Weighted likelihood methods for robust fitting of wrapped models for p-torus data

Abstract

We consider, robust estimation of wrapped models to multivariate circular data that are points on the surface of a p-torus based on the weighted likelihood methodology. Robust model fitting is achieved by a set of weighted likelihood estimating equations, based on the computation of data dependent weights aimed to down-weight anomalous values, such as unexpected directions that do not share the main pattern of the bulk of the data. Weighted likelihood estimating equations with weights evaluated on the torus or obtained after unwrapping the data onto the Euclidean space are proposed and compared. Asymptotic properties and robustness features of the estimators under study have been studied, whereas their finite sample behavior has been investigated by Monte Carlo numerical experiment and real data examples.

Appendices

Appendix A: MLE for wrapped unimodal elliptically symmetric distributions

Let us consider the circular model

$$\begin{aligned} m^\circ (\varvec{y};\, \varvec{\mu }, \Sigma ) = \sum _{\varvec{j} \in {\mathbb {Z}}^p} m(\varvec{y} + 2\pi \varvec{j};\, \varvec{\mu }, \Sigma ), \end{aligned}$$

where

$$\begin{aligned} m(\varvec{x};\, \varvec{\theta }) \propto \vert \Sigma \vert ^{-1/2} h\left( (\varvec{x} - \varvec{\mu })^\top \Sigma ^{-1} (\varvec{x} - \varvec{\mu })\right) \end{aligned}$$

is a unimodal elliptically symmetric distribution. The log-likelihood function based on an i.i.d. sample $\varvec{y}_1, \ldots , \varvec{y}_n$ is

$$\begin{aligned} \ell ^\circ (\varvec{\mu }, \Sigma )&= \sum _{i=1}^n \log m^\circ (\varvec{y}_i;\, \varvec{\mu }, \Sigma ) \\&= \sum _{i=1}^n \log \sum _{\varvec{j} \in {\mathbb {Z}}^p} m(\varvec{y}_i + 2\pi \varvec{j};\, \varvec{\mu }, \Sigma ) \\&\quad \propto \sum _{i=1}^n \log \sum _{\varvec{j} \in {\mathbb {Z}}^p} \vert \Sigma \vert ^{-\frac{1}{2}} h\left[ (\varvec{y}_i + 2\pi \varvec{j} - \varvec{\mu })^\top \Sigma ^{-1} (\varvec{y}_i + 2\pi \varvec{j} - \varvec{\mu })\right] \\&= \frac{n}{2} \log \vert \Sigma ^{-1} \vert + \sum _{i=1}^n \log \sum _{\varvec{j} \in {\mathbb {Z}}^p} h\left[ {\text {tr}}\left( (\varvec{y}_i + 2\pi \varvec{j} - \varvec{\mu }) (\varvec{y}_i + 2\pi \varvec{j} - \varvec{\mu })^\top \Sigma ^{-1} \right) \right] \\ \end{aligned}$$

Recall that for given square matrices A and B, both symmetric and positive definite we have that

• 1. $\nabla _{A} {\text {tr}}(BA) = B^\top$,

• 2. $\nabla _A \log (\vert A \vert ) = \left( A^{-1}\right) ^\top$,

• 3. $\nabla _{\varvec{x}} (\varvec{x}^\top A \varvec{x}) = 2 A \varvec{x}$.

Let $d_{i\varvec{j}}(\varvec{\mu },\Sigma ) = (\varvec{y}_i + 2\pi \varvec{j} - \varvec{\mu })^\top \Sigma ^{-1}(\varvec{y}_i + 2\pi \varvec{j} - \varvec{\mu })$. Taking the derivatives w.r.t. $\varvec{\mu }$ and $\Sigma ^{-1}$, the likelihood equations are

$$\begin{aligned} \nabla _{\varvec{\mu }} \ell ^\circ (\varvec{\mu }, \Sigma )&= \sum _{i=1}^n \nabla _{\varvec{\mu }} \log \sum _{\varvec{j} \in {\mathbb {Z}}^p} h(d_{i\varvec{j}}(\varvec{\mu },\Sigma )) \\&= \sum _{i=1}^n \frac{\sum _{\varvec{j} \in {\mathbb {Z}}^p} \nabla _{\varvec{\mu }} h(d_{i\varvec{j}}(\varvec{\mu },\Sigma ))}{\sum _{\varvec{k} \in {\mathbb {Z}}^p} h(d_{i\varvec{k}}(\varvec{\mu },\Sigma ))} \\&= 2 \sum _{i=1}^n \frac{\sum _{\varvec{j} \in {\mathbb {Z}}^p} h^\prime (d_{i\varvec{j}}(\varvec{\mu },\Sigma )) \Sigma ^{-1} (\varvec{y}_i + 2\pi \varvec{j} - \varvec{\mu })}{\sum _{\varvec{k} \in {\mathbb {Z}}^p} h(d_{i\varvec{k}}(\varvec{\mu },\Sigma ))} \end{aligned}$$

