Theory

Theory the gips is based on

The package is based on the article [1]. There the math behind the package is precisely demonstrated, and all the theorems are proven.

In this vignette, we would like to give a gentle introduction. We want to point out all the most important results from this work from the user’s point of view. We will also show examples of those results in the gips package.

library(gips)

As mentioned in the abstract, the outline of the paper is to “derive the distribution of the maximum likelihood estimate of the covariance parameter \(\Sigma\) (…)” and then to “perform Bayesian model selection in the class of complete Gaussian models invariant by the action of a subgroup of the symmetric group (…)”. Those ideas are implemented in the gips package.

Basic definitions

Let \(V=\{1,\ldots,p\}\) be a finite index set, and for every \(i\in V\), \(Z^{(i)}=(Z_1^{(i)},\ldots, Z_p^{(i)})^\top\) be a multivariate random variable following a centered Gaussian model \(\mathrm{N}_p(0,\Sigma)\), and let \(Z^{(1)},\ldots, Z^{(n)}\) be an i.i.d. (independent and identically distributed) sample from this distribution. Name the whole sample \(Z = (Z^{(1)},\ldots, Z^{(n)})\).

Let \(\mathfrak{S}_p\) denote the symmetric group on \(V\), that is, the set of all permutations on \(\{1,\ldots,p\}\) with function composition as the group operation. Let \(\Gamma\) be an arbitrary subgroup of \(\mathfrak{S}_p\). The model \(\mathrm{N}_p(0,\Sigma)\) is said to be invariant under the action of \(\Gamma\) if for all \(g\in \Gamma\), \(g\cdot\Sigma\cdot g^\top=\Sigma\) (here, we identify a permutation \(g\) with its permutation matrix).

For a subgroup \(\Gamma \subset \mathfrak{S}_p\), we define the colored space, i.e. the space of symmetric matrices invariant under \(\Gamma\), \[\mathcal{Z}_{\Gamma} := \{S \in \mathrm{Sym}(p;\mathbb{R})\colon S_{i,j} = S_{\sigma(i),\sigma(j)} \text{ for all }\sigma \in \Gamma\mbox{ for all }i,j\in V\},\] and the colored cone of positive definite matrices valued in \(\mathcal{Z}_{\Gamma}\), \[\mathcal{P}_{\Gamma} := \mathcal{Z}_{\Gamma} \cap \mathrm{Sym}^+(p;\mathbb{R}).\]

Block Decomposition - [1], Theorem 1

The main theoretical result in this theory (Theorem 1 in [1]) states that given a permutation subgroup \(\Gamma\) there exists an orthogonal matrix \(U_\Gamma\) such that all the symmetric matrices \(S\in\mathcal{Z}_\Gamma\) can be transformed into block-diagonal form.

The exact form of blocks depends on so-called structure constants \((k_i,d_i,r_i)_{i=1}^L\). It is worth pointing out that constants \(k = d\) for cyclic group \(\Gamma = \left<\sigma\right>\) and that gips searches within cyclic subgroups only.

Examples

p <- 6
S <- matrix(c(1.1,0.9,0.8,0.7,0.8,0.9,
              0.9,1.1,0.9,0.8,0.7,0.8,
              0.8,0.9,1.1,0.9,0.8,0.7,
              0.7,0.8,0.9,1.1,0.9,0.8,
              0.8,0.7,0.8,0.9,1.1,0.9,
              0.9,0.8,0.7,0.8,0.9,1.1), nrow = p)

S is a symmetric matrix invariant under the group \(\Gamma = \left<(1,2,3,4,5,6)\right>\).

g_perm <- gips_perm("(1,2,3,4,5,6)", p)
U_Gamma <- prepare_orthogonal_matrix(g_perm)

block_decomposition <- t(U_Gamma) %*% S %*% U_Gamma
round(block_decomposition, 5)
#>      [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,]  5.2  0.0  0.0  0.0  0.0  0.0
#> [2,]  0.0  0.5  0.0  0.0  0.0  0.0
#> [3,]  0.0  0.0  0.5  0.0  0.0  0.0
#> [4,]  0.0  0.0  0.0  0.1  0.0  0.0
#> [5,]  0.0  0.0  0.0  0.0  0.1  0.0
#> [6,]  0.0  0.0  0.0  0.0  0.0  0.2

The transformed matrix is in the block-diagonal form of [1], Theorem 1. Blank entries are off-block entries and equal to 0. The result was rounded to the 5th place after the decimal to hide the inaccuracies of floating point arithmetic.

