Source-linked AI summary
Copula Processes
Andrew Gordon Wilson, Zoubin Ghahramani
TL;DR
Copula processes address how to model dependencies among arbitrarily many random variables independently of their marginals, and the paper applies this idea in GCPV to predict latent standard deviations. The authors report that GCPV can outperform GARCH on simulated and financial data, while noting a normal, zero-mean conditional observation assumption that can be relaxed.
Problem
Copula applications typically use bivariate copulas, while the paper seeks a process describing dependencies among arbitrarily many variables independently of their marginal distributions.
Source: §1 · p. 1Method
Results
GCPV can outperform GARCH and is competitive on financial data that especially suits GARCH.
Source: §1 · p. 2Takeaways & Limitations
GCPV can handle missing data, incorporate covariates beyond time, use varied covariance functions, and provide insight through its parameters.
Source: §7 · p. 9The observation model assumes zero-mean normality conditional on variance, although the paper says these assumptions can be relaxed and are not central.
Source: §4 · p. 4
Abstract
from arXiv · showhide
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 both methods comparable. We also 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.
1 Introduction
The paper introduces copula processes to model dependencies among arbitrarily many variables independently of their marginal distributions, and presents GCPV as a stochastic-volatility application. It reports that GCPV can outperform GARCH and remain competitive on financial data.
1 Introduction
- A copula process describes dependencies among arbitrarily many random variables independently of their marginal distributions.The paper develops GCPV as an example and a Bayesian framework for learning marginal distributions and dependency structure.Source: §1 · p. 1
- Stochastic-volatility models predict standard deviations in heteroscedastic sequences, where random variables have different variances.The introduction illustrates increasing measurement variance as a rocket moves farther from Earth and notes the relevance of heteroscedasticity in econometrics.Source: §1 · p. 1
- GCPV can outperform GARCH and is competitive on financial data that especially suits GARCH.GARCH is described as a strong predictor of equity-index and currency-exchange returns.Source: §1 · p. 2
2 Copulas
Copulas separate a multivariate distribution’s dependence structure from its marginal distributions. Sklar’s theorem expresses the joint distribution through the marginals and a copula.
2 Copulas
- Transforming each variable through its marginal CDF yields uniform variables whose joint CDF is the n-copula.The copula represents the dependencies among the transformed variables.Source: §2 · p. 2
- Sklar’s theorem writes a joint distribution as a copula applied to its marginal CDF values.The theorem states that a copula exists for an n-dimensional distribution with specified marginals.Source: §2 · p. 2
- When all marginals are continuous, the copula is unique; otherwise, it is unique on the product of their ranges.Source: §2 · p. 2
- The bivariate Gaussian copula uses a correlated Gaussian CDF on inverse-standard-Gaussian-CDF-transformed inputs.Substituting marginal CDF values produces a joint distribution with those marginals and a Gaussian dependency structure.Source: §2 · p. 2
3 Copula Processes
A copula process generates arbitrarily many variables with chosen marginals and dependencies by transforming Gaussian-process draws. The paper defines this framework and applies it to stochastic volatility through GCPV.
3 Copula Processes
- Transforming Gaussian-process draws through a Gaussian CDF and then chosen inverse CDFs yields variables with different marginals but shared underlying dependencies.The paper illustrates uniform and beta marginals as examples.Source: §3 · p. 3
- A copula process generates samples from arbitrarily many random variables with arbitrary marginals and desired dependencies.Source: §3 · p. 3
- The formal definition constructs finite-dimensional distributions by applying the base process’s joint distribution to quasi-inverse-transformed marginal CDF values.The quasi-inverse of the base marginal CDF is used in the definition.Source: §3 · p. 3
- A Gaussian copula process is a copula process whose base measure is a Gaussian process; a mapping of its variables is a Gaussian process.The paper gives the mapped process mean and covariance function as part of its notation.Source: §3 · p. 3
- GCPV applies a warped Gaussian process to learn latent-variable marginals and dependency structure and make predictions.The inference is on latent variables rather than observations, requiring approximations.Source: §3 · p. 3
4 Gaussian Copula Process Volatility
GCPV models latent standard deviations as a Gaussian copula process, learning their dependence and marginal distributions to predict volatility at unobserved times. Its observation model assumes zero-mean normality conditional on variance, though the paper says these assumptions can be relaxed.