and

$$\begin{aligned} \nabla _{\Sigma ^{-1}} \ell ^\circ (\varvec{\mu }, \Sigma )&= \frac{n}{2} \Sigma ^\top + \sum _{i=1}^n \nabla _{\Sigma ^{-1}} \log \sum _{\varvec{j} \in {\mathbb {Z}}^p} h(d_{i\varvec{j}}(\varvec{\mu },\Sigma )) \\&= \frac{n}{2} \Sigma + \sum _{i=1}^n \frac{\sum _{\varvec{j} \in {\mathbb {Z}}^p} \nabla _{\Sigma ^{-1}} h(d_{i\varvec{j}}(\varvec{\mu },\Sigma ))}{\sum _{\varvec{k} \in {\mathbb {Z}}^p} h(d_{i\varvec{k}}(\varvec{\mu },\Sigma ))} \\&= \frac{n}{2} \Sigma + \sum _{i=1}^n \frac{\sum _{\varvec{j} \in {\mathbb {Z}}^p} h^\prime (d_{i\varvec{j}}(\varvec{\mu },\Sigma )) (\varvec{y}_i + 2\pi \varvec{j}-\varvec{\mu }) (\varvec{y}_i + 2\pi \varvec{j}-\varvec{\mu })^\top }{\sum _{\varvec{k} \in {\mathbb {Z}}^p} h(d_{i\varvec{k}}(\varvec{\mu },\Sigma ))}, \\ \end{aligned}$$

where $h^\prime (d) = \partial h(d)/\partial d$. Let

$$\begin{aligned} v_{i\varvec{j}} = \frac{h^\prime (d_{i\varvec{j}}(\varvec{\mu },\Sigma ))}{\sum _{\varvec{k} \in {\mathbb {Z}}^p} h(d_{i\varvec{k}}(\varvec{\mu },\Sigma ))} . \end{aligned}$$

then, the MLE $(\hat{\varvec{\mu }}, {{\hat{\Sigma }}})$ is the solution to the (set of) fixed point equations

$$\begin{aligned} \varvec{\mu }&= \frac{\sum _{i=1}^n \sum _{\varvec{j} \in {\mathbb {Z}}^p} v_{i\varvec{j}}(\varvec{y}_i + 2\pi \varvec{j})}{\sum _{i=1}^n \sum _{\varvec{k} \in {\mathbb {Z}}^p} v_{i\varvec{k}}} \\ \Sigma&= -\frac{2}{n} \sum _{i=1}^n \sum _{\varvec{j} \in {\mathbb {Z}}^p} v_{i\varvec{j}} (\varvec{y}_i + 2\pi \varvec{j}-\varvec{\mu }) (\varvec{y}_i + 2\pi \varvec{j}-\varvec{\mu })^\top . \end{aligned}$$

The WN distribution corresponds to $h(t) = \exp \left( -\frac{t}{2} \right)$. Since, $h^\prime (t) = -\frac{1}{2} h(d)$ then

$$\begin{aligned} v_{i\varvec{j}} = -\frac{1}{2} \frac{h(d_{i\varvec{j}})}{\sum _{\varvec{k} \in {\mathbb {Z}}^p} h(d_{i\varvec{k}})} = -\frac{1}{2} \frac{m(\varvec{y}_i + 2\pi \varvec{j};\, \varvec{\mu }, \Sigma )}{\sum _{\varvec{k} \in {\mathbb {Z}}^p} m(\varvec{y}_i + 2\pi \varvec{k};\, \varvec{\mu }, \Sigma )} . \end{aligned}$$

