nips nips2010 nips2010-53 knowledge-graph by maker-knowledge-mining

53 nips-2010-Copula Bayesian Networks

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Author: Gal Elidan

Abstract: We present the Copula Bayesian Network model for representing multivariate continuous distributions, while taking advantage of the relative ease of estimating univariate distributions. Using a novel copula-based reparameterization of a conditional density, joined with a graph that encodes independencies, our model offers great ﬂexibility in modeling high-dimensional densities, while maintaining control over the form of the univariate marginals. We demonstrate the advantage of our framework for generalization over standard Bayesian networks as well as tree structured copula models for varied real-life domains that are of substantially higher dimension than those typically considered in the copula literature. 1

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1 il Abstract We present the Copula Bayesian Network model for representing multivariate continuous distributions, while taking advantage of the relative ease of estimating univariate distributions. [sent-3, score-0.244]

2 Using a novel copula-based reparameterization of a conditional density, joined with a graph that encodes independencies, our model offers great ﬂexibility in modeling high-dimensional densities, while maintaining control over the form of the univariate marginals. [sent-4, score-0.255]

3 We demonstrate the advantage of our framework for generalization over standard Bayesian networks as well as tree structured copula models for varied real-life domains that are of substantially higher dimension than those typically considered in the copula literature. [sent-5, score-1.407]

4 In contrast, aside from the normal representation, few univariate distributions have a convenient multivariate generalization. [sent-8, score-0.333]

5 Copulas [23] offer a general framework for constructing multivariate distributions using any given (or estimated) univariate marginals and a copula function C that links these marginals. [sent-10, score-1.042]

6 The importance of copulas is rooted in Sklar’s theorem [29] that states that any multivariate distribution can be represented as a copula function of its marginals. [sent-11, score-1.015]

7 The constructive converse is important from a modeling perspective as it allows us to separate the choice of the marginals and that of the dependence structure which is expressed in C. [sent-12, score-0.218]

8 In practice, copula constructions often lead to signiﬁcant improvement in density estimation. [sent-15, score-0.816]

9 Accordingly, there has been a dramatic growth of academic and practical interest in copulas in recent years, with applications ranging from mainstream ﬁnancial risk assessment and actuarial analysis (e. [sent-16, score-0.26]

10 Despite the generality of the framework, constructing high-dimensional copulas is difﬁcult, and much of the research involves only the bivariate case. [sent-22, score-0.338]

11 Several works have attempted to overcome this difﬁculty by suggesting innovative ways in which bivariate copulas can be combined to form workable copulas of higher dimensions. [sent-23, score-0.579]

12 These attempts, however, are either limited to hierarchical [26] or mixture of trees [14] compositions, or rely on a recursive construction of conditional bivariate copulas [1, 3, 17] that is somewhat elaborate for high dimensions. [sent-24, score-0.413]

13 In this widely used framework, a graph structure encodes independencies which imply a decomposition of the joint density into local terms (the density of each variable conditioned on its 1 parents). [sent-27, score-0.395]

14 Furthermore, aside from the case of the normal distribution, the form of the univariate marginal is neither under control nor is it typically known. [sent-33, score-0.269]

15 Our goal is to construct ﬂexible multivariate continuous distributions that maintain desired marginals while accommodating tens and hundreds of variables, or more. [sent-34, score-0.21]

16 We present Copula Bayesian Networks (CBNs), an elegant marriage between the copula and the Bayesian network frameworks. [sent-35, score-0.726]

17 Differently, we rely on local copula functions and an explicit globally shared parameterization of the univariate densities. [sent-37, score-0.874]

18 This allows us to retain the ﬂexibility of BNs, while offering control over the form of the marginals, resulting in substantially improved multivariate densities (see Section 7 for a discussion of the related works of Kirshner [14] and Liu et al. [sent-38, score-0.172]

19 At the heart of our approach is a novel reparameterization of a conditional density using a copula quotient. [sent-40, score-0.846]

20 With this construction, we prove a parallel to the BN factorization theorem: a decomposition of the joint density according to the structure of the graph implies a decomposition of the joint copula. [sent-41, score-0.243]

21 Conversely, a product of local copula-based quotient terms is a valid multivariate copula. [sent-42, score-0.163]

22 This result provides us with a ﬂexible modeling tool where joint densities are constructed via a composition of local copulas and marginal densities. [sent-43, score-0.446]

23 We learn the structure and parameters of a CBN for three varied real-life domains that are of a signiﬁcantly higher dimension than typically reported in the copula literature. [sent-46, score-0.696]

24 Using standard copula functions, we show that in all cases our approach leads to consistent and signiﬁcant improvement in generalization when compared to standard BN models as well as a tree-structured copula model. [sent-47, score-1.342]

25 A copula function [23, 29] links marginal distributions to form a multivariate one. [sent-56, score-0.819]

26 A copula function C : [0, 1]N → [0, 1] is a joint distribution function C(u1 , . [sent-62, score-0.695]

27 , xN ) be any multivariate distribution over real-valued random variables, then there exists a copula function such that F (x1 , . [sent-72, score-0.755]

28 Thus, copulas are “distribution-generating” functions that allow us to separate the choice of the univariate marginals and that of the dependence structure expressed in the copula function C, often resulting in an effective real-valued construction. [sent-81, score-1.226]

29 These extensions are orthogonal to our work and to maintain clarity we focus here on the continuous case 2 Figure 1: Samples from the 2dimensional normal copula density using a correlation matrix with a unit diagonal and an off-diagonal coefﬁcient of 0. [sent-86, score-0.853]

30 5 8 4 6 3 4 2 2 1 0 0 2 1 4 1 0 1 2 3 4 5 Normal(1, 1) marginals 2 0 2 4 6 8 Mix of Gaussians marginals N ∂ F (x) To derive the joint density f (x) = ∂x1 . [sent-89, score-0.338]

