`gips`

is based onThe 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.

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.

The theory derived in [1] is for general group invariance, while in this package, we only consider invariance under cyclic groups. This allows for massive simplifications. This simplified version is comprehensibly set out in [2].

Let \(V=\{1,\ldots,p\}\) be a finite index set, and for every \(i\in \{1, \dots, n\}\), \(Z^{(i)}=(Z_1^{(i)},\ldots, Z_p^{(i)})^\top\) 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 \(\mathcal{Z}_{\Gamma}\), \[\mathcal{P}_{\Gamma} := \mathcal{Z}_{\Gamma} \cap \mathrm{Sym}^+(p;\mathbb{R}).\]

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.

```
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. Notice that, for example, the [2,3] is not blank regardless of being 0. This is because it is a part of the block-diagonal form but happens to have a value of 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`

: \[\forall_{i\in\{1,2\},j\in\{3,4,5,6\}}\text{block_decomposition}[i,j]
= 0\]

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.

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.

We will abbreviate the notation: when the \(\Gamma = \left< c \right>\) is a cyclic group of a permutation \(c\), we will write \(\pi_{c}(S) := \pi_{\Gamma}(S) = \pi_{\left< c \right>}(S)\).

Let `S`

be any symmetric square matrix:

```
round(S, 2)
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 137.51 -16.21 10.03 0.16 -24.35 -17.42
#> [2,] -16.21 34.08 -10.62 15.93 12.23 -2.74
#> [3,] 10.03 -10.62 35.47 3.10 -3.81 -9.60
#> [4,] 0.16 15.93 3.10 26.74 7.71 -13.51
#> [5,] -24.35 12.23 -3.81 7.71 26.00 -7.24
#> [6,] -17.42 -2.74 -9.60 -13.51 -7.24 16.77
```

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

```
S_projected <- project_matrix(S, perm = "(1,2)(3,4,5,6)")
round(S_projected, 2)
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 85.80 -16.21 -0.28 -3.91 -0.28 -3.91
#> [2,] -16.21 85.80 -3.91 -0.28 -3.91 -0.28
#> [3,] -0.28 -3.91 26.25 -1.51 -8.66 -1.51
#> [4,] -3.91 -0.28 -1.51 26.25 -1.51 -8.66
#> [5,] -0.28 -3.91 -8.66 -1.51 26.25 -1.51
#> [6,] -3.91 -0.28 -1.51 -8.66 -1.51 26.25
```

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]`

.

`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:

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.

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

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

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

), see [1], Sec. 3.4.

Footnote: Actually, for \(\Gamma =
\{id\}\), \(\delta > 0\)
parameters are theoretically correct. In `gips`

, we want this
to be defined for all cyclic groups \(\Gamma\), so we restrict \(\delta > 1\). Refer to the [1].

`gips`

technical detailsIn `gips`

, \(\delta\) is
named `delta`

, and \(D\) is
named `D_matrix`

. By default, they are set to \(3\) and `diag(d, p)`

,
respectively, where `d = mean(diag(S))`

. However, it is worth
running the procedure for several parameters `D_matrix`

of
form \(d\cdot diag(p)\) for positive
constant \(d\). Small \(d\) (compared to the data) favors small
structures. Large \(d\) will “forget”
the data.

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

.

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 Posteriori (MAP).

The space of all permutations is enormous for bigger \(p\) (in our experiments, \(p\ge 10\) is too big). In such a big space, estimating the MAP is more reasonable than calculating 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", package="gips")`

or its pkgdown
page.

Let’s say we have this data, `Z`

. It has dimension \(p=6\) and only \(4\) observations. 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 362 posteriori calculations;
#> - is 133.158 times more likely than the () permutation.
S_projected <- project_matrix(S, g_map)
```

We see the posterior probability [1,(30)] has the biggest
value for the permutation \((1,2,3,4,5,6)\). It was over 100 times
bigger than for the trivial \(\text{id} =
(1)(2)\ldots(p)\) permutation. We interpret that under the
assumptions (centered Gaussian), it is over 100 times more reasonable to
assume the data `Z`

was drawn from model \(\mathrm{N}_p(0,\text{S_projected})\) than
from model \(\mathrm{N}_p(0,\text{S})\).

One may be interested in Akaike’s An Information Criterion (AIC) or Schwarz’s Bayesian Information Criterion (BIC) of the found model. Those are defined based on log-Likelihood:

\[\log L\left(\Sigma; Z^{(1)},\ldots,Z^{(n)}\right) = \sum_{i=1}^n \left(- \frac{p}{2}\log (2\pi) - \frac{1}{2}\log\left( \det\left( \Sigma\right)\right) - \frac12 {Z^{(i)}}^\top \Sigma^{-1} Z^{(i)}\right)= \]

\[- \frac{np}{2}\log (2\pi) - \frac{n}{2}\log\left( \det\left( \Sigma\right)\right) - \frac{n}2\mathrm{tr}(\Sigma^{-1} S),\] where \(S = \frac{1}{n} \sum_{i=1}^n Z^{(i)}\cdot {Z^{(i)}}^\top\).

The MLE of \(\Sigma\) in a model invariant under \(\Gamma=c\) is \(\hat{\Sigma} = \pi_{c}(S)\). Further, for every \(c\) we have \(\mathrm{tr}(\pi_{c}(S)^{-1} \cdot S) = p\), so:

\[\log L\left(\pi_{c}(S); Z^{(1)},\ldots,Z^{(n)}\right) = - \frac{np}{2}\log (2\pi) - \frac{n}{2}\log\left( \det\left( \pi_{c}(S)\right)\right) - \frac{np}2\]

which can be calculated by `logLik.gips()`

.

Then AIC and BIC are defined by:

\[AIC = 2 \cdot (\dim M) -2 \log L(\pi_{c}(S))\] \[BIC = (\log n) \cdot (\dim M) -2 \log L(\pi_{c}(S))\]

A smaller value of the criteria for a given model indicates a better fit.

Those can be calculated by `AIC.gips()`

and
`BIC.gips()`

.

When the mean was estimated, we have \(S = \frac{1}{n-1} \sum_{i=1}^n (Z^{(i)} - \bar{Z})\cdot ({Z^{(i)} - \bar{Z})}^\top\), where \(\bar{Z} = \frac{1}{n} \sum_{i=1}^n Z^{(i)}\). Then in the \(\log L\) we use \(n-1\) in stead of \(n\). Definitions of AIC and BIC stay the same.

Consider an example similar to one in the **Bayesian model
selection** section:

Let’s say we have this data, `Z`

. It has dimension \(p=6\) and \(7\) observations. 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] 7 6
number_of_observations <- nrow(Z) # 7
p <- ncol(Z) # 6
S <- (t(Z) %*% Z) / number_of_observations
g <- gips(S, number_of_observations, was_mean_estimated = FALSE)
g_map <- find_MAP(g, optimizer = "brute_force")
#> ================================================================================
```

```
AIC(g)
#> [1] 64.19906
AIC(g_map) # this is smaller, so this is better
#> [1] 62.99751
BIC(g)
#> [1] 63.06318
BIC(g_map) # this is smaller, so this is better
#> [1] 62.78115
```

We will consider a `g_map`

better model both in terms of
the AIC and the BIC.

[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

[2] “Learning permutation symmetries with gips in R” by
`gips`

developers Adam Chojecki, Paweł Morgen, and Bartosz
Kołodziejek, available on arXiv:2307.00790.