Manual test

Purpose

This validation reconstructs the fitted parameters of gipslda(), gipsqda(), and gipsmultqda() independently. The reference calculation uses only base R, stats, and the public gips API. It does not call any gipsDA helper or internal function.

The executable version is tests/testthat/manual/test-manual-parameter-reconstruction.R. The formulas and implementation notes are kept beside the calculations in that test so that differences between the reference and package implementations remain reviewable.

Mock data in $\mathbb{R}^3$

Three Gaussian classes are generated with sample sizes 9, 10, and 11. Their means are

$$\mu_A=(-1.5,0.2,0.8),\quad \mu_B=(0.6,1.7,-0.5),\quad \mu_C=(1.4,-1.0,1.2).$$

Each class has a different positive-definite covariance matrix:

$$\Sigma_A = \begin{pmatrix} 1.20&0.35&0.10\\ 0.35&0.80&-0.15\\ 0.10&-0.15&0.65 \end{pmatrix},\quad \Sigma_B = \begin{pmatrix} 0.75&-0.20&0.18\\ -0.20&1.35&0.30\\ 0.18&0.30&0.90 \end{pmatrix},$$

$$\Sigma_C = \begin{pmatrix} 1.05&0.25&-0.22\\ 0.25&0.70&0.12\\ -0.22&0.12&1.25 \end{pmatrix}.$$

With a fixed seed, standard-normal matrices $Z_g$ are transformed as

$$X_g = Z_g\,\operatorname{chol}(\Sigma_g) + \mu_g.$$

The unequal class sizes test the empirical priors and the sample-size vector used by the joint projection. Three dimensions keep brute-force permutation search small and deterministic.

Covariance projection

For every reference projection, the test constructs a gips object directly:

search <- gips::gips(
  empirical_covariances,
  sample_sizes,
  was_mean_estimated = TRUE
)
search <- gips::find_MAP(
  search,
  optimizer = "BF",
  show_progress_bar = FALSE
)
permutation <- search[[1L]]
projected <- lapply(
  empirical_covariances,
  gips::project_matrix,
  permutation
)

When only one covariance matrix is projected, that matrix is passed directly to gips::gips(). The brute-force optimizer enumerates the permutations in $\mathbb{R}^3$, avoiding Monte Carlo variation.

The projected covariance $S$ is regularized only if its eigenvalue $\lambda$ nearest zero satisfies $|\lambda| < 0.05$. In that case,

$$s=\frac{0.05-\lambda}{1-0.05}, \qquad S^*=\frac{S+sI}{1+s}.$$

This transformation is implemented explicitly in the test.

LDA reconstruction

Let $M_g$ denote the sample mean of class $g$, and let $E_i=x_i-M_{g_i}$ be the within-class residuals. Define

$$D=\operatorname{diag}\left( \frac{1}{\sqrt{\operatorname{diag}(\operatorname{var}(E))}} \right).$$

For $n$ observations and $G$ classes, the covariance supplied to gips is calculated manually as

$$S_W=\frac{n}{n-G}\operatorname{cov}(ED).$$

After MAP projection and regularization, write $S_W^*=V\operatorname{diag}(d)V^\mathsf{T}$. The whitening transform is

$$W=DV\operatorname{diag}(d^{-1/2}).$$

Using the empirical priors $\pi_g=n_g/n$ and $\bar{x}=\sum_g\pi_gM_g$, the between-class matrix is

$$B_g= \sqrt{\frac{n\pi_g}{G-1}}\,(M_g-\bar{x})W.$$

If $B=U_B\operatorname{diag}(d_B)V_B^\mathsf{T}$, the stored discriminant scaling is $WV_B$, truncated using the same numerical rank rule as the public model. The test independently compares:

• empirical priors, class counts, means, labels, and sample size;
• the MAP permutation;
• between-class singular values;
• the discriminant scaling after resolving arbitrary SVD column signs.

Separate QDA reconstruction

For each class, the sample covariance

$$S_g=\operatorname{cov}(X_g)$$

is projected in a separate brute-force gips search. The total sample size $n$ is supplied to each search, matching the estimator’s statistical parameterization. If

$$S_g^*=V_g\operatorname{diag}(d_g)V_g^\mathsf{T},$$

the reference parameters are

$$L_g=V_g\operatorname{diag}(d_g^{-1/2}), \qquad \log|S_g^*|=\sum_j\log d_{gj}.$$

Because singular vectors can change sign without changing the model, the test compares $L_gL_g^\mathsf{T}$, the precision matrix used by QDA, rather than the signs of individual columns.

Joint QDA reconstruction

The joint model starts from the same class covariances but passes $(S_A,S_B,S_C)$ and the sample-size vector $(9,10,11)$ to one gips search. Its single MAP permutation is then used to project every covariance. The scaling matrices and log determinants are calculated from the resulting matrices using the QDA equations above.

This distinguishes gipsmultqda() from gipsqda(): the former estimates one shared symmetry, while the latter estimates each class symmetry independently.

Assertions

The three public fitting functions are called only after all reference parameters have been computed. Numerical quantities are compared with tolerance $10^{-10}$. Discrete quantities, dimensions, labels, sample size, and MAP permutations are compared exactly. The QDA scaling checks use precision matrices, and the LDA scaling check aligns SVD signs before applying the tolerance.

The test therefore validates the complete set of fitted numerical parameters, apart from the recorded call, against a calculation that is independent of the package implementation.