Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 9

Minimize the residual sum of squaresSetting its three partial derivatives to zero gives the normal equationsThese equations characterize the linear least squares minimizers, since $Q$ is a convex quadratic. A unique coefficient vector additionally requires full column rank of the design matrix.

The centered regressors make the first normal equation read $n{\hat{\beta }}_0={\sum }_i{Y}_i$. DefineThe remaining equations are $A{\hat{\beta }}_1+B{\hat{\beta }}_2=u$ and $B{\hat{\beta }}_1+C{\hat{\beta }}_2=v$. Provided $D=AC-B^2>0$, matrix inversion givesBy the Cauchy-Schwarz inequality, $D\ge 0$. If $D=0$, the two regressor columns are linearly dependent, and their separate slopes are not uniquely identifiable. One still has well-defined linear least squares fitted values, but cannot use the displayed divisions by $D$.

If the two regressors are strongly associated, the data may provide little support for varying one while holding the other fixed. Multicollinearity makes the separate slope estimates unstable, and exact linear dependence makes them unidentifiable. A conditional slope may also be an extrapolation beyond combinations actually observed.

Furthermore, a slope in a linear regression model is not automatically a causal effect. A causal interpretation needs appropriate design or assumptions excluding confounding; independence of the errors across observations and zero mean alone do not establish those assumptions. The coefficients describe the specified linear mean relation, but changing a regressor by intervention can have a different interpretation.

Principal component analysis finds orthogonal directions of greatest sample variation, ordered by the variance explained. It can summarize correlated variables, reveal lower-dimensional structure, and provide a reduced representation by retaining only the first few principal components. Centering is standard; scaling may also be appropriate when the variables have incomparable units.

Let $X_c$ be the column-centered data matrix. Its sample covariance matrix is $S=X_c^T{X}_c/(n-1)$. The loadings matrix $V=(v_1,\dots ,v_p)$ has orthonormal eigenvectors of $S$ as columns, ordered by nonincreasing eigenvalues:Each principal component loading specifies the coefficients of a principal direction in the original variables. Signs are arbitrary, and equal eigenvalues permit orthogonal changes of basis within their eigenspaces.

The principal component scores are the projections of centered observations onto the loading directions:If the symbol $X$ already denotes centered data, this simplifies to $Z=XV$. The principal component loadings are directions; the principal component scores are coordinates of individual observations in those directions.

Here both regressor columns are already centered, so $Z=XV$ and $X=Z{V}^T$. Writing $\beta =({\beta }_1,{\beta }_2{)}^T$, the linear predictor is $X\beta =Z{V}^T\beta$. ThusThe intercept stays ${\beta }_0$, because centering made the transformation a rotation of the regressor columns without an added translation.

The new score columns are orthogonal, since $Z^{T}Z=V^T{X}^{T}XV$ is diagonal. This decouples the slope normal equations; with positive eigenvalues and the stated error covariance, the two estimated score coefficients are uncorrelated. It also makes variance explained and possible dimension reduction explicit.

The trade-off is interpretation: each principal component is a mixture of the original variables, so its regression coefficient is less directly connected to changing one original input. Keeping both components is an invertible reparameterization and gives exactly the same fitted values and residual sum of squares. It does not repair a zero eigenvalue or remove instability associated with a very small one. Dropping a low-variance component may improve numerical stability and reduce estimator variance, but changes the model and can discard predictive information, since principal component analysis uses predictor variance rather than the response. This is the distinction between a full rotation and principal component regression with truncation.