3. Empirical Checkerboard Copula with known margins

Preliminary notations

First of all, for a certain dimension of the data at hand \(d\) and a certain checkerboard parameter \(m\), let’s consider the ensemble of multi-indexes \(\mathcal{I} = \{\mathbf{i} = (i_1,..,i_d) \subset \{1,...,m \}^d\}\) which indexes the boxes :

\[B_{\mathbf{i}} = \left]\frac{\mathbf{i}-1}{m},\frac{\mathbf{i}}{m}\right]\]

partitioning the space \(\mathbb{I}^d = [0,1]^d\). Denote the set of thoose boxes by \(\mathcal{B}_\mathcal{I} = \left\{B_{\mathbf{i}}, \mathbf{i} \in \mathcal{I}\right\}\). Furthermore, let’s choose \(p\) dimensions that would be assigned to a known copula, by setting : \(J \subset \{1,...,d\}, |J| = p\) and let’s define proper projections for the boxes :

\[B^J_{\mathbf{i}} = \{ \mathbf{x} \in [0,1]^p, x_j \in \left]\frac{i_j-1}{m},\frac{i_j}{m}\right] \forall j \in J \}\]

\[B^{-J}_{\mathbf{i}} = \{\mathbf{x} \in [0,1]^p, x_j \in \left]\frac{i_j-1}{m},\frac{i_j}{m}\right] \forall j \notin J \}\]

such that \(B_{\mathbf{i}} = B^J_{\mathbf{i}} \times B^{-J}_{\mathbf{i}}\) for all \(\mathbf{i} \in \mathcal{I}\). The tensor product is here understood taking dimensions of \(\mathbb{I}^d = [0,1]^d\) in the right order such that \(B^J_{\mathbf{i}}\) dimensions end up in place of dimensions indexed by \(J\). Think of dimensions as re-ordered such that \(J = \{1,...,p\}\).

Let now \(\lambda\) be the dimension-unspecified Lebesgue measure on any power of \(\mathbb{R}\), that is :

\[\forall d \in \mathbb{N}, \forall \mathbf{x},\mathbf{y} \in \mathbb{R}^p, \lambda(\left[\mathbf{x},\mathbf{y}\right]) = \prod\limits_{p=1}^{d} (y_i - x_i)\]

Let furthermore \(\mu^J\) be a copula measure of dimension \(p\), corresponding to the known multivariate margin associated to marginals in \(J\). Let also \(\mu\) and \(\hat{\mu}\) be dimension-unspecific version of the true copula measure of the sample at hand and (respectively) the classical Deheuvels empirical copula, that is :

• For \(n\) i.i.d observation of the copula of dimension \(d\), let \(\forall i \in \{1,...,d\}, \, R_i^1,...,R_i^d\) be the marginal ranks for the variable \(i\).
• \(\forall \mathbf{x} \in \mathcal{I}^d, \text{ let } \hat{\mu}([0,x]) = \frac{1}{n} \sum\limits_{k=1}^n \mathbb{1}_{R_1^k\le x_1,...,R_d^k\le x_d}\)

We are now ready to define the empirical checkerboard copula with known margins.

Definition, estimation and simulation procedures

The empirical copula with known margins is the copula that correspond to the following simulation procedure.

• Simulate a sample from the known sub-copula \(\mu^J\), of dimension \(p\), through any avaliable method (depends on the known copula model). Let \(B_{\mathbf{i}}^J\) be the (projected) box containing this sample.
• Sample one box \(B_{\mathbf{i}}\) among all boxes with projection \(B_{\mathbf{i}}^J\) with probability weights :
  • \(\frac{\hat{\mu}(B_{\mathbf{i}})}{\hat{\mu}(B_{\mathbf{i}}^J)}\) if the projected box \(B_{\mathbf{i}}^J\) contains one or more (empirical) data point, that is \(\hat{\mu}(B_{\mathbf{i}}^J) \neq 0\)
  • \(\frac{\lambda(B_{\mathbf{i}})}{\lambda(B_{\mathbf{i}}^J)}\) otherwise.
• Simulate uniformly from \(B_{\mathbf{i}}^{-J}\)

This algorithm simulates first the known part of the model (dimensions in \(J\)), and then, conditionally, the checkerboard part, ensuring that the known copula is respected. The downside of this behavior is that the checkerboard part may have points outside standard checkerboard boxes, making this part of the copula less sparse than a true checkerboard. But is does become sparser as soon as the data fits the known margins. On the other hand, this algorithm allows for a lot of flexibility, mainly in the following points :

• The “grid” given by \(\mathcal{B}_\mathcal{I}\) can be taken more arbitrarily than \(m^d\) boxes of same volume, as soon as it’s a partition of \(\mathbb{I}^d\).
• The known copula is not restricted at all and can be chose among all \(p\)-dimensionnal copulas.
• The estimation of the checkerboard part can be turn into a more flexible patchwork construction, by changing the independence copula for an other one inside the boxes. See Durante2013,Durante2015,Durante2015a

We are now going to define properly the measure associated to this simulation procedure, a.k.a the empirical checkerboard copula with known margins. Let \(\nu\) be this measure and let \(\mathbf{U}\) be a random vector drawn from it. Then \(\forall \mathbf{x} \in \mathbb{I}^d\), following the above procedure, we have :

\[\nu([0,\mathbf{x}]) = \sum\limits_{{\mathbf{i}} \in \mathcal{I}} \mathbb{P}(\mathbf{U}^{-J} \in B_{\mathbf{i}}^{-J} \cap [0,\mathbf{x}^{-J}] | \mathbf{U}^{J} \in B_{\mathbf{i}}^{J} \cap [0,\mathbf{x}^{J}]) \mathbb{P}(\mathbf{U}^{J} \in B_{\mathbf{i}}^{J} \cap [0,\mathbf{x}^{J}])\]

While the unconditional term is easy to handle since it’s the measure associated with the known copula, \(\mathbb{P}(\mathbf{U}^{J} \in B_{\mathbf{i}}^{J} \cap [0,\mathbf{x}^{J}]) = \mu^J(B_{\mathbf{i}}^{J} \cap [0,\mathbf{x}^{J}])\), the conditional term can be treated according to the algorithm : it will be \(\frac{\lambda(B_{\mathbf{i}}^{-J} \cap [0,\mathbf{x}^{-J}])}{\lambda(B_{\mathbf{i}}^{-J})}\) inside a box chosen with probability conditional on \(\hat{\mu}(B_{\mathbf{i}}^J) \neq 0\). We finally get the following definition :

The empirical checkerboard copula with parameter \(m\) and with set of known margins \(J\) following the measure \(\mu^J\) is the copula corresponding to the measure \(\nu\) given by :

\[ \nu([0,x]) = \sum\limits_{{\mathbf{i}} \in \mathcal{I}} \mu^J(B_{\mathbf{i}}^{J} \cap [0,\mathbf{x}^{J}]) \frac{\lambda(B_{\mathbf{i}}^{-J} \cap [0,\mathbf{x}^{-J}])}{\lambda(B_{\mathbf{i}}^{-J})} \left[ \frac{\hat{\mu}(B_{\mathbf{i}})}{\hat{\mu}(B_{\mathbf{i}}^J)} \mathbb{1}_{\hat{\mu}(B_{\mathbf{i}}^J) \neq 0} + \frac{\lambda(B_{\mathbf{i}})}{\lambda(B_{\mathbf{i}}^J)} \mathbb{1}_{\hat{\mu}(B_{\mathbf{i}}^J) = 0} \right] \]

The next section will discuss the current implementation of this copula;