31 ∂xN from the copula construction, assuming F has N-order partial derivatives (true almost everywhere when F is continuous), and using the chain rule, we have f (x) = ∂ N C(F(x1 ), . [sent-92, score-0.671]

32 , F(xN )), is called the copula density function. [sent-104, score-0.777]

33 3: A simple copula widely explored in the ﬁnancial community is the Gaussian copula constructed directly by inverting Sklar’s theorem [7] C({F(xi )}) = ΦΣ Φ−1 (F(x1 )), . [sent-108, score-1.342]

34 To get a sense of the power of copulas, Figure 1 shows samples generated from this copula using two different families of univariate marginals. [sent-112, score-0.813]

35 3 Copula Bayesian Networks (CBNs) As in the copula framework, our goal is to model real-valued multivariate distributions while taking advantage of the relative ease of one dimensional estimation. [sent-114, score-0.795]

36 To achieve this goal, we will construct multivariate copulas that are a composition of local copulas that follow the structure of the graph. [sent-116, score-0.689]

37 , yK }, be a conditional density function and let f (x) be the marginal density of X. [sent-124, score-0.286]

38 Then there exists a copula density function c(F(x), F(y1 ), . [sent-125, score-0.777]

39 The converse is also true, for any copula density function c, Rc (F(x), F(y1 ), . [sent-150, score-0.822]

40 , F(yK ))f (x) deﬁnes a valid conditional density function. [sent-153, score-0.158]

41 Recall that c() is itself an N -order derivative of the copula function so computing our denominator is no more difﬁcult than computing c(). [sent-158, score-0.671]

42 Indeed, for the majority of existing copula functions, both have an explicit form. [sent-159, score-0.671]

43 In contrast, the integral term depends both on the copula form and the univariate marginal, and is generally difﬁcult to compute. [sent-160, score-0.813]

44 3 Proof: From the basic properties of cumulative distribution functions, we have that for any copula function C(1, F(y1 ), . [sent-161, score-0.7]

45 , yk ) and thus, using the derivative chain rule, f (y) = ∂ K C(1, F(y1 ), . [sent-167, score-0.192]

46 (1) we have that there exists a copula density for which f (x, y1 , . [sent-181, score-0.777]

47 It follows that there exists a copula for which f (x | y) = c(F(x), F(y1 ), . [sent-188, score-0.671]

48 The implications of this result will underlie our construction: any copula density function c(x, y1 , . [sent-214, score-0.777]

49 , yK ), together with f (x), can be used to parameterize a conditional density f (x | y). [sent-217, score-0.161]

50 The converse is also true: if I(G) hold in fX (x) then the density decomposes according to G (see [16], theorems 3. [sent-226, score-0.188]

51 These results form the basis for the BN model [25] where a joint density is constructed via a composition of local conditional densities. [sent-229, score-0.222]

52 Let G be a directed acyclic graph over X , and let fX (x) be parameterized via a joint copula density fX (x) = c(F(x1 ), . [sent-234, score-0.846]

53 If fX (x) decomposes according to G then the copula density c(F(x1 ), . [sent-238, score-0.814]

54 , F(xN )) = i where ci is a local copula that depends only on the value of Xi and its parents in G. [sent-244, score-0.841]

55 , F(paiki ))} be a set of strictly positive copula densities associated with the nodes of G that have at least one parent. [sent-260, score-0.731]

56 , F(xN )) = i is a valid copula density c(F(x1 ), . [sent-264, score-0.797]

57 It is important to note that the local copulas do not need to agree on the non-univariate marginals of overlapping variables. [sent-269, score-0.403]

58 This is a result of the fact that each copula ci only appears as part of a quotient term which is used to parameterize a conditional density. [sent-270, score-0.746]

59 This gives us the freedom to mix and match local copulas of different types. [sent-271, score-0.324]

60 Equally important is the fact that aside from the univariate densities, we do not need to concern ourselves with any marginal constraints when estimating the parameters of these local copulas functions. [sent-272, score-0.511]

61 3 A Multivariate Copula Model We are now ready to construct a joint density given univariate marginals by properly composing local terms and without worrying about global coherence: Deﬁnition 3. [sent-274, score-0.415]

62 ΘC is a set of local copula densities functions ci (F(xi ), {F(paik )}) that are associated with the nodes of G that have at least one parent. [sent-276, score-0.77]

63 5 : A Copula Bayesian Network deﬁnes a valid joint density fX (x) whose marginal distributions are parameterized by Θf and where the independence statements I(G) hold. [sent-281, score-0.238]

64 The main difference between the CBN model and a regular BN, aside from a novel choice for the local conditional parameterization, is in the shared global component that has the explicit semantics of the univariate marginals. [sent-282, score-0.241]

65 Concretely, the CBN model allows us to decompose the problem of representing a multivariate distribution with given (or estimated) univariate marginals into many local problems that, depending on the structure of G, can be substantially smaller in dimension. [sent-283, score-0.422]

66 For each family of Xi and its parents we are still faced with the problem of choosing an appropriate local copula. [sent-284, score-0.17]

67 In this work we simply limit ourselves to copulas that have convenient multivariate form, but any of the recently suggested methods for constructing multivariate copulas functions (see Section 6) can also be used. [sent-285, score-0.707]

68 In either case, limiting ourselves to a smaller number of variables (a node and its parents) makes the construction of the local copula substantially easier than the construction of the full copula over X . [sent-286, score-1.508]

69 Importantly, as in the case of BNs, our construction of a joint copula density that decomposes over the graph structure G also facilitates efﬁcient parameter estimation and model selection (structure learning), as we brieﬂy discuss in the next section. [sent-287, score-0.948]

70 , F(paiki [m])) While this objective appears to fully decompose according to the structure of G, each marginal distribution F(xi ) actually appears in several local copula terms (of Xi and its children in G). [sent-293, score-0.777]

71 Given F(xi ), we can then estimate the parameters of each local copula independently of the others. [sent-295, score-0.71]

