Imagine that you are the Chief Chocolate Inspector of a chocolate factory. Your job is to ensure that the days production of chocolate by the factory is of good quality. Only upon your recommendation will the factory be shipping out its days worth of chocolate production to its distribution points.
The chocolate factory consists of three adjacent rooms connected together by a conveyor belt/assembly line.
The first room is where the chocolate pieces are produced and initially placed onto the conveyor belt. The factory produces a variety of decadent chocolate types simultaneously and randomly places each type onto the conveyor belt.
The second adjoining room is where each of the individual chocolate pieces get wrapped up. Afterwards, you are given a choice whether to choose that particular piece for taste inspection, or ignore it and wait for the next one.
The final adjoining room is where the chocolate pieces get packaged together in boxes and loaded onto the delivery trucks. The trucks only move upon your decision at the end of the day.
Your goal, as the Chief Inspector, is to notify the truck drivers if the packages are suitable for shipping in as efficient manner as possible. Armed with a special selection bin to hold pieces of chocolate, you decide to choose uniformly random pieces of chocolate out the total pieces of chocolate produced by the factory throughout the day, where . Afterwards, you base your taste decision on this random subset.
Unfortunately, you have no idea as to how many chocolate entities will be passing by on any given day. Some days, the amount of chocolate produced by the factory is very high, on others it is pretty low. In other words, you do not know what is beforehand.
While you could wait to make the sampling after the days production is done; you don’t want to hold up the assembly line making these quality control choices. “I Love Lucy” has shown what kind of hijinks can happen if this process goes awry.
What to do?
One solution to this problem might be to roll a die as each chocolate piece passes you. If the die rolls even, hold the chocolate for inspection in the selection bin. Otherwise, let the chocolate pass through. If the selection bin is full when choosing a chocolate piece, simply replace the oldest chocolate piece in the bin with the new one.
In this naïve approach, one is biased against keeping the earliest chocolate samples with the ones selected later in the day. To resolve this issue, you can use a loaded die or get more “Dungeons & Dragons”-esque by using more sophisticated dice and rules as the day progresses.
Then with some luck, you could get a uniform sampling of the days production of chocolate.
But there is a better way…
A Smarter Way
Suppose you could assign a uniform random (i.i.d of course!) number from to each piece of chocolate you encounter on the assembly line.
Now, to obtain a sampling of the days production of chocolate, simply choose the chocolates corresponding to the lowest random numbers linked with them. Each chocolate piece has an equal chance of being in the lowest choices, so the sampling is uniform.
With the ubiquitous computing power available these days, you could substitute using dice with a iPhone or Android app to help out. Each time you see a new chocolate piece, generate a random number with your app. Hold onto the chocolate pieces that correspond to the lowest random numbers. If you encounter a new chocolate piece that is linked with a random number that is lower than the largest random number associated chocolate already in your selection bin, simply replace the old one with the new one.
By the end of the process you now have uniform samples of the entire days chocolate production, and you never needed to hold onto more than those items. Pretty clever, eh!
This approach could be optimized even further. For example, you could preemptively generate a large list of random numbers; identify the lowest ones; and afterwards directly choose those chocolate pieces as they appear off the production line and bypass the rest.
Amazingly, you could sample the entire days production of chocolates, , in less than time. According to this fellow, you can make the process efficient.
These techniques of sampling a set of unknown size through one pass are called reservoir algorithms.
The “Real” World
While most of us are not chocolate inspectors, we are all nowadays swamped in a “sea of data”, from science and business to personal tracking. Often times these raw data sets come in large chunks. One data-mining technique to get a better handle of the data is sampling it. Sometimes less is more. Now chocolate pieces become record sets and the assembly line becomes a large file.
Reservoir algorithms can be a useful tool in one’s data science & engineering endeavors.