My purpose of paper is to show that sometimes correlation between parameters can be by chance and not by causality.
Definition:
Pearson correlation coefficient (usually called merely ‘correlation coefficient’) measures a linear relationship between two variables. It’s worth noticing Pearson correlation coefficient reflects an extent of linear relationship (not exponential or quadratic relationship).
The range of the correlation coefficient is [-1.0,…,1.0], the more the coefficient close to 1 or -1 the more linear relationship found between two variables. If the coefficient is zero or nearby then no linear correlation is detected.
Use case from video compression
You are required to estimate a linear relationship between a scene complexity (e.g. the percentage of edge pixels) and the generated frame size, where an encoder is configured with constant QP (otherwise if Rate Control is applied frame sizes are similar in magnitudes. In case with Rate Control it’s worth checking a relationship between video quality and complexity).
You expect the following: more complex scene more bits are produced. To check the relationship you choose 10 clips (or scenes), the complexity of each scene is pre-computed. Then you apply the encoder on each excerpt and calculate Pearson correlation coefficient and plot the scatter graph (schematically):
Let’s assume that the correlation coefficient r = 0.5.
Now the question is raised – maybe, no linear relation exists and we get r=0.5 by chance?
What’s a confidence (or a probability) of getting r=0.5 provided that no linear relationship exists?
Or in statistics jargon, is the correlation coefficient r=0.5 is significant or not?
Significance testing of Pearson correlation coefficient is based on t-student distribution with (n-2) degrees of freedom, where ‘n’ is the number of samples (in our case 10):
We test two hypothesis:
- Null Hypothesis: complexities and frame sizes are linearly unrelated, i.e. r = 0
- Alternative Hypothesis: complexities and frame sizes are in someway related, r ≠ 0 (two-sided) or r>0 (one-sided, i.e. a positive linear relationship conjectured).
So, for our case with r=0.5 and n = 10 (8 degrees of freedom) we get t-value ~1.64.
According to t-student table for 8 degrees of freedom the confidence level that there is a linear relationship between complexities and frame sizes is around 85%:
Usually the confidence of 85% is not sufficient to reject the null hypothesis. It’s common rule: the null hypothesis can be rejected if the confidence is above 95%
Even if we consider the alternative hypothesis with positive linear relationship the confidence is ~93%.
Conclusion
The confidence 85% of a linear relationship between complexities and frame sizes is not sufficient to assert that a correlation between these two variables exist.
23+ years’ programming and theoretical experience in the computer science fields such as video compression, media streaming and artificial intelligence (co-author of several papers and patents).
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