72 We estimate the univariate densities using a standard normal kernel-based approach [24]. [sent-296, score-0.259]

73 In this work we consider two of the simplest and most commonly used copula functions. [sent-297, score-0.671]

74 Models compared: Sigmoid BN; CBN with a uniform correlation normal copula (single parameter); CBN with a full normal copula (0. [sent-338, score-1.475]

75 For the Gaussian copula with a full covariance matrix, a reasonably effective and substantially more efﬁcient method is based on the relationship between the copula function and Kendall’s Tau dependence measure [19]. [sent-345, score-1.394]

76 For lack of space, further details for both of these copulas are provided in the supplementary material. [sent-346, score-0.26]

77 For CBNs, to demonstrate the ﬂexibility of our framework, we consider the three local copula functions discussed in Section 4: fully parametrized Normal copula; the same copula with a single correlation parameter and unit diagonal (UnifCorr); Frank’s single parameter Archimedean copula. [sent-353, score-1.4]

78 We use standard normal kernel density estimation for the univariate densities. [sent-354, score-0.305]

79 For all datasets, the copula based models offer a clear gain in training performance as well as in generalization on unseen test instances. [sent-369, score-0.671]

80 In fact, even Frank’s single parameter Archimedean copula which is constrained by the fact that all of its K-marginals are equal [23], is superior to the BN model. [sent-371, score-0.671]

81 To understand the role of the univariate marginals, we start with the no dependency network (0 on x-axis), where the advantage of CBNs is solely due to the use of ﬂexible univariate marginals. [sent-376, score-0.327]

82 As expected, this is not the case when we constrain the CBN model to have normal marginals (Normal-UnifCorr) and when the domain is sufﬁciently complex (Crime). [sent-378, score-0.161]

83 Finally, the Normal-UnifCorr CBN model, despite the forced normal marginals, does not lead to overly complex structures as it is constrained by the simplicity of the copula function (single parameter). [sent-382, score-0.728]

84 For the challenging Crime dataset, the differences are more pronounced: both the linear and non-linear BN models almost saturate the limit of 4 parents per variable, while the Kernel-UnifCorr copula model requires, on average, less than half the number of parents to achieve superior performance. [sent-383, score-0.933]

85 We recall that the CBN model uses a simple normal copula so that the advantage is solely rooted in the distortion of the input to the copula created by the kernel-based univariate representation. [sent-387, score-1.559]

86 With more expressive copulas we can expect further qualitative and quantitative advantages. [sent-388, score-0.26]

87 01 Physical density Physical density 1 2 10 7 0. [sent-402, score-0.212]

88 005 Physical density Physical density Figure 4: Demonstration of the dependency learned for the Wine dataset for two variable pairs. [sent-409, score-0.212]

89 985 1 0 1 2 3 4 5 6 7 Residual sugar Related Work For lack of space we do not discuss direct multivariate copula constructions (e. [sent-416, score-0.848]

90 The Vine model [3] relies on a recursive construction of bivariate copulas to parameterize a multivariate one. [sent-419, score-0.466]

91 Kirshner [14] uses the copula product operator of Darsow et al. [sent-428, score-0.671]

92 7 Discussion and Future Work We presented Copula Bayesian Networks, a marriage between the Bayesian network and copula frameworks. [sent-436, score-0.726]

93 Building on a novel reparameterization of the conditional density, our model offers great ﬂexibility in modeling high-dimensional continuous distribution while offering control over the form of the univariate marginals. [sent-437, score-0.211]

94 First, our framework allows us to ﬂexibly “mix and match” local copulas and univariate densities of any form. [sent-440, score-0.501]

95 Third, we leverage on existing machinery to perform model selection in signiﬁcantly higher dimensions than typically considered in the copula literature. [sent-442, score-0.696]

96 Thus, our work opens the door for numerous applications where the ﬂexibility of copulas is needed but could not be previously utilized. [sent-443, score-0.26]

97 The gap between train and test performance for CBNs motivates the development of model selection scores tailored to the copula framework (e. [sent-445, score-0.671]

98 Modeling dependence with copulas and applications to risk management. [sent-491, score-0.284]

99 Asymmetric CAPM dependence for large dimensions: The canonical vine autoregressive copula model. [sent-513, score-0.744]

100 The vine copula method for representing high dimensional dependent distributions: Applications to continuous belief nets. [sent-542, score-0.72]

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same-paper 1 0.99999982 53 nips-2010-Copula Bayesian Networks

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2 0.31212968 54 nips-2010-Copula Processes

