipation shows, they may be huge.

I really like how Horace has plotted penetration; it makes a fine point about adoption…history. I’m very concerned about the implicit presumptions embedded in simple trending that price/performance will continue to improve as it has, that acculturation is fairly steady, that loyalty will continue to tie users to a product category when the NEXT big thing surprises us next week, etc.

Both to Apple and analysts, the real interest is what drives these trends, either along the current lines, or up/down from them. Even momentum investors need a better explanation than that the stock will keep going up because it has been.

• JaneDoe12

First, thank you VERY much for taking the time to make the 3 charts! And you’re right, the log-log plots have evenly spaced points.

I didn’t quite understand Horace’s 3rd chart before you commented. I missed the fact that the green line fits the blue so closely because the predictive formula is superbly accurate. Also, if iPhone sales continue to parallel the overall market, then at the end of 2016 Apple’s P/(1-P) will be about 1.8, which would make US penetration about 64%.

• You’re welcome, and thank you for your comments motivating me to work on simulating the data which was quite fun.

I’m sure you’ve seen Horace’s follow up post where he extends the iPhone projection and ends up with 68% penetration by February 2017, so your estimate of 64% by end of ’16 seems right.

I went to the latest comScore report (October) and grabbed the overall smartphone penetration of 62.5% (call this big P) and iPhone’s share of that of 40.6%, so iPhone’s penetration of all US mobile users is (40.6%)*(62.5%)=25.4% (call this small p). So we have P/(1-P) =1.67, ln(1.67)=0.51 for all smartphones and p/(1-p)=0.34, ln(0.34)=-1.08. Let’s compute the ratio 0.34/1.67=20.4%.

Now Horace’s predictive formula for the whole market says that this ln(P/(1-P) will grow linearly at the rate of one unit every 22.5 months. Your end of ’16 is 38 months after this latest October ’13 point, so the predicted ln() value would be 38/22.5=1.69 higher. For the whole smartphone market this is 0.51+1.69=2.20, then P/(1-P)=e^2.2=9.02, and finally P=9.02/(1+9.02)=90.0%.

If iPhone increases at the same, “parallel” logistic rate as the market, then the new ln() value currently at -1.08 would also increase by the same 38/22.5 amount in the 38 months, so it’d be at -1.08+1.69=0.61, then p/(1-p)=e^0.61=1.84 (you accurately said about 1.8), and finally p=1.84/2.84=64.8%. That ratio we computed before would then be 1.84/9.02=20.4% exactly the same value as before.

To summarize, assuming the logistic holds steady for the whole market in the next 3 years (nothing guarantees this), and iPhone keeps growing “in parallel” (no guarantee either), then the prediction is that iPhone’s share of the US smartphone market would be 64.8%/90%= 72% by the end of 2016.

In general the assumption that iPhone grows (1) “in parallel” to the (2) “logistic” market means that at any time:

ln(p/(1-p))=(constant displacement)+ln(P/(1-P))
p/(1-p)=(constant ratio)*P/(1-P)

This constant ratio is the one we’ve computed above as 20.4%. If we approximate this to 1/5, a nice straightforward solution from the above equation for iPhone’s share of smartphones p/P is 1/(5-4P) for any given total smartphone penetration P at any point in time (but again remember the two non-guaranteed assumptions).

• JaneDoe12

Daniel, I followed you all the way up to the end where p = 1/(5-4P). I don’t see how you got that, but please don’t take the time to explain because I probably won’t understand you. You are an amazing mathematician and I appreciate the time you’ve spent so far!

I hope the iPhone continues to grow in parallel to the whole market. And I read some good news today that bolsters my hope. Over the holidays, Apple was awarded a patent for an embedded heart monitor. [1] The main purpose appears to be for authorization. From the patent:

This is directed to an electronic device having an integrated sensor for detecting a user’s cardiac activity and cardiac electrical signals. …Using the detected signals, the electronic device can identify or authenticate the user and perform an operation based on the identity of the user. In some embodiments, the electronic device can determine the user’s mood from the cardiac signals and provide data related to the user’s mood.

Mood ring features aside, having a heart rate monitor could be an interesting addition to Apple’s devices. Exercise enthusiasts can use it to effortlessly monitor their heart rate. Biofeedback apps would most likely proliferate like wildfire. Apps could possibly become available that could diagnose more straightforward arrhythmia’s like atrial fibrillation, heart block, premature ventricular contractions (PVCs), and ventricular tachycardia. The legal liabilities for applications like these would be an interesting discussion.

This is good news that’s fun to think about.

1. Requires a subscription. Ed Ponsi. Back to Andrew on APPL. RealMoney(dot)com. Jan 2, 2014.

2. Barad J. Apple Files Patent for iPhone Embedded Heart Monitor. May 2, 2010. http://www.medgadget.com/2010/05/apple_files_patent_for_iphone_heart_monitor.html

• Hi Jane. Thanks for the kind words. I must make clear that I’m not a true mathematician (the stuff involved in real modern mathematics is just too nuts, inhuman even). My academic background is in engineering.

Sorry for omitting some of the algebra to get to that last formula. Notice I wanted to solve for p/P, the ratio of iPhone penetration to all smartphones penetration in the US, or iPhone’s share of smartphones, not just p.