and the estimating equations simplify to

$$\begin{aligned} \varvec{\mu }&= \frac{1}{n} \sum _{i=1}^n \sum _{\varvec{j} \in {\mathbb {Z}}^p} \omega _{i\varvec{j}} (\varvec{y}_i + 2\pi \varvec{j}) \\ \Sigma&= \frac{1}{n} \sum _{i=1}^n \sum _{\varvec{j} \in {\mathbb {Z}}^p} \omega _{i\varvec{j}} (\varvec{y}_i + 2\pi \varvec{j} - \varvec{\mu })(\varvec{y}_i + 2\pi \varvec{j} - \varvec{\mu })^\top \ . \\ \end{aligned}$$

with

$$\begin{aligned} \omega _{i\varvec{j}} = \frac{m(\varvec{y}_i + 2\pi \varvec{j};\, \varvec{\mu }, \Sigma )}{\sum _{\varvec{k} \in {\mathbb {Z}}^p} m(\varvec{y}_i + 2\pi \varvec{k};\, \varvec{\mu }, \Sigma )} . \end{aligned}$$

Appendix B: EM algorithm for WN estimation

Given, an i.i.d. sample $(\varvec{y}_1, \ldots , \varvec{y}_n)$ from a WN distribution, in the EM algorithm the wrapping coefficients $\varvec{j}$ are considered as latent variables and the observed torus data $\varvec{y}_i$s as being incomplete, that is $\varvec{y}_i$ assumed to be one component of the pair $(\varvec{y}_i,\varvec{\omega }_i)$, where $\varvec{\omega }_i=(\omega _{i\varvec{j}}: \varvec{j} \in {\mathbb {Z}}^p)$ is the associated latent wrapping coefficients label vector. Then, the MLE for $\varvec{\theta } = (\varvec{\mu }, \Sigma )$ is the result of the EM algorithm based on the complete log-likelihood function

$$\begin{aligned} \ell _c(\varvec{\theta })=\sum _{i=1}^n \sum _{\varvec{j} \in {\mathbb {Z}}^p} \omega _{i\varvec{j}} \log m(\varvec{y}_i + 2 \pi \varvec{j};\, \varvec{\theta }) . \end{aligned}$$

(25)

In the expectation step (E-step), we evaluate the conditional expectation of (25) given the observed data and the current parameters value $\varvec{\theta }$ by computing the conditional probability that $\varvec{y}_i$ has $\varvec{j}$ as wrapping coefficients vector, that is

$$\begin{aligned} \omega _{i\varvec{j}}=\frac{m(\varvec{y}_i + 2 \pi \varvec{j};\, \varvec{\theta })}{\sum _{\varvec{k} \in {\mathbb {Z}}^p} m(\varvec{y}_i + 2 \pi \varvec{k};\, \varvec{\theta })}, \forall \varvec{j} \in {\mathbb {Z}}^p . \end{aligned}$$

parameters estimation is carried out in the maximization step (M-step) solving the set of (complete) likelihood equations

$$\begin{aligned} \sum _{i=1}^n \sum _{\varvec{j} \in {\mathbb {Z}}^p} \omega _{i\varvec{j}} u(\varvec{y}_i + 2 \pi \varvec{j};\, \varvec{\theta }) = \varvec{0} \, \end{aligned}$$

with $u(\varvec{y}_i + 2 \pi \varvec{j};\, \varvec{\theta })=\nabla _{\varvec{\theta }} \log m(\varvec{y}+ 2 \pi \varvec{j};\, \varvec{\theta })$. An alternative estimation strategy can be based on a CEM algorithm leading to an approximated solution. At each iteration, a Classification step (C-step) is performed after the E-step, that provides crispy assignments. Let

$$\begin{aligned} \hat{\varvec{j}}_i= \textrm{argmax}_{\varvec{j} \in {\mathbb {Z}}^p} \omega _{i\varvec{j}}, \end{aligned}$$

then, set $\omega _{i\varvec{j}} = 1$ when $\varvec{j}=\hat{\varvec{j}}_i$, $\omega _{i\varvec{j}} = 0$ otherwise. As a result, the torus data $\varvec{y}_i$ are unwrapped to (fitted) linear data $\hat{\varvec{x}}_i = \varvec{y}_i + 2\pi \hat{\varvec{j}}_i$. It is easy to see that the M-step simplifies to

$$\begin{aligned} \sum _{i=1}^n u(\hat{\varvec{x}}_{i};\, \varvec{\theta }) = \varvec{0} . \end{aligned}$$

both the procedures are iterated until some convergence criterion is fulfilled, that could be based on the changes in the likelihood or in fitted parameter values (Nodehi et al. 2021).