Copula Processes

We define a copula process which describes the dependencies between arbitrarily many random variables independently of their marginal distributions. As an example, we develop a stochastic volatility model, Gaussian Copula Process Volatility (GCPV), to predict the latent standard deviations of a sequence of random variables. To make predictions we use Bayesian inference, with the Laplace approximation, and with Markov chain Monte Carlo as an alternative. We find our model can outperform GARCH on simulated and financial data. And unlike GARCH, GCPV can easily handle missing data, incorporate covariates other than time, and model a rich class of covariance structures. Imagine measuring the distance of a rocket as it leaves Earth, and wanting to know how these measurements correlate with one another. How much does the value of the measurement at fifteen minutes depend on the measurement at five minutes? Once we’ve learned this correlation structure, suppose we want to compare it to the dependence between measurements of the rocket’s velocity. To do this, it is convenient to separate dependence from the marginal distributions of our measurements. At any given time, a rocket’s distance from Earth could have a Gamma distribution, while its velocity could have a Gaussian distribution. And separating dependence from marginal distributions is precisely what a copula function does. While copulas have recently become popular, especially in financial applications [1, 2], as Nelsen [3] writes, “the study of copulas and the role they play in probability, statistics, and stochastic processes is a subject still in its infancy. There are many open problems. . . ” Typically only bivariate (and recently trivariate) copulas are being used and studied. In our introductory example, we are interested in learning the correlations in different stochastic processes, and comparing them. It would therefore be useful to have a copula process, which can describe the dependencies between arbitrarily many random variables independently of their marginal distributions. We define such a process. And as an example, we develop a stochastic volatility model, Gaussian Copula Process Volatility (GCPV). In doing so, we provide a Bayesian framework for the learning the marginal distributions and dependency structure of what we call a Gaussian copula process. The volatility of a random variable is its standard deviation. Stochastic volatility models are used to predict the volatilities of a heteroscedastic sequence – a sequence of random variables with different variances, like distance measurements of a rocket as it leaves the Earth. As the rocket gets further away, the variance on the measurements increases. Heteroscedasticity is especially important in econometrics; the returns on equity indices, like the S&P 500, or on currency exchanges, are heteroscedastic. Indeed, in 2003, Robert Engle won the Nobel Prize in economics “for methods of analyzing economic time series with time-varying volatility”. GARCH [4], a generalized version of Engle’s ARCH, is arguably unsurpassed for predicting the volatility of returns on equity indices and currency exchanges [5, 6, 7]. GCPV can outperform GARCH, and is competitive on financial data that especially suits GARCH [8, 9, 10]. Before discussing GCPV, we first introduce copulas and the copula process. For a review of Gaussian processes, see Rasmussen and Williams [11]. ∗http://mlg.eng.cam.ac.uk/andrew †Also at the machine learning department at Carnegie Mellon University.