Author: Andrew Wilson, Zoubin Ghahramani

Abstract: 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. 1 1 Copulas Copulas are important because they separate the dependency structure between random variables from their marginal distributions. Intuitively, we can describe the dependency structure of any multivariate joint distribution H(x1 , . . . , xn ) = P (X1 ≤ x1 , . . . Xn ≤ xn ) through a two step process. First we take each univariate random variable Xi and transform it through its cumulative distribution function (cdf) Fi to get Ui = Fi (Xi ), a uniform random variable. We then express the dependencies between these transformed variables through the n-copula C(u1 , . . . , un ). Formally, an n-copula C : [0, 1]n → [0, 1] is a multivariate cdf with uniform univariate marginals: C(u1 , u2 , . . . , un ) = P (U1 ≤ u1 , U2 ≤ u2 , . . . , Un ≤ un ), where U1 , U2 , . . . , Un are standard uniform random variables. Sklar [12] precisely expressed our intuition in the theorem below. Theorem 1.1. Sklar’s theorem Let H be an n-dimensional distribution function with marginal distribution functions F1 , F2 , . . . , Fn . Then there exists an n-copula C such that for all (x1 , x2 , . . . , xn ) ∈ [−∞, ∞]n , H(x1 , x2 , . . . , xn ) = C(F1 (x1 ), F2 (x2 ), . . . , Fn (xn )) = C(u1 , u2 , . . . , un ). (1) If F1 , F2 , . . . , Fn are all continuous then C is unique; otherwise C is uniquely determined on Range F1 × Range F2 × · · · × Range Fn . Conversely, if C is an n-copula and F1 , F2 , . . . , Fn are distribution functions, then the function H is an n-dimensional distribution function with marginal distribution functions F1 , F2 , . . . , Fn . (−1) As a corollary, if Fi (u) = inf{x : F (x) ≥ u}, the quasi-inverse of Fi , then for all u1 , u2 , . . . , un ∈ [0, 1]n , (−1) C(u1 , u2 , . . . , un ) = H(F1 (−1) (u1 ), F2 (−1) (u2 ), . . . , Fn (un )). (2) In other words, (2) can be used to construct a copula. For example, the bivariate Gaussian copula is deﬁned as C(u, v) = Φρ (Φ−1 (u), Φ−1 (v)), (3) where Φρ is a bivariate Gaussian cdf with correlation coefﬁcient ρ, and Φ is the standard univariate Gaussian cdf. Li [2] popularised the bivariate Gaussian copula, by showing how it could be used to study ﬁnancial risk and default correlation, using credit derivatives as an example. By substituting F (x) for u and G(y) for v in equation (3), we have a bivariate distribution H(x, y), with a Gaussian dependency structure, and marginals F and G. Regardless of F and G, the resulting H(x, y) can still be uniquely expressed as a Gaussian copula, so long as F and G are continuous. It is then a copula itself that captures the underlying dependencies between random variables, regardless of their marginal distributions. For this reason, copulas have been called dependence functions [13, 14]. Nelsen [3] contains an extensive discussion of copulas. 2 Copula Processes Imagine choosing a covariance function, and then drawing a sample function at some ﬁnite number of points from a Gaussian process. The result is a sample from a collection of Gaussian random variables, with a dependency structure encoded by the speciﬁed covariance function. Now, suppose we transform each of these values through a univariate Gaussian cdf, such that we have a sample from a collection of uniform random variables. These uniform random variables also have this underlying Gaussian process dependency structure. One might call the resulting values a draw from a Gaussian-Uniform Process. We could subsequently put these values through an inverse beta cdf, to obtain a draw from what could be called a Gaussian-Beta Process: the values would be a sample from beta random variables, again with an underlying Gaussian process dependency structure. We could also transform the uniform values with different inverse cdfs, which would give a sample from different random variables, with dependencies encoded by the Gaussian process. The above procedure is a means to generate samples from arbitrarily many random variables, with arbitrary marginal distributions, and desired dependencies. It is an example of how to use what we call a copula process – in this case, a Gaussian copula process, since a Gaussian copula describes the dependency structure of a ﬁnite number of samples. We now formally deﬁne a copula process. 2 Deﬁnition 2.1. Copula Process Let {Wt } be a collection of random variables indexed by t ∈ T , with marginal distribution functions Ft , and let Qt = Ft (Wt ). Further, let µ be a stochastic process measure with marginal distribution functions Gt , and joint distribution function H. Then Wt is copula process distributed with base measure µ, or Wt ∼ CP(µ), if and only if for all n ∈ N, ai ∈ R, n P( (−1) {Gti (Qti ) ≤ ai }) = Ht1 ,t2 ,...,tn (a1 , a2 , . . . , an ). (4) i=1 (−1) Each Qti ∼ Uniform(0, 1), and Gti is the quasi-inverse of Gti , as previously deﬁned. Deﬁnition 2.2. Gaussian Copula Process Wt is Gaussian copula process distributed if it is copula process distributed and the base measure µ is a Gaussian process. If there is a mapping Ψ such that Ψ(Wt ) ∼ GP(m(t), k(t, t )), then we write Wt ∼ GCP(Ψ, m(t), k(t, t )). For example, if we have Wt ∼ GCP with m(t) = 0 and k(t, t) = 1, then in the deﬁnition of a copula process, Gt = Φ, the standard univariate Gaussian cdf, and H is the usual GP joint distribution function. Supposing this GCP is a Gaussian-Beta process, then Ψ = Φ−1 ◦ FB , where FB is a univariate Beta cdf. One could similarly deﬁne other copula processes. We described generally how a copula process can be used to generate samples of arbitrarily many random variables with desired marginals and dependencies. We now develop a speciﬁc and practical application of this framework. We introduce a stochastic volatility model, Gaussian Copula Process Volatility (GCPV), as an example of how to learn the joint distribution of arbitrarily many random variables, the marginals of these random variables, and to make predictions. To do this, we ﬁt a Gaussian copula process by using a type of Warped Gaussian Process [15]. However, our methodology varies substantially from Snelson et al. [15], since we are doing inference on latent variables as opposed to observations, which is a much greater undertaking that involves approximations, and we are doing so in a different context. 