Let’s see the other example:

p <- 6
S <- matrix(c(1.2,0.9,0.9,0.4,0.2,0.1,
              0.9,1.2,0.9,0.1,0.4,0.2,
              0.9,0.9,1.2,0.2,0.1,0.4,
              0.4,0.1,0.2,1.2,0.9,0.9,
              0.2,0.4,0.1,0.9,1.2,0.9,
              0.1,0.2,0.4,0.9,0.9,1.2), nrow = p)

Now, S is a symmetric matrix invariant under the group \(\Gamma = \left<(1,2,3)(4,5,6)\right>\).

g_perm <- gips_perm("(1,2,3)(4,5,6)", p)
U_Gamma <- prepare_orthogonal_matrix(g_perm)

block_decomposition <- t(U_Gamma) %*% S %*% U_Gamma
round(block_decomposition, 5)
#>      [,1] [,2]   [,3]    [,4]    [,5]   [,6]
#> [1,]  3.0  0.7 0.0000  0.0000  0.0000 0.0000
#> [2,]  0.7  3.0 0.0000  0.0000  0.0000 0.0000
#> [3,]  0.0  0.0 0.3000  0.0000  0.2500 0.0866
#> [4,]  0.0  0.0 0.0000  0.3000 -0.0866 0.2500
#> [5,]  0.0  0.0 0.2500 -0.0866  0.3000 0.0000
#> [6,]  0.0  0.0 0.0866  0.2500  0.0000 0.3000

Again, this result is in accordance with [1], Theorem 1. Notice the zeros in block_decomposition: \[\forall_{i\in\{1,2\},j\in\{3,4,5,6\}}\text{block_decomposition}[i,j] = 0\]

Project Matrix - [1, Eq. (6)]

One can also take any symmetric square matrix S and find the orthogonal projection on \(\mathcal{Z}_{\Gamma}\), the space of matrices invariant under the given permutation:

\[\pi_\Gamma(S) := \frac{1}{|\Gamma|}\sum_{\sigma\in\Gamma}\sigma\cdot S\cdot\sigma^\top\]

The projected matrix is the element of the cone \(\pi_\Gamma(S)\in\mathcal{Z}_{\Gamma}\), which means: \[\forall_{i,j\in \{1,\ \dots,\ p\}} \pi_\Gamma(S)[i,j] = \pi_\Gamma(S)[\sigma(i),\sigma(j)] \text{ for all }\sigma\in\Gamma\]

So it has some identical elements.

Trivial case

Note that for \(\Gamma = \{\text{id}\} = \{(1)(2)\dots(p)\}\) we have \(\pi_{\{\text{id}\}}(S) = S\).

So no additional assumptions are made; thus the standard covariance estimator is the best we can do.

Example

Let S be any symmetric square matrix:

round(S, 2)
#>        [,1]  [,2]   [,3]  [,4]   [,5]  [,6]
#> [1,] 108.99  3.44 -14.41 14.90 -15.28 -5.78
#> [2,]   3.44 27.71  -3.45  2.56   4.81 -4.46
#> [3,] -14.41 -3.45  42.59  8.15  -7.57 -0.95
#> [4,]  14.90  2.56   8.15 29.94  -1.52  0.83
#> [5,] -15.28  4.81  -7.57 -1.52  10.85  5.56
#> [6,]  -5.78 -4.46  -0.95  0.83   5.56 27.04

One can project this matrix, for example, on \(\Gamma = \left<(1,2)(3,4,5,6)\right>\):

S_projected <- project_matrix(S, perm = "(1,2)(3,4,5,6)")
round(S_projected, 2)
#>       [,1]  [,2]  [,3]  [,4]  [,5]  [,6]
#> [1,] 68.35  3.44 -7.90  2.62 -7.90  2.62
#> [2,]  3.44 68.35  2.62 -7.90  2.62 -7.90
#> [3,] -7.90  2.62 27.60  2.81 -3.37  2.81
#> [4,]  2.62 -7.90  2.81 27.60  2.81 -3.37
#> [5,] -7.90  2.62 -3.37  2.81 27.60  2.81
#> [6,]  2.62 -7.90  2.81 -3.37  2.81 27.60

Notice in the S_projected matrix, there are identical elements according to the equation from the beginning of this section. For example, S_projected[1,1] = S_projected[2,2].

\(C_\sigma\) and n0

It is a well-known fact that without additional assumptions, the Maximum Likelihood Estimator (MLE) of the covariance matrix in the Gaussian model exists if and only if \(n \ge p\). However, if the additional assumption is added as the covariance matrix is invariant under permutation \(\sigma\), then the sample size \(n\) required for the MLE to exist is lower than \(p\). It is equal to the number of cycles, denoted hereafter by \(C_\sigma\).

For example, if the permutation \(\sigma = (1,2,3,4,5,6)\) is discovered by the find_MAP() function, then there is a single cycle in it \(C_\sigma = 1\). Therefore a single observation would be enough to estimate a covariance matrix with project_matrix(). If the permutation \(\sigma = (1,2)(3,4,5,6)\) is discovered, then \(C_\sigma = 2\) and so 2 observations would be enough.