4 Gaussian Copula Process Volatility
- Given observations with latent standard deviations, the goal is to learn their correlation structure and predict the distribution of a standard deviation at an unrealised time.Source: §4 · p. 4
- GCPV models standard deviations through a monotonic warping of latent Gaussian-process values, so each standard deviation is g(f_i, ω).The finite-dimensional standard-deviation sequences have a Gaussian copula.Source: §4 · p. 4
- Learning the warping function learns the marginal distribution, F(a) = Φ(g^-1(a)), while g^-1(σ_t) is modeled by a Gaussian process.The model also allows the warping function to vary with time, assuming marginals are stationary over small time periods.Source: §4 · p. 4
- Observations are assumed normally distributed with zero mean conditional on variance, but the paper says these assumptions can be relaxed and are not central.Source: §4 · p. 4
5 Predictions with GCPV
GCPV predicts latent standard deviations by combining a Gaussian-process prior with a warping function. It uses Laplace inference or MCMC to make predictions and learn model parameters.
5 Predictions with GCPV
- For a new time, GCPV infers the predictive distribution of its latent standard deviation by sampling latent function values and transforming them through the warp.The model also learns covariance hyperparameters by maximizing the marginal likelihood.Source: §5 · pp. 4–5
5.1 Laplace Approximation
- The Laplace method approximates the latent posterior with a Gaussian, then uses it for predictions and approximate marginal-likelihood optimization.It finds the posterior mode with a modified Newton method; the reported per-iteration cost is O(n^3).Source: §5 · p. 5, §5.1 · pp. 5–6
5.2 Markov chain Monte Carlo
- MCMC samples the latent posterior using Elliptical Slice Sampling, then maps predictive samples through the warping function to obtain standard-deviation predictions.The sampler jointly updates the latent variables and has no free parameters.Source: §5.2 · p. 6
5.3 Warping Function
- GCPV maps Gaussian-process function values to positive standard deviations through a one-to-one warping function whose parameters can be learned.The paper also compares this flexible warp with the unparametrized exponential warp.Source: §5.3 · pp. 6–7
6 Experiments
The experiments compare GCPV with GARCH on simulated volatility sequences and financial exchange returns. GCPV and GARCH are competitive on the financial data, while the learned warp improves on an unparametrized warp and Laplace inference is comparable to MCMC.
6 Experiments
- The experiments predict latent standard deviations at observed and unobserved times, evaluate predictions with MSE, and use squared observations as a proxy when true variance is unknown.The financial study uses a rolling 120-day window, makes 1-, 7-, and 30-day forecasts, and retrains every 7 days; its historical predictions are assessed on their training data.Source: §6 · pp. 7–8, §6.2 · p. 8, §6.1 · p. 8
6.1 Simulations
- In simulations, GCPV captures volatility dependencies more accurately than GARCH and outperforms it on abrupt jumps, especially in low-volatility regions.GCPV is mostly robust as jump peaks grow and troughs shrink, while GARCH suffers more.Source: §6.1 · p. 8
6.2 Financial Data
- GCPV and GARCH are competitive for forecasting currency exchange returns; a learned warp outperforms an unparametrized warp, and Laplace inference is comparable to MCMC in accuracy and speed.The authors report the same inference comparison for simulations and recommend whichever method is more convenient to implement.Source: §6.2 · p. 9, Table 1 · p. 9
7 Discussion
GCPV separates dependencies among volatilities from their marginal distributions, learning marginals with a time-changing warping function. It can outperform GARCH and offers flexible covariance modeling and other advantages over GARCH.
7 Discussion
- A rolling-window warping function lets GCPV learn volatility marginals without assuming they are stationary.An unparametrized exponential warp would incorrectly assume marginally Gaussian log volatilities.Source: §7 · p. 9
- GCPV handles missing data, incorporates non-time covariates, supports varied covariance functions, and offers parameters that provide insight into volatility sources.The periodic covariance function is identified as well suited to cyclical high-frequency returns.Source: §7 · p. 9
- The copula process represents dependence among arbitrarily many random variables independently of their marginal distributions, extending beyond the often-used bivariate copulas.Source: §7 · p. 9
- GCPV can outperform GARCH, and its learned DMGBP volatility marginal is similar to Gamma(4.15,0.00045).Figure 1d compares the learned marginal density with the Gamma density.Source: §7 · p. 9, Figure 1 · p. 10