3 Gaussian Copula Process Volatility Assume we have a sequence of observations y = (y1 , . . . , yn ) at times t = (t1 , . . . , tn ) . The observations are random variables with different latent standard deviations. We therefore have n unobserved standard deviations, σ1 , . . . , σn , and want to learn the correlation structure between these standard deviations, and also to predict the distribution of σ∗ at some unrealised time t∗ . We model the standard deviation function as a Gaussian copula process: σt ∼ GCP(g −1 , 0, k(t, t )). (5) f (t) ∼ GP(m(t) = 0, k(t, t )) σ(t) = g(f (t), ω) (6) (7) y(t) ∼ N (0, σ 2 (t)), (8) Speciﬁcally, where g is a monotonic warping function, parametrized by ω. For each of the observations y = (y1 , . . . , yn ) we have corresponding GP latent function values f = (f1 , . . . , fn ) , where σ(ti ) = g(fi , ω), using the shorthand fi to mean f (ti ). σt ∼ GCP, because any ﬁnite sequence (σ1 , . . . , σp ) is distributed as a Gaussian copula: P (σ1 ≤ a1 , . . . , σp ≤ ap ) = P (g −1 (σ1 ) ≤ g −1 (a1 ), . . . , g −1 (σp ) ≤ g −1 (ap )) = ΦΓ (g −1 (a1 ), . . . , g −1 −1 (ap )) = ΦΓ (Φ = ΦΓ (Φ −1 (F (a1 )), . . . , Φ (u1 ), . . . , Φ −1 −1 (9) (F (ap ))) (up )) = C(u1 , . . . , up ), where Φ is the standard univariate Gaussian cdf (supposing k(t, t) = 1), ΦΓ is a multivariate Gaussian cdf with covariance matrix Γij = cov(g −1 (σi ), g −1 (σj )), and F is the marginal distribution of 3 each σi . In (5), we have Ψ = g −1 , because it is g −1 which maps σt to a GP. The speciﬁcation in (5) is equivalently expressed by (6) and (7). With GCPV, the form of g is learned so that g −1 (σt ) is best modelled by a GP. By learning g, we learn the marginal of each σ: F (a) = Φ(g −1 (a)) for a ∈ R. Recently, a different sort of ‘kernel copula process’ has been used, where the marginals of the variables being modelled are not learned [16].1 Further, we also consider a more subtle and ﬂexible form of our model, where the function g itself is indexed by time: g = gt (f (t), ω). We only assume that the marginal distributions of σt are stationary over ‘small’ time periods, and for each of these time periods (5)-(7) hold true. We return to this in the ﬁnal discussion section. Here we have assumed that each observation, conditioned on knowing its variance, is normally distributed with zero mean. This is a common assumption in heteroscedastic models. The zero mean and normality assumptions can be relaxed and are not central to this paper. 4 Predictions with GCPV Ultimately, we wish to infer p(σ(t∗ )|y, z), where z = {θ, ω}, and θ are the hyperparameters of the GP covariance function. To do this, we sample from p(f∗ |y, z) = p(f∗ |f , θ)p(f |y, z)df (10) and then transform these samples by g. Letting (Cf )ij = δij g(fi , ω)2 , where δij is the Kronecker delta, Kij = k(ti , tj ), (k∗ )i = k(t∗ , ti ), we have p(f |y, z) = N (f ; 0, K)N (y; 0, Cf )/p(y|z), p(f∗ |f , θ) = N (k∗ K −1 f , k(t∗ , t∗ ) − k∗ K −1 (11) k∗ ). (12) We also wish to learn z, which we can do by ﬁnding the z that maximizes the marginal likelihood, ˆ p(y|z) = p(y|f , ω)p(f |θ)df . (13) Unfortunately, for many functions g, (10) and (13) are intractable. Our methods of dealing with this can be used in very general circumstances, where one has a Gaussian process prior, but an (optionally parametrized) non-Gaussian likelihood. We use the Laplace approximation to estimate p(f |y, z) as a Gaussian. Then we can integrate (10) for a Gaussian approximation to p(f∗ |y, z), which we sample from to make predictions of σ∗ . Using Laplace, we can also ﬁnd an expression for an approximate marginal likelihood, which we maximize to determine z. Once we have found z with Laplace, we use Markov chain Monte Carlo to sample from p(f∗ |y, z), and compare that to using Laplace to sample from p(f∗ |y, z). In the supplement we relate this discussion to (9). 4.1 Laplace Approximation The goal is to approximate (11) with a Gaussian, so that we can evaluate (10) and (13) and make predictions. In doing so, we follow Rasmussen and Williams [11] in their treatment of Gaussian process classiﬁcation, except we use a parametrized likelihood, and modify Newton’s method. First, consider as an objective function the logarithm of an unnormalized (11): s(f |y, z) = log p(y|f , ω) + log p(f |θ). (14) ˆ The Laplace approximation uses a second order Taylor expansion about the f which maximizes ˆ, for which we use (14), to ﬁnd an approximate objective s(f |y, z). So the ﬁrst step is to ﬁnd f ˜ Newton’s method. The Newton update is f new = f − ( s(f ))−1 s(f ). Differentiating (14), s(f |y, z) = s(f |y, z) = where W is the diagonal matrix − 1 log p(y|f , ω) − K −1 f log p(y|f , ω) − K −1 = −W − K −1 , log p(y|f , ω). Note added in proof : Also, for a very recent related model, see Rodr´guez et al. [17]. ı 4 (15) (16) If the likelihood function p(y|f , ω) is not log concave, then W may have negative entries. Vanhatalo et al. [18] found this to be problematic when doing Gaussian process regression with a Student-t ˆ likelihood. They instead use an expectation-maximization (EM) algorithm for ﬁnding f , and iterate ordered rank one Cholesky updates to evaluate the Laplace approximate marginal likelihood. But EM can converge slowly, especially near a local optimum, and each of the rank one updates is vulnerable to numerical instability. With a small modiﬁcation of Newton’s method, we often get close to ˆ quadratic convergence for ﬁnding f , and can evaluate the Laplace approximate marginal likelihood in a numerically stable fashion, with no approximate Cholesky factors, and optimal computational requirements. Some comments are in the supplementary material but, in short, we use an approximate negative Hessian, − s ≈ M + K −1 , which is guaranteed to be positive deﬁnite, since M is formed on each iteration by zeroing the negative entries of W . For stability, we reformulate our 1 1 1 1 optimization in terms of B = I + M 2 KM 2 , and let Q = M 2 B −1 M 2 , b = M f + log p(y|f ), a = b − QKb. Since (K −1 + M )−1 = K − KQK, the Newton update becomes f new = Ka. ˆ With