In the previous post, we introduced the notion of leverage in linear regression. If we have a response vector and design matrix , the hat matrix is defined as , and the leverage of data point is the th diagonal entry of , which we denote by . It is so called because it is a measure of the influence that has on its own prediction : . The higher the leverage, the more influence has on its own prediction.
It turns out that leverage must satisfy the following bounds:
for all .
This is easy to prove using the following 2 facts:
- Note that , i.e. is idempotent.
- Note that , i.e. is symmetric.
Since , using the symmetry of it follows that
- Since the RHS is a sum of square numbers, .
- Since the second term on the RHS is a sum of square numbers, , and since , we have .
There is also a constraint on the sum of leverages which is easy to derive. By the cyclic property of the trace operator,