Despacito is the most viewed video on Youtube: An introduction to integral calculus

July 11^th, 2017

Despacito vs See you again… Goodbye Gangnam style.

On April 20^th I was in Milan. That night, the first word I heard when I arrived at the Naviglio Pavese was, “Despacito.” It amused me, but when I heard it again and again in the groups of people that crossed by my side, I realized something strange was happening, and I was not mistaken. That day, Luis Fonsi had surpassed one billion views on Youtube… The next day, I traveled to Munich, and when I arrived the first thing I did was to ask a German friend if she knew the song, she liked rock so she said no, but two months later when I spoke to her again, she told me that she could no longer tolerate it, she has been hearing the song everywhere. Exactly two months later, on June 20^th, Despacito had reached 2 billion views on YouTube. Today, July 11^th, 2017, the ranking of the most viewed Youtube videos is as follows:

• See You Again – Wiz Khalifa
• Gangnam Style – PSY
• Sorry – Justin Bieber
• Uptown Funk – Bruno Mars
• Despacito – Luis Fonsi

Gangnam Style has just lost the throne! However, the question is, how far can Despacito get? The answer lies in mathematics, and that is why today, banks, insurance companies, among other financial groups, are obsessed recruiting mathematicians because they are the new oracles of the world. Able to predict the future and generate strong money gains with their probabilistic stochastic models.  In this article we won’t go deep in the theory of probability or mathematics in finance, but we will give some insight about them. To begin, in the following graph we show the evolution of the songs in the last six days (or 144 hours), starting on July 4^th at 00:00 and culminating on July 9^th at 23:59 PM (UTC +2).In the vertical axis we show the number of Youtube views and in the horizontal axis the time interval expressed in hours. Clearly, we observe that as time evolves, Luis Fonsi grows much more than the others. Moreover, Wiz Khalifa is very close to surpass PSY. Maybe you cannot see it well, so here we show a zoom of PSY vs Wiz Khalifa:We have to recall that the graph shows the evolution of Youtube views from July 4^th – July 9^th, that is, when PSY was in the first position. Why I’m convinced from the graph that Wiz Khalifa will surpass PSY? The truth is that I should not be; I’m simply predicting the future based on a “mathematical hunch” where I study the historical behavior of the number of Youtube views to project future data. In order to do so, it is necessary to estimate the slope of each line using the ordinary least squares method, and then, consider the trendline equations (given in the form f(x) = mx + b) in order to predict the future in which my hypothesis will occur. In simple terms, we will extend our lines in the graph and see where they intersect, with this we can deduce the exact moment in which Wiz Khalifa surpasses PSY and Despacito becomes the video with most views on Youtube. Mathematically, e.g., if the trendline equations of Wiz Khalifa (w) and PSY (p) are given by:

f(x_w) = m_wx_w + b_w
f(xp) = m_px_p + b_p

The exact moment in which Wiz Khalifa dethrones PSY is given by f(x_w) = f(xp). We will not make this cumbersome so here we show the final result (UTC +2):

• Wiz Khalifa dethrones PSY on July 10^th at 2 PM.
• Luis Fonsi beats Bruno Mars on July 14^th at 8:30 AM.
• Luis Fonsi beats Justin Bieber on July 18^th at 9:30 AM.
• Luis Fonsi surpasses Khalifa and Despacito becomes the video with most views on Youtube on July 31^st at 6:30 AM.

Can you guess what would happen if instead of talking about songs the graph would show the value of company shares? Many, if not all, would invest in Luis Fonsi’s company because they would be convinced that beginning August these shares will become the most expensive ones, and therefore will generate greater profits. However it is not as easy as it seems, because in the financial world the graphs are more complex and depend on many factors, both endogenous and exogenous. In this sense, the most sophisticated models that predict the future behavior of an event consider variables as stochastic, that is, random variables (non-deterministic) exposed to external shocks that can change the “normal” course of history. In this example, the number of views is a random variable, because I do not know exactly how many people will watch the Youtube video one hour after I publish this article, but I can dare to say a number with a certain level of significance, that is, the prediction I make is subjected to a margin of error that depends on the model I use. In the example outlined at the beginning, where we estimated the slopes of the lines, we omitted a number of exogenous factors that affect the number of Youtube views, such as people’s routine, internet access, each individual’s diminishing marginal utility (e.g., the fact that songs soon or later become annoying), among other factors. How long does it take to get bored of listening the same song? We do not know, each song has different behaviors. We only know that at some point, we will get bored of listening the same thing. If we want a slightly more sophisticated model, where we can use additional variables, statistical concepts such as “Bayesian Theory” and “Time Series” are needed. Yes, sounds complicated.