these updates we ﬁnd f and get an expression for s which we use to approximate (13) and ˜ (11). The approximate marginal likelihood q(y|z) is given by exp(˜)df . Taking its logarithm, s 1ˆ 1 ˆ log q(y|z) = − f af + log p(y|f ) − log |Bf |, (17) ˆ ˆ 2 2 ˆ ˆ where Bf is B evaluated at f , and af is a numerically stable evaluation of K −1 f . ˆ ˆ To learn the parameters z, we use conjugate gradient descent to maximize (17) with respect to z. ˆ ˆ Since f is a function of z, we initialize z, and update f every time we vary z. Once we have found an optimum z , we can make predictions. By exponentiating s, we ﬁnd a Gaussian approximation to ˆ ˜ ˆ the posterior (11), q(f |y, z) = N (f , K − KQK). The product of this approximate posterior with p(f∗ |f ) is Gaussian. Integrating this product, we approximate p(f∗ |y, z) as ˆ q(f∗ |y, z) = N (k∗ log p(y|f ), k(t∗ , t∗ ) − k∗ Qk∗ ). (18) Given n training observations, the cost of each Newton iteration is dominated by computing the cholesky decomposition of B, which takes O(n3 ) operations. The objective function typically changes by less than 10−6 after 3 iterations. Once Newton’s method has converged, it takes only O(1) operations to draw from q(f∗ |y, z) and make predictions. 4.2 Markov chain Monte Carlo We use Markov chain Monte Carlo (MCMC) to sample from (11), so that we can later sample from p(σ∗ |y, z) to make predictions. Sampling from (11) is difﬁcult, because the variables f are strongly coupled by a Gaussian process prior. We use a new technique, Elliptical Slice Sampling [19], and ﬁnd it extremely effective for this purpose. It was speciﬁcally designed to sample from posteriors with correlated Gaussian priors. It has no free parameters, and jointly updates every element of f . For our setting, it is over 100 times as fast as axis aligned slice sampling with univariate updates. To make predictions, we take J samples of p(f |y, z), {f 1 , . . . , f J }, and then approximate (10) as a mixture of J Gaussians: J 1 p(f∗ |f i , θ). (19) p(f∗ |y, z) ≈ J i=1 Each of the Gaussians in this mixture have equal weight. So for each sample of f∗ |y, we uniformly choose a random p(f∗ |f i , θ) and draw a sample. In the limit J → ∞, we are sampling from the exact p(f∗ |y, z). Mapping these samples through g gives samples from p(σ∗ |y, z). After one O(n3 ) and one O(J) operation, a draw from (19) takes O(1) operations. 4.3 Warping Function The warping function, g, maps fi , a GP function value, to σi , a standard deviation. Since fi can take any value in R, and σi can take any non-negative real value, g : R → R+ . For each fi to correspond to a unique deviation, g must also be one-to-one. We use K g(x, ω) = aj log[exp[bj (x + cj )] + 1], j=1 5 aj , bj > 0. (20) This is monotonic, positive, inﬁnitely differentiable, asymptotic towards zero as x → −∞, and K tends to ( j=1 aj bj )x as x → ∞. In practice, it is useful to add a small constant to (20), to avoid rare situations where the parameters ω are trained to make g extremely small for certain inputs, at the expense of a good overall ﬁt; this can happen when the parameters ω are learned by optimizing a likelihood. A suitable constant could be one tenth the absolute value of the smallest nonzero observation. By inferring the parameters of the warping function, or distributions of these parameters, we are learning a transformation which will best model σt with a Gaussian process. The more ﬂexible the warping function, the more potential there is to improve the GCPV ﬁt – in other words, the better we can estimate the ‘perfect’ transformation. To test the importance of this ﬂexibility, we also try a simple unparametrized warping function, g(x) = ex . In related work, Goldberg et al. [20] place a GP prior on the log noise level in a standard GP regression model on observations, except for inference they use Gibbs sampling, and a high level of ‘jitter’ for conditioning. Once g is trained, we can infer the marginal distribution of each σ: F (a) = Φ(g −1 (a)), for a ∈ R. This suggests an alternate way to initialize g: we can initialize F as a mixture of Gaussians, and then map through Φ−1 to ﬁnd g −1 . Since mixtures of Gaussians are dense in the set of probability distributions, we could in principle ﬁnd the ‘perfect’ g using an inﬁnite mixture of Gaussians [21]. 5 Experiments In our experiments, we predict the latent standard deviations σ of observations y at times t, and also σ∗ at unobserved times t∗ . To do this, we use two versions of GCPV. The ﬁrst variant, which we simply refer to as GCPV, uses the warping function (20) with K = 1, and squared exponential covariance function, k(t, t ) = A exp(−(t−t )2 /l2 ), with A = 1. The second variant, which we call GP-EXP, uses the unparametrized warping function ex , and the same covariance function, except the amplitude A is a trained hyperparameter. The other hyperparameter l is called the lengthscale of the covariance function. The greater l, the greater the covariance between σt and σt+a for a ∈ R. We train hyperparameters by maximizing the Laplace approximate log marginal likelihood (17). We then sample from p(f∗ |y) using the Laplace approximation (18). We also do this using MCMC (19) with J = 10000, after discarding a previous 10000 samples of p(f |y) as burn-in. We pass 2 these samples of f∗ |y through g and g 2 to draw from p(σ∗ |y) and p(σ∗ |y), and compute the sample mean and variance of σ∗ |y. We use the sample mean as a point predictor, and the sample variance for error bounds on these predictions, and we use 10000 samples to compute these quantities. For GCPV we use Laplace and MCMC for inference, but for GP-EXP we only use Laplace. We compare predictions to GARCH(1,1), which has been shown in extensive and recent reviews to be competitive with other GARCH variants, and more sophisticated models [5, 6, 7]. GARCH(p,q) speciﬁes y(t) ∼ p 2 2 N (0, σ 2 (t)), and lets the variance be a deterministic function of the past: σt = a0 + i=1 ai yt−i + q 2 j=1 bj σt−j . We use the Matlab Econometrics Toolbox implementation of GARCH, where the parameters a0 , ai and bj are estimated using a constrained maximum likelihood. We make forecasts of volatility, and we predict historical