{"title": "Poisson-Randomized Gamma Dynamical Systems", "book": "Advances in Neural Information Processing Systems", "page_first": 782, "page_last": 793, "abstract": "This paper presents the Poisson-randomized gamma dynamical system (PRGDS), a model for sequentially observed count tensors that encodes a strong inductive bias toward sparsity and burstiness. The PRGDS is based on a new motif in Bayesian latent variable modeling, an alternating chain of discrete Poisson and continuous gamma latent states that is analytically convenient and computationally tractable. This motif yields closed-form complete conditionals for all variables by way of the Bessel distribution and a novel discrete distribution that we call the shifted confluent hypergeometric distribution. We draw connections to closely related models and compare the PRGDS to these models in studies of real-world count data sets of text, international events, and neural spike trains. We find that a sparse variant of the PRGDS, which allows the continuous gamma latent states to take values of exactly zero, often obtains better predictive performance than other models and is uniquely capable of inferring latent structures that are highly localized in time.", "full_text": "Poisson-Randomized Gamma Dynamical Systems\n\nAaron Schein\n\nData Science Institute\nColumbia University\n\nScott W. Linderman\nDepartment of Statistics\n\nStanford University\n\nMingyuan Zhou\n\nMcCombs School of Business\nUniversity of Texas at Austin\n\nDavid M. Blei\n\nDepartment of Statistics\n\nColumbia University\n\nHanna Wallach\nMicrosoft Research\n\nNew York, NY\n\nAbstract\n\nThis paper presents the Poisson-randomized gamma dynamical system (PRGDS), a\nmodel for sequentially observed count tensors that encodes a strong inductive bias\ntoward sparsity and burstiness. The PRGDS is based on a new motif in Bayesian\nlatent variable modeling, an alternating chain of discrete Poisson and continuous\ngamma latent states that is analytically convenient and computationally tractable.\nThis motif yields closed-form complete conditionals for all variables by way of the\nBessel distribution and a novel discrete distribution that we call the shifted con\ufb02uent\nhypergeometric distribution. We draw connections to closely related models and\ncompare the PRGDS to these models in studies of real-world count data sets of\ntext, international events, and neural spike trains. We \ufb01nd that a sparse variant of\nthe PRGDS, which allows the continuous gamma latent states to take values of\nexactly zero, often obtains better predictive performance than other models and is\nuniquely capable of inferring latent structures that are highly localized in time.\n\n1\n\nIntroduction\n\nPolitical scientists routinely analyze event counts of the number of times country i took action a\ntoward country j during time step t [1]. Such data can be represented as a sequence of count tensors\nY (1), . . . , Y (T ) each of which contains the V\u00d7V\u00d7A event counts for that time step for every combina-\ntion of V sender countries, V receivers, and A action types. International event data sets exhibit \u201ccom-\nplex dependence structures\u201d [2] like coalitions of countries and bursty temporal dynamics. These de-\npendence structures violate the independence assumptions of the regression-based methods that politi-\ncal scientists have traditionally used to test theories of international relations [3\u20135]. Political scientists\nhave therefore advocated for using latent variable models to infer unobserved structures as a way of\ncontrolling for them [6]. This approach motivates interpretable yet expressive models that are capable\nof capturing a variety of complex dependence structures. Recent work has applied tensor factorization\nmethods to international event data sets [7\u201311] to infer coalition structures among countries and topic\nstructures among actions; however, these methods assume that the sequentially observed count tensors\nare exchangeable, thereby failing to capture the bursty temporal dynamics inherent to such data sets.\nSequentially observed count tensors present unique statistical challenges because they tend to be bursty\n[12], high-dimensional, and sparse [13, 14]. There are few models that are tailored to the challenging\nproperties of both time series and count tensors. In recent years, Poisson factorization has emerged\nas a framework for modeling count matrices [15\u201320] and tensors [13, 21, 9]. Although factorization\nmethods generally scale with the size of the matrix or tensor, many Poisson factorization models\nyield inference algorithms that scale linearly with the number of non-zero entries. This property\nallows researchers to ef\ufb01ciently infer latent structures from massive tensors, provided these tensors\nare sparse; however, this property is unique to a subset of Poisson factorization models that only posit\n\n33rd Conference on Neural Information Processing Systems (NeurIPS 2019), Vancouver, Canada.\n\n\f(a) Poisson\u2013gamma dynamical systems [22]\n\n(b) Poisson-randomized gamma dynamical systems\n\nFigure 1: Left: The PGDS imposes dependencies directly between the gamma latent states, preventing\nclosed-form complete conditionals. Right: The PRGDS (this paper) breaks these dependencies with discrete\nPoisson latent states\u2014doing so yields closed-form conditionals for all variables without data augmentation.\n\nnon-negative prior distributions, which are dif\ufb01cult to chain in state-space models for time series. Hier-\narchical compositions of non-negative priors\u2014notably, gamma and Dirichlet distributions\u2014typically\nintroduce non-conjugate dependencies that require innovative approaches to posterior inference.\nThis paper \ufb01lls a gap in the literature between Poisson factorization models that are tractable\u2014i.e.,\nyielding closed-form complete conditionals that make inference algorithms easy to derive\u2014and those\nthat are expressive\u2014i.e., capable of capturing a variety of complex dependence structures. To do\nso, we introduce an alternating chain of discrete Poisson and continuous gamma latent states, a new\nmodeling motif that is analytically convenient and computationally tractable. We rely on this motif\nto construct the Poisson-randomized gamma dynamical system (PRGDS), a model for sequentially\nobserved count tensors that is tractable, expressive, and ef\ufb01cient. The PRGDS is closely related to the\nPoisson\u2013gamma dynamical system (PGDS) [22], a recently introduced model for dynamic count ma-\ntrices, that is based on non-conjugate chains of gamma states. These chains are intractable; thus, poste-\nrior inference in the PGDS relies on sophisticated data augmentation schemes that are cumbersome to\nderive and impose unnatural restrictions on the priors over other variables. In contrast, the PRGDS in-\ntroduces intermediate Poisson states that break the intractable dependencies between the gamma states\n(see Fig. 1). Although this motif is only semi-conjugate, it is tractable, yielding closed-form complete\nconditionals for the Poisson states by way of the little-known Bessel distribution [23] and a novel\ndiscrete distribution that we derive and call the shifted con\ufb02uent hypergeometric (SCH) distribution.\nWe study the inductive bias of the PRGDS by comparing its smoothing and forecasting performance\nto that of the PGDS and two other baselines on a range of real-world count data sets of text, interna-\ntional events, and neural spike data. For smoothing, we \ufb01nd that the PRGDS performs better than or\nsimilarly to the PGDS; for forecasting, we \ufb01nd the converse relationship. Both models outperform the\nother baselines. Using a speci\ufb01c hyperparameter setting, the PRGDS permits the continuous gamma\nlatent states to take values of exactly zero, thereby encoding a unique inductive bias tailored to sparsity\nand burstiness. We \ufb01nd that this sparse variant always obtains better smoothing and forecasting perfor-\nmance than the non-sparse variant. We also \ufb01nd that this sparse variant yields a qualitatively broader\nrange of latent structures\u2014speci\ufb01cally, bursty latent structures that are highly localized in time.\n\n2 Poisson-randomized gamma dynamical systems (PRGDS)\n\nNotation. Consider a data set of sequentially observed count tensors Y (1), . . . , Y (T ), each of\ni \u2208{0, 1, 2, . . .} in the tth tensor is subscripted by a multi-index\nwhich has M modes. An entry y(t)\ni \u2261 (i1, . . . , iM ) that indexes into the M modes of the tensor. As an example, the event count of\nthe number of times country i took action a toward country j during time step t can be written as\ni where the multi-index corresponds to the sender, receiver, and action type\u2014i.e., i = (i, j, a).\ny(t)\nGenerative process. The PRGDS is a form of canonical polyadic decomposition [24] that assumes\n\ny(t)\n\ni \u223c Pois(cid:16)\u03c1(t)\n\nK(cid:88)k=1\n\n\u03bbk \u03b8(t)\nk\n\nM(cid:89)m=1\n\n\u03c6(m)\n\nkim(cid:17),\n\n2\n\n(1)\n\n\u2713(3)AAACLHicbVDLSsNAFJ34rPVVdekmWMS6KYkKuiy4cVnBPqCJZTK9bYdOHszcCDXkE/wVN271L9yIuHXpNzhNs9C2B4Y5nHMvnHu8SHCFlvVhLC2vrK6tFzaKm1vbO7ulvf2mCmPJoMFCEcq2RxUIHkADOQpoRxKo7wloeaPrid96AKl4GNzhOALXp4OA9zmjqKVu6cTxQtFTY19/iYNDQJreJ45Pcah8KgTIpHJ+mqbdUtmqWhnMeWLnpExy1LulH6cXstiHAJmgSnVsK0I3oRI5E5AWnVhBRNmIDqCjaUB9UG6SHZSax1rpmf1Q6hegmal/NxLqq0lmPZklnfUm4kJPglD8ERZ5nRj7V27CgyhGCNg0RD8WJobmpDmzxyUwFGNNKJNc32GyIZWUoe63qAuyZ+uYJ82zqm1V7duLcq2SV1Ugh+SIVIhNLkmN3JA6aRBGnsgLeSVvxrPxbnwaX9PRJSPfOSD/YHz/Ar25qVs=AAACLHicbVDLSsNAFJ34rPVVdekmWMS6KYkKuiy4cVnBPqCJZTK9bYdOHszcCDXkE/wVN271L9yIuHXpNzhNs9C2B4Y5nHMvnHu8SHCFlvVhLC2vrK6tFzaKm1vbO7ulvf2mCmPJoMFCEcq2RxUIHkADOQpoRxKo7wloeaPrid96AKl4GNzhOALXp4OA9zmjqKVu6cTxQtFTY19/iYNDQJreJ45Pcah8KgTIpHJ+mqbdUtmqWhnMeWLnpExy1LulH6cXstiHAJmgSnVsK0I3oRI5E5AWnVhBRNmIDqCjaUB9UG6SHZSax1rpmf1Q6hegmal/NxLqq0lmPZklnfUm4kJPglD8ERZ5nRj7V27CgyhGCNg0RD8WJobmpDmzxyUwFGNNKJNc32GyIZWUoe63qAuyZ+uYJ82zqm1V7duLcq2SV1Ugh+SIVIhNLkmN3JA6aRBGnsgLeSVvxrPxbnwaX9PRJSPfOSD/YHz/Ar25qVs=AAACLHicbVDLSsNAFJ34rPVVdekmWMS6KYkKuiy4cVnBPqCJZTK9bYdOHszcCDXkE/wVN271L9yIuHXpNzhNs9C2B4Y5nHMvnHu8SHCFlvVhLC2vrK6tFzaKm1vbO7ulvf2mCmPJoMFCEcq2RxUIHkADOQpoRxKo7wloeaPrid96AKl4GNzhOALXp4OA9zmjqKVu6cTxQtFTY19/iYNDQJreJ45Pcah8KgTIpHJ+mqbdUtmqWhnMeWLnpExy1LulH6cXstiHAJmgSnVsK0I3oRI5E5AWnVhBRNmIDqCjaUB9UG6SHZSax1rpmf1Q6hegmal/NxLqq0lmPZklnfUm4kJPglD8ERZ5nRj7V27CgyhGCNg0RD8WJobmpDmzxyUwFGNNKJNc32GyIZWUoe63qAuyZ+uYJ82zqm1V7duLcq2SV1Ugh+SIVIhNLkmN3JA6aRBGnsgLeSVvxrPxbnwaX9PRJSPfOSD/YHz/Ar25qVs=AAACLHicbVDLSsNAFJ34rPVVdekmWMS6KYkKuiy4cVnBPqCJZTK9bYdOHszcCDXkE/wVN271L9yIuHXpNzhNs9C2B4Y5nHMvnHu8SHCFlvVhLC2vrK6tFzaKm1vbO7ulvf2mCmPJoMFCEcq2RxUIHkADOQpoRxKo7wloeaPrid96AKl4GNzhOALXp4OA9zmjqKVu6cTxQtFTY19/iYNDQJreJ45Pcah8KgTIpHJ+mqbdUtmqWhnMeWLnpExy1LulH6cXstiHAJmgSnVsK0I3oRI5E5AWnVhBRNmIDqCjaUB9UG6SHZSax1rpmf1Q6hegmal/NxLqq0lmPZklnfUm4kJPglD8ERZ5nRj7V27CgyhGCNg0RD8WJobmpDmzxyUwFGNNKJNc32GyIZWUoe63qAuyZ+uYJ82zqm1V7duLcq2SV1Ugh+SIVIhNLkmN3JA6aRBGnsgLeSVvxrPxbnwaX9PRJSPfOSD/YHz/Ar25qVs=Y(2)AAACJ3icbVDLSsNAFJ34rPUVdekmtAp1U5Ii6LLgxmUF+5Amlsnkph06eTAzEWrI3l9x41b/wp3o0h/wG5ymWWjbA8MczrkXzj1uzKiQpvmlrayurW9slrbK2zu7e/v6wWFHRAkn0CYRi3jPxQIYDaEtqWTQizngwGXQdcdXU7/7AFzQKLyVkxicAA9D6lOCpZIGesV2I+aJSaC+9C67T+0Ay5EIMGPA01rjLMsGetWsmzmMRWIVpIoKtAb6j+1FJAkglIRhIfqWGUsnxVxSwiAr24mAGJMxHkJf0RAHIJw0vyUzTpXiGX7E1Qulkat/N1IciGlcNZknnfem4lKPAxP0EZZ5/UT6l05KwziREJJZCD9hhoyMaWmGRzkQySaKYMKpusMgI8wxkarasirImq9jkXQadcusWzfn1eZJUVUJHaMKqiELXaAmukYt1EYEPaEX9IretGftXfvQPmejK1qxc4T+Qfv+BUuopws=AAACJ3icbVDLSsNAFJ34rPUVdekmtAp1U5Ii6LLgxmUF+5Amlsnkph06eTAzEWrI3l9x41b/wp3o0h/wG5ymWWjbA8MczrkXzj1uzKiQpvmlrayurW9slrbK2zu7e/v6wWFHRAkn0CYRi3jPxQIYDaEtqWTQizngwGXQdcdXU7/7AFzQKLyVkxicAA9D6lOCpZIGesV2I+aJSaC+9C67T+0Ay5EIMGPA01rjLMsGetWsmzmMRWIVpIoKtAb6j+1FJAkglIRhIfqWGUsnxVxSwiAr24mAGJMxHkJf0RAHIJw0vyUzTpXiGX7E1Qulkat/N1IciGlcNZknnfem4lKPAxP0EZZ5/UT6l05KwziREJJZCD9hhoyMaWmGRzkQySaKYMKpusMgI8wxkarasirImq9jkXQadcusWzfn1eZJUVUJHaMKqiELXaAmukYt1EYEPaEX9IretGftXfvQPmejK1qxc4T+Qfv+BUuopws=AAACJ3icbVDLSsNAFJ34rPUVdekmtAp1U5Ii6LLgxmUF+5Amlsnkph06eTAzEWrI3l9x41b/wp3o0h/wG5ymWWjbA8MczrkXzj1uzKiQpvmlrayurW9slrbK2zu7e/v6wWFHRAkn0CYRi3jPxQIYDaEtqWTQizngwGXQdcdXU7/7AFzQKLyVkxicAA9D6lOCpZIGesV2I+aJSaC+9C67T+0Ay5EIMGPA01rjLMsGetWsmzmMRWIVpIoKtAb6j+1FJAkglIRhIfqWGUsnxVxSwiAr24mAGJMxHkJf0RAHIJw0vyUzTpXiGX7E1Qulkat/N1IciGlcNZknnfem4lKPAxP0EZZ5/UT6l05KwziREJJZCD9hhoyMaWmGRzkQySaKYMKpusMgI8wxkarasirImq9jkXQadcusWzfn1eZJUVUJHaMKqiELXaAmukYt1EYEPaEX9IretGftXfvQPmejK1qxc4T+Qfv+BUuopws=AAACJ3icbVDLSsNAFJ34rPUVdekmtAp1U5Ii6LLgxmUF+5Amlsnkph06eTAzEWrI3l9x41b/wp3o0h/wG5ymWWjbA8MczrkXzj1uzKiQpvmlrayurW9slrbK2zu7e/v6wWFHRAkn0CYRi3jPxQIYDaEtqWTQizngwGXQdcdXU7/7AFzQKLyVkxicAA9D6lOCpZIGesV2I+aJSaC+9C67T+0Ay5EIMGPA01rjLMsGetWsmzmMRWIVpIoKtAb6j+1FJAkglIRhIfqWGUsnxVxSwiAr24mAGJMxHkJf0RAHIJw0vyUzTpXiGX7E1Qulkat/N1IciGlcNZknnfem4lKPAxP0EZZ5/UT6l05KwziREJJZCD9hhoyMaWmGRzkQySaKYMKpusMgI8wxkarasirImq9jkXQadcusWzfn1eZJUVUJHaMKqiELXaAmukYt1EYEPaEX9IretGftXfvQPmejK1qxc4T+Qfv+BUuopws=Y(3)AAACJ3icbVDLSsNAFJ34rPUVdekmtAp1UxIVdFlw47KCfUgby2Ry0w6dPJiZCDVk76+4cat/4U506Q/4DU7SLLTtgWEO59wL5x4nYlRI0/zSlpZXVtfWSxvlza3tnV19b78twpgTaJGQhbzrYAGMBtCSVDLoRhyw7zDoOOOrzO88ABc0DG7lJALbx8OAepRgqaSBXuk7IXPFxFdfcpfeJ30fy5HwMWPAk9rZSZoO9KpZN3MY88QqSBUVaA70n74bktiHQBKGhehZZiTtBHNJCYO03I8FRJiM8RB6igbYB2En+S2pcawU1/BCrl4gjVz9u5FgX2Rx1WSedNbLxIUeByboIyzyerH0Lu2EBlEsISDTEF7MDBkaWWmGSzkQySaKYMKpusMgI8wxkarasirImq1jnrRP65ZZt27Oq42joqoSOkQVVEMWukANdI2aqIUIekIv6BW9ac/au/ahfU5Hl7Ri5wD9g/b9C01Lpww=AAACJ3icbVDLSsNAFJ34rPUVdekmtAp1UxIVdFlw47KCfUgby2Ry0w6dPJiZCDVk76+4cat/4U506Q/4DU7SLLTtgWEO59wL5x4nYlRI0/zSlpZXVtfWSxvlza3tnV19b78twpgTaJGQhbzrYAGMBtCSVDLoRhyw7zDoOOOrzO88ABc0DG7lJALbx8OAepRgqaSBXuk7IXPFxFdfcpfeJ30fy5HwMWPAk9rZSZoO9KpZN3MY88QqSBUVaA70n74bktiHQBKGhehZZiTtBHNJCYO03I8FRJiM8RB6igbYB2En+S2pcawU1/BCrl4gjVz9u5FgX2Rx1WSedNbLxIUeByboIyzyerH0Lu2EBlEsISDTEF7MDBkaWWmGSzkQySaKYMKpusMgI8wxkarasirImq1jnrRP65ZZt27Oq42joqoSOkQVVEMWukANdI2aqIUIekIv6BW9ac/au/ahfU5Hl7Ri5wD9g/b9C01Lpww=AAACJ3icbVDLSsNAFJ34rPUVdekmtAp1UxIVdFlw47KCfUgby2Ry0w6dPJiZCDVk76+4cat/4U506Q/4DU7SLLTtgWEO59wL5x4nYlRI0/zSlpZXVtfWSxvlza3tnV19b78twpgTaJGQhbzrYAGMBtCSVDLoRhyw7zDoOOOrzO88ABc0DG7lJALbx8OAepRgqaSBXuk7IXPFxFdfcpfeJ30fy5HwMWPAk9rZSZoO9KpZN3MY88QqSBUVaA70n74bktiHQBKGhehZZiTtBHNJCYO03I8FRJiM8RB6igbYB2En+S2pcawU1/BCrl4gjVz9u5FgX2Rx1WSedNbLxIUeByboIyzyerH0Lu2EBlEsISDTEF7MDBkaWWmGSzkQySaKYMKpusMgI8wxkarasirImq1jnrRP65ZZt27Oq42joqoSOkQVVEMWukANdI2aqIUIekIv6BW9ac/au/ahfU5Hl7Ri5wD9g/b9C01Lpww=AAACJ3icbVDLSsNAFJ34rPUVdekmtAp1UxIVdFlw47KCfUgby2Ry0w6dPJiZCDVk76+4cat/4U506Q/4DU7SLLTtgWEO59wL5x4nYlRI0/zSlpZXVtfWSxvlza3tnV19b78twpgTaJGQhbzrYAGMBtCSVDLoRhyw7zDoOOOrzO88ABc0DG7lJALbx8OAepRgqaSBXuk7IXPFxFdfcpfeJ30fy5HwMWPAk9rZSZoO9KpZN3MY88QqSBUVaA70n74bktiHQBKGhehZZiTtBHNJCYO03I8FRJiM8RB6igbYB2En+S2pcawU1/BCrl4gjVz9u5FgX2Rx1WSedNbLxIUeByboIyzyerH0Lu2EBlEsISDTEF7MDBkaWWmGSzkQySaKYMKpusMgI8wxkarasirImq1jnrRP65ZZt27Oq42joqoSOkQVVEMWukANdI2aqIUIekIv6BW9ac/au/ahfU5Hl7Ri5wD9g/b9C01Lpww=Y(1)AAACJ3icbVDLSsNAFJ3UV62vqEs3oVWom5KIoMuCG5cV7EPaWCaTm3bo5MHMRKghe3/FjVv9C3eiS3/Ab3CSZqFtDwxzOOdeOPc4EaNCmuaXVlpZXVvfKG9WtrZ3dvf0/YOOCGNOoE1CFvKegwUwGkBbUsmgF3HAvsOg60yuMr/7AFzQMLiV0whsH48C6lGCpZKGenXghMwVU199yV16nwx8LMfCx4wBT+rWaZoO9ZrZMHMYi8QqSA0VaA31n4EbktiHQBKGhehbZiTtBHNJCYO0MogFRJhM8Aj6igbYB2En+S2pcaIU1/BCrl4gjVz9u5FgX2Rx1WSedN7LxKUeByboIyzz+rH0Lu2EBlEsISCzEF7MDBkaWWmGSzkQyaaKYMKpusMgY8wxkaraiirImq9jkXTOGpbZsG7Oa83joqoyOkJVVEcWukBNdI1aqI0IekIv6BW9ac/au/ahfc5GS1qxc4j+Qfv+BUoFpwo=AAACJ3icbVDLSsNAFJ3UV62vqEs3oVWom5KIoMuCG5cV7EPaWCaTm3bo5MHMRKghe3/FjVv9C3eiS3/Ab3CSZqFtDwxzOOdeOPc4EaNCmuaXVlpZXVvfKG9WtrZ3dvf0/YOOCGNOoE1CFvKegwUwGkBbUsmgF3HAvsOg60yuMr/7AFzQMLiV0whsH48C6lGCpZKGenXghMwVU199yV16nwx8LMfCx4wBT+rWaZoO9ZrZMHMYi8QqSA0VaA31n4EbktiHQBKGhehbZiTtBHNJCYO0MogFRJhM8Aj6igbYB2En+S2pcaIU1/BCrl4gjVz9u5FgX2Rx1WSedN7LxKUeByboIyzz+rH0Lu2EBlEsISCzEF7MDBkaWWmGSzkQyaaKYMKpusMgY8wxkaraiirImq9jkXTOGpbZsG7Oa83joqoyOkJVVEcWukBNdI1aqI0IekIv6BW9ac/au/ahfc5GS1qxc4j+Qfv+BUoFpwo=AAACJ3icbVDLSsNAFJ3UV62vqEs3oVWom5KIoMuCG5cV7EPaWCaTm3bo5MHMRKghe3/FjVv9C3eiS3/Ab3CSZqFtDwxzOOdeOPc4EaNCmuaXVlpZXVvfKG9WtrZ3dvf0/YOOCGNOoE1CFvKegwUwGkBbUsmgF3HAvsOg60yuMr/7AFzQMLiV0whsH48C6lGCpZKGenXghMwVU199yV16nwx8LMfCx4wBT+rWaZoO9ZrZMHMYi8QqSA0VaA31n4EbktiHQBKGhehbZiTtBHNJCYO0MogFRJhM8Aj6igbYB2En+S2pcaIU1/BCrl4gjVz9u5FgX2Rx1WSedN7LxKUeByboIyzz+rH0Lu2EBlEsISCzEF7MDBkaWWmGSzkQyaaKYMKpusMgY8wxkaraiirImq9jkXTOGpbZsG7Oa83joqoyOkJVVEcWukBNdI1aqI0IekIv6BW9ac/au/ahfc5GS1qxc4j+Qfv+BUoFpwo=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AAACJ3icbVDLSsNAFJ3UV62vqEs3oVWom5KIoMuCG5cV7EPaWCaTm3bo5MHMRKghe3/FjVv9C3eiS3/Ab3CSZqFtDwxzOOdeOPc4EaNCmuaXVlpZXVvfKG9WtrZ3dvf0/YOOCGNOoE1CFvKegwUwGkBbUsmgF3HAvsOg60yuMr/7AFzQMLiV0whsH48C6lGCpZKGenXghMwVU199yV16nwx8LMfCx4wBT+rWaZoO9ZrZMHMYi8QqSA0VaA31n4EbktiHQBKGhehbZiTtBHNJCYO0MogFRJhM8Aj6igbYB2En+S2pcaIU1/BCrl4gjVz9u5FgX2Rx1WSedN7LxKUeByboIyzz+rH0Lu2EBlEsISCzEF7MDBkaWWmGSzkQyaaKYMKpusMgY8wxkaraiirImq9jkXTOGpbZsG7Oa83joqoyOkJVVEcWukBNdI1aqI0IekIv6BW9ac/au/ahfc5GS1qxc4j+Qfv+BUoFpwo=AAACJ3icbVDLSsNAFJ3UV62vqEs3oVWom5KIoMuCG5cV7EPaWCaTm3bo5MHMRKghe3/FjVv9C3eiS3/Ab3CSZqFtDwxzOOdeOPc4EaNCmuaXVlpZXVvfKG9WtrZ3dvf0/YOOCGNOoE1CFvKegwUwGkBbUsmgF3HAvsOg60yuMr/7AFzQMLiV0whsH48C6lGCpZKGenXghMwVU199yV16nwx8LMfCx4wBT+rWaZoO9ZrZMHMYi8QqSA0VaA31n4EbktiHQBKGhehbZiTtBHNJCYO0MogFRJhM8Aj6igbYB2En+S2pcaIU1/BCrl4gjVz9u5FgX2Rx1WSedN7LxKUeByboIyzz+rH0Lu2EBlEsISCzEF7MDBkaWWmGSzkQyaaKYMKpusMgY8wxkaraiirImq9jkXTOGpbZsG7Oa83joqoyOkJVVEcWukBNdI1aqI0IekIv6BW9ac/au/ahfc5GS1qxc4j+Qfv+BUoFpwo=AAACJ3icbVDLSsNAFJ3UV62vqEs3oVWom5KIoMuCG5cV7EPaWCaTm3bo5MHMRKghe3/FjVv9C3eiS3/Ab3CSZqFtDwxzOOdeOPc4EaNCmuaXVlpZXVvfKG9WtrZ3dvf0/YOOCGNOoE1CFvKegwUwGkBbUsmgF3HAvsOg60yuMr/7AFzQMLiV0whsH48C6lGCpZKGenXghMwVU199yV16nwx8LMfCx4wBT+rWaZoO9ZrZMHMYi8QqSA0VaA31n4EbktiHQBKGhehbZiTtBHNJCYO0MogFRJhM8Aj6igbYB2En+S2pcaIU1/BCrl4gjVz9u5FgX2Rx1WSedN7LxKUeByboIyzz+rH0Lu2EBlEsISCzEF7MDBkaWWmGSzkQyaaKYMKpusMgY8wxkaraiirImq9jkXTOGpbZsG7Oa83joqoyOkJVVEcWukBNdI1aqI0IekIv6BW9ac/au/ahfc5GS1qxc4j+Qfv+BUoFpwo=AAACJ3icbVDLSsNAFJ3UV62vqEs3oVWom5KIoMuCG5cV7EPaWCaTm3bo5MHMRKghe3/FjVv9C3eiS3/Ab3CSZqFtDwxzOOdeOPc4EaNCmuaXVlpZXVvfKG9WtrZ3dvf0/YOOCGNOoE1CFvKegwUwGkBbUsmgF3HAvsOg60yuMr/7AFzQMLiV0whsH48C6lGCpZKGenXghMwVU199yV16nwx8LMfCx4wBT+rWaZoO9ZrZMHMYi8QqSA0VaA31n4EbktiHQBKGhehbZiTtBHNJCYO0MogFRJhM8Aj6igbYB2En+S2pcaIU1/BCrl4gjVz9u5FgX2Rx1WSedN7LxKUeByboIyzz+rH0Lu2EBlEsISCzEF7MDBkaWWmGSzkQyaaKYMKpusMgY8wxkaraiirImq9jkXTOGpbZsG7Oa83joqoyOkJVVEcWukBNdI1aqI0IekIv6BW9ac/au/ahfc5GS1qxc4j+Qfv+BUoFpwo=AAACJ3icbVDLSsNAFJ3UV62vqEs3oVWom5KIoMuCG5cV7EPaWCaTm3bo5MHMRKghe3/FjVv9C3eiS3/Ab3CSZqFtDwxzOOdeOPc4EaNCmuaXVlpZXVvfKG9WtrZ3dvf0/YOOCGNOoE1CFvKegwUwGkBbUsmgF3HAvsOg60yuMr/7AFzQMLiV0whsH48C6lGCpZKGenXghMwVU199yV16nwx8LMfCx4wBT+rWaZoO9ZrZMHMYi8QqSA0VaA31n4EbktiHQBKGhehbZiTtBHNJCYO0MogFRJhM8Aj6igbYB2En+S2pcaIU1/BCrl4gjVz9u5FgX2Rx1WSedN7LxKUeByboIyzz+rH0Lu2EBlEsISCzEF7MDBkaWWmGSzkQyaaKYMKpusMgY8wxkaraiirImq9jkXTOGpbZsG7Oa83joqoyOkJVVEcWukBNdI1aqI0IekIv6BW9ac/au/ahfc5GS1qxc4j+Qfv+BUoFpwo=AAACJ3icbVDLSsNAFJ3UV62vqEs3oVWom5KIoMuCG5cV7EPaWCaTm3bo5MHMRKghe3/FjVv9C3eiS3/Ab3CSZqFtDwxzOOdeOPc4EaNCmuaXVlpZXVvfKG9WtrZ3dvf0/YOOCGNOoE1CFvKegwUwGkBbUsmgF3HAvsOg60yuMr/7AFzQMLiV0whsH48C6lGCpZKGenXghMwVU199yV16nwx8LMfCx4wBT+rWaZoO9ZrZMHMYi8QqSA0VaA31n4EbktiHQBKGhehbZiTtBHNJCYO0MogFRJhM8Aj6igbYB2En+S2pcaIU1/BCrl4gjVz9u5FgX2Rx1WSedN7LxKUeByboIyzz+rH0Lu2EBlEsISCzEF7MDBkaWWmGSzkQyaaKYMKpusMgY8wxkaraiirImq9jkXTOGpbZsG7Oa83joqoyOkJVVEcWukBNdI1aqI0IekIv6BW9ac/au/ahfc5GS1qxc4j+Qfv+BUoFpwo=\u21e7AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBahp5KIoMeCF48R7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZemAqujet+O6WNza3tnfJuZW//4PCoenzS1kmmGLZYIhLVDalGwSW2DDcCu6lCGocCO+Hkdu53nlBpnshHM00xiOlI8ogzaqz00Pf5oFpzG+4CZJ14BalBAX9Q/eoPE5bFKA0TVOue56YmyKkynAmcVfqZxpSyCR1hz1JJY9RBvjh1Ri6sMiRRomxJQxbq74mcxlpP49B2xtSM9ao3F//zepmJboKcyzQzKNlyUZQJYhIy/5sMuUJmxNQSyhS3txI2pooyY9Op2BC81ZfXSfuy4bkN7/6q1qwXcZThDM6hDh5cQxPuwIcWMBjBM7zCmyOcF+fd+Vi2lpxi5hT+wPn8ARdLjZM=AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBahp5KIoMeCF48R7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZemAqujet+O6WNza3tnfJuZW//4PCoenzS1kmmGLZYIhLVDalGwSW2DDcCu6lCGocCO+Hkdu53nlBpnshHM00xiOlI8ogzaqz00Pf5oFpzG+4CZJ14BalBAX9Q/eoPE5bFKA0TVOue56YmyKkynAmcVfqZxpSyCR1hz1JJY9RBvjh1Ri6sMiRRomxJQxbq74mcxlpP49B2xtSM9ao3F//zepmJboKcyzQzKNlyUZQJYhIy/5sMuUJmxNQSyhS3txI2pooyY9Op2BC81ZfXSfuy4bkN7/6q1qwXcZThDM6hDh5cQxPuwIcWMBjBM7zCmyOcF+fd+Vi2lpxi5hT+wPn8ARdLjZM=AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBahp5KIoMeCF48R7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZemAqujet+O6WNza3tnfJuZW//4PCoenzS1kmmGLZYIhLVDalGwSW2DDcCu6lCGocCO+Hkdu53nlBpnshHM00xiOlI8ogzaqz00Pf5oFpzG+4CZJ14BalBAX9Q/eoPE5bFKA0TVOue56YmyKkynAmcVfqZxpSyCR1hz1JJY9RBvjh1Ri6sMiRRomxJQxbq74mcxlpP49B2xtSM9ao3F//zepmJboKcyzQzKNlyUZQJYhIy/5sMuUJmxNQSyhS3txI2pooyY9Op2BC81ZfXSfuy4bkN7/6q1qwXcZThDM6hDh5cQxPuwIcWMBjBM7zCmyOcF+fd+Vi2lpxi5hT+wPn8ARdLjZM=AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBahp5KIoMeCF48R7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZemAqujet+O6WNza3tnfJuZW//4PCoenzS1kmmGLZYIhLVDalGwSW2DDcCu6lCGocCO+Hkdu53nlBpnshHM00xiOlI8ogzaqz00Pf5oFpzG+4CZJ14BalBAX9Q/eoPE5bFKA0TVOue56YmyKkynAmcVfqZxpSyCR1hz1JJY9RBvjh1Ri6sMiRRomxJQxbq74mcxlpP49B2xtSM9ao3F//zepmJboKcyzQzKNlyUZQJYhIy/5sMuUJmxNQSyhS3txI2pooyY9Op2BC81ZfXSfuy4bkN7/6q1qwXcZThDM6hDh5cQxPuwIcWMBjBM7zCmyOcF+fd+Vi2lpxi5hT+wPn8ARdLjZM=AAAB/XicbVDLSsNAFL2pr1pf8bFzM1iErkoigi4LblxWsA9oQplMJu3QySTMTIQair/ixoUibv0Pd/6NkzYLbb0wzOGce5gzJ0g5U9pxvq3K2vrG5lZ1u7azu7d/YB8edVWSSUI7JOGJ7AdYUc4E7WimOe2nkuI44LQXTG4KvfdApWKJuNfTlPoxHgkWMYK1oYb2iRckPFTT2Fy5x40xxLOhXXeaznzQKnBLUIdy2kP7ywsTksVUaMKxUgPXSbWfY6kZ4XRW8zJFU0wmeEQHBgocU+Xn8/QzdG6YEEWJNEdoNGd/O3IcqyKg2YyxHqtlrSD/0waZjq79nIk001SQxUNRxpFOUFEFCpmkRPOpAZhIZrIiMsYSE20Kq5kS3OUvr4LuRdN1mu7dZb3VKOuowimcQQNcuIIW3EIbOkDgEZ7hFd6sJ+vFerc+FqsVq/Qcw5+xPn8AO4uVoQ==AAAB/XicbVDLSsNAFL2pr1pf8bFzM1iErkoigi4LblxWsA9oQplMJu3QySTMTIQair/ixoUibv0Pd/6NkzYLbb0wzOGce5gzJ0g5U9pxvq3K2vrG5lZ1u7azu7d/YB8edVWSSUI7JOGJ7AdYUc4E7WimOe2nkuI44LQXTG4KvfdApWKJuNfTlPoxHgkWMYK1oYb2iRckPFTT2Fy5x40xxLOhXXeaznzQKnBLUIdy2kP7ywsTksVUaMKxUgPXSbWfY6kZ4XRW8zJFU0wmeEQHBgocU+Xn8/QzdG6YEEWJNEdoNGd/O3IcqyKg2YyxHqtlrSD/0waZjq79nIk001SQxUNRxpFOUFEFCpmkRPOpAZhIZrIiMsYSE20Kq5kS3OUvr4LuRdN1mu7dZb3VKOuowimcQQNcuIIW3EIbOkDgEZ7hFd6sJ+vFerc+FqsVq/Qcw5+xPn8AO4uVoQ==AAAB/XicbVDLSsNAFL2pr1pf8bFzM1iErkoigi4LblxWsA9oQplMJu3QySTMTIQair/ixoUibv0Pd/6NkzYLbb0wzOGce5gzJ0g5U9pxvq3K2vrG5lZ1u7azu7d/YB8edVWSSUI7JOGJ7AdYUc4E7WimOe2nkuI44LQXTG4KvfdApWKJuNfTlPoxHgkWMYK1oYb2iRckPFTT2Fy5x40xxLOhXXeaznzQKnBLUIdy2kP7ywsTksVUaMKxUgPXSbWfY6kZ4XRW8zJFU0wmeEQHBgocU+Xn8/QzdG6YEEWJNEdoNGd/O3IcqyKg2YyxHqtlrSD/0waZjq79nIk001SQxUNRxpFOUFEFCpmkRPOpAZhIZrIiMsYSE20Kq5kS3OUvr4LuRdN1mu7dZb3VKOuowimcQQNcuIIW3EIbOkDgEZ7hFd6sJ+vFerc+FqsVq/Qcw5+xPn8AO4uVoQ==AAAB/XicbVDLSsNAFL2pr1pf8bFzM1iErkoigi4LblxWsA9oQplMJu3QySTMTIQair/ixoUibv0Pd/6NkzYLbb0wzOGce5gzJ0g5U9pxvq3K2vrG5lZ1u7azu7d/YB8edVWSSUI7JOGJ7AdYUc4E7WimOe2nkuI44LQXTG4KvfdApWKJuNfTlPoxHgkWMYK1oYb2iRckPFTT2Fy5x40xxLOhXXeaznzQKnBLUIdy2kP7ywsTksVUaMKxUgPXSbWfY6kZ4XRW8zJFU0wmeEQHBgocU+Xn8/QzdG6YEEWJNEdoNGd/O3IcqyKg2YyxHqtlrSD/0waZjq79nIk001SQxUNRxpFOUFEFCpmkRPOpAZhIZrIiMsYSE20Kq5kS3OUvr4LuRdN1mu7dZb3VKOuowimcQQNcuIIW3EIbOkDgEZ7hFd6sJ+vFerc+FqsVq/Qcw5+xPn8AO4uVoQ==\u2713(1)AAACLHicbVDLSsNAFJ3UV62vqks3wSLWTUlE0GXBjcsK9gFNLZPJbTt08mDmRqghn+CvuHGrf+FGxK1Lv8FJm4W2PTDM4Zx74dzjRoIrtKwPo7Cyura+UdwsbW3v7O6V9w9aKowlgyYLRSg7LlUgeABN5CigE0mgviug7Y6vM7/9AFLxMLjDSQQ9nw4DPuCMopb65VPHDYWnJr7+EgdHgDS9Txyf4kj5VAiQSdU+S9N+uWLVrCnMRWLnpEJyNPrlH8cLWexDgExQpbq2FWEvoRI5E5CWnFhBRNmYDqGraUB9UL1kelBqnmjFMweh1C9Ac6r+3Uior7LMenKadN7LxKWeBKH4IyzzujEOrnoJD6IYIWCzEINYmBiaWXOmxyUwFBNNKJNc32GyEZWUoe63pAuy5+tYJK3zmm3V7NuLSr2aV1UkR+SYVIlNLkmd3JAGaRJGnsgLeSVvxrPxbnwaX7PRgpHvHJJ/ML5/AbpzqVk=AAACLHicbVDLSsNAFJ3UV62vqks3wSLWTUlE0GXBjcsK9gFNLZPJbTt08mDmRqghn+CvuHGrf+