Contents
What is the relationship between independent, correlation and orthogonality?
Unlike that independent is a stronger concept of uncorrelated, i.e., independent will lead to uncorrelated, (non-)orthogonal and (un)correlated can happen at the same time. I am being the TA of probability this semester, so I make a short video about Independence, Correlation, Orthogonality.
Why is it simpler to use a non-orthogonal basis?
So the main reason why using an orthogonal basis is simpler is that you (want to) keep your obvious metric. Changing to a non-orthogonal base while keeping the orthogonal metric will make your formulas more complicated.
When is the vector no longer orthogonal to y?
The vector is no longer orthogonal to Y. If two variables are uncorrelated they are orthogonal and if two variables are orthogonal, they are uncorrelated. Correlation and orthogonality are simply different, though equivalent — algebraic and geometric — ways of expressing the notion of linear independence.
When is there no correlation between two variables?
When there is no relationship between two variables this is known as a zero correlation. For example their is no relationship between the amount of tea drunk and level of intelligence. A correlation can be expressed visually.
When do two independent random variables become orthogonal?
And no, orthogonal does not mean that the two variables are independent. Further, independence does not imply orthogonality. But we can state the following: two independent random variables are orthogonal if, and only if, at least one of the two R.V. has mean: .
When is the correlation coefficient of a variable orthogonal?
Therefore this correlation coefficient can be seen as a distance function, which is when the variables are orthogonal or , with maximum value and minimum value . At this point, notice that if linearly depends on : then we obtain that the correlation coefficient is equal to or , depending on the sign of the coefficients.
Why are the regressors in ANOVA mutually orthogonal?
If the original regressors are mutually orthogonal and one applies orthogonal contrasts, the new regressors are mutually orthogonal, too. This ensures that the set of contrasts can be seen as describing a decomposition of variance, e.g. into main effects and interactions, the idea underlying ANOVA.