volatility. By ‘historical volatility’ we mean the volatility at observed time points, or between these points. Uncovering historical volatility is important. It could, for instance, be used to study what causes ﬂuctuations in the stock market, or to understand physical systems. To evaluate our model, we use the Mean Squared Error (MSE) between the true variance, or proxy for the truth, and the predicted variance. Although likelihood has advantages, we are limited in space, and we wish to harmonize with the econometrics literature, and other assessments of volatility models, where MSE is the standard. In a similar assessment of volatility models, Brownlees et al. [7] found that MSE and quasi-likelihood rankings were comparable. When the true variance is unknown we follow Brownlees et al. [7] and use squared observations as a proxy for the truth, to compare our model to GARCH.2 The more observations, the more reliable these performance estimates will be. However, not many observations (e.g. 100) are needed for a stable ranking of competing models; in Brownlees et al. [7], the rankings derived from high frequency squared observations are similar to those derived using daily squared observations. 2 Since each observation y is assumed to have zero mean and variance σ 2 , E[y 2 ] = σ 2 . 6 5.1 Simulations We simulate observations from N (0, σ 2 (t)), using σ(t) = sin(t) cos(t2 ) + 1, at t = (0, 0.02, 0.04, . . . , 4) . We call this data set TRIG. We also simulate using a standard deviation that jumps from 0.1 to 7 and back, at times t = (0, 0.1, 0.2, . . . , 6) . We call this data set JUMP. To forecast, we use all observations up until the current time point, and make 1, 7, and 30 step ahead predictions. So, for example, in TRIG we start by observing t = 0, and make forecasts at t = 0.02, 0.14, 0.60. Then we observe t = 0, 0.02 and make forecasts at t = 0.04, 0.16, 0.62, and so on, until all data points have been observed. For historical volatility, we predict the latent σt at the observation times, which is safe since we are comparing to the true volatility, which is not used in training; the results are similar if we interpolate. Figure 1 panels a) and b) show the true volatility for TRIG and JUMP respectively, alongside GCPV Laplace, GCPV MCMC, GP-EXP Laplace, and GARCH(1,1) predictions of historical volatility. Table 1 shows the results for forecasting and historical volatility. In panel a) we see that GCPV more accurately captures the dependencies between σ at different times points than GARCH: if we manually decrease the lengthscale in the GCPV covariance function, we can replicate the erratic GARCH behaviour, which inaccurately suggests that the covariance between σt and σt+a decreases quickly with increases in a. We also see that GCPV with an unparametrized exponential warping function tends to overestimates peaks and underestimate troughs. In panel b), the volatility is extremely difﬁcult to reconstruct or forecast – with no warning it will immediately and dramatically increase or decrease. This behaviour is not suited to a smooth squared exponential covariance function. Nevertheless, GCPV outperforms GARCH, especially in regions of low volatility. We also see this in panel a) for t ∈ (1.5, 2). GARCH is known to respond slowly to large returns, and to overpredict volatility [22]. In JUMP, the greater the peaks, and the smaller the troughs, the more GARCH suffers, while GCPV is mostly robust to these changes. 5.2 Financial Data The returns on the daily exchange rate between the Deutschmark (DM) and the Great Britain Pound (GBP) from 1984 to 1992 have become a benchmark for assessing the performance of GARCH models [8, 9, 10]. This exchange data, which we refer to as DMGBP, can be obtained from www.datastream.com, and the returns are calculated as rt = log(Pt+1 /Pt ), where Pt is the number of DM to GBP on day t. The returns are assumed to have a zero mean function. We use a rolling window of the previous 120 days of returns to make 1, 7, and 30 day ahead volatility forecasts, starting at the beginning of January 1988, and ending at the beginning of January 1992 (659 trading days). Every 7 days, we retrain the parameters of GCPV and GARCH. Every time we retrain parameters, we predict historical volatility over the past 120 days. The average MSE for these historical predictions is given in Table 1, although they should be observed with caution; unlike with the simulations, the DMGBP historical predictions are trained using the same data they are assessed on. In Figure 1c), we see that the GARCH one day ahead forecasts are lifted above the GCPV forecasts, but unlike in the simulations, they are now operating on a similar lengthscale. This suggests that GARCH could still be overpredicting volatility, but that GCPV has adapted its estimation of how σt and σt+a correlate with one another. Since GARCH is suited to this ﬁnancial data set, it is reassuring that GCPV predictions have a similar time varying structure. Overall, GCPV and GARCH are competitive with one another for forecasting currency exchange returns, as seen in Table 1. Moreover, a learned warping function g outperforms an unparametrized one, and a full Laplace solution is comparable to using MCMC for inference, in accuracy and speed. This is also true for the simulations. Therefore we recommend whichever is more convenient to implement. 