To understand the basics, let’s focus on Despacito. This song was released on January 12^th, 2017. 97 days later it reached a billion views on YouTube, which means that on average it accumulated 10 million views per day from January 12^th to April 20^th. With this data, can we infer what will happen next? Not necessarily, because the behavior is stochastic, then there are exogenous factors that can alter trends. One of these factors has a name: Justin Bieber. On April 17^th, he altered the probabilistic data with his Despacito remix. This alteration gave a positive shock to the number of views, because Justin Bieber, due to the number of fans he has worldwide, transformed the probabilistic growth of Fonsi, who became more popular in english-speaking countries. This does not imply that Fonsi could not have done it himself, perhaps yes, but it was Bieber the external variable that gave a push to the song. On June 20^th, Despacito reached 2 billion views, that is, in 60 days (from April 20^th to June 20^th) it accumulated 16.7 million views per day. Now, what is happening in the last weeks (until July 9^th)? The following histogram shows us:

Although the song can be considered old, Fonsi, within the last weeks, is enjoying its highest peak, reaching on average 21 million views per day, with peaks beyond 24 million views on weekends. Can Fonsi grow beyond these values? Probably no. Although the calculations are not shown here, so as not to make the discussion cumbersome, one can estimate that Fonsi has reached stable values and is likely to decrease “despacito” and “poquito a poquito” in the coming weeks; nonetheless, there will always be exogenous factors that will alter the result.

To finalize, it is important to mention that in the last days (until July 9^th) the Youtube views are growing as follow:

Because these are weekend values, they correspond, on average, to the highest values that each singer can achieve. These data enable us to forecast the trends of Khalifa and Fonsi, the future leaders of Youtube. Removing exogenous factors, we can intuitively guess that in the near future W. Khalifa will drop to 2.3 million views per day because these are the stable values ​​of Justin Bieber and Bruno Mars. In the end, these results put “See You Again” as the most viewed video on Youtube in the month of July.

In our first analysis we said that W. Khalifa would reach the first place on July 10^th at 2PM (UTC +2). Officially he did it at 11:30PM (UTC +2). This confirms all what we have just said, it is nice though to realize that in the most basic analysis, our prediction remained very close to reality. Additionally, we mentioned that Luis Fonsi would surpass W. Khalifa on July 31^st. This won’t be the case because, as we have said, Fonsi has reached stable values, declining “poquito a poquito” in the coming weeks. So, what will happen on July 31^st? Most likely, Luis Fonsi will step on the feet of W. Khalifa, perturbing fans of both singers, this perturbation will become an important exogenous variable in the month of August, as we don’t know what fans may be capable of.

The pending question would be, which song will become the first to reach 3 billion views?

The curious thing about this question is that if we use the graphs shown at the beginning, on July 31^st W. Khalifa and L. Fonsi would reach 2 964 829 089 views. Meaning that they would require 35 million views to reach the historical number of 3 billion views. This can be achieved by Luis Fonsi in a couple of days if he keeps a trend of 21 million views per day. That is, on August 2^nd, at best, Despacito would become the first video clip to reach that number. It would be a historic moment in the internet world… but it won’t be the case. As mentioned, Fonsi will hardly keep his growth unchanged until July 31^st, which means, and here is the funny thing, that it may happen that W. Khalifa and L. Fonsi together will reach 3 billion views at a later date.

Granting to Wiz Khalifa a lower bound of 2.3 million views per day (as given from the previous table), it can be deduced that he will reach 3 billion views, in the worst case, on August 24^th. Therefore, we can conclude,

Everything seems to indicate that in August Luis Fonsi and W. Khalifa will make history, with a high probability on the threshold of the second week, this is, finalizing the first one and starting the second one.

Christian Ortiz, PhD.