Random sequences are a bunch of random numbers arranged one after the other.
Something like this : 4, 6, 10, -6, -2, 1
Seems straightforward right? But, what if we wanted to bring some order to these sequences, while still retaining their randomness?
To get a sense of what this means, take a look at these charts :
Each chart is a plotted random sequence: The value of the number represented on the Y axis, and the position of the number of the sequence on the X axis.
(You can even REGENERATE them)
Even though each chart is random (in the sense that the next value cannot be guessed from any of the other values), the second chart seems much less random than the first, and the third much less than the second.
How are these sequences created and what can they be used for?
Derivatives of random sequences
The first chart looks like an absolute mess. Almost like those static radio signals you see in movies. This chart is made from a truly random sequence. This means that each value in the sequence is selected randomly, and independently. To generate this type of sequence, just roll some dice and note down the sequence of values that appear.
Let’s call this type of sequence a random position sequence (you’ll understand why in a bit).
The second chart looks a bit jagged, but much more ordered than the first one. In fact, it actually looks a lot like a chart of stock prices throughout the week
We’ll name this type of sequence a random velocity sequence. The method to generate this sequence is surprisingly simple : The value of each point in a “random velocity” type sequence is the sum of all previous points of a “random position” type sequence. So, if you have a random position sequence of :
[ 2, -1, 3, 9, -4, 3, 1 ]
- The first number in our random velocity sequence would be
2(since it’s the first element).
- The second number would be
2 + (-1), which is
- The third number would be
2 + (-1) + 3, which is
- …and so on
In the end we would get a random velocity sequence of :
[ 2, 1, 4, 13, 9, 12, 13 ]
The last chart looks much smoother than the other two, although with a few slopes here and there. Appearance wise, this looks similar to the slope of a hill or mountain.
To generate this type of a sequence, we perform the same action of summation, but this time on a random velocity sequence. Looking at the random velocity sequence in the last section, the new sequence we would get after summation would be :
[ 2, 3, 7, 20, 29, 41, 54 ]
We will call this type of sequence a “random force” type sequence.
Random sequences in real life
There’s a reason we named the above sequences as random force, velocity, and distance. To understand this better, we need to take a little deviation into physics and calculus.
If we have a time series function for the velocity of any object, we can find its position time series function by integrating the velocity function over time. It’s ok if you don’t know what this is, but what you should know, is that the discrete time equivalent of integration is summation. The exact same summation that was performed to obtain the “random velocity” and “random force” type sequences.
Similarly, the acceleration (which is proportional to force, for an object with a given mass) discrete time series is found by summation of its velocity sequence.
Now, the reason for these names should be more clear:
- The “random position” sequence describes the position of an object with respect to time, when its position is random.
- The “random velocity” sequence describes the position of an object with respect to time, when its velocity is random.
- The “random force” sequence describes the position of an object with respect to time, when its acceleration (or force for constant mass) is random.
Let’s attach these concepts to something relatable to really understand them…
Visualizing two-dimensional movement
We will now attempt to visualize the movement of an ant (or any other insect of your choice) in two dimensions. To do this, we first generate a “position × time” graph. This is generated for all 3 types of random sequences (position, velocity, and acceleration). The first graph is for the x coordinate of the ant, and the second graph is for the y coordinate of the ant.
First, lets start with random position :
This is nuts! It’s quite obvious that this does not look like how an ant would move around in real life (because ants cannot teleport). Let’s move on and model its movement with a random velocity type sequence :
This seems a little more realistic than the motion described by random position, but our ant is quite jittery, and its motion makes it look like its shivering.
Things do get better when we plot the motion described by coordinates generated by a random acceleration sequence :
Now, things start to look more realistic. The motion is smooth, and represents actual movement, even though the movement we are seeing is completely derived from random values.
It’s not a coincidence that the random force sequence more accurately depicts real movement. If you think about the real world, there is almost no object whose position is “random”. Rather, its position is affected by its velocity at any point in time, and its velocity is affected by the forces acting on it.
So, when you want to give something random motion, you don’t simply randomize its position. You randomize the thing that changes its position, by randomizing the thing that changes its change in position.
This can also be applied to almost any dynamic quantity, like stock prices or electricity, and can be extended in multiple dimensions (just like how two dimensional motion was simulated) to simulate multidimensional quantities (say, if we wanted to add a third dimension + direction orientation) to our existing simulation.
The source code for the chart generation and animation can be found here