6 Discussion We deﬁned a copula process, and as an example, developed a stochastic volatility model, GCPV, which can outperform GARCH. With GCPV, the volatility σt is distributed as a Gaussian Copula Process, which separates the modelling of the dependencies between volatilities at different times from their marginal distributions – arguably the most useful property of a copula. Further, GCPV ﬁts the marginals in the Gaussian copula process by learning a warping function. If we had simply chosen an unparametrized exponential warping function, we would incorrectly be assuming that the log 7 Table 1: MSE for predicting volatility. Data set Model Historical 1 step 7 step 30 step TRIG GCPV (LA) GCPV (MCMC) GP-EXP GARCH 0.0953 0.0760 0.193 0.938 0.588 0.622 0.646 1.04 0.951 0.979 1.36 1.79 1.71 1.76 1.15 5.12 JUMP GCPV (LA) GCPV (MCMC) GP-EXP GARCH 0.588 1.21 1.43 1.88 0.891 0.951 1.76 1.58 1.38 1.37 6.95 3.43 1.35 1.35 14.7 5.65 GCPV (LA) GCPV (MCMC) GP-EXP GARCH 2.43 2.39 2.52 2.83 3.00 3.00 3.20 3.03 3.08 3.08 3.46 3.12 3.17 3.17 5.14 3.32 ×103 DMGBP ×10−9 TRIG JUMP DMGBP 20 DMGBP 0.015 600 Probability Density 3 1 Volatility Volatility Volatility 15 2 10 0.01 0.005 5 0 0 1 2 Time (a) 3 4 0 0 2 4 0 6 Time (b) 0 200 400 Days (c) 600 400 200 0 0 0.005 σ (d) 0.01 Figure 1: Predicting volatility and learning its marginal pdf. For a) and b), the true volatility, and GCPV (MCMC), GCPV (LA), GP-EXP, and GARCH predictions, are shown respectively by a thick green line, a dashed thick blue line, a dashed black line, a cyan line, and a red line. a) shows predictions of historical volatility for TRIG, where the shade is a 95% conﬁdence interval about GCPV (MCMC) predictions. b) shows predictions of historical volatility for JUMP. In c), a black line and a dashed red line respectively show GCPV (LA) and GARCH one day ahead volatility forecasts for DMGBP. In d), a black line and a dashed blue line respectively show the GCPV learned marginal pdf of σt in DMGBP and a Gamma(4.15,0.00045) pdf. volatilities are marginally Gaussian distributed. Indeed, for the DMGBP data, we trained the warping function g over a 120 day period, and mapped its inverse through the univariate standard Gaussian cdf Φ, and differenced, to estimate the marginal probability density function (pdf) of σt over this period. The learned marginal pdf, shown in Figure 1d), is similar to a Gamma(4.15,0.00045) distribution. However, in using a rolling window to retrain the parameters of g, we do not assume that the marginals of σt are stationary; we have a time changing warping function. While GARCH is successful, and its simplicity is attractive, our model is also simple and has a number of advantages. We can effortlessly handle missing data, we can easily incorporate covariates other than time (like interest rates) in our covariance function, and we can choose from a rich class of covariance functions – squared exponential, Brownian motion, Mat´ rn, periodic, etc. In fact, the e volatility of high frequency intradaily returns on equity indices and currency exchanges is cyclical [23], and GCPV with a periodic covariance function is uniquely well suited to this data. And the parameters of GCPV, like the covariance function lengthscale, or the learned warping function, provide insight into the underlying source of volatility, unlike the parameters of GARCH. Finally, copulas are rapidly becoming popular in applications, but often only bivariate copulas are being used. With our copula process one can learn the dependencies between arbitrarily many random variables independently of their marginal distributions. We hope the Gaussian Copula Process Volatility model will encourage other applications of copula processes. More generally, we hope our work will help bring together the machine learning and econometrics communities. Acknowledgments: Thanks to Carl Edward Rasmussen and Ferenc Husz´ r for helpful conversaa tions. AGW is supported by an NSERC grant. 8 References [1] Paul Embrechts, Alexander McNeil, and Daniel Straumann. Correlation and dependence in risk management: Properties and pitfalls. In Risk Management: Value at risk and beyond, pages 176–223. Cambridge University Press, 1999. [2] David X. Li. On default correlation: A copula function approach. Journal of Fixed Income, 9(4):43–54, 2000. [3] Roger B. Nelsen. An Introduction to Copulas. Springer Series in Statistics, second edition, 2006. [4] Tim Bollerslev. Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31 (3):307–327, 1986. [5] Ser-Huang Poon and Clive W.J. Granger. Practical issues in forecasting volatility. Financial Analysts Journal, 61(1):45–56, 2005. [6] Peter Reinhard Hansen and Asger Lunde. A forecast comparison of volatility models: Does anything beat a GARCH(1,1). Journal of Applied Econometrics, 20(7):873–889, 2005. [7] Christian T. Brownlees, Robert F. Engle, and Bryan T. Kelly. A practical guide to volatility forecasting through calm and storm, 2009. Available at SSRN: http://ssrn.com/abstract=1502915. [8] T. Bollerslev and E. Ghysels. Periodic autoregressive conditional heteroscedasticity. Journal of Business and Economic Statistics, 14:139–151, 1996. [9] B.D. McCullough and C.G. Renfro. Benchmarks and software standards: A case study of GARCH procedures. Journal of Economic and Social Measurement, 25:59–71, 1998. [10] C. Brooks, S.P. Burke, and G. Persand. Benchmarks and the accuracy of GARCH model estimation. International Journal of Forecasting, 17:45–56, 2001. [11] Carl Edward Rasmussen and Christopher K.I. Williams. Gaussian processes for Machine Learning. The MIT Press, 2006. ` [12] Abe Sklar. Fonctions de r´ partition a n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris, 8: e 229–231, 1959. [13] P Deheuvels. Caract´ isation compl` te des lois extrˆ mes multivari´ s et de la convergence des types e e e e extrˆ mes. Publications de l’Institut de Statistique de l’Universit´ de Paris, 23:1–36, 1978. e e [14] G Kimeldorf and A Sampson. Uniform representations of bivariate distributions. Communications in Statistics, 4:617–627, 1982. [15] Edward Snelson, Carl Edward Rasmussen, and Zoubin Ghahramani. Warped Gaussian Processes. In NIPS, 2003. [16] Sebastian Jaimungal and Eddie K.H. Ng. Kernel-based Copula processes. In ECML PKDD, 2009. [17] A. Rodr´guez, D.B. Dunson, and A.E. Gelfand. Latent stick-breaking processes. Journal of the American ı Statistical Association, 105(490):647–659, 2010. [18] Jarno Vanhatalo, Pasi Jylanki, and Aki Vehtari. Gaussian process regression with Student-t likelihood. In NIPS, 2009. [19] Iain Murray, Ryan Prescott Adams, and David J.C. MacKay. Elliptical Slice Sampling. In AISTATS, 2010. [20] Paul W. Goldberg, Christopher K.I. Williams, and Christopher M. Bishop. Regression with inputdependent noise: A Gaussian process treatment. In NIPS, 1998. [21] Carl Edward Rasmussen. The Inﬁnite Gaussian Mixture Model. In NIPS, 2000. [22] Ruey S. Tsay. Analysis of Financial Time Series. John Wiley & Sons, 2002. [23] Torben G. 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