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“Bonjour Monsieur!” Noise and Dynamics in Data.

Noise and dynamics are two words with a precise mathematical meaning. For now though, lets imagine the colloquial meaning of noise or more specifically, let’s consider the noise in a classroom.

Firstly, as it’s the weekend, it would probably be silent. So lets think about a regular term-time weekday; Monday afternoon. There are two classes – 1.30pm until 2.30pm, 2.30pm until 3.30pm, then the bell will ring and silence will fall as everyone runs off for ice-cream (lets say it’s summer and above 17 degrees!)

From 2pm, lots of children enter the classroom, all talking at once. The class has just come back from lunch and the teacher is yet to enter the room. There is a fairly low level of murmuring, playground conversations are continuing. Three minutes of this pass until the teacher enters the room. Hence we expect 3 minutes of noise before the continuous sound signature of a class in process.

Let’s say it’s French class, and when the teacher finally enters, all those bright eyed, eager 12 year olds are trained to stand up and chime ‘Bonjour Monsieur’ in unison, before throwing themselves back down onto a wooden chair ready for a vocabulary test.

Assuming this what happens, the noise has been predictably interrupted by an outside source (exogenous)- the teacher entering the room. We can visualise this as below in Figure 1:Figure_1The class continues until 3pm, at which point the children file quietly out of the room to their next lesson, and the next hour cycle begins. Every hour of each Monday afternoon, we expect a similar noise signature, indicating the start of the lesson.

At 3pm we get a different scenario- the teacher is running 8 minutes and 23 seconds late. The corridor conversation continues, a pupil in the middle row tells and a joke, and there is laughter. The pupils at the front have to raise their voices to carry on gossiping, so the next joke is told louder and gradually all noise in the room is amplified.

There are a few possible endings here:

  1. Adjusting to the higher amplitude of chatter, the noise of the class settles to a higher level than before.
  2. The noise gets higher and higher, until the teacher next door enters and silences the class by shouting. They wait silently for their teacher.

Figure 2 shows us scenario b) followed by the physics lesson:

Figure_2Dissecting this gives us a signature for a Physics lesson and the French lesson, seen in Figure 3a:Figure_3aA signal for noise during standard waiting, a signal that the teacher is late, and the threshold at which the teacher next door comes in to shout is illustrated in Figure 3b:Figure_3bHere’s how we break this down into components according to non-linear dynamics:

Component 1: Internal Feedbacks

The rise in noise due to the delayed teacher comes from within the class. The joke occurs solely within the classroom, so this is a feedback ‘from within’ (endogenous).

Component 2: External Perturbations

The class requires the perturbation of the teacher arriving on time to prevent it from moving to an elevated level. The teacher arriving late is due to external factors rather than what is happening in the room. So we say this is an event ‘from outside’ (exogenous).

Component 3: System to System Feedbacks

If it passes a threshold, then the teacher arrives from next door and the noise is silenced. The teacher coming over from next door is due to the state of the classroom, so this is a dynamic triggered by the breach in noise threshold.

The Physics teacher and the French teacher both bound the noise, the Physics teacher by his timing, the French teacher by his tolerance.

Final thought towards reality:

Lets assume the classes are actually happening next door to each other at the same time. It’s a warm day (remember the ice-creams), so the doors to both rooms are open, and due to confidentiality we can only record from the corridor.

We pick up both classroom signatures at once. Furthermore, when the French teacher comes in to shout at the Physics class, the French class murmur to themselves, quickly conferring on whether it is ‘un pomme de terre’ or ‘une pomme de terre’.

Look at Figure 4:

Figure_4

Extracting those signals just got a bit harder. Welcome to the life of a data scientist.