FGxK1Lv8FJm4W2PTDM4Zx74dzjRoIrtKwPo7Cyura+UdwsbW3v7O6V9w9aKowlgyYLRSg7LlUgeABN5CigE0mgviug7Y6vM7/9AFLxMLjDSQQ9nw4DPuCMopb65VPHDYWnJr7+EgdHgDS9Txyf4kj5VAiQSdU+S9N+uWLVrCnMRWLnpEJyNPrlH8cLWexDgExQpbq2FWEvoRI5E5CWnFhBRNmYDqGraUB9UL1kelBqnmjFMweh1C9Ac6r+3Uior7LMenKadN7LxKWeBKH4IyzzujEOrnoJD6IYIWCzEINYmBiaWXOmxyUwFBNNKJNc32GyEZWUoe63pAuy5+tYJK3zmm3V7NuLSr2aV1UkR+SYVIlNLkmd3JAGaRJGnsgLeSVvxrPxbnwaX7PRgpHvHJJ/ML5/AbpzqVk=AAACLHicbVDLSsNAFJ3UV62vqks3wSLWTUlE0GXBjcsK9gFNLZPJbTt08mDmRqghn+CvuHGrf+FGxK1Lv8FJm4W2PTDM4Zx74dzjRoIrtKwPo7Cyura+UdwsbW3v7O6V9w9aKowlgyYLRSg7LlUgeABN5CigE0mgviug7Y6vM7/9AFLxMLjDSQQ9nw4DPuCMopb65VPHDYWnJr7+EgdHgDS9Txyf4kj5VAiQSdU+S9N+uWLVrCnMRWLnpEJyNPrlH8cLWexDgExQpbq2FWEvoRI5E5CWnFhBRNmYDqGraUB9UL1kelBqnmjFMweh1C9Ac6r+3Uior7LMenKadN7LxKWeBKH4IyzzujEOrnoJD6IYIWCzEINYmBiaWXOmxyUwFBNNKJNc32GyEZWUoe63pAuy5+tYJK3zmm3V7NuLSr2aV1UkR+SYVIlNLkmd3JAGaRJGnsgLeSVvxrPxbnwaX7PRgpHvHJJ/ML5/AbpzqVk=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AAACIXicbZC7SgNBFIbPeo0x6mprsxjE2IRdGy0FG8sI5gLZGGYnJ8mQ2QszZ4W47CP4Kja2+hY2Ir6Cz+DkUmiSA8P8/P8M/OcLEik0ue6ntba+sbm1Xdgp7pb29g/sw1JDx6niWOexjFUrYBqliLBOgiS2EoUsDCQ2g9HNJG8+otIiju5pnGAnZINI9AVnZKyufeYHsezpcWiuzKchEssfMj9kNNQhkxJVVvHO87xrl92qOx1nWXhzUYb51Lr2j9+LeRpiRFwyrduem1AnY4oEl5gX/VRjwviIDbBtZMRC1J1sulDunBqn5/RjZU5EztT9+yNjoZ50Ni+nTRezibkyUyi1eMJVWTul/lUnE1GSEkZ8VqKfSodiZ0LO6QmFnOTYCMaVMHs4fMgU42T4Fg0gbxHHsmhcVD236t25UIBjOIEKeHAJ13ALNagDh2d4hTd4t16sD+trhnLNmjM9gn9jff8CfLun3A==AAACLHicbVC7TsNAEDzzDOEVoKSxiBChiWwaKCPRUAaJPKTYROfLOjnlfLbu1kjB8ifwKzS08Bc0CNFS8g1cHgUkGel0o5ldaXaCRHCNjvNhrayurW9sFraK2zu7e/ulg8OmjlPFoMFiEat2QDUILqGBHAW0EwU0CgS0guH12G89gNI8lnc4SsCPaF/ykDOKRuqWzrwgFj09isyXeTgApPl95kUUBzqiQoDKKu55nndLZafqTGAvEndGymSGerf04/VilkYgkQmqdcd1EvQzqpAzAXnRSzUklA1pHzqGShqB9rPJQbl9apSeHcbKPIn2RP27kdFIjzObyUnSeW8sLvUUCM0fYZnXSTG88jMukxRBsmmIMBU2xva4ObvHFTAUI0MoU9zcYbMBVZSh6bdoCnLn61gkzYuq61TdW6dcq8yqKpBjckIqxCWXpEZuSJ00CCNP5IW8kjfr2Xq3Pq2v6eiKNds5Iv9gff8CuTOpVQ==AAACLHicbVDLSsNAFJ3UV62vqks3wSLWTUlE0GXBjcsK9gFNLZPJbTt08mDmRqghn+CvuHGrf+FGxK1Lv8FJm4W2PTDM4Zx74dzjRoIrtKwPo7Cyura+UdwsbW3v7O6V9w9aKowlgyYLRSg7LlUgeABN5CigE0mgviug7Y6vM7/9AFLxMLjDSQQ9nw4DPuCMopb65VPHDYWnJr7+EgdHgDS9Txyf4kj5VAiQSdU+S9N+uWLVrCnMRWLnpEJyNPrlH8cLWexDgExQpbq2FWEvoRI5E5CWnFhBRNmYDqGraUB9UL1kelBqnmjFMweh1C9Ac6r+3Uior7LMenKadN7LxKWeBKH4IyzzujEOrnoJD6IYIWCzEINYmBiaWXOmxyUwFBNNKJNc32GyEZWUoe63pAuy5+tYJK3zmm3V7NuLSr2aV1UkR+SYVIlNLkmd3JAGaRJGnsgLeSVvxrPxbnwaX7PRgpHvHJJ/ML5/AbpzqVk=AAACLHicbVDLSsNAFJ3UV62vqks3wSLWTUlE0GXBjcsK9gFNLZPJbTt08mDmRqghn+CvuHGrf+FGxK1Lv8FJm4W2PTDM4Zx74dzjRoIrtKwPo7Cyura+UdwsbW3v7O6V9w9aKowlgyYLRSg7LlUgeABN5CigE0mgviug7Y6vM7/9AFLxMLjDSQQ9nw4DPuCMopb65VPHDYWnJr7+EgdHgDS9Txyf4kj5VAiQSdU+S9N+uWLVrCnMRWLnpEJyNPrlH8cLWexDgExQpbq2FWEvoRI5E5CWnFhBRNmYDqGraUB9UL1kelBqnmjFMweh1C9Ac6r+3Uior7LMenKadN7LxKWeBKH4IyzzujEOrnoJD6IYIWCzEINYmBiaWXOmxyUwFBNNKJNc32GyEZWUoe63pAuy5+tYJK3zmm3V7NuLSr2aV1UkR+SYVIlNLkmd3JAGaRJGnsgLeSVvxrPxbnwaX7PRgpHvHJJ/ML5/AbpzqVk=AAACLHicbVDLSsNAFJ3UV62vqks3wSLWTUlE0GXBjcsK9gFNLZPJbTt08mDmRqghn+CvuHGrf+FGxK1Lv8FJm4W2PTDM4Zx74dzjRoIrtKwPo7Cyura+UdwsbW3v7O6V9w9aKowlgyYLRSg7LlUgeABN5CigE0mgviug7Y6vM7/9AFLxMLjDSQQ9nw4DPuCMopb65VPHDYWnJr7+EgdHgDS9Txyf4kj5VAiQSdU+S9N+uWLVrCnMRWLnpEJyNPrlH8cLWexDgExQpbq2FWEvoRI5E5CWnFhBRNmYDqGraUB9UL1kelBqnmjFMweh1C9Ac6r+3Uior7LMenKadN7LxKWeBKH4IyzzujEOrnoJD6IYIWCzEINYmBiaWXOmxyUwFBNNKJNc32GyEZWUoe63pAuy5+tYJK3zmm3V7NuLSr2aV1UkR+SYVIlNLkmd3JAGaRJGnsgLeSVvxrPxbnwaX7PRgpHvHJJ/ML5/AbpzqVk=AAACLHicbVDLSsNAFJ3UV62vqks3wSLWTUlE0GXBjcsK9gFNLZPJbTt08mDmRqghn+CvuHGrf+FGxK1Lv8FJm4W2PTDM4Zx74dzjRoIrtKwPo7Cyura+UdwsbW3v7O6V9w9aKowlgyYLRSg7LlUgeABN5CigE0mgviug7Y6vM7/9AFLxMLjDSQQ9nw4DPuCMopb65VPHDYWnJr7+EgdHgDS9Txyf4kj5VAiQSdU+S9N+uWLVrCnMRWLnpEJyNPrlH8cLWexDgExQpbq2FWEvoRI5E5CWnFhBRNmYDqGraUB9UL1kelBqnmjFMweh1C9Ac6r+3Uior7LMenKadN7LxKWeBKH4IyzzujEOrnoJD6IYIWCzEINYmBiaWXOmxyUwFBNNKJNc32GyEZWUoe63pAuy5+tYJK3zmm3V7NuLSr2aV1UkR+SYVIlNLkmd3JAGaRJGnsgLeSVvxrPxbnwaX7PRgpHvHJJ/ML5/AbpzqVk=AAACLHicbVDLSsNAFJ3UV62vqks3wSLWTUlE0GXBjcsK9gFNLZPJbTt08mDmRqghn+CvuHGrf+FGxK1Lv8FJm4W2PTDM4Zx74dzjRoIrtKwPo7Cyura+UdwsbW3v7O6V9w9aKowlgyYLRSg7LlUgeABN5CigE0mgviug7Y6vM7/9AFLxMLjDSQQ9nw4DPuCMopb65VPHDYWnJr7+EgdHgDS9Txyf4kj5VAiQSdU+S9N+uWLVrCnMRWLnpEJyNPrlH8cLWexDgExQpbq2FWEvoRI5E5CWnFhBRNmYDqGraUB9UL1kelBqnmjFMweh1C9Ac6r+3Uior7LMenKadN7LxKWeBKH4IyzzujEOrnoJD6IYIWCzEINYmBiaWXOmxyUwFBNNKJNc32GyEZWUoe63pAuy5+tYJK3zmm3V7NuLSr2aV1UkR+SYVIlNLkmd3JAGaRJGnsgLeSVvxrPxbnwaX7PRgpHvHJJ/ML5/AbpzqVk=AAACLHicbVDLSsNAFJ3UV62vqks3wSLWTUlE0GXBjcsK9gFNLZPJbTt08mDmRqghn+CvuHGrf+FGxK1Lv8FJm4W2PTDM4Zx74dzjRoIrtKwPo7Cyura+UdwsbW3v7O6V9w9aKowlgyYLRSg7LlUgeABN5CigE0mgviug7Y6vM7/9AFLxMLjDSQQ9nw4DPuCMopb65VPHDYWnJr7+EgdHgDS9Txyf4kj5VAiQSdU+S9N+uWLVrCnMRWLnpEJyNPrlH8cLWexDgExQpbq2FWEvoRI5E5CWnFhBRNmYDqGraUB9UL1kelBqnmjFMweh1C9Ac6r+3Uior7LMenKadN7LxKWeBKH4IyzzujEOrnoJD6IYIWCzEINYmBiaWXOmxyUwFBNNKJNc32GyEZWUoe63pAuy5+tYJK3zmm3V7NuLSr2aV1UkR+SYVIlNLkmd3JAGaRJGnsgLeSVvxrPxbnwaX7PRgpHvHJJ/ML5/AbpzqVk=\u2713(2)AAACLHicbVDLSsNAFJ34rPVVdekmWMS6KUkRdFlw47KCfUBTy2Ry2w6dPJi5EWrIJ/grbtzqX7gRcevSb3DSZqFtDwxzOOdeOPe4keAKLevDWFldW9/YLGwVt3d29/ZLB4ctFcaSQZOFIpQdlyoQPIAmchTQiSRQ3xXQdsfXmd9+AKl4GNzhJIKeT4cBH3BGUUv90pnjhsJTE19/iYMjQJreJ45PcaR8KgTIpFI7T9N+qWxVrSnMRWLnpExyNPqlH8cLWexDgExQpbq2FWEvoRI5E5AWnVhBRNmYDqGraUB9UL1kelBqnmrFMweh1C9Ac6r+3Uior7LMenKadN7LxKWeBKH4IyzzujEOrnoJD6IYIWCzEINYmBiaWXOmxyUwFBNNKJNc32GyEZWUoe63qAuy5+tYJK1a1baq9u1FuV7JqyqQY3JCKsQml6RObkiDNAkjT+SFvJI349l4Nz6Nr9noipHvHJF/ML5/AbwWqVo=AAACLHicbVDLSsNAFJ34rPVVdekmWMS6KUkRdFlw47KCfUBTy2Ry2w6dPJi5EWrIJ/grbtzqX7gRcevSb3DSZqFtDwxzOOdeOPe4keAKLevDWFldW9/YLGwVt3d29/ZLB4ctFcaSQZOFIpQdlyoQPIAmchTQiSRQ3xXQdsfXmd9+AKl4GNzhJIKeT4cBH3BGUUv90pnjhsJTE19/iYMjQJreJ45PcaR8KgTIpFI7T9N+qWxVrSnMRWLnpExyNPqlH8cLWexDgExQpbq2FWEvoRI5E5AWnVhBRNmYDqGraUB9UL1kelBqnmrFMweh1C9Ac6r+3Uior7LMenKadN7LxKWeBKH4IyzzujEOrnoJD6IYIWCzEINYmBiaWXOmxyUwFBNNKJNc32GyEZWUoe63qAuy5+tYJK1a1baq9u1FuV7JqyqQY3JCKsQml6RObkiDNAkjT+SFvJI349l4Nz6Nr9noipHvHJF/ML5/AbwWqVo=AAACLHicbVDLSsNAFJ34rPVVdekmWMS6KUkRdFlw47KCfUBTy2Ry2w6dPJi5EWrIJ/grbtzqX7gRcevSb3DSZqFtDwxzOOdeOPe4keAKLevDWFldW9/YLGwVt3d29/ZLB4ctFcaSQZOFIpQdlyoQPIAmchTQiSRQ3xXQdsfXmd9+AKl4GNzhJIKeT4cBH3BGUUv90pnjhsJTE19/iYMjQJreJ45PcaR8KgTIpFI7T9N+qWxVrSnMRWLnpExyNPqlH8cLWexDgExQpbq2FWEvoRI5E5AWnVhBRNmYDqGraUB9UL1kelBqnmrFMweh1C9Ac6r+3Uior7LMenKadN7LxKWeBKH4IyzzujEOrnoJD6IYIWCzEINYmBiaWXOmxyUwFBNNKJNc32GyEZWUoe63qAuy5+tYJK1a1baq9u1FuV7JqyqQY3JCKsQml6RObkiDNAkjT+SFvJI349l4Nz6Nr9noipHvHJF/ML5/AbwWqVo=AAACLHicbVDLSsNAFJ34rPVVdekmWMS6KUkRdFlw47KCfUBTy2Ry2w6dPJi5EWrIJ/grbtzqX7gRcevSb3DSZqFtDwxzOOdeOPe4keAKLevDWFldW9/YLGwVt3d29/ZLB4ctFcaSQZOFIpQdlyoQPIAmchTQiSRQ3xXQdsfXmd9+AKl4GNzhJIKeT4cBH3BGUUv90pnjhsJTE19/iYMjQJreJ45PcaR8KgTIpFI7T9N+qWxVrSnMRWLnpExyNPqlH8cLWexDgExQpbq2FWEvoRI5E5AWnVhBRNmYDqGraUB9UL1kelBqnmrFMweh1C9Ac6r+3Uior7LMenKadN7LxKWeBKH4IyzzujEOrnoJD6IYIWCzEINYmBiaWXOmxyUwFBNNKJNc32GyEZWUoe63qAuy5+tYJK1a1baq9u1FuV7JqyqQY3JCKsQml6RObkiDNAkjT+SFvJI349l4Nz6Nr9noipHvHJF/ML5/AbwWqVo=AAAB7XicbVDLSgNBEOyNrxhfUY9eBoOQU9gVQY8BLx4jmAckS+idzCZjZnaWmVkhhPyDFw+KePV/vPk3TpI9aGJBQ1HVTXdXlApurO9/e4WNza3tneJuaW//4PCofHzSMirTlDWpEkp3IjRM8IQ1LbeCdVLNUEaCtaPx7dxvPzFtuEoe7CRlocRhwmNO0Tqp1RuilNgvV/yavwBZJ0FOKpCj0S9/9QaKZpIllgo0phv4qQ2nqC2ngs1KvcywFOkYh6zraIKSmXC6uHZGLpwyILHSrhJLFurviSlKYyYycp0S7cisenPxP6+b2fgmnPIkzSxL6HJRnAliFZm/TgZcM2rFxBGkmrtbCR2hRmpdQCUXQrD68jppXdYCvxbcX1Xq1TyOIpzBOVQhgGuowx00oAkUHuEZXuHNU96L9+59LFsLXj5zCn/gff4AfxmO+w==AAAB7XicbVDLSgNBEOyNrxhfUY9eBoOQU9gVQY8BLx4jmAckS+idzCZjZnaWmVkhhPyDFw+KePV/vPk3TpI9aGJBQ1HVTXdXlApurO9/e4WNza3tneJuaW//4PCofHzSMirTlDWpEkp3IjRM8IQ1LbeCdVLNUEaCtaPx7dxvPzFtuEoe7CRlocRhwmNO0Tqp1RuilNgvV/yavwBZJ0FOKpCj0S9/9QaKZpIllgo0phv4qQ2nqC2ngs1KvcywFOkYh6zraIKSmXC6uHZGLpwyILHSrhJLFurviSlKYyYycp0S7cisenPxP6+b2fgmnPIkzSxL6HJRnAliFZm/TgZcM2rFxBGkmrtbCR2hRmpdQCUXQrD68jppXdYCvxbcX1Xq1TyOIpzBOVQhgGuowx00oAkUHuEZXuHNU96L9+59LFsLXj5zCn/gff4AfxmO+w==AAAB7XicbVDLSgNBEOyNrxhfUY9eBoOQU9gVQY8BLx4jmAckS+idzCZjZnaWmVkhhPyDFw+KePV/vPk3TpI9aGJBQ1HVTXdXlApurO9/e4WNza3tneJuaW//4PCofHzSMirTlDWpEkp3IjRM8IQ1LbeCdVLNUEaCtaPx7dxvPzFtuEoe7CRlocRhwmNO0Tqp1RuilNgvV/yavwBZJ0FOKpCj0S9/9QaKZpIllgo0phv4qQ2nqC2ngs1KvcywFOkYh6zraIKSmXC6uHZGLpwyILHSrhJLFurviSlKYyYycp0S7cisenPxP6+b2fgmnPIkzSxL6HJRnAliFZm/TgZcM2rFxBGkmrtbCR2hRmpdQCUXQrD68jppXdYCvxbcX1Xq1TyOIpzBOVQhgGuowx00oAkUHuEZXuHNU96L9+59LFsLXj5zCn/gff4AfxmO+w==AAAB7XicbVDLSgNBEOyNrxhfUY9eBoOQU9gVQY8BLx4jmAckS+idzCZjZnaWmVkhhPyDFw+KePV/vPk3TpI9aGJBQ1HVTXdXlApurO9/e4WNza3tneJuaW//4PCofHzSMirTlDWpEkp3IjRM8IQ1LbeCdVLNUEaCtaPx7dxvPzFtuEoe7CRlocRhwmNO0Tqp1RuilNgvV/yavwBZJ0FOKpCj0S9/9QaKZpIllgo0phv4qQ2nqC2ngs1KvcywFOkYh6zraIKSmXC6uHZGLpwyILHSrhJLFurviSlKYyYycp0S7cisenPxP6+b2fgmnPIkzSxL6HJRnAliFZm/TgZcM2rFxBGkmrtbCR2hRmpdQCUXQrD68jppXdYCvxbcX1Xq1TyOIpzBOVQhgGuowx00oAkUHuEZXuHNU96L9+59LFsLXj5zCn/gff4AfxmO+w==discreteAAAB73icdVDLSgNBEJyNrxhfUY9eBqPgaZmNmsct4MVjBPOAZAmzk04yZHZ2nZkVQshPePGgiFd/x5t/42wSQUULGoqqbrq7glhwbQj5cDIrq2vrG9nN3Nb2zu5efv+gqaNEMWiwSESqHVANgktoGG4EtGMFNAwEtILxVeq37kFpHslbM4nBD+lQ8gFn1Fip3eeaKTDQyxeIe1699EgRE7dUJtVqSiqlSvmiiD2XzFFAS9R7+fduP2JJCNIwQbXueCQ2/pQqw5mAWa6baIgpG9MhdCyVNATtT+f3zvCpVfp4EClb0uC5+n1iSkOtJ2FgO0NqRvq3l4p/eZ3EDCr+lMs4MSDZYtEgEdhEOH0e97kCZsTEEsoUt7diNqKKMmMjytkQvj7F/5Nm0fWI690UC7WTZRxZdISO0RnyUBnV0DWqowZiSKAH9ISenTvn0XlxXhetGWc5c4h+wHn7BL3MkFU=AAAB73icdVDLSgNBEJyNrxhfUY9eBqPgaZmNmsct4MVjBPOAZAmzk04yZHZ2nZkVQshPePGgiFd/x5t/42wSQUULGoqqbrq7glhwbQj5cDIrq2vrG9nN3Nb2zu5efv+gqaNEMWiwSESqHVANgktoGG4EtGMFNAwEtILxVeq37kFpHslbM4nBD+lQ8gFn1Fip3eeaKTDQyxeIe1699EgRE7dUJtVqSiqlSvmiiD2XzFFAS9R7+fduP2JJCNIwQbXueCQ2/pQqw5mAWa6baIgpG9MhdCyVNATtT+f3zvCpVfp4EClb0uC5+n1iSkOtJ2FgO0NqRvq3l4p/eZ3EDCr+lMs4MSDZYtEgEdhEOH0e97kCZsTEEsoUt7diNqKKMmMjytkQvj7F/5Nm0fWI690UC7WTZRxZdISO0RnyUBnV0DWqowZiSKAH9ISenTvn0XlxXhetGWc5c4h+wHn7BL3MkFU=AAAB73icdVDLSgNBEJyNrxhfUY9eBqPgaZmNmsct4MVjBPOAZAmzk04yZHZ2nZkVQshPePGgiFd/x5t/42wSQUULGoqqbrq7glhwbQj5cDIrq2vrG9nN3Nb2zu5efv+gqaNEMWiwSESqHVANgktoGG4EtGMFNAwEtILxVeq37kFpHslbM4nBD+lQ8gFn1Fip3eeaKTDQyxeIe1699EgRE7dUJtVqSiqlSvmiiD2XzFFAS9R7+fduP2JJCNIwQbXueCQ2/pQqw5mAWa6baIgpG9MhdCyVNATtT+f3zvCpVfp4EClb0uC5+n1iSkOtJ2FgO0NqRvq3l4p/eZ3EDCr+lMs4MSDZYtEgEdhEOH0e97kCZsTEEsoUt7diNqKKMmMjytkQvj7F/5Nm0fWI690UC7WTZRxZdISO0RnyUBnV0DWqowZiSKAH9ISenTvn0XlxXhetGWc5c4h+wHn7BL3MkFU=AAAB73icdVDLSgNBEJyNrxhfUY9eBqPgaZmNmsct4MVjBPOAZAmzk04yZHZ2nZkVQshPePGgiFd/x5t/42wSQUULGoqqbrq7glhwbQj5cDIrq2vrG9nN3Nb2zu5efv+gqaNEMWiwSESqHVANgktoGG4EtGMFNAwEtILxVeq37kFpHslbM4nBD+lQ8gFn1Fip3eeaKTDQyxeIe1699EgRE7dUJtVqSiqlSvmiiD2XzFFAS9R7+fduP2JJCNIwQbXueCQ2/pQqw5mAWa6baIgpG9MhdCyVNATtT+f3zvCpVfp4EClb0uC5+n1iSkOtJ2FgO0NqRvq3l4p/eZ3EDCr+lMs4MSDZYtEgEdhEOH0e97kCZsTEEsoUt7diNqKKMmMjytkQvj7F/5Nm0fWI690UC7WTZRxZdISO0RnyUBnV0DWqowZiSKAH9ISenTvn0XlxXhetGWc5c4h+wHn7BL3MkFU=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AAAB/XicdVDLSgMxFM34rPU1PnZuglVwVTJStN0V3LisYB/QDiWTZtrQzIPkjliH4q+4caGIW//DnX9jph1BRQ8EDufec3Pv8WIpNBDyYS0sLi2vrBbWiusbm1vb9s5uS0eJYrzJIhmpjkc1lyLkTRAgeSdWnAae5G1vfJHV2zdcaRGF1zCJuRvQYSh8wSgYqW/v94DfggpSEYKiDKgxTvt2iZTPahXHqWJSJjMY4pBalVSwkysllKPRt997g4glAQ+BSap11yExuClVIJiZV+wlmseUjemQdw0NacC1m862n+JjowywHynzQsAz9bsjpYHWk8AznQGFkf5dy8S/at0E/KprDosT4CGbf+QnEkOEsyjwQCjOQE4MoUwJsytmI5qlYAIrmhC+LsX/k9apiaXsXFVK9aM8jgI6QIfoBDnoHNXRJWqgJmLoDj2gJ/Rs3VuP1ov1Om9dsHLPHvoB6+0T3YaWCQ==AAAB/XicdVDLSgMxFM34rPU1PnZuglVwVTJStN0V3LisYB/QDiWTZtrQzIPkjliH4q+4caGIW//DnX9jph1BRQ8EDufec3Pv8WIpNBDyYS0sLi2vrBbWiusbm1vb9s5uS0eJYrzJIhmpjkc1lyLkTRAgeSdWnAae5G1vfJHV2zdcaRGF1zCJuRvQYSh8wSgYqW/v94DfggpSEYKiDKgxTvt2iZTPahXHqWJSJjMY4pBalVSwkysllKPRt997g4glAQ+BSap11yExuClVIJiZV+wlmseUjemQdw0NacC1m862n+JjowywHynzQsAz9bsjpYHWk8AznQGFkf5dy8S/at0E/KprDosT4CGbf+QnEkOEsyjwQCjOQE4MoUwJsytmI5qlYAIrmhC+LsX/k9apiaXsXFVK9aM8jgI6QIfoBDnoHNXRJWqgJmLoDj2gJ/Rs3VuP1ov1Om9dsHLPHvoB6+0T3YaWCQ==AAAB/XicdVDLSgMxFM34rPU1PnZuglVwVTJStN0V3LisYB/QDiWTZtrQzIPkjliH4q+4caGIW//DnX9jph1BRQ8EDufec3Pv8WIpNBDyYS0sLi2vrBbWiusbm1vb9s5uS0eJYrzJIhmpjkc1lyLkTRAgeSdWnAae5G1vfJHV2zdcaRGF1zCJuRvQYSh8wSgYqW/v94DfggpSEYKiDKgxTvt2iZTPahXHqWJSJjMY4pBalVSwkysllKPRt997g4glAQ+BSap11yExuClVIJiZV+wlmseUjemQdw0NacC1m862n+JjowywHynzQsAz9bsjpYHWk8