To get this \(C_\sigma\) number in gips, one can call summary() on the appropriate gips object:

g1 <- gips(S, n, perm = "(1,2,3,4,5,6)", was_mean_estimated = FALSE)
summary(g1)$n0
#> [1] 1
g2 <- gips(S, n, perm = "(1,2)(3,4,5,6)", was_mean_estimated = FALSE)
summary(g2)$n0
#> [1] 2

This is called n0 and not \(C_\sigma\), because it is increased by 1 when the mean was estimated:

S <- cov(Z)
g1 <- gips(S, n, perm = "(1,2,3,4,5,6)", was_mean_estimated = TRUE)
summary(g1)$n0
#> [1] 2
g2 <- gips(S, n, perm = "(1,2)(3,4,5,6)", was_mean_estimated = TRUE)
summary(g2)$n0
#> [1] 3

Bayesian model selection

When one has the data matrix Z, one would like to know if it has a hidden structure of dependencies between features. Luckily, the paper demonstrates a way how to find it.

General workflow

  1. Choose the prior distribution on \(\Gamma\) and \(\Sigma\).
  2. Calculate the posteriori distribution (up to a normalizing constant) by formula [1], (30).
  3. Use the Metropolis-Hastings algorithm to find the permutation with the biggest value of the posterior probability \(\mathbb{P}(\Gamma|Z)\).

Details on the prior distribution

The considered prior distribution of \(\Gamma\) and \(K=\Sigma^{-1}\):

  1. \(\Gamma\) is uniformly distributed on the set of all cyclic subgroups of \(\mathfrak{S}_p\).
  2. \(K\) given \(\Gamma\) follows the Diaconis-Ylvisaker conjugate prior distribution with parameters \(\delta\) (real number, \(\delta > 2\)) and \(D\) (symmetric, positive definite square matrix of the same size as S), see [1], Sec. 3.4.

Footnote: Actually in some cases smaller (but still positive) values for \(\delta\) parameter are theoretically correct. Refer to the [1].

gips technical details

In gips, \(\delta\) is named delta, and \(D\) is named D_matrix. By default, they are set to \(3\) and diag(p), respectively. However, it is worth running the procedure for several parameters D_matrix of form \(C\cdot diag(p)\) for positive constant \(C\). Large \(C\) (compared to the data) favors small groups.

One can calculate the logarithm of formula (30) with function log_posteriori_of_gips().

Interpretation

When all assumptions are met, the formula (30) puts a number on each permutation’s cyclic group. The bigger its value, the more likely the data was drawn from that model.

When one finds the permutations group \(c_{\text{max}}\) that maximizes (30), \[c_{\text{map}} = \operatorname{arg\,max}_{c\in\mathfrak{S}_p} \mathbb{P}\left(\Gamma=c|Z^{(1)},\ldots,Z^{(n)}\right)\]

one can reasonably assume the data \(Z\) was drawn from the model \[\mathrm{N}_p(0,\pi_{c_{\text{map}}}(S))\]

where \(S = \frac{1}{n} \sum_{i=1}^n Z^{(i)}\cdot {Z^{(i)}}^\top\)

In such a case, we call \(c_{\text{map}}\) the Maximum A Posteri (MAP).

Finding the MAP Estimator

The space of all permutations is enormous for bigger \(p\) (\(p\ge9\)). It is more reasonable to estimate the MAP in such a big space than to calculate it precisely.

Metropolis-Hastings algorithm suggested by the authors of [1] is a natural way to do it. To see the discussion on it and other options available in gips, see vignette("Optimizers") or its pkgdown page.

Example

Let’s say we have this data, Z. It has only \(4\) observations, and its dimension is \(p=6\). Let’s assume Z was drawn from the normal distribution with the mean \((0,0,0,0,0,0)\). We want to estimate the covariance matrix:

dim(Z)
#> [1] 4 6
number_of_observations <- nrow(Z) # 4
p <- ncol(Z) # 6

# Calculate the covariance matrix from the data (assume the mean is 0):
S <- (t(Z) %*% Z) / number_of_observations

# Make the gips object out of data:
g <- gips(S, number_of_observations, was_mean_estimated = FALSE)

g_map <- find_MAP(g, optimizer = "brute_force")
#> ================================================================================
print(g_map)
#> The permutation (1,2,3,4,5,6)
#>  - was found after 720 log_posteriori calculations
#>  - is 157.903480698963 times more likely than the starting, () permutation.

S_projected <- project_matrix(S, g_map[[1]])

We see the posterior probability [1,(30)] has the biggest value for the permutation \((1,2,3,4,5,6)\). It was over 150 times bigger than for the trivial \(\text{id} = (1)(2)\ldots(p)\) permutation. We interpret that under the assumptions (centered Gaussian), it is over 150 times more reasonable to assume the data Z was drawn from model \(\mathrm{N}_p(0,\text{S_projected})\) than from model \(\mathrm{N}_p(0,\text{S})\).

References

[1] Piotr Graczyk, Hideyuki Ishi, Bartosz Kołodziejek, Hélène Massam. “Model selection in the space of Gaussian models invariant by symmetry.” The Annals of Statistics, 50(3) 1747-1774 June 2022. arXiv link; DOI: 10.1214/22-AOS2174