Start from here:
p/(1-p)=(constant ratio)*P/(1-P)

I showed how this “constant” ratio was at 20.4% this last October. It will remain roughly constant as long as iPhone (p) stays logistically “parallel” to all smartphones (P).

I thought I’d approximate this 20.4% (0.204) to 0.2=1/5, so we have:
p/(1-p)=(1/5)*P/(1-P)
(1-p)/p=5(1-P)/P [inverses are equal]
1/p-1=5/P-5 [distribute and simplify on both sides]
1/p=5/P-4 [add 1 to both sides]
1/p=(5-4P)/P [factor 1/P on the right]
p=P/(5-4P) [inverses are equal]
p/P=1/(5-4P) [divide by P on both sides]

And there you have it.

Thanks for the patent info, very interesting.

• Whoops, for some reason that image got uploaded twice. It’s just 3 big charts side by side not 6, and below each chart there’s 2 sort of “sparklines” for the residuals analysis. Apologies for any confusion.

• JaneDoe12

Oops, I goofed-up my explanation. This is what I meant to say:

On log-log, the line would be parabolic. When x is between 0.2 – 0.6, the points would be more evenly spaced. But when it’s outside that range, the points get further apart.

• Ernest Stefan Matyus

Given that there is no data present on either ends of the chart I wouldn’t worry about clumping too much.

• Visually, the clumping may be a mere inconvenience. But analytically it violates one of the standard linear regression requirements (homoscedasticity) for valid inference. Simply regressing the log of the variables fixes this (although other issues remain). See my latest comment below in reply to JaneDoe12.

• N8nnc

The article appears truncated:
“The parallel behavior of iPhone and the overall market remains s”

• Hi, thanks. I fixed it.

• tfd2

i really wish we could fast forward a few quarters and see what happens to android – i know extrapolation is wrong, but it certainly looks like android is going to arc over, and if apple keeps following the overall market, it’s going to be in the lead. of course, this will somehow be spun as bad for apple.

• Kevin

So if Apple’s growth continues as it is now, then there will be around 173 million iPhone subscribers, around the viewership of the 2012 Super Bowl. Does that mean we can expect the US iPhone advertising business to be around \$11 billion? http://usatoday30.usatoday.com/money/economy/story/2012-02-04/cnbc-super-bowl-spending/52939344/1

• normm

Love these graphs, Horace! But since you’re willing to extrapolate the overall logistic growth, you should also be willing to say out loud what’s happening with Apple. If Apple’s straight line continues (and it certainly looks pretty straight!) they will have a 75% share of all cell phone subscribers in the US in another four years.

It seems to me that the Android curve mainly reflects manufacturer behavior rather than customer behavior. I wonder if you could predict the shape of that curve based on penetration among the N cellphone manufacturers of switching to making low-end Android smartphones rather than feature phones or Symbian?

• obarthelemy

The wonders of extrapolation ^^

• Sacto_Joe

As I said on 2.0 recently, I think that’s a good point, and one generally overlooked. Apple started into the mobile phone business pretty much from scratch, and has been building its production capacity ever since. But others, for example Samsung, were in place as manufacturers in volume. Their job, especially with a free OS and their willingness to rip off the ideas of others, was far, far easier than Apple’s job.

• charly

You act like being inspired by others is wrong and it is not like Apple isn’t inspired by others.

• DocNo42

Inspired is one thing. Outright theft is another. Guess which one Samsung was convicted of?

• Ian Ollmann

Horace, does the picture change if we remove the “barely functioning smartphones” sold as feature phones from smartphones? I realize your logistic plot may not end at 100%, anymore.

This would seem to weed out a lot of distracting data.

• Ian Ollmann

Norm captured this sentiment below in another way with his manufacturer preference observation. That is, when looking at consumer behavior it might be more useful to look at platform products with app store, OTA updates, etc. vs. standalone limited functionality devices, to understand where customers are going. The actual usage experience and job to be done of the “barely functioning smartphone” seems not a lot different from feature phones, so they probably should be lumped together.

• Vincent bowry

Jargon Alert! Sorry Horace but this might as well have been written in Ancient Macedonian if you don’t define your terms within the report.

• This is one of a long series of posts on the subject. It is by no means a “report”.

• sharrestom

Horace,

Your “Updated US Smartphone Saturation Forecast” post was linked in MacRumors through another link via Forbes writer Tim Worstall, who commented on that post.

I posted a direct link, but most Macrumors readers are commenting “eyes wide shut” as per usual. I apologize in advance if there is a tsunami of ill considered posts to follow.

• victor

Hi horace,
As always your thoughts have been very provocative. Here my comments on apples share predicton when the market reach saturation:

1_ I believe its too risk to predict the share of one individual player based on the data. Technology s curve theory explains the technology adoption not the individual products based on the new technology
2_ the data till now reflects the past and cant be extrapolated to the future without a theory. those individual curves will change if the basis of competition changes and that can be predicted by integration and modularization theory.

My disbelief on apples share prediction would fall if you could caractherize the i phone as a totaly different technology that has its own s curve, doing totaly different jobs from the others.