AznQGFkf5dy8S/at0E/KprDosT4CGbf+QnEkOEsyjwQCjOQE4MoUwJsytmI5qlYAIrmhC+LsX/k9apiaXsXFVK9aM8jgI6QIfoBDnoHNXRJWqgJmLoDj2gJ/Rs3VuP1ov1Om9dsHLPHvoB6+0T3YaWCQ==tractableAAAB8HicbVDLSsNAFL2pr1pfVZduglVwVZJudFlw47KCfUgbymQ6aYfOTMLMjVBCv8KNC0Xc+jnu/BsnbRbaemDgcM65zL0nTAQ36HnfTmljc2t7p7xb2ds/ODyqHp90TJxqyto0FrHuhcQwwRVrI0fBeolmRIaCdcPpbe53n5g2PFYPOEtYIMlY8YhTglZ6RE0oEhseVmte3VvAXSd+QWpQoDWsfg1GMU0lU0gFMabvewkGGdHIqWDzyiA1LCF0Ssasb6kikpkgWyw8dy+tMnKjWNun0F2ovycyIo2ZydAmJcGJWfVy8T+vn2J0E2RcJSkyRZcfRalwMXbz690R14yimFlCqOZ2V5dOSF6C7ahiS/BXT14nnUbd9+r+faPWvCjqKMMZnMMV+HANTbiDFrSBgoRneIU3RzsvzrvzsYyWnGLmFP7A+fwB90OQZg==AAAB8HicbVDLSsNAFL2pr1pfVZduglVwVZJudFlw47KCfUgbymQ6aYfOTMLMjVBCv8KNC0Xc+jnu/BsnbRbaemDgcM65zL0nTAQ36HnfTmljc2t7p7xb2ds/ODyqHp90TJxqyto0FrHuhcQwwRVrI0fBeolmRIaCdcPpbe53n5g2PFYPOEtYIMlY8YhTglZ6RE0oEhseVmte3VvAXSd+QWpQoDWsfg1GMU0lU0gFMabvewkGGdHIqWDzyiA1LCF0Ssasb6kikpkgWyw8dy+tMnKjWNun0F2ovycyIo2ZydAmJcGJWfVy8T+vn2J0E2RcJSkyRZcfRalwMXbz690R14yimFlCqOZ2V5dOSF6C7ahiS/BXT14nnUbd9+r+faPWvCjqKMMZnMMV+HANTbiDFrSBgoRneIU3RzsvzrvzsYyWnGLmFP7A+fwB90OQZg==AAAB8HicbVDLSsNAFL2pr1pfVZduglVwVZJudFlw47KCfUgbymQ6aYfOTMLMjVBCv8KNC0Xc+jnu/BsnbRbaemDgcM65zL0nTAQ36HnfTmljc2t7p7xb2ds/ODyqHp90TJxqyto0FrHuhcQwwRVrI0fBeolmRIaCdcPpbe53n5g2PFYPOEtYIMlY8YhTglZ6RE0oEhseVmte3VvAXSd+QWpQoDWsfg1GMU0lU0gFMabvewkGGdHIqWDzyiA1LCF0Ssasb6kikpkgWyw8dy+tMnKjWNun0F2ovycyIo2ZydAmJcGJWfVy8T+vn2J0E2RcJSkyRZcfRalwMXbz690R14yimFlCqOZ2V5dOSF6C7ahiS/BXT14nnUbd9+r+faPWvCjqKMMZnMMV+HANTbiDFrSBgoRneIU3RzsvzrvzsYyWnGLmFP7A+fwB90OQZg==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AAAB5XicbVBNS8NAEJ3Ur1qrVq9eFovgqSRe9Ch48VjBtJU2lM120y7d3YTdiVBCf4UXD4r4l7z5b9y0PWjrg4HHezPMzIszKSz6/rdX2dre2d2r7tcO6odHx42TesemuWE8ZKlMTS+mlkuheYgCJe9lhlMVS96Np3el333mxopUP+Is45GiYy0SwSg66QkNZUhd87DR9Fv+AmSTBCvShBXaw8bXYJSyXHGNTFJr+4GfYVRQg4JJPq8NcsszyqZ0zPuOaqq4jYrFwXNy4ZQRSVLjSiNZqL8nCqqsnanYdSqKE7vuleJ/Xj/H5CYqhM5y5JotFyW5JJiS8nsyEoYzlDNHKDPC3UrYhJYhuIxqLoRg/eVN0rlqBX4rePChCmdwDpcQwDXcwj20IQQGCl7gDd494716H8u4Kt4qt1P4A+/zB8bFjys=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AAAB8HicbVDLSsNAFL2pr1pfVZduglVwVZJudFlw47KCfUgbymQ6aYfOTMLMjVBCv8KNC0Xc+jnu/BsnbRbaemDgcM65zL0nTAQ36HnfTmljc2t7p7xb2ds/ODyqHp90TJxqyto0FrHuhcQwwRVrI0fBeolmRIaCdcPpbe53n5g2PFYPOEtYIMlY8YhTglZ6RE0oEhseVmte3VvAXSd+QWpQoDWsfg1GMU0lU0gFMabvewkGGdHIqWDzyiA1LCF0Ssasb6kikpkgWyw8dy+tMnKjWNun0F2ovycyIo2ZydAmJcGJWfVy8T+vn2J0E2RcJSkyRZcfRalwMXbz690R14yimFlCqOZ2V5dOSF6C7ahiS/BXT14nnUbd9+r+faPWvCjqKMMZnMMV+HANTbiDFrSBgoRneIU3RzsvzrvzsYyWnGLmFP7A+fwB90OQZg==AAAB8HicbVDLSsNAFL2pr1pfVZduglVwVZJudFlw47KCfUgbymQ6aYfOTMLMjVBCv8KNC0Xc+jnu/BsnbRbaemDgcM65zL0nTAQ36HnfTmljc2t7p7xb2ds/ODyqHp90TJxqyto0FrHuhcQwwRVrI0fBeolmRIaCdcPpbe53n5g2PFYPOEtYIMlY8YhTglZ6RE0oEhseVmte3VvAXSd+QWpQoDWsfg1GMU0lU0gFMabvewkGGdHIqWDzyiA1LCF0Ssasb6kikpkgWyw8dy+tMnKjWNun0F2ovycyIo2ZydAmJcGJWfVy8T+vn2J0E2RcJSkyRZcfRalwMXbz690R14yimFlCqOZ2V5dOSF6C7ahiS/BXT14nnUbd9+r+faPWvCjqKMMZnMMV+HANTbiDFrSBgoRneIU3RzsvzrvzsYyWnGLmFP7A+fwB90OQZg==AAAB8HicbVDLSsNAFL2pr1pfVZduglVwVZJudFlw47KCfUgbymQ6aYfOTMLMjVBCv8KNC0Xc+jnu/BsnbRbaemDgcM65zL0nTAQ36HnfTmljc2t7p7xb2ds/ODyqHp90TJxqyto0FrHuhcQwwRVrI0fBeolmRIaCdcPpbe53n5g2PFYPOEtYIMlY8YhTglZ6RE0oEhseVmte3VvAXSd+QWpQoDWsfg1GMU0lU0gFMabvewkGGdHIqWDzyiA1LCF0Ssasb6kikpkgWyw8dy+tMnKjWNun0F2ovycyIo2ZydAmJcGJWfVy8T+vn2J0E2RcJSkyRZcfRalwMXbz690R14yimFlCqOZ2V5dOSF6C7ahiS/BXT14nnUbd9+r+faPWvCjqKMMZnMMV+HANTbiDFrSBgoRneIU3RzsvzrvzsYyWnGLmFP7A+fwB90OQZg==AAAB8HicbVDLSsNAFL2pr1pfVZduglVwVZJudFlw47KCfUgbymQ6aYfOTMLMjVBCv8KNC0Xc+jnu/BsnbRbaemDgcM65zL0nTAQ36HnfTmljc2t7p7xb2ds/ODyqHp90TJxqyto0FrHuhcQwwRVrI0fBeolmRIaCdcPpbe53n5g2PFYPOEtYIMlY8YhTglZ6RE0oEhseVmte3VvAXSd+QWpQoDWsfg1GMU0lU0gFMabvewkGGdHIqWDzyiA1LCF0Ssasb6kikpkgWyw8dy+tMnKjWNun0F2ovycyIo2ZydAmJcGJWfVy8T+vn2J0E2RcJSkyRZcfRalwMXbz690R14yimFlCqOZ2V5dOSF6C7ahiS/BXT14nnUbd9+r+faPWvCjqKMMZnMMV+HANTbiDFrSBgoRneIU3RzsvzrvzsYyWnGLmFP7A+fwB90OQZg==AAAB8HicbVDLSsNAFL2pr1pfVZduglVwVZJudFlw47KCfUgbymQ6aYfOTMLMjVBCv8KNC0Xc+jnu/BsnbRbaemDgcM65zL0nTAQ36HnfTmljc2t7p7xb2ds/ODyqHp90TJxqyto0FrHuhcQwwRVrI0fBeolmRIaCdcPpbe53n5g2PFYPOEtYIMlY8YhTglZ6RE0oEhseVmte3VvAXSd+QWpQoDWsfg1GMU0lU0gFMabvewkGGdHIqWDzyiA1LCF0Ssasb6kikpkgWyw8dy+tMnKjWNun0F2ovycyIo2ZydAmJcGJWfVy8T+vn2J0E2RcJSkyRZcfRalwMXbz690R14yimFlCqOZ2V5dOSF6C7ahiS/BXT14nnUbd9+r+faPWvCjqKMMZnMMV+HANTbiDFrSBgoRneIU3RzsvzrvzsYyWnGLmFP7A+fwB90OQZg==\u2713(3)AAACLHicbVDLSsNAFJ34rPVVdekmWMS6KYkKuiy4cVnBPqCJZTK9bYdOHszcCDXkE/wVN271L9yIuHXpNzhNs9C2B4Y5nHMvnHu8SHCFlvVhLC2vrK6tFzaKm1vbO7ulvf2mCmPJoMFCEcq2RxUIHkADOQpoRxKo7wloeaPrid96AKl4GNzhOALXp4OA9zmjqKVu6cTxQtFTY19/iYNDQJreJ45Pcah8KgTIpHJ+mqbdUtmqWhnMeWLnpExy1LulH6cXstiHAJmgSnVsK0I3oRI5E5AWnVhBRNmIDqCjaUB9UG6SHZSax1rpmf1Q6hegmal/NxLqq0lmPZklnfUm4kJPglD8ERZ5nRj7V27CgyhGCNg0RD8WJobmpDmzxyUwFGNNKJNc32GyIZWUoe63qAuyZ+uYJ82zqm1V7duLcq2SV1Ugh+SIVIhNLkmN3JA6aRBGnsgLeSVvxrPxbnwaX9PRJSPfOSD/YHz/Ar25qVs=AAACLHicbVDLSsNAFJ34rPVVdekmWMS6KYkKuiy4cVnBPqCJZTK9bYdOHszcCDXkE/wVN271L9yIuHXpNzhNs9C2B4Y5nHMvnHu8SHCFlvVhLC2vrK6tFzaKm1vbO7ulvf2mCmPJoMFCEcq2RxUIHkADOQpoRxKo7wloeaPrid96AKl4GNzhOALXp4OA9zmjqKVu6cTxQtFTY19/iYNDQJreJ45Pcah8KgTIpHJ+mqbdUtmqWhnMeWLnpExy1LulH6cXstiHAJmgSnVsK0I3oRI5E5AWnVhBRNmIDqCjaUB9UG6SHZSax1rpmf1Q6hegmal/NxLqq0lmPZklnfUm4kJPglD8ERZ5nRj7V27CgyhGCNg0RD8WJobmpDmzxyUwFGNNKJNc32GyIZWUoe63qAuyZ+uYJ82zqm1V7duLcq2SV1Ugh+SIVIhNLkmN3JA6aRBGnsgLeSVvxrPxbnwaX9PRJSPfOSD/YHz/Ar25qVs=AAACLHicbVDLSsNAFJ34rPVVdekmWMS6KYkKuiy4cVnBPqCJZTK9bYdOHszcCDXkE/wVN271L9yIuHXpNzhNs9C2B4Y5nHMvnHu8SHCFlvVhLC2vrK6tFzaKm1vbO7ulvf2mCmPJoMFCEcq2RxUIHkADOQpoRxKo7wloeaPrid96AKl4GNzhOALXp4OA9zmjqKVu6cTxQtFTY19/iYNDQJreJ45Pcah8KgTIpHJ+mqbdUtmqWhnMeWLnpExy1LulH6cXstiHAJmgSnVsK0I3oRI5E5AWnVhBRNmIDqCjaUB9UG6SHZSax1rpmf1Q6hegmal/NxLqq0lmPZklnfUm4kJPglD8ERZ5nRj7V27CgyhGCNg0RD8WJobmpDmzxyUwFGNNKJNc32GyIZWUoe63qAuyZ+uYJ82zqm1V7duLcq2SV1Ugh+SIVIhNLkmN3JA6aRBGnsgLeSVvxrPxbnwaX9PRJSPfOSD/YHz/Ar25qVs=AAACLHicbVDLSsNAFJ34rPVVdekmWMS6KYkKuiy4cVnBPqCJZTK9bYdOHszcCDXkE/wVN271L9yIuHXpNzhNs9C2B4Y5nHMvnHu8SHCFlvVhLC2vrK6tFzaKm1vbO7ulvf2mCmPJoMFCEcq2RxUIHkADOQpoRxKo7wloeaPrid96AKl4GNzhOALXp4OA9zmjqKVu6cTxQtFTY19/iYNDQJreJ45Pcah8KgTIpHJ+mqbdUtmqWhnMeWLnpExy1LulH6cXstiHAJmgSnVsK0I3oRI5E5AWnVhBRNmIDqCjaUB9UG6SHZSax1rpmf1Q6hegmal/NxLqq0lmPZklnfUm4kJPglD8ERZ5nRj7V27CgyhGCNg0RD8WJobmpDmzxyUwFGNNKJNc32GyIZWUoe63qAuyZ+uYJ82zqm1V7duLcq2SV1Ugh+SIVIhNLkmN3JA6aRBGnsgLeSVvxrPxbnwaX9PRJSPfOSD/YHz/Ar25qVs=h(1)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(2)AAACJ3icbVDLSsNAFJ34rPUVdekmtAp1U5Ii6LLgxmUF+5Amlsnkph06eTAzEWrI3l9x41b/wp3o0h/wG5ymWWjbA8MczrkXzj1uzKiQpvmlrayurW9slrbK2zu7e/v6wWFHRAkn0CYRi3jPxQIYDaEtqWTQizngwGXQdcdXU7/7AFzQKLyVkxicAA9D6lOCpZIGesV2I+aJSaC+9C67T+0Ay5EIMGPA01rjLMsGetWsmzmMRWIVpIoKtAb6j+1FJAkglIRhIfqWGUsnxVxSwiAr24mAGJMxHkJf0RAHIJw0vyUzTpXiGX7E1Qulkat/N1IciGlcNZknnfem4lKPAxP0EZZ5/UT6l05KwziREJJZCD9hhoyMaWmGRzkQySaKYMKpusMgI8wxkarasirImq9jkXQadcusWzfn1eZJUVUJHaMKqiELXaAmukYt1EYEPaEX9IretGftXfvQPmejK1qxc4T+Qfv+BUuopws=AAACJ3icbVDLSsNAFJ34rPUVdekmtAp1U5Ii6LLgxmUF+5Amlsnkph06eTAzEWrI3l9x41b/wp3o0h/wG5ymWWjbA8MczrkXzj1uzKiQpvmlrayurW9slrbK2zu7e/v6wWFHRAkn0CYRi3jPxQIYDaEtqWTQizngwGXQdcdXU7/7AFzQKLyVkxicAA9D6lOCpZIGesV2I+aJSaC+9C67T+0Ay5EIMGPA01rjLMsGetWsmzmMRWIVpIoKtAb6j+1FJAkglIRhIfqWGUsnxVxSwiAr24mAGJMxHkJf0RAHIJw0vyUzTpXiGX7E1Qulkat/N1IciGlcNZknnfem4lKPAxP0EZZ5/UT6l05KwziREJJZCD9hhoyMaWmGRzkQySaKYMKpusMgI8wxkarasirImq9jkXQadcusWzfn1eZJUVUJHaMKqiELXaAmukYt1EYEPaEX9IretGftXfvQPmejK1qxc4T+Qfv+BUuopws=AAACJ3icbVDLSsNAFJ34rPUVdekmtAp1U5Ii6LLgxmUF+5Amlsnkph06eTAzEWrI3l9x41b/wp3o0h/wG5ymWWjbA8MczrkXzj1uzKiQpvmlrayurW9slrbK2zu7e/v6wWFHRAkn0CYRi3jPxQIYDaEtqWTQizngwGXQdcdXU7/7AFzQKLyVkxicAA9D6lOCpZIGesV2I+aJSaC+9C67T+0Ay5EIMGPA01rjLMsGetWsmzmMRWIVpIoKtAb6j+1FJAkglIRhIfqWGUsnxVxSwiAr24mAGJMxHkJf0RAHIJw0vyUzTpXiGX7E1Qulkat/N1IciGlcNZknnfem4lKPAxP0EZZ5/UT6l05KwziREJJZCD9hhoyMaWmGRzkQySaKYMKpusMgI8wxkarasirImq9jkXQadcusWzfn1eZJUVUJHaMKqiELXaAmukYt1EYEPaEX9IretGftXfvQPmejK1qxc4T+Qfv+BUuopws=AAACJ3icbVDLSsNAFJ34rPUVdekmtAp1U5Ii6LLgxmUF+5Amlsnkph06eTAzEWrI3l9x41b/wp3o0h/wG5ymWWjbA8MczrkXzj1uzKiQpvmlrayurW9slrbK2zu7e/v6wWFHRAkn0CYRi3jPxQIYDaEtqWTQizngwGXQdcdXU7/7AFzQKLyVkxicAA9D6lOCpZIGesV2I+aJSaC+9C67T+0Ay5EIMGPA01rjLMsGetWsmzmMRWIVpIoKtAb6j+1FJAkglIRhIfqWGUsnxVxSwiAr24mAGJMxHkJf0RAHIJw0vyUzTpXiGX7E1Qulkat/N1IciGlcNZknnfem4lKPAxP0EZZ5/UT6l05KwziREJJZCD9hhoyMaWmGRzkQySaKYMKpusMgI8wxkarasirImq9jkXQadcusWzfn1eZJUVUJHaMKqiELXaAmukYt1EYEPaEX9IretGftXfvQPmejK1qxc4T+Qfv+BUuopws=Y(3)AAACJ3icbVDLSsNAFJ34rPUVdekmtAp1UxIVdFlw47KCfUgby2Ry0w6dPJiZCDVk76+4cat/4U506Q/4DU7SLLTtgWEO59wL5x4nYlRI0/zSlpZXVtfWSxvlza3tnV19b78twpgTaJGQhbzrYAGMBtCSVDLoRhyw7zDoOOOrzO88ABc0DG7lJALbx8OAepRgqaSBXuk7IXPFxFdfcpfeJ30fy5HwMWPAk9rZSZoO9KpZN3MY88QqSBUVaA70n74bktiHQBKGhehZZiTtBHNJCYO03I8FRJiM8RB6igbYB2En+S2pcawU1/BCrl4gjVz9u5FgX2Rx1WSedNbLxIUeByboIyzyerH0Lu2EBlEsISDTEF7MDBkaWWmGSzkQySaKYMKpusMgI8wxkarasirImq1jnrRP65ZZt27Oq42joqoSOkQVVEMWukANdI2aqIUIekIv6BW9ac/au/ahfU5Hl7Ri5wD9g/b9C01Lpww=AAACJ3icbVDLSsNAFJ34rPUVdekmtAp1UxIVdFlw47KCfUgby2Ry0w6dPJiZCDVk76+4cat/4U506Q/4DU7SLLTtgWEO59wL5x4nYlRI0/zSlpZXVtfWSxvlza3tnV19b78twpgTaJGQhbzrYAGMBtCSVDLoRhyw7zDoOOOrzO88ABc0DG7lJALbx8OAepRgqaSBXuk7IXPFxFdfcpfeJ30fy5HwMWPAk9rZSZoO9KpZN3MY88QqSBUVaA70n74bktiHQBKGhehZZiTtBHNJCYO03I8FRJiM8RB6igbYB2En+S2pcawU1/BCrl4gjVz9u5FgX2Rx1WSedNbLxIUeByboIyzyerH0Lu2EBlEsISDTEF7MDBkaWWmGSzkQySaKYMKpusMgI8wxkarasirImq1jnrRP65ZZt27Oq42joqoSOkQVVEMWukANdI2aqIUIekIv6BW9ac/au/ahfU5Hl7Ri5wD9g/b9C01Lpww=AAACJ3icbVDLSsNAFJ34rPUVdekmtAp1UxIVdFlw47KCfUgby2Ry0w6dPJiZCDVk76+4cat/4U506Q/4DU7SLLTtgWEO59wL5x4nYlRI0/zSlpZXVtfWSxvlza3tnV19b78twpgTaJGQhbzrYAGMBtCSVDLoRhyw7zDoOOOrzO88ABc0DG7lJALbx8OAepRgqaSBXuk7IXPFxFdfcpfeJ30fy5HwMWPAk9rZSZoO9KpZN3MY88QqSBUVaA70n74bktiHQBKGhehZZiTtBHNJCYO03I8FRJiM8RB6igbYB2En+S2pcawU1/BCrl4gjVz9u5FgX2Rx1WSedNbLxIUeByboIyzyerH0Lu2EBlEsISDTEF7MDBkaWWmGSzkQySaKYMKpusMgI8wxkarasirImq1jnrRP65ZZt27Oq42joqoSOkQVVEMWukANdI2aqIUIekIv6BW9ac/au/ahfU5Hl7Ri5wD9g/b9C01Lpww=AAACJ3icbVDLSsNAFJ34rPUVdekmtAp1UxIVdFlw47KCfUgby2Ry0w6dPJiZCDVk76+4cat/4U506Q/4DU7SLLTtgWEO59wL5x4nYlRI0/zSlpZXVtfWSxvlza3tnV19b78twpgTaJGQhbzrYAGMBtCSVDLoRhyw7zDoOOOrzO88ABc0DG7lJALbx8OAepRgqaSBXuk7IXPFxFdfcpfeJ30fy5HwMWPAk9rZSZoO9KpZN3MY88QqSBUVaA70n74bktiHQBKGhehZZiTtBHNJCYO03I8FRJiM8RB6igbYB2En+S2pcawU1/BCrl4gjVz9u5FgX2Rx1WSedNbLxIUeByboIyzyerH0Lu2EBlEsISDTEF7MDBkaWWmGSzkQySaKYMKpusMgI8wxkarasirImq1jnrRP65ZZt27Oq42joqoSOkQVVEMWukANdI2aqIUIekIv6BW9ac/au/ahfU5Hl7Ri5wD9g/b9C01Lpww=h(2)AAACIHicbVDLSsNAFJ3UV62vqEs3oUWom5IUQZcFNy4r2Ac0sUwmN+3QyYOZiVBD9n6HH+BWP8GduNQf8DecpFlo64FhDufcy733uDGjQprmp1ZZW9/Y3Kpu13Z29/YP9MOjvogSTqBHIhbxoYsFMBpCT1LJYBhzwIHLYODOrnJ/cA9c0Ci8lfMYnABPQupTgqWSxnrddiPmiXmgvnSa3aV2gOVUBJgx4GmzfZZlY71htswCxiqxStJAJbpj/dv2IpIEEErCsBAjy4ylk2IuKWGQ1exEQIzJDE9gpGiIAxBOWtySGadK8Qw/4uqF0ijU3x0pDkS+rqosNl32cvFfjwMT9GF5vPQvnZSGcSIhJIvpfsIMGRl5WoZHORDJ5opgwqk6wCBTzDGRKtOaSsZazmGV9Nsty2xZN+eNTrPMqIpOUB01kYUuUAddoy7qIYIe0TN6Qa/ak/amvWsfi9KKVvYcoz/Qvn4AOcOkkg==AAACIHicbVDLSsNAFJ3UV62vqEs3oUWom5IUQZcFNy4r2Ac0sUwmN+3QyYOZiVBD9n6HH+BWP8GduNQf8DecpFlo64FhDufcy733uDGjQprmp1ZZW9/Y3Kpu13Z29/YP9MOjvogSTqBHIhbxoYsFMBpCT1LJYBhzwIHLYODOrnJ/cA9c0Ci8lfMYnABPQupTgqWSxnrddiPmiXmgvnSa3aV2gOVUBJgx4GmzfZZlY71htswCxiqxStJAJbpj/dv2IpIEEErCsBAjy4ylk2IuKWGQ1exEQIzJDE9gpGiIAxBOWtySGadK8Qw/4uqF0ijU3x0pDkS+rqosNl32cvFfjwMT9GF5vPQvnZSGcSIhJIvpfsIMGRl5WoZHORDJ5opgwqk6wCBTzDGRKtOaSsZazmGV9Nsty2xZN+eNTrPMqIpOUB01kYUuUAddoy7qIYIe0TN6Qa/ak/amvWsfi9KKVvYcoz/Qvn4AOcOkkg==AAACIHicbVDLSsNAFJ3UV62vqEs3oUWom5IUQZcFNy4r2Ac0sUwmN+3QyYOZiVBD9n6HH+BWP8GduNQf8DecpFlo64FhDufcy733uDGjQprmp1ZZW9/Y3Kpu13Z29/YP9MOjvogSTqBHIhbxoYsFMBpCT1LJYBhzwIHLYODOrnJ/cA9c0Ci8lfMYnABPQupTgqWSxnrddiPmiXmgvnSa3aV2gOVUBJgx4GmzfZZlY71htswCxiqxStJAJbpj/dv2IpIEEErCsBAjy4ylk2IuKWGQ1exEQIzJDE9gpGiIAxBOWtySGadK8Qw/4uqF0ijU3x0pDkS+rqosNl32cvFfjwMT9GF5vPQvnZSGcSIhJIvpfsIMGRl5WoZHORDJ5opgwqk6wCBTzDGRKtOaSsZazmGV9Nsty2xZN+eNTrPMqIpOUB01kYUuUAddoy7qIYIe0TN6Qa/ak/amvWsfi9KKVvYcoz/Qvn4AOcOkkg==AAACIHicbVDLSsNAFJ3UV62vqEs3oUWom5IUQZcFNy4r2Ac0sUwmN+3QyYOZiVBD9n6HH+BWP8GduNQf8DecpFlo64FhDufcy733uDGjQprmp1ZZW9/Y3Kpu13Z29/YP9MOjvogSTqBHIhbxoYsFMBpCT1LJYBhzwIHLYODOrnJ/cA9c0Ci8lfMYnABPQupTgqWSxnrddiPmiXmgvnSa3aV2gOVUBJgx4GmzfZZlY71htswCxiqxStJAJbpj/dv2IpIEEErCsBAjy4ylk2IuKWGQ1exEQIzJDE9gpGiIAxBOWtySGadK8Qw/4uqF0ijU3x0pDkS+rqosNl32cvFfjwMT9GF5vPQvnZSGcSIhJIvpfsIMGRl5WoZHORDJ5opgwqk6wCBTzDGRKtOaSsZazmGV9Nsty2xZN+eNTrPMqIpOUB01kYUuUAddoy7qIYIe0TN6Qa/ak/amvWsfi9KKVvYcoz/Qvn4AOcOkkg==Y(1)AAACJ3icbVDLSsNAFJ3UV62vqEs3oVWom5KIoMuCG5cV7EPaWCaTm3bo5MHMRKghe3/FjVv9C3eiS3/Ab3CSZqFtDwxzOOdeOPc4EaNCmuaXVlpZXVvfKG9WtrZ3dvf0/YOOCGNOoE1CFvKegwUwGkBbUsmgF3HAvsOg60yuMr/7AFzQMLiV0whsH48C6lGCpZKGenXghMwVU199yV16nwx8LMfCx4wBT+rWaZoO9ZrZMHMYi8QqSA0VaA31n4EbktiHQBKGhehbZiTtBHNJCYO0MogFRJhM8Aj6igbYB2En+S2pcaIU1/BCrl4gjVz9u5FgX2Rx1WSedN7LxKUeByboIyzz+rH0Lu2EBlEsISCzEF7MDBkaWWmGSzkQyaaKYMKpusMgY8wxkaraiirImq9jkXTOGpbZsG7Oa83joqoyOkJVVEcWukBNdI1aqI0IekIv6BW9ac/au/ahfc5GS1qxc4j+Qfv+BUoFpwo=AAACJ3icbVDLSsNAFJ3UV62vqEs3oVWom5KIoMuCG5cV7EPaWCaTm3bo5MHMRKghe3/FjVv9C3eiS3/Ab3CSZqFtDwxzOOdeOPc4EaNCmuaXVlpZXVvfKG9WtrZ3dvf0/YOOCGNOoE1CFvKegwUwGkBbUsmgF3HAvsOg60yuMr/7AFzQMLiV0whsH48C6lGCpZKGenXghMwVU199yV16nwx8LMfCx4wBT+rWaZoO9ZrZMHMYi8QqSA0VaA31n4EbktiHQBKGhehbZiTtBHNJCYO0MogFRJhM8Aj6igbYB2En+S2pcaIU1/BCrl4gjVz9u5FgX2Rx1WSedN7LxKUeByboIyzz+rH0Lu2EBlEsISCzEF7MDBkaWWmGSzkQyaaKYMKpusMgY8wxkaraiirImq9jkXTOGpbZsG7Oa83joqoyOkJVVEcWukBNdI1aqI0IekIv6BW9ac/au/ahfc5GS1qxc4j+Qfv+BUoFpwo=AAACJ3icbVDLSsNAFJ3UV62vqEs3oVWom5KIoMuCG5cV7EPaWCaTm3bo5MHMRKghe3/FjVv9C3eiS3/Ab3CSZqFtDwxzOOdeOPc4EaNCmuaXVlpZXVvfKG9WtrZ3dvf0/YOOCGNOoE1CFvKegwUwGkBbUsmgF3HAvsOg60yuMr/7AFzQMLiV0whsH48C6lGCpZKGenXghMwVU199yV16nwx8LMfCx4wBT+rWaZoO9ZrZMHMYi8QqSA0VaA31n4EbktiHQBKGhehbZiTtBHNJCYO0MogFRJhM8Aj6igbYB2En+S2pcaIU1/BCrl4gjVz9u5FgX2Rx1WSedN7LxKUeByboIyzz+rH0Lu2EBlEsISCzEF7MDBkaWWmGSzkQyaaKYMKpusMgY8wxkaraiirImq9jkXTOGpbZsG7Oa83joqoyOkJVVEcWukBNdI1aqI0IekIv6BW9ac/au/ahfc5GS1qxc4j+Qfv+BUoFpwo=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AAACJ3icbVDLSsNAFJ3UV62vqEs3oVWom5KIoMuCG5cV7EPaWCaTm3bo5MHMRKghe3/FjVv9C3eiS3/Ab3CSZqFtDwxzOOdeOPc4EaNCmuaXVlpZXVvfKG9WtrZ3dvf0/YOOCGNOoE1CFvKegwUwGkBbUsmgF3HAvsOg60yuMr/7AFzQMLiV0whsH48C6lGCpZKGenXghMwVU199yV16nwx8LMfCx4wBT+rWaZoO9ZrZMHMYi8QqSA0VaA31n4EbktiHQBKGhehbZiTtBHNJCYO0MogFRJhM8Aj6igbYB2En+S2pcaIU1/BCrl4gjVz9u5FgX2Rx1WSedN7LxKUeByboIyzz+rH0Lu2EBlEsISCzEF7MDBkaWWmGSzkQyaaKYMKpusMgY8wxkaraiirImq9jkXTOGpbZsG7Oa83joqoyOkJVVEcWukBNdI1aqI0IekIv6BW9ac/au/ahfc5GS1qxc4j+Qfv+BUoFpwo=AAACJ3icbVDLSsNAFJ3UV62vqEs3oVWom5KIoMuCG5cV7EPaWCaTm3bo5MHMRKghe3/FjVv9C3eiS3/Ab3CSZqFtDwxzOOdeOPc4EaNCmuaXVlpZXVvfKG9WtrZ3dvf0/YOOCGNOoE1CFvKegwUwGkBbUsmgF3HAvsOg60yuMr/7AFzQMLiV0whsH48C6lGCpZKGenXghMwVU199yV16nwx8LMfCx4wBT+rWaZoO9ZrZMHMYi8QqSA0VaA31n4EbktiHQBKGhehbZiTtBHNJCYO0MogFRJhM8Aj6igbYB2En+S2pcaIU1/BCrl4gjVz9u5FgX2Rx1WSedN7LxKUeByboIyzz+rH0Lu2EBlEsISCzEF7MDBkaWWmGSzkQyaaKYMKpusMgY8wxkaraiirImq9jkXTOGpbZsG7Oa83joqoyOkJVVEcWukBNdI1aqI0IekIv6BW9ac/au/ahfc5GS1qxc4j+Qfv+BUoFpwo=AAACJ3icbVDLSsNAFJ3UV62vqEs3oVWom5KIoMuCG5cV7EPaWCaTm3bo5MHMRKghe3/FjVv9C3eiS3/Ab3CSZqFtDwxzOOdeOPc4EaNCmuaXVlpZXVvfKG9WtrZ3dvf0/YOOCGNOoE1CFvKegwUwGkBbUsmgF3HAvsOg60yuMr/7AFzQMLiV0whsH48C6lGCpZKGenXghMwVU199yV16nwx8LMfCx4wBT+rWaZoO9ZrZMHMYi8QqSA0VaA31n4EbktiHQBKGhehbZiTtBHNJCYO0MogFRJhM8Aj6igbYB2En+S2pcaIU1/BCrl4gjVz9u5FgX2Rx1WSedN7LxKUeByboIyzz+rH0Lu2EBlEsISCzEF7MDBkaWWmGSzkQyaaKYMKpusMgY8wxkaraiirImq9jkXTOGpbZsG7Oa83joqoyOkJVVEcWukBNdI1aqI0IekIv6BW9ac/au/ahfc5GS1qxc4j+Qfv+BUoFpwo=AAACJ3icbVDLSsNAFJ3UV62vqEs3oVWom5KIoMuCG5cV7EPaWCaTm3bo5MHMRKghe3/FjVv9C3eiS3/Ab3CSZqFtDwxzOOdeOPc4EaNCmuaXVlpZXVvfKG9WtrZ3dvf0/YOOCGNOoE1CFvKegwUwGkBbUsmgF3HAvsOg60yuMr/7AFzQMLiV0whsH48C6lGCpZKGenXghMwVU199yV16nwx8LMfCx4wBT+rWaZoO9ZrZMHMYi8QqSA0VaA31n4EbktiHQBKGhehbZiTtBHNJCYO0MogFRJhM8Aj6igbYB2En+S2pcaIU1/BCrl4gjVz9u5FgX2Rx1WSedN7LxKUeByboIyzz+rH0Lu2EBlEsISCzEF7MDBkaWWmGSzkQyaaKYMKpusMgY8wxkaraiirImq9jkXTOGpbZsG7Oa83joqoyOkJVVEcWukBNdI1aqI0IekIv6BW9ac/au/ahfc5GS1qxc4j+Qfv+BUoFpwo=AAACJ3icbVDLSsNAFJ3UV62vqEs3oVWom5KIoMuCG5cV7EPaWCaTm3bo5MHMRKghe3/FjVv9C3eiS3/Ab3CSZqFtDwxzOOdeOPc4EaNCmuaXVlpZXVvfKG9WtrZ3dvf0/YOOCGNOoE1CFvKegwUwGkBbUsmgF3HAvsOg60yuMr/7AFzQMLiV0whsH48C6lGCpZKGenXghMwVU199yV16nwx8LMfCx4wBT+rWaZoO9ZrZMHMYi8QqSA0VaA31n4EbktiHQBKGhehbZiTtBHNJCYO0MogFRJhM8Aj6igbYB2En+S2pcaIU1/BCrl4gjVz9u5FgX2Rx1WSedN7LxKUeByboIyzz+rH0Lu2EBlEsISCzEF7MDBkaWWmGSzkQyaaKYMKpusMgY8wxkaraiirImq9jkXTOGpbZsG7Oa83joqoyOkJVVEcWukBNdI1aqI0IekIv6BW9ac/au/ahfc5GS1qxc4j+Qfv+BUoFpwo=AAACJ3icbVDLSsNAFJ3UV62vqEs3oVWom5KIoMuCG5cV7EPaWCaTm3bo5MHMRKghe3/FjVv9C3eiS3/Ab3CSZqFtDwxzOOdeOPc4EaNCmuaXVlpZXVvfKG9WtrZ3dvf0/YOOCGNOoE1CFvKegwUwGkBbUsmgF3HAvsOg60yuMr/7AFzQMLiV0whsH48C6lGCpZKGenXghMwVU199yV16nwx8LMfCx4wBT+rWaZoO9ZrZMHMYi8QqSA0VaA31n4EbktiHQBKGhehbZiTtBHNJCYO0MogFRJhM8Aj6igbYB2En+S2pcaIU1/BCrl4gjVz9u5FgX2Rx1WSedN7LxKUeByboIyzz+rH0Lu2EBlEsISCzEF7MDBkaWWmGSzkQyaaKYMKpusMgY8wxkaraiirImq9jkXTOGpbZsG7Oa83joqoyOkJVVEcWukBNdI1aqI0IekIv6BW9ac/au/ahfc5GS1qxc4j+Qfv+BUoFpwo=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\u21e7AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBahp5KIoMeCF48R7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZemAqujet+O6WNza3tnfJuZW//4PCoenzS1kmmGLZYIhLVDalGwSW2DDcCu6lCGocCO+Hkdu53nlBpnshHM00xiOlI8ogzaqz00Pf5oFpzG+4CZJ14BalBAX9Q/eoPE5bFKA0TVOue56YmyKkynAmcVfqZxpSyCR1hz1JJY9RBvjh1Ri6sMiRRomxJQxbq74mcxlpP49B2xtSM9ao3F//zepmJboKcyzQzKNlyUZQJYhIy/5sMuUJmxNQSyhS3txI2pooyY9Op2BC81ZfXSfuy4bkN7/6q1qwXcZThDM6hDh5cQxPuwIcWMBjBM7zCmyOcF+fd+Vi2lpxi5hT+wPn8ARdLjZM=AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBahp5KIoMeCF48R7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZemAqujet+O6WNza3tnfJuZW//4PCoenzS1kmmGLZYIhLVDalGwSW2DDcCu6lCGocCO+Hkdu53nlBpnshHM00xiOlI8ogzaqz00Pf5oFpzG+4CZJ14BalBAX9Q/eoPE5bFKA0TVOue56YmyKkynAmcVfqZxpSyCR1hz1JJY9RBvjh1Ri6sMiRRomxJQxbq74mcxlpP49B2xtSM9ao3F//zepmJboKcyzQzKNlyUZQJYhIy/5sMuUJmxNQSyhS3txI2pooyY9Op2BC81ZfXSfuy4bkN7/6q1qwXcZThDM6hDh5cQxPuwIcWMBjBM7zCmyOcF+fd+Vi2lpxi5hT+wPn8ARdLjZM=AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBahp5KIoMeCF48R7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZemAqujet+O6WNza3tnfJuZW//4PCoenzS1kmmGLZYIhLVDalGwSW2DDcCu6lCGocCO+Hkdu53nlBpnshHM00xiOlI8ogzaqz00Pf5oFpzG+4CZJ14BalBAX9Q/eoPE5bFKA0TVOue56YmyKkynAmcVfqZxpSyCR1hz1JJY9RBvjh1Ri6sMiRRomxJQxbq74mcxlpP49B2xtSM9ao3F//zepmJboKcyzQzKNlyUZQJYhIy/5sMuUJmxNQSyhS3txI2pooyY9Op2BC81ZfXSfuy4bkN7/6q1qwXcZThDM6hDh5cQxPuwIcWMBjBM7zCmyOcF+fd+Vi2lpxi5hT+wPn8ARdLjZM=AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBahp5KIoMeCF48R7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZemAqujet+O6WNza3tnfJuZW//4PCoenzS1kmmGLZYIhLVDalGwSW2DDcCu6lCGocCO+Hkdu53nlBpnshHM00xiOlI8ogzaqz00Pf5oFpzG+4CZJ14BalBAX9Q/eoPE5bFKA0TVOue56YmyKkynAmcVfqZxpSyCR1hz1JJY9RBvjh1Ri6sMiRRomxJQxbq74mcxlpP49B2xtSM9ao3F//zepmJboKcyzQzKNlyUZQJYhIy/5sMuUJmxNQSyhS3txI2pooyY9Op2BC81ZfXSfuy4bkN7/6q1qwXcZThDM6hDh5cQxPuwIcWMBjBM7zCmyOcF+fd+Vi2lpxi5hT+wPn8ARdLjZM=h(3)AAACIHicbVDLSsNAFJ3UV62vqEs3oUWom5KooMuCG5cV7AOaWCaTm3bo5MHMRKghe7/DD3Crn+BOXOoP+BtO0iy09cAwh3Pu5d573JhRIU3zU6usrK6tb1Q3a1vbO7t7+v5BT0QJJ9AlEYv4wMUCGA2hK6lkMIg54MBl0HenV7nfvwcuaBTeylkMToDHIfUpwVJJI71uuxHzxCxQXzrJ7lI7wHIiAswY8LR5dpJlI71htswCxjKxStJAJToj/dv2IpIEEErCsBBDy4ylk2IuKWGQ1exEQIzJFI9hqGiIAxBOWtySGcdK8Qw/4uqF0ijU3x0pDkS+rqosNl30cvFfjwMT9GFxvPQvnZSGcSIhJPPpfsIMGRl5WoZHORDJZopgwqk6wCATzDGRKtOaSsZazGGZ9E5bltmybs4b7WaZURUdoTpqIgtdoDa6Rh3URQQ9omf0gl61J+1Ne9c+5qUVrew5RH+gff0AO1+kkw==AAACIHicbVDLSsNAFJ3UV62vqEs3oUWom5KooMuCG5cV7AOaWCaTm3bo5MHMRKghe7/DD3Crn+BOXOoP+BtO0iy09cAwh3Pu5d573JhRIU3zU6usrK6tb1Q3a1vbO7t7+v5BT0QJJ9AlEYv4wMUCGA2hK6lkMIg54MBl0HenV7nfvwcuaBTeylkMToDHIfUpwVJJI71uuxHzxCxQXzrJ7lI7wHIiAswY8LR5dpJlI71htswCxjKxStJAJToj/dv2IpIEEErCsBBDy4ylk2IuKWGQ1exEQIzJFI9hqGiIAxBOWtySGcdK8Qw/4uqF0ijU3x0pDkS+rqosNl30cvFfjwMT9GFxvPQvnZSGcSIhJPPpfsIMGRl5WoZHORDJZopgwqk6wCATzDGRKtOaSsZazGGZ9E5bltmybs4b7WaZURUdoTpqIgtdoDa6Rh3URQQ9omf0gl61J+1Ne9c+5qUVrew5RH+gff0AO1+kkw==AAACIHicbVDLSsNAFJ3UV62vqEs3oUWom5KooMuCG5cV7AOaWCaTm3bo5MHMRKghe7/DD3Crn+BOXOoP+BtO0iy09cAwh3Pu5d573JhRIU3zU6usrK6tb1Q3a1vbO7t7+v5BT0QJJ9AlEYv4wMUCGA2hK6lkMIg54MBl0HenV7nfvwcuaBTeylkMToDHIfUpwVJJI71uuxHzxCxQXzrJ7lI7wHIiAswY8LR5dpJlI71htswCxjKxStJAJToj/dv2IpIEEErCsBBDy4ylk2IuKWGQ1exEQIzJFI9hqGiIAxBOWtySGcdK8Qw/4uqF0ijU3x0pDkS+rqosNl30cvFfjwMT9GFxvPQvnZSGcSIhJPPpfsIMGRl5WoZHORDJZopgwqk6wCATzDGRKtOaSsZazGGZ9E5bltmybs4b7WaZURUdoTpqIgtdoDa6Rh3URQQ9omf0gl61J+1Ne9c+5qUVrew5RH+gff0AO1+kkw==AAACIHicbVDLSsNAFJ3UV62vqEs3oUWom5KooMuCG5cV7AOaWCaTm3bo5MHMRKghe7/DD3Crn+BOXOoP+BtO0iy09cAwh3Pu5d573JhRIU3zU6usrK6tb1Q3a1vbO7t7+v5BT0QJJ9AlEYv4wMUCGA2hK6lkMIg54MBl0HenV7nfvwcuaBTeylkMToDHIfUpwVJJI71uuxHzxCxQXzrJ7lI7wHIiAswY8LR5dpJlI71htswCxjKxStJAJToj/dv2IpIEEErCsBBDy4ylk2IuKWGQ1exEQIzJFI9hqGiIAxBOWtySGcdK8Qw/4uqF0ijU3x0pDkS+rqosNl30cvFfjwMT9GFxvPQvnZSGcSIhJPPpfsIMGRl5WoZHORDJZopgwqk6wCATzDGRKtOaSsZazGGZ9E5bltmybs4b7WaZURUdoTpqIgtdoDa6Rh3URQQ9omf0gl61J+1Ne9c+5qUVrew5RH+gff0AO1+kkw==AAAB/XicbVDLSsNAFL2pr1pf8bFzM1iErkoigi4LblxWsA9oQplMJu3QySTMTIQair/ixoUibv0Pd/6NkzYLbb0wzOGce5gzJ0g5U9pxvq3K2vrG5lZ1u7azu7d/YB8edVWSSUI7JOGJ7AdYUc4E7WimOe2nkuI44LQXTG4KvfdApWKJuNfTlPoxHgkWMYK1oYb2iRckPFTT2Fy5x40xxLOhXXeaznzQKnBLUIdy2kP7ywsTksVUaMKxUgPXSbWfY6kZ4XRW8zJFU0wmeEQHBgocU+Xn8/QzdG6YEEWJNEdoNGd/O3IcqyKg2YyxHqtlrSD/0waZjq79nIk001SQxUNRxpFOUFEFCpmkRPOpAZhIZrIiMsYSE20Kq5kS3OUvr4LuRdN1mu7dZb3VKOuowimcQQNcuIIW3EIbOkDgEZ7hFd6sJ+vFerc+FqsVq/Qcw5+xPn8AO4uVoQ==AAAB/XicbVDLSsNAFL2pr1pf8bFzM1iErkoigi4LblxWsA9oQplMJu3QySTMTIQair/ixoUibv0Pd/6NkzYLbb0wzOGce5gzJ0g5U9pxvq3K2vrG5lZ1u7azu7d/YB8edVWSSUI7JOGJ7AdYUc4E7WimOe2nkuI44LQXTG4KvfdApWKJuNfTlPoxHgkWMYK1oYb2iRckPFTT2Fy5x40xxLOhXXeaznzQKnBLUIdy2kP7ywsTksVUaMKxUgPXSbWfY6kZ4XRW8zJFU0wmeEQHBgocU+Xn8/QzdG6YEEWJNEdoNGd/O3IcqyKg2YyxHqtlrSD/0waZjq79nIk001SQxUNRxpFOUFEFCpmkRPOpAZhIZrIiMsYSE20Kq5kS3OUvr4LuRdN1mu7dZb3VKOuowimcQQNcuIIW3EIbOkDgEZ7hFd6sJ+vFerc+FqsVq/Qcw5+xPn8AO4uVoQ==AAAB/XicbVDLSsNAFL2pr1pf8bFzM1iErkoigi4LblxWsA9oQplMJu3QySTMTIQair/ixoUibv0Pd/6NkzYLbb0wzOGce5gzJ0g5U9pxvq3K2vrG5lZ1u7azu7d/YB8edVWSSUI7JOGJ7AdYUc4E7WimOe2nkuI44LQXTG4KvfdApWKJuNfTlPoxHgkWMYK1oYb2iRckPFTT2Fy5x40xxLOhXXeaznzQKnBLUIdy2kP7ywsTksVUaMKxUgPXSbWfY6kZ4XRW8zJFU0wmeEQHBgocU+Xn8/QzdG6YEEWJNEdoNGd/O3IcqyKg2YyxHqtlrSD/0waZjq79nIk001SQxUNRxpFOUFEFCpmkRPOpAZhIZrIiMsYSE20Kq5kS3OUvr4LuRdN1mu7dZb3VKOuowimcQQNcuIIW3EIbOkDgEZ7hFd6sJ+vFerc+FqsVq/Qcw5+xPn8AO4uVoQ==AAAB/XicbVDLSsNAFL2pr1pf8bFzM1iErkoigi4LblxWsA9oQplMJu3QySTMTIQair/ixoUibv0Pd/6NkzYLbb0wzOGce5gzJ0g5U9pxvq3K2vrG5lZ1u7azu7d/YB8edVWSSUI7JOGJ7AdYUc4E7WimOe2nkuI44LQXTG4KvfdApWKJuNfTlPoxHgkWMYK1oYb2iRckPFTT2Fy5x40xxLOhXXeaznzQKnBLUIdy2kP7ywsTksVUaMKxUgPXSbWfY6kZ4XRW8zJFU0wmeEQHBgocU+Xn8/QzdG6YEEWJNEdoNGd/O3IcqyKg2YyxHqtlrSD/0waZjq79nIk001SQxUNRxpFOUFEFCpmkRPOpAZhIZrIiMsYSE20Kq5kS3OUvr4LuRdN1mu7dZb3VKOuowimcQQNcuIIW3EIbOkDgEZ7hFd6sJ+vFerc+FqsVq/Qcw5+xPn8AO4uVoQ==\u2713(1)AAACLHicbVDLSsNAFJ3UV62vqks3wSLWTUlE0GXBjcsK9gFNLZPJbTt08mDmRqghn+CvuHGrf+FGxK1Lv8FJm4W2PTDM4Zx74dzjRoIrtKwPo7Cyura+UdwsbW3v7O6V9w9aKowlgyYLRSg7LlUgeABN5CigE0mgviug7Y6vM7/9AFLxMLjDSQQ9nw4DPuCMopb65VPHDYWnJr7+EgdHgDS9Txyf4kj5VAiQSdU+S9N+uWLVrCnMRWLnpEJyNPrlH8cLWexDgExQpbq2FWEvoRI5E5CWnFhBRNmYDqGraUB9UL1kelBqnmjFMweh1C9Ac6r+3Uior7LMenKadN7LxKWeBKH4IyzzujEOrnoJD6IYIWCzEINYmBiaWXOmxyUwFBNNKJNc32GyEZWUoe63pAuy5+tYJK3zmm3V7NuLSr2aV1UkR+SYVIlNLkmd3JAGaRJGnsgLeSVvxrPxbnwaX7PRgpHvHJJ/ML5/AbpzqVk=AAACLHicbVDLSsNAFJ3UV62vqks3wSLWTUlE0GXBjcsK9gFNLZPJbTt08mDmRqghn+CvuHGrf+FGxK1Lv8FJm4W2PTDM4Zx74dzjRoIrtKwPo7Cyura+UdwsbW3v7O6V9w9aKowlgyYLRSg7LlUgeABN5CigE0mgviug7Y6vM7/9AFLxMLjDSQQ9nw4DPuCMopb65VPHDYWnJr7+EgdHgDS9Txyf4kj5VAiQSdU+S9N+uWLVrCnMRWLnpEJyNPrlH8cLWexDgExQpbq2FWEvoRI5E5CWnFhBRNmYDqGraUB9UL1kelBqnmjFMweh1C9Ac6r+3Uior7LMenKadN7LxKWeBKH4IyzzujEOrnoJD6IYIWCzEINYmBiaWXOmxyUwFBNNKJNc32GyEZWUoe63pAuy5+tYJK3zmm3V7NuLSr2aV1UkR+SYVIlNLkmd3JAGaRJGnsgLeSVvxrPxbnwaX7PRgpHvHJJ/ML5/AbpzqVk=AAACLHicbVDLSsNAFJ3UV62vqks3wSLWTUlE0GXBjcsK9gFNLZPJbTt08mDmRqghn+CvuHGrf+FGxK1Lv8FJm4W2PTDM4Zx74dzjRoIrtKwPo7Cyura+UdwsbW3v7O6V9w9aKowlgyYLRSg7LlUgeABN5CigE0mgviug7Y6vM7/9AFLxMLjDSQQ9nw4DPuCMopb65VPHDYWnJr7+EgdHgDS9Txyf4kj5VAiQSdU+S9N+uWLVrCnMRWLnpEJyNPrlH8cLWexDgExQpbq2FWEvoRI5E5CWnFhBRNmYDqGraUB9UL1kelBqnmjFMweh1C9Ac6r+3Uior7LMenKadN7LxKWeBKH4IyzzujEOrnoJD6IYIWCzEINYmBiaWXOmxyUwFBNNKJNc32GyEZWUoe63pAuy5+tYJK3zmm3V7NuLSr2aV1UkR+SYVIlNLkmd3JAGaRJGnsgLeSVvxrPxbnwaX7PRgpHvHJJ/ML5/AbpzqVk=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AAACIXicbZC7SgNBFIbPeo0x6mprsxjE2IRdGy0FG8sI5gLZGGYnJ8mQ2QszZ4W47CP4Kja2+hY2Ir6Cz+DkUmiSA8P8/P8M/OcLEik0ue6ntba+sbm1Xdgp7pb29g/sw1JDx6niWOexjFUrYBqliLBOgiS2EoUsDCQ2g9HNJG8+otIiju5pnGAnZINI9AVnZKyufeYHsezpcWiuzKchEssfMj9kNNQhkxJVVvHO87xrl92qOx1nWXhzUYb51Lr2j9+LeRpiRFwyrduem1AnY4oEl5gX/VRjwviIDbBtZMRC1J1sulDunBqn5/RjZU5EztT9+yNjoZ50Ni+nTRezibkyUyi1eMJVWTul/lUnE1GSEkZ8VqKfSodiZ0LO6QmFnOTYCMaVMHs4fMgU42T4Fg0gbxHHsmhcVD236t25UIBjOIEKeHAJ13ALNagDh2d4hTd4t16sD+trhnLNmjM9gn9jff8CfLun3A==AAACLHicbVC7TsNAEDzzDOEVoKSxiBChiWwaKCPRUAaJPKTYROfLOjnlfLbu1kjB8ifwKzS08Bc0CNFS8g1cHgUkGel0o5ldaXaCRHCNjvNhrayurW9sFraK2zu7e/ulg8OmjlPFoMFiEat2QDUILqGBHAW0EwU0CgS0guH12G89gNI8lnc4SsCPaF/ykDOKRuqWzrwgFj09isyXeTgApPl95kUUBzqiQoDKKu55nndLZafqTGAvEndGymSGerf04/VilkYgkQmqdcd1EvQzqpAzAXnRSzUklA1pHzqGShqB9rPJQbl9apSeHcbKPIn2RP27kdFIjzObyUnSeW8sLvUUCM0fYZnXSTG88jMukxRBsmmIMBU2xva4ObvHFTAUI0MoU9zcYbMBVZSh6bdoCnLn61gkzYuq61TdW6dcq8yqKpBjckIqxCWXpEZuSJ00CCNP5IW8kjfr2Xq3Pq2v6eiKNds5Iv9gff8CuTOpVQ==AAACLHicbVDLSsNAFJ3UV62vqks3wSLWTUlE0GXBjcsK9gFNLZPJbTt08mDmRqghn+CvuHGrf+FGxK1Lv8FJm4W2PTDM4Zx74dzjRoIrtKwPo7Cyura+UdwsbW3v7O6V9w9aKowlgyYLRSg7LlUgeABN5CigE0mgviug7Y6vM7/9AFLxMLjDSQQ9nw4DPuCMopb65VPHDYWnJr7+EgdHgDS9Txyf4kj5VAiQSdU+S9N+uWLVrCnMRWLnpEJyNPrlH8cLWexDgExQpbq2FWEvoRI5E5CWnFhBRNmYDqGraUB9UL1kelBqnmjFMweh1C9Ac6r+3Uior7LMenKadN7LxKWeBKH4IyzzujEOrnoJD6IYIWCzEINYmBiaWXOmxyUwFBNNKJNc32GyEZWUoe63pAuy5+tYJK3zmm3V7NuLSr2aV1UkR+SYVIlNLkmd3JAGaRJGnsgLeSVvxrPxbnwaX7PRgpHvHJJ/ML5/AbpzqVk=AAACLHicbVDLSsNAFJ3UV62vqks3wSLWTUlE0GXBjcsK9gFNLZPJbTt08mDmRqghn+CvuHGrf+FGxK1Lv8FJm4W2PTDM4Zx74dzjRoIrtKwPo7Cyura+UdwsbW3v7O6V9w9aKowlgyYLRSg7LlUgeABN5CigE0mgviug7Y6vM7/9AFLxMLjDSQQ9nw4DPuCMopb65VPHDYWnJr7+EgdHgDS9Txyf4kj5VAiQSdU+S9N+uWLVrCnMRWLnpEJyNPrlH8cLWexDgExQpbq2FWEvoRI5E5CWnFhBRNmYDqGraUB9UL1kelBqnmjFMweh1C9Ac6r+3Uior7LMenKadN7LxKWeBKH4IyzzujEOrnoJD6IYIWCzEINYmBiaWXOmxyUwFBNNKJNc32GyEZWUoe63pAuy5+tYJK3zmm3V7NuLSr2aV1UkR+SYVIlNLkmd3JAGaRJGnsgLeSVvxrPxbnwaX7PRgpHvHJJ/ML5/AbpzqVk=AAACLHicbVDLSsNAFJ3UV62vqks3wSLWTUlE0GXBjcsK9gFNLZPJbTt08mDmRqghn+CvuHGrf+FGxK1Lv8FJm4W2PTDM4Zx74dzjRoIrtKwPo7Cyura+UdwsbW3v7O6V9w9aKowlgyYLRSg7LlUgeABN5CigE0mgviug7Y6vM7/9AFLxMLjDSQQ9nw4DPuCMopb65VPHDYWnJr7+EgdHgDS9Txyf4kj5VAiQSdU+S9N+uWLVrCnMRWLnpEJyNPrlH8cLWexDgExQpbq2FWEvoRI5E5CWnFhBRNmYDqGraUB9UL1kelBqnmjFMweh1C9Ac6r+3Uior7LMenKadN7LxKWeBKH4IyzzujEOrnoJD6IYIWCzEINYmBiaWXOmxyUwFBNNKJNc32GyEZWUoe63pAuy5+tYJK3zmm3V7NuLSr2aV1UkR+SYVIlNLkmd3JAGaRJGnsgLeSVvxrPxbnwaX7PRgpHvHJJ/ML5/AbpzqVk=AAACLHicbVDLSsNAFJ3UV62vqks3wSLWTUlE0GXBjcsK9gFNLZPJbTt08mDmRqghn+CvuHGrf+FGxK1Lv8FJm4W2PTDM4Zx74dzjRoIrtKwPo7Cyura+UdwsbW3v7O6V9w9aKowlgyYLRSg7LlUgeABN5CigE0mgviug7Y6vM7/9AFLxMLjDSQQ9nw4DPuCMopb65VPHDYWnJr7+EgdHgDS9Txyf4kj5VAiQSdU+S9N+uWLVrCnMRWLnpEJyNPrlH8cLWexDgExQpbq2FWEvoRI5E5CWnFhBRNmYDqGraUB9UL1kelBqnmjFMweh1C9Ac6r+3Uior7LMenKadN7LxKWeBKH4IyzzujEOrnoJD6IYIWCzEINYmBiaWXOmxyUwFBNNKJNc32GyEZWUoe63pAuy5+tYJK3zmm3V7NuLSr2aV1UkR+SYVIlNLkmd3JAGaRJGnsgLeSVvxrPxbnwaX7PRgpHvHJJ/ML5/AbpzqVk=AAACLHicbVDLSsNAFJ3UV62vqks3wSLWTUlE0GXBjcsK9gFNLZPJbTt08mDmRqghn+CvuHGrf+FGxK1Lv8FJm4W2PTDM4Zx74dzjRoIrtKwPo7Cyura+UdwsbW3v7O6V9w9aKowlgyYLRSg7LlUgeABN5CigE0mgviug7Y6vM7/9AFLxMLjDSQQ9nw4DPuCMopb65VPHDYWnJr7+EgdHgDS9Txyf4kj5VAiQSdU+S9N+uWLVrCnMRWLnpEJyNPrlH8cLWexDgExQpbq2FWEvoRI5E5CWnFhBRNmYDqGraUB9UL1kelBqnmjFMweh1C9Ac6r+3Uior7LMenKadN7LxKWeBKH4IyzzujEOrnoJD6IYIWCzEINYmBiaWXOmxyUwFBNNKJNc32GyEZWUoe63pAuy5+tYJK3zmm3V7NuLSr2aV1UkR+SYVIlNLkmd3JAGaRJGnsgLeSVvxrPxbnwaX7PRgpHvHJJ/ML5/AbpzqVk=AAACLHicbVDLSsNAFJ3UV62vqks3wSLWTUlE0GXBjcsK9gFNLZPJbTt08mDmRqghn+CvuHGrf+FGxK1Lv8FJm4W2PTDM4Zx74dzjRoIrtKwPo7Cyura+UdwsbW3v7O6V9w9aKowlgyYLRSg7LlUgeABN5CigE0mgviug7Y6vM7/9AFLxMLjDSQQ9nw4DPuCMopb65VPHDYWnJr7+EgdHgDS9Txyf4kj5VAiQSdU+S9N+uWLVrCnMRWLnpEJyNPrlH8cLWexDgExQpbq2FWEvoRI5E5CWnFhBRNmYDqGraUB9UL1kelBqnmjFMweh1C9Ac6r+3Uior7LMenKadN7LxKWeBKH4IyzzujEOrnoJD6IYIWCzEINYmBiaWXOmxyUwFBNNKJNc32GyEZWUoe63pAuy5+tYJK3zmm3V7NuLSr2aV1UkR+SYVIlNLkmd3JAGaRJGnsgLeSVvxrPxbnwaX7PRgpHvHJJ/ML5/AbpzqVk=\u2713(2)AAACLHicbVDLSsNAFJ34rPVVdekmWMS6KUkRdFlw47KCfUBTy2Ry2w6dPJi5EWrIJ/grbtzqX7gRcevSb3DSZqFtDwxzOOdeOPe4keAKLevDWFldW9/YLGwVt3d29/ZLB4ctFcaSQZOFIpQdlyoQPIAmchTQiSRQ3xXQdsfXmd9+AKl4GNzhJIKeT4cBH3BGUUv90pnjhsJTE19/iYMjQJreJ45PcaR8KgTIpFI7T9N+qWxVrSnMRWLnpExyNPqlH8cLWexDgExQpbq2FWEvoRI5E5AWnVhBRNmYDqGraUB9UL1kelBqnmrFMweh1C9Ac6r+3Uior7LMenKadN7LxKWeBKH4IyzzujEOrnoJD6IYIWCzEINYmBiaWXOmxyUwFBNNKJNc32GyEZWUoe63qAuy5+tYJK1a1baq9u1FuV7JqyqQY3JCKsQml6RObkiDNAkjT+SFvJI349l4Nz6Nr9noipHvHJF/ML5/AbwWqVo=AAACLHicbVDLSsNAFJ34rPVVdekmWMS6KUkRdFlw47KCfUBTy2Ry2w6dPJi5EWrIJ/grbtzqX7gRcevSb3DSZqFtDwxzOOdeOPe4keAKLevDWFldW9/YLGwVt3d29/ZLB4ctFcaSQZOFIpQdlyoQPIAmchTQiSRQ3xXQdsfXmd9+AKl4GNzhJIKeT4cBH3BGUUv90pnjhsJTE19/iYMjQJreJ45PcaR8KgTIpFI7T9N+qWxVrSnMRWLnpExyNPqlH8cLWexDgExQpbq2FWEvoRI5E5AWnVhBRNmYDqGraUB9UL1kelBqnmrFMweh1C9Ac6r+3Uior7LMenKadN7LxKWeBKH4IyzzujEOrnoJD6IYIWCzEINYmBiaWXOmxyUwFBNNKJNc32GyEZWUoe63qAuy5+tYJK1a1baq9u1FuV7JqyqQY3JCKsQml6RObkiDNAkjT+SFvJI349l4Nz6Nr9noipHvHJF/ML5/AbwWqVo=AAACLHicbVDLSsNAFJ34rPVVdekmWMS6KUkRdFlw47KCfUBTy2Ry2w6dPJi5EWrIJ/grbtzqX7gRcevSb3DSZqFtDwxzOOdeOPe4keAKLevDWFldW9/YLGwVt3d29/ZLB4ctFcaSQZOFIpQdlyoQPIAmchTQiSRQ3xXQdsfXmd9+AKl4GNzhJIKeT4cBH3BGUUv90pnjhsJTE19/iYMjQJreJ45PcaR8KgTIpFI7T9N+qWxVrSnMRWLnpExyNPqlH8cLWexDgExQpbq2FWEvoRI5E5AWnVhBRNmYDqGraUB9UL1kelBqnmrFMweh1C9Ac6r+3Uior7LMenKadN7LxKWeBKH4IyzzujEOrnoJD6IYIWCzEINYmBiaWXOmxyUwFBNNKJNc32GyEZWUoe63qAuy5+tYJK1a1baq9u1FuV7JqyqQY3JCKsQml6RObkiDNAkjT+SFvJI349l4Nz6Nr9noipHvHJF/ML5/AbwWqVo=AAACLHicbVDLSsNAFJ34rPVVdekmWMS6KUkRdFlw47KCfUBTy2Ry2w6dPJi5EWrIJ/grbtzqX7gRcevSb3DSZqFtDwxzOOdeOPe4keAKLevDWFldW9/YLGwVt3d29/ZLB4ctFcaSQZOFIpQdlyoQPIAmchTQiSRQ3xXQdsfXmd9+AKl4GNzhJIKeT4cBH3BGUUv90pnjhsJTE19/iYMjQJreJ45PcaR8KgTIpFI7T9N+qWxVrSnMRWLnpExyNPqlH8cLWexDgExQpbq2FWEvoRI5E5AWnVhBRNmYDqGraUB9UL1kelBqnmrFMweh1C9Ac6r+3Uior7LMenKadN7LxKWeBKH4IyzzujEOrnoJD6IYIWCzEINYmBiaWXOmxyUwFBNNKJNc32GyEZWUoe63qAuy5+tYJK1a1baq9u1FuV7JqyqQY3JCKsQml6RObkiDNAkjT+SFvJI349l4Nz6Nr9noipHvHJF/ML5/AbwWqVo=AAAB7XicbVDLSgNBEOyNrxhfUY9eBoOQU9gVQY8BLx4jmAckS+idzCZjZnaWmVkhhPyDFw+KePV/vPk3TpI9aGJBQ1HVTXdXlApurO9/e4WNza3tneJuaW//4PCofHzSMirTlDWpEkp3IjRM8IQ1LbeCdVLNUEaCtaPx7dxvPzFtuEoe7CRlocRhwmNO0Tqp1RuilNgvV/yavwBZJ0FOKpCj0S9/9QaKZpIllgo0phv4qQ2nqC2ngs1KvcywFOkYh6zraIKSmXC6uHZGLpwyILHSrhJLFurviSlKYyYycp0S7cisenPxP6+b2fgmnPIkzSxL6HJRnAliFZm/TgZcM2rFxBGkmrtbCR2hRmpdQCUXQrD68jppXdYCvxbcX1Xq1TyOIpzBOVQhgGuowx00oAkUHuEZXuHNU96L9+59LFsLXj5zCn/gff4AfxmO+w==AAAB7XicbVDLSgNBEOyNrxhfUY9eBoOQU9gVQY8BLx4jmAckS+idzCZjZnaWmVkhhPyDFw+KePV/vPk3TpI9aGJBQ1HVTXdXlApurO9/e4WNza3tneJuaW//4PCofHzSMirTlDWpEkp3IjRM8IQ1LbeCdVLNUEaCtaPx7dxvPzFtuEoe7CRlocRhwmNO0Tqp1RuilNgvV/yavwBZJ0FOKpCj0S9/9QaKZpIllgo0phv4qQ2nqC2ngs1KvcywFOkYh6zraIKSmXC6uHZGLpwyILHSrhJLFurviSlKYyYycp0S7cisenPxP6+b2fgmnPIkzSxL6HJRnAliFZm/TgZcM2rFxBGkmrtbCR2hRmpdQCUXQrD68jppXdYCvxbcX1Xq1TyOIpzBOVQhgGuowx00oAkUHuEZXuHNU96L9+59LFsLXj5zCn/gff4AfxmO+w==AAAB7XicbVDLSgNBEOyNrxhfUY9eBoOQU9gVQY8BLx4jmAckS+idzCZjZnaWmVkhhPyDFw+KePV/vPk3TpI9aGJBQ1HVTXdXlApurO9/e4WNza3tneJuaW//4PCofHzSMirTlDWpEkp3IjRM8IQ1LbeCdVLNUEaCtaPx7dxvPzFtuEoe7CRlocRhwmNO0Tqp1RuilNgvV/yavwBZJ0FOKpCj0S9/9QaKZpIllgo0phv4qQ2nqC2ngs1KvcywFOkYh6zraIKSmXC6uHZGLpwyILHSrhJLFurviSlKYyYycp0S7cisenPxP6+b2fgmnPIkzSxL6HJRnAliFZm/TgZcM2rFxBGkmrtbCR2hRmpdQCUXQrD68jppXdYCvxbcX1Xq1TyOIpzBOVQhgGuowx00oAkUHuEZXuHNU96L9+59LFsLXj5zCn/gff4AfxmO+w==AAAB7XicbVDLSgNBEOyNrxhfUY9eBoOQU9gVQY8BLx4jmAckS+idzCZjZnaWmVkhhPyDFw+KePV/vPk3TpI9aGJBQ1HVTXdXlApurO9/e4WNza3tneJuaW//4PCofHzSMirTlDWpEkp3IjRM8IQ1LbeCdVLNUEaCtaPx7dxvPzFtuEoe7CRlocRhwmNO0Tqp1RuilNgvV/yavwBZJ0FOKpCj0S9/9QaKZpIllgo0phv4qQ2nqC2ngs1KvcywFOkYh6zraIKSmXC6uHZGLpwyILHSrhJLFurviSlKYyYycp0S7cisenPxP6+b2fgmnPIkzSxL6HJRnAliFZm/TgZcM2rFxBGkmrtbCR2hRmpdQCUXQrD68jppXdYCvxbcX1Xq1TyOIpzBOVQhgGuowx00oAkUHuEZXuHNU96L9+59LFsLXj5zCn/gff4AfxmO+w==\fk represents the activation of the kth component at time step t. Each component represents a\nwhere \u03b8(t)\nfor each mode m. For international\ndependence structure in the data set by way of a factor vector \u03c6(m)\nkV ) represents the rate at which each of the V\nevents, the \ufb01rst factor vector \u03c6(1)\ncountries acts as a sender in the kth component while the second factor vector \u03c6(2)\nk represents the rate\nat which each country acts as a receiver. The weights \u03bbk and \u03c1(t) represent the scales of component\nk and time step t. The PRGDS is stationary if \u03c1(t) = \u03c1. We posit the following conjugate priors:\n\nk1, . . . , \u03c6(1)\n\nk = (\u03c6(1)\n\nk\n\n(2)\nThe PRGDS is characterized by an alternating chain of discrete and continuous latent states. The\ncontinuous states \u03b8(1)\nas follows:\n\nk \u223c Dir(a0, . . . , a0).\nevolve via the intermediate discrete states h(1)\n\n\u03c1(t) \u223c Gam (a0, b0)\n\nand \u03c6(m)\n\nk , . . . , h(T )\nk\n\nk , . . . , \u03b8(T )\nk\n\n\u03b8(t)\n\nk \u223c Gam(cid:0)\u0001(\u03b8)\n\n0 +h(t)\n\nk , \u03c4(cid:1) and h(t)\n\nk \u223c Pois(cid:16)\u03c4\n\nK(cid:88)k2=1\n\n\u03c0kk2 \u03b8(t\u22121)\n\nk2 (cid:17),\n\n(3)\n\nk\n\nk = \u03bbk to be the per-component weight from Eq. (1).\n\nk is conditionally gamma distributed with rate \u03c4 and shape equal to h(t)\n\nwhere we de\ufb01ne \u03b8(0)\nIn other words, the\nPRGDS assumes that \u03b8(t)\nk plus\nhyperparameter \u0001(\u03b8)\n0 \u2265 0. We adopt the convention that a gamma random variable will be zero, almost\nsurely, if its shape is zero. Therefore, setting \u0001(\u03b8)\n0 = 0 de\ufb01nes a sparse variant of the PRGDS, where\nthe gamma latent state \u03b8(t)\nk = 0.\nThe transition weight \u03c0kk2 in Eq. (3) represents how strongly component k2 excites component k\nat the next time step. We view these weights collectively as a K\u00d7K transition matrix \u03a0 and impose\nDirichlet priors over the columns of this matrix. We also place a gamma prior over concentration\nparameter \u03c4. This prior is conjugate to the gamma and Poisson distributions in which it appears:\n\nk takes the value of exactly zero provided h(t)\n\nk = 0\u2014i.e., \u03b8(t)\n\na.s.\n= 0 if h(t)\n\nis analogous to \u0001(\u03b8)\n\n\u03c4 \u223c Gam (\u03b10, \u03b10) and \u03c0k \u223c Dir (a0, . . . , a0) such that(cid:80)K\nK + gk, \u03b2(cid:17) and gk \u223c Pois(cid:0) \u03b3\nK(cid:1) ,\nand \u03b2 \u223c Gam (\u03b10, \u03b10) .\n0 , \u0001(\u03bb)\n\n(4)\nFor the per-component weights \u03bb1, . . . , \u03bbK, we use a hierarchical prior with a similar \ufb02avor to Eq. (3):\n\u03bbk \u223c Gam(cid:16) \u0001(\u03bb)\n(5)\n0 . Finally, we use the following gamma priors, which are both conjugate:\n\u03b3 \u223c Gam (a0, b0)\n(6)\nThe PRGDS has \ufb01ve \ufb01xed hyperparameters: \u0001(\u03b8)\n0 , \u03b10, a0, and b0. For the empirical studies in\n\u00a7 5, we set a0 = b0 = 0.01 to de\ufb01ne weakly informative gamma and Dirichlet priors and set \u03b10 = 10\nto de\ufb01ne a gamma prior that promotes values close to 1; we consider \u0001(\u03b8)\n0 = 1.\nProperties. In Eq. (5), both \u0001(\u03bb)\n0 and \u03b3 are divided by the number of components K. This means that\nas the number of components grows K \u2192\u221e, the expected sum of the weights remains \ufb01nite and \ufb01xed:\n(7)\n\n0 \u2208 {0, 1} and set \u0001(\u03bb)\n\nwhere \u0001(\u03bb)\n0\n\n\u03c0k1k = 1.\n\nE [\u03bbk] =\n\nk1\n\n0\n\nK + \u03b3\n\n0\n\n\u221e(cid:88)k=1(cid:0) \u0001(\u03bb)\n\nK + E [gk](cid:1)\u03b2\u22121 =\n\n0\n\n\u221e(cid:88)k=1(cid:0) \u0001(\u03bb)\n\nK(cid:1)\u03b2\u22121 =(cid:0)\u0001(\u03bb)\n\n0 + \u03b3(cid:1)\u03b2\u22121.\n\nThis prior encodes an inductive bias toward small values of \u03bbk and may be interpreted as the \ufb01nite\ntruncation of a novel Bayesian nonparametric process. A small value of \u03bbk shrinks the Poisson rates\nk . As a result, this prior encourages the PRGDS to only\nof both y(t)\ninfer components that are both predictive of the data and useful for capturing the temporal dynamics.