Wanna predict the future? Math can do that!

Have you ever wondered how to predict outcomes of events? By looking at past trends in, say, consumer habits at a convenience store, we can use matrices to predict if a new product is set to be successful in the long run, and all we need is a set of data collected over a short span of time. For example, you manage the release of a new product line of energy drinks at your local convenience store and wish to predict, early in production, if your product is going to earn a fair market share. By using the matrix operations, you can foretell how successful it will be (assuming consumer habits do not change, which they usually don't). It's called the transition matrix and here's how it works:

You collect information about purchases made for your product and its competitor. Let's call them Product A (Your product) and Product B (the competitor). I managed to collect this information for a span of 3 days: (I have converted the percentages into decimals for simpler calculations)

	Product A	Product B
Market Share	30% (0.3)	70% (0.7)

Now, upon surveying the manager and reviewing the inventory logs, I have collected the following information on purchases made on energy drinks; 20% of customers who bought Product A switched to Product B for their next purchase. This means that 80% of the customers who bought Product A remained with the same product (100-20=80). I also found that 60% of customers who purchased Product B switched to Product A for their next purchase. This means that 40% of the customers who purchased Product B remained with the same product (100-60=40):

	Product A	Product B
Product A	80% (0.8)	20% (0.2)
Product B	60% (0.6)	40% (0.4)

Notice that the elements in the rows add up to 100%. This is ALWAYS true.

By using this data in table 2 as a reference, we can standardise this by making it constant assuming customers retain their habits. After converting the above into a matrix form, we call this the transition matrix. Note that it is a square matrix. This is ALWAYS true (since the labels for rows must be the same as labels for columns)

If we then multiply the above data for current market share by the transition matrix (change), we can predict the market share after one round of purchases on energy drinks. Ensure table one converts into a row matrix so that the matrix dimensions match for multiplication.

We have now calculated the new market share after one round of purchases. What happens for the next round? We multiply by the transition matrix (change matrix) again…

What happens after the third round? We multiply by the transition matrix again…

What do you notice? We are merely multiplying the initial market share by exponents of the transition matrix to find the share after the nth round:

By plugging in a value for ‘n’, we can predict the market share after ‘n’ rounds (the share after the 5^th round can be calculated by plugging in 5 where ‘n’ is in the above equation)

If you repeat this process, you will notice that the market share begins to stabilize. The change becomes so small from one round to the next that it becomes insignificant. It’s called the stabilization point, and is ALWAYS present for transitional matrices. For the above example, the market share stabilizes at 75% for product A and 25% for product B after the 5^th round. It stays the same even after the 100^th round. It is said to be Stable.

To conclude, we have been able to successfully predict the market share of energy drinks for a long time by only using data from a small span of time. ie: We have predicted the future. Pretty cool, eh?