Orthogonal Decomposition: Any vector in $V$ can be uniquely decomposed into components in a subspace $W$ and its orthogonal complement $W^{\perp}$. This enables the definition of the orthogonal projection operator $P$, which is self-adjoint and idempotent.
Matrix Representations of Projections: The projection matrices $P, Q$, and $R$ clearly demonstrate how projection operators work in terms of standard basis transformations. The sum $Q+R=I$ confirms that every vector decomposes into components in $W$ and $W^{\perp}$.
Connection to Statistics: Correlation is naturally interpreted as the cosine of the angle between two vectors. Partial correlation, in turn, corresponds to the cosine of the angle between their projections onto a hyperplane orthogonal to a third variable.
Geometric Intuition of Partial Correlation: By removing the effect of a confounding variable $z$, the residual vectors $r_x$ and $r_y$ give a clearer picture of the direct relationship between $x$ and $y$. This aligns well with the interpretation of partial correlation in regression analysis.
We can apply the concepts of orthogonal projection and partial correlation to a real-world dataset, such as economic indicators or health metrics. Let's take an example from finance: the relationship between stock returns, market returns, and inflation rates.
Scenario: We have a dataset where:
We want to:
We'll use NumPy and SciPy to compute projections and partial correlations.
We simulate stock returns, market returns, and inflation rates.
We project stock returns and market returns onto the hyperplane orthogonal to inflation.