\nThe marginal expectation of \u03b8(t) = (\u03b8(t)\n\nK ) takes the form of a linear dynamical system:\n\nand the \ufb01rst discrete latent state h(0)\n\n1 , . . . , \u03b8(t)\n\ni\n\n\u221e(cid:88)k=1\n\nThis is because E(cid:2)\u03b8(t)\n\nE(cid:2)\u03b8(t) | \u03b8(t\u22121)(cid:3) = E(cid:2)E(cid:2)\u03b8(t) | h(t\u22121)(cid:3)(cid:3) = \u0001(\u03b8)\nk (cid:3) =(cid:0)\u0001(\u03b8)\n0 +\u03c4(cid:80)K\n\nk (cid:3)(cid:1)\u03c4\u22121 =(cid:0)\u0001(\u03b8)\n\n0 +E(cid:2)h(t)\n\n0 \u03c4\u22121 + \u03a0 \u03b8(t\u22121).\nk2=1 \u03c0kk2 \u03b8(t\u22121)\n\nk2 (cid:1)\u03c4\u22121 by iterated expec-\n\ntation. Concentration parameter \u03c4 appears in both the Poisson and gamma distributions in Eq. (3).\nIt contributes to the variance of the PRGDS, while simultaneously canceling out of the expectation\nin Eq. (8), except for its role in the additive term \u0001(\u03b8)\nFinally, we can analytically marginalize out all of the discrete Poisson latent states to obtain a purely\ncontinuous dynamical system. When \u0001(\u03b8)\n\n0 > 0, this dynamical system can be written as follows:\n\n0 \u03c4\u22121, which itself disappears when \u0001(\u03b8)\n\n0 = 0.\n\n(8)\n\n\u03b8(t)\n\nk \u223c RG1(cid:16)\u0001(\u03b8)\n\n0 , \u03c4\n\nK(cid:88)k2=1\n\n\u03c0kk2\u03b8(t\u22121)\n\nk2\n\n, \u03c4(cid:17),\n\n(9)\n\nwhere RG1 denotes the randomized gamma distribution of the \ufb01rst type [23, 25]. When \u0001(\u03b8)\n0 = 0, the\ndynamical system can be written in terms of a limiting form of the RG1. We describe the RG1 in Fig. 2.\n\n3\n\n\fFigure 2: The randomized gamma distribution of the \ufb01rst type (RG1) [23, 25] has support \u03b8 > 0 and is de\ufb01ned\nby three parameters: \u0001, \u03bb, \u03b2 > 0. Its PDF is displayed in the \ufb01gure; I\u0001\u22121(\u00b7) is the modi\ufb01ed Bessel function of the\n\ufb01rst kind [26]. When \u0001 < 1 (left), the RG1 resembles a soft \u201cspike-and-slab\u201d distribution; when \u0001 \u2265 1 (middle\nand right), it resembles a more-dispersed form of the gamma distribution. The Poisson-randomized gamma distri-\nbution [27], which includes zeros in its support (i.e., \u03b8 \u2265 0), is a limiting case of the RG1 that occurs when \u0001\u2192 0.\n\n3 Related work\n\nThe PRGDS is closely related to the Poisson\u2013gamma dynamical system (PGDS) [22]. In the PGDS,\n\n\u03b8(t)\n\nk \u223c Gam(cid:16)\u03c4\n\n\u03c0kk2\u03b8(t\u22121)\n\nk2\n\ni\n\n(10)\n\nin Eq. (1). To encourage parsimony, the PGDS instead draws \u03bbk \u223c Gam( \u03b3\n\n, \u03c4(cid:17) such that E(cid:2)\u03b8(t) | \u03b8(t\u22121)(cid:3) = \u03a0 \u03b8(t\u22121).\n\nK(cid:88)k2=1\nThe PGDS imposes non-conjugate dependencies directly between the gamma latent states. The\ncomplete conditional P (\u03b8(t)\nk |\u2212) is not available in closed form, and posterior inference relies on\na sophisticated data augmentation scheme. The PRGDS instead introduces intermediate Poisson\nstates that break the intractable dependencies between the gamma states; we visualize this in\nFig. 1. Although the Poisson distribution is not a conjugate prior for the gamma rate, this motif\nis still tractable, yielding the complete conditional P (h(t)\nk |\u2212) in closed form, as we explain\nin \u00a7 4. The PGDS is limited by the data augmentation scheme that it relies on for posterior\ninference\u2014speci\ufb01cally, this augmentation scheme does not allow \u03bbk to appear in the Poisson rate\nof y(t)\nK , \u03b2) and then uses\nthese per-component weights to shrink the transition matrix \u03a0. This approach introduces additional\nintractable dependencies that require a different data augmentation scheme for posterior inference.\nFinally, the data augmentation schemes additionally require that each factor vector \u03c6(m)\nand each\ncolumn \u03c0k of the transition matrix are Dirichlet distributed. We note that although we also use\nDirichlet distributions in this paper, this is a choice rather than a requirement imposed by the PRGDS.\nThe PGDS and its \u201cdeep\u201d variants [28, 29] generalize gamma process dynamic Poisson factor analysis\n(GP-DPFA) [30], which assumes a simple random walk \u03b8(t)\nand Koeppl is also closely related [31]. These models belong to a line of work exploring the \u201caugment-\nand-conquer\u201d data augmentation scheme [32] for posterior inference in hierarchies of gamma variables\nchained via their shapes and linked to Poisson observations. Beyond models for time series, this motif\ncan be used to build belief networks [33]. An alternative approach is to chain gamma variables via\ntheir rates\u2014e.g., \u03b8(t) \u223c Gam (a, \u03b8(t\u22121)). This motif is conjugate and tractable, and has been applied\nto models for time series [34\u201336] and deep belief networks [37]. However, unlike the shape, the rate\ncontributes to the variance of the gamma quadratically. Rate chains can therefore be highly volatile.\nMore broadly, gamma shape and rate chains are examples of non-negative chains. Such chains\nare especially well motivated in the context of Poisson factorization, which is particularly ef\ufb01cient\nwhen only non-negative prior distributions are used. In general, Poisson factorization assumes\nthat each observed count yi is drawn from a Poisson distribution with a latent rate \u00b5i that is some\nfunction of the model parameters\u2014i.e., yi \u223c Pois (\u00b5i). When the rate is linear\u2014i.e., \u00b5i =(cid:80)K\nk=1 \u00b5ik\u2014\nPoisson factorization is allocative [38] and admits a latent source representation [16, 18], where yi (cid:44)\n(cid:80)K\nk=1 yik is de\ufb01ned to be the sum of K latent sources yi1, . . . , yiK and yik \u223c Pois (\u00b5ik). Conditioning\non the latent sources often induces conditional independencies that, in turn, facilitate closed-form,\nef\ufb01cient, and parallelizable posterior inference. The \ufb01rst step in either MCMC or variational inference\n\n, c(t)(cid:1); the model of Yang\n\nk \u223cGam(cid:0)\u03b8(t\u22121)\n\nk\n\nk\n\n4\n\n051015\u27130.00.20.40.60.81.01.2P(\u2713|\u270f,,=1)\u270f=0.5051015\u2713\u270f=1051015\u2713\u270f=4=4=2=0.5RG1(\u2713;\u270f,,)= r\u2713!\u270f1e\u2713I\u270f1\u21e32p\u2713\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\fis therefore to update each latent source from its complete conditional, which is multinomial [39]:\n\n(cid:0)(yi1, . . . , yiK)| \u2212(cid:1) \u223c Multinom (yi, (\u00b5i1, . . . , \u00b5iK)) ,\n\n(11)\nwhere the normalization of the non-negative rates \u00b5i1, . . . , \u00b5iK into a probability vector is left\na.s.\n= 0\u2014\nimplicit. When the observed count is zero\u2014i.e., yi = 0\u2014the sources are also zero\u2014i.e., yik\nand no computation is required to update them. As a result, any Poisson factorization model that\nadmits a latent source representation scales linearly with only the non-zero entries. This property\nis indispensable when modeling count tensors which typically contain exponentially more zeros\nthan non-zeros [40]. We emphasize that although the PRGDS and PGDS are substantively different\nmodels, they are both instances of allocative Poisson factorization, so the time complexity of posterior\ninference for both models is the same and equal to O (SK) where S is the number of non-zero entries.\nBecause a latent source representation is only available when the rate \u00b5i is a linear function of\nthe model parameters and, by de\ufb01nition of the Poisson distribution, the rate must be non-negative,\nef\ufb01cient Poisson factorization is only possible with non-negative priors. Modeling time series and\nother complex dependence structures via ef\ufb01cient Poisson factorization therefore requires developing\nnovel motifs that exclude the Gaussian priors that researchers have traditionally relied on for analytic\nconvenience and tractability. For example, the Poisson linear dynamical system [41\u201343] links the\nwidely used Gaussian linear dynamical system [44, 45] to Poisson observations via an exponential link\n\nfunction\u2014i.e., \u00b5i = exp ((cid:80)k \u00b7\u00b7\u00b7). This approach, which is based on the generalized linear model\n\n[46], relies on a non-linear link function and therefore does not admit a latent source representation.\nAnother approach is to use log-normal priors, as in dynamic Poisson factorization [47]; however, the\nlog-normal is not conjugate to the Poisson distribution and does not yield closed-form conditionals.\nThere is also a long tradition of autoregressive models for time series of counts, including variational\nautoregressive models [48] and models that are based on the Hawkes process [49\u201352]. This approach\navoids the challenge of constructing tractable state-space models from non-negative priors by\nmodeling temporal correlations directly between the observed counts. However, for high-dimensional\ndata, such as sequentially observed count tensors, an autoregressive approach is often impractical.\n\n4 Posterior inference\n\nIteratively re-sampling each latent variable in the PRGDS from its complete conditional constitutes\na Gibbs sampling algorithm. The complete conditionals for all variables are immediately available in\nclosed form without data augmentation. We provide conditionals for the variables with non-standard\npriors below; the remaining conditionals are in the supplementary material. The PRGDS is based on\na new motif in Bayesian latent variable modeling. We introduce the motif in its general form, derive\nits conditionals, and then use these to obtain the closed-form complete conditionals for the PRGDS.\n\n4.1 Poisson\u2013gamma\u2013Poisson chains\n\nConsider the following model of count m involving variables \u03b8 and h and \ufb01xed c1, c2, c3, \u0001(\u03b8)\n\n0 > 0:\n(12)\nThis model is semi-conjugate. The gamma prior over \u03b8 is conjugate to the Poisson and its posterior is\n(13)\nThe Poisson prior over h is not conjugate to the gamma; however, despite this, the posterior of h\nis still available in closed form by way of the Bessel distribution [23], which we de\ufb01ne in Fig. 3(a):\n\nm \u223c Pois (\u03b8c3) , \u03b8 \u223c Gam(cid:0)\u0001(\u03b8)\n(cid:0)\u03b8 | \u2212(cid:1) \u223c Gam(cid:0)\u0001(\u03b8)\n(cid:0)h| \u2212(cid:1) \u223c Bes(cid:0)\u0001(\u03b8)\n\n0 +h, c2(cid:1) , and h \u223c Pois (c1) .\n0 +h + m, c2 + c3(cid:1) .\n0 \u22121, 2(cid:112)\u03b8 c2 c1(cid:1).\n\n(14)\nThe Bessel distribution can be sampled ef\ufb01ciently [53]; our Cython implementation is available\nonline.1 Provided that \u0001(\u03b8)\n0 > 0, sampling \u03b8 and h iteratively from Eqs. (13) and (14) constitutes a\na.s.\n= 0 if h = 0, and vice versa. As\nvalid Markov chain for posterior inference. When \u0001(\u03b8)\na result, this Markov chain has an absorbing condition at h = 0 and violates detailed balance. In this\ncase, we must therefore sample h with \u03b8 marginalized out. Toward that end, we prove Theorem 1.\n\n0 = 0, though, \u03b8\n\n1https://github.com/aschein/PRGDS\n\n5\n\n\f(a) Bessel distribution [23]\n\n(b) Shifted con\ufb02uent hypergeometric (SCH) distribution\n\nFigure 3: Two discrete distributions that arise as posteriors in Poisson\u2013gamma\u2013Poisson chains.\n\n0 = 0,\u2212\\\u03b8) (cid:44)(cid:82) P (h, \u03b8 | \u0001(\u03b8)\nTheorem 1: The incomplete conditional P (h| \u0001(\u03b8)\n(h|\u2212\\\u03b8) \u223c(cid:40)Pois(cid:0) c1 c2\nc3+c2(cid:1)\nc3+c2(cid:1)\nSCH(cid:0)m, c1 c2\n\nif m = 0\notherwise,\n\n0 = 0,\u2212) d\u03b8 is\n\n(15)\n\nwhere SCH denotes the shifted con\ufb02uent hypergeometric distribution. We describe the SCH in\nFig. 3(b) and provide further information in the supplementary material, including the derivation\nof its PMF, PGF, and mode, along with details of how we sample from it and the proof for Theorem 1.