We always hear about this vector of data VS this other vector of data being independent from each other, or uncorrelated, etc, and while it is easy to come across the math regarding those two concepts, I want to tie them into examples from real-life, and also find ways to measure this relationship.
From this stand point, I am looking for examples of two signals that are of the following combinations: (I will start with some):
Two signals that are independent AND (necessarily) uncorrelated:
- The noise from a car engine (call it $v_1[n]$) and your voice ($v_2[n]$) as you are talking.
- A recording of humidity every day ($v_1[n]$) and the dow-jones index ($v_2[n]$).
Q1) How would you measure/prove that they are independent with those two vectors in hand? We know that independence means that the product of their pdfs equals their joint pdf, and thats great, but with those two vectors in hand, how does one prove their independence?
- Two signals that are NOT independent, but still uncorrelated:
Q2) I cant think of any examples here ... what would some examples be? I know we can measure correlatedness by taking the cross-correlation of two such vectors, but how would we prove that they are also NOT independent?
- Two signals that are correlated:
- A vector measuring an opera singer's voice in the main hall, $v_1[n]$, while someone records her voice from somewhere inside the building, say in the rehearsal room ($v_2[n]$).
- If you continuously measured your heart rate in your car, ($v_1[n]$), and also measured the intensity of blue lights impinging on your rear windshield ($v_2[n]$)... I am guessing those would be very correlated... :-)
Q3) Related to q2, but in the case of measuring cross-correlation from this empirical stand point, is it enough to look at the dot product of those vectors (since that is the value at the peak of their cross-correlation)? Why would we care about other values in the cross-corr function?
Thanks again, the more examples given the better for building of the intuition!