\n\n4.2 Closed-form complete conditionals for the PRGDS\n\nThe PRGDS admits a latent source representation, so the \ufb01rst step of posterior inference is therefore\n\ni\n\nm=1\u03c6(m)\n\n)K\n\nkk2\n\nk1=1 h(t+1)\n\n(16)\n\n(17)\n\nk\u00b7 , (\u03c0kk2\u03b8(t\u22121)\n\nk2\n\n)K\n\nk \u2261 h(t)\n\nk2=1 h(t)\nkk2\n\n,\n\nik )K\n\n(cid:0)(y(t)\n\nk\u00b7 =(cid:80)K\n\nk=1(cid:17) .\nkim(cid:1)K\n\n,(cid:0)\u03bbk \u03b8(t)\nk (cid:81)M\n\nk under its latent source representation\u2014i.e., h(t)\n\nTo derive the conditional for \u03b8(t)\ntivity, the column sum h(t+1)\n\nk we aggregate the Poisson variables that depend on it. By Poisson addi-\n\n. The complete conditional of the kth row of counts, when conditioned on their sum h(t)\n\nWe may similarly represent h(t)\nwhere h(t)\nsumming over a mode. In this case, h(t)\ncounts h(t)\nkk2\n\nk=1 | \u2212(cid:1) \u223c Multinom(cid:16)y(t)\nkk2 \u223cPois(cid:0)\u03c4 \u03c0kk2 \u03b8(t\u22121)\nk2 (cid:1). When notationally convenient, we use dot-notation (\u201c\u00b7\u201d) to denote\nk\u00b7 denotes the sum of the kth row of the K\u00d7K matrix of latent\nk\u00b7 , is\nk2=1 | \u2212(cid:1) \u223c Multinom(cid:0)h(t)\n(cid:0)(h(t)\n\u00b7k =(cid:80)K\nis distributed as y(t)\u00b7k \u223cPois(cid:0)\u03b8(t)\nk \u03c1(t)\u03bbk(cid:81)M\n(cid:0)\u03b8(t)\nk | \u2212(cid:1) \u223c Gam(cid:0)\u0001(\u03b8)\nk\u00b7 | \u2212(cid:1) \u223c Bessel(cid:16)\u0001(\u03b8)\n(cid:0)h(t)\nk (cid:1) \u223c(cid:26)Pois(\u03b6 (t)\n\nk2=1(cid:1) .\nk \u03c4 \u03c0\u00b7k(cid:1) and similarly y(t)\u00b7k\n\u00b7k \u223cPois(cid:0)\u03b8(t)\nk\u00b7 (cid:1). The count m(t)\nk , \u03c4 + \u03c4 \u03c0\u00b7k + \u03c1(t)\u03bbk(cid:81)M\nk2 (cid:17).\n\u03c4 2(cid:80)K\n\nk and is also Poisson distributed. By gamma\u2013Poisson conjugacy, the conditional of \u03b8(t)\n\n0 > 0, we apply the identity in Eq. (14) and sample h(t)\n\nWhen \u0001(\u03b8)\n\n0 = 0, we instead apply Theorem 1 to sample h(t)\n\nk )\nSCH(m(t)\nk , \u03b6 (t)\n\nk = 0\nk ) otherwise\n\nif m(t)\n\nk1k is distributed as h(t+1)\nm=1 \u03c6(m)\n\nk\u00b7 , where m(t)\n\nk is analogous to m in Eq. (15):\n\nk isolates all depen-\nk is\n\n0 \u22121, 2(cid:113)\u03b8(t)\n\nk \u03c4 2(cid:80)K\n\nk\u00b7 from its complete conditional:\n\nThe complete conditionals for \u03bbk and gk follow from applying the same Poisson\u2013gamma\u2013Poisson\nidentities, while the complete conditionals for \u03b3, \u03b2, \u03c6(m)\n\nk , \u03c0k, and \u03c4 all follow from conjugacy.\n\nk2=1 \u03c0kk2 \u03b8(t\u22121)\n\n(cid:81)M\n\nk2\nm=1 \u03c6(m)\nk\u00b7\n\n.\n\n(cid:44) h(t+1)\n\n\u00b7k +y(t)\n\nk\n\nm=1\u03c6(m)\n\nk\u00b7 (cid:1).\n\nk\u00b7 |\u2212\\\u03b8(t)\n\n(cid:0)h(t)\n\n(18)\n\n(19)\n\n(20)\n\nwhere \u03b6 (t)\nk\n\n(cid:44)\n\n\u03c4 +\u03c4 \u03c0\u00b7k+\u03c1(t)\u03bbk\n\ndence on \u03b8(t)\n\nWhen \u0001(\u03b8)\n\n0 +h(t)\n\nk\u00b7 + m(t)\n\nk2=1\u03c0kk2\u03b8(t\u22121)\n\n6\n\n020406080100h0.000.050.100.150.200.250.30P(h|v,a)v=-0.5,a=30v=-0.8,a=10v=1,a=50v=2,a=7v=10,a=150v=400,a=300Bes(h;v,a)=(a2)2h+vh!(v+h+1)Iv(a)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120406080100h0.000.050.100.150.200.250.300.35P(h|m,\u21e3)m=1,\u21e3=1m=1,\u21e3=10m=1,\u21e3=50m=10,\u21e3=1m=1000,\u21e3=1m=45,\u21e3=45SCH(h;m,\u21e3)=(h+m+1)!(h+1)!h!m!\u21e3h11F1(m+1;2;\u21e3)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\f(a) Matrix empirical studies (originally described by Schein et al. [22])\nFigure 4: The smoothing performance (top row) or forecasting performance (bottom row) of each model is\nquanti\ufb01ed by its information gain over a non-dynamic baseline (BPTF [9]), where higher values are better.\n\n(b) Tensor empirical studies\n\n5 Empirical studies\n\nAs explained in the previous section, the Poisson\u2013gamma\u2013Poisson motif of the PRGDS (see \u00a7 4.1)\nyields a more tractable (see Fig. 1) and \ufb02exible (see \u00a7 3) model than previous models. This motif\nalso encodes a unique inductive bias tailored to sparsity and burstiness that we test by comparing the\nPRGDS to the PGDS (described in \u00a7 3). As we can see by comparing Eqs. (9) and (10), comparing\nthese models isolates the impact of the Poisson\u2013gamma\u2013Poisson motif. Because the PGDS was pre-\nviously introduced to model a T \u00d7V matrix Y of sequentially observed V -dimensional count vectors\ny(1), . . . , y(T ), we generalize the PGDS to M-mode tensors and provide derivations of its complete\nconditionals in the supplementary material. Our Cython implementation of this generalized PGDS\n(and the PRGDS) is available online. We also compare the variant of the PRGDS with \u0001(\u03b8)\n0 = 1 to the\nvariant with \u0001(\u03b8)\n0 = 0, which allows the continuous gamma latent states to take values of exactly zero.\nSetup. Our empirical studies all have the following setup. For each data set Y (1), . . . , Y (T ), the\ncounts Y (t) in randomly selected time steps are held out. Additionally, the counts in the last\ntwo time steps are always held out. Each model is \ufb01t to the data set using independent MCMC\nchains that impute the heldout counts and, ultimately, return a set of posterior samples of the la-\ntent variables. We distinguish the task of predicting the counts in intermediate time steps, known\nas smoothing, from the task of predicting the counts in the last two time steps, known as fore-\ncasting. To quantify the performance of each model, we use the S posterior samples returned\nby the independent chains to approximate the information rate [54] of the heldout counts\u2014i.e.,\nR(\u2206) = \u2212 1\n(de\ufb01ned in Eq. (1)) computed from the\nheldout counts and \u00b5(t)\nsth posterior sample. The information rate quanti\ufb01es the average number of nats needed to compress\neach heldout count; it is equivalent to log perplexity [55] and to the negative of log pointwise predic-\ntive density (LPPD) [56]. In each study, we also \ufb01t Bayesian Poisson tensor factorization (BPTF)\n[9], a non-dynamic baseline that assumes that the count tensors at different time steps are i.i.d.\u2014i.e.,\ni \u223cPois (\u00b5i). For each model, we then report the information gain over BPTF, where higher values\ny(t)\nare better, which we compute by subtracting the information rate of the model from that of BPTF.\nMatrices. We \ufb01rst replicated the empirical studies of Schein et al. [22]. These studies followed the\nsetup described above and compared the PGDS to GP-DPFA [30], a simple dynamic baseline (de-\nscribed in \u00a7 3). The matrices in these studies were based on three text data sets\u2014NeurIPS papers [57],\nDBLP abstracts [58], and State of the Union (SOTU) speeches [59]\u2014where y(t)\nv is the number of\ntimes word v occurs in time step t, and two international event data sets\u2014GDELT [60] and ICEWS\n[61]\u2014where y(t)\nv is the number of times sender\u2013receiver pair v interacted during time step t. We used\nthe matrices and heldout time steps, along with the posterior samples for both PGDS and GP-DPFA,\noriginally obtained by Schein et al. [22]. We then \ufb01t the PRGDS using the MCMC settings that they\ndescribe. In this matrix setting, BPTF reduces to y(t)\nv \u223cPois(\u00b5v), where v indexes a single mode, and\n\u00b5v cannot be meaningfully factorized. We therefore posited a conjugate gamma prior over \u00b5v directly\nand drew exact posterior samples to compute the information rate. We depict the results in Fig. 4(a).\n\ni,s(cid:1)(cid:105), where \u2206 is the set of multi-indices of the\n\n|\u2206|(cid:80)(t,i)\u2208\u2206 log(cid:104) 1\nS(cid:80)S\n\ni,s is the expectation of heldout count y(t)\ni\n\ns=1 Pois(cid:0)y(t)\n\ni\n\n; \u00b5(t)\n\n7\n\n0.00.51.01.52.0SMOOTHINGInformation gainover BPTF (nats)GDELT0.00.51.0ICEWS012NeurIPS0.00.20.4DBLP0.000.050.100.15SOTU0.000.250.500.751.00FORECASTINGInformation gainover BPTF (nats) 0.00.20.40257100.00.10.20.30.00.10.2GP-DPFAPGDSPRGDS \u03b5(\u03b8)0=0PRGDS \u03b5(\u03b8)0=10.000.020.040.06GDELT0.0000.0020.0040.0060.0080.010ICEWS0.000.010.020.030.04Macaques0.0000.0250.0500.0750.1000.1250.0000.0010.0020.0030.0040.000.020.040.06\f(a) We visualize two components inferred by a sparse variant of the PRGDS (i.e., \u0001(\u03b8)\n0 = 0) from the ICEWS data\nset of international events. The blue component was also inferred by the other models while the red component\nwas not. The red component is speci\ufb01c to South Sudan, as revealed by visualizing the largest values of the\nsender and receiver factor vectors (bottom row, red). South Sudan was not a country until July 2011 when it\ngained independence from Sudan. The gamma states (top row, red) are therefore sparse\u2014i.e., \u03b8(t)\nk = 0 in 94%\nof time steps (months) prior to July 2011 and in 83% of the time steps overall. In contrast, the blue component\nrepresents Southeast Asian relations, which are active in all time steps. The sparse variant can infer both temporally\npersistent latent structures (e.g., blue), as well as bursty latent structures that are highly localized in time (e.g., red).\n\n(b) We visualize components inferred by the PRGDS from the macaque motor cortex data set. The components\ninferred by a sparse variant (i.e., \u0001(\u03b8)\n0 = 0) are bursty and highly localized in time (left), suggesting that neurons\nmay be tuned to speci\ufb01c periods of the trial. The K\u00d7T gamma latent states for this variant of the PRGDS are\nsparse (middle, white cells correspond to \u03b8(t)\nk = 0). The components (rows) are sorted by the time step in which\nk occurred, so the banded structure indicates that each component is only active for a short duration.\nthe largest \u03b8(t)\nIn contrast, the components inferred by the non-sparse variant (i.e., \u0001(\u03b8)\n0 = 1) are active in all time steps (right).\n\nFigure 5: The PRGDS is capable of inferring latent structures that are highly localized in time.\n\ni\n\nTensors. We used two international event data sets\u2014GDELT and ICEWS\u2014where y(t)\nis the\na\u2212\u2192j\nnumber of times country i took action a toward country j during time step t. Each data set consists\nof a sequence of count tensors, each of which contains the V \u00d7 V \u00d7 A event counts for that time\nstep, where V = 249 countries and A = 20 action types. For both data sets, we used months as\ntime steps. For GDELT, we considered the date range 2003\u20132008, yielding T = 72; for ICEWS, we\nconsidered the date range 1995\u20132013, yielding T = 228. We also used a data set of multi-neuronal\nspike train recordings of macaque monkey motor cortexes [62, 63]. In this data set, a count y(t)\nij is\nthe number of times neuron i spiked in trial j during time step t. These counts form a sequence of\nN\u00d7V matrices, where N = 100 is the number of neurons and V = 1, 716 is the number of trials. We\nused 20-millisecond intervals as time steps, yielding T = 162. For each data set, we created three\nrandom masks, each corresponding to six heldout time steps in the range [2, T\u22122]. We \ufb01t each model\nto each data set and mask using two independent chains of 4,000 MCMC iterations, saving every 50th\nposterior sample after the \ufb01rst 1,000 iterations to compute the information rate. We also \ufb01t BPTF using\nvariational inference as described by Schein et al. [9], and then sampled from the \ufb01tted variational\nposterior to compute the information rate. Following Schein et al. [22], we set K = 100 for all models.\nWe depict the results in Fig. 4(b), where the error bars re\ufb02ect variability across the random masks.\nQuantitative results. In all sixteen studies, the dynamic models outperform BPTF. In all but one\nstudy, the PGDS and a sparse variant of the PRGDS (i.e., \u0001(\u03b8)\n0 = 0) outperform the other models. For\nsmoothing, the PRGDS performs better than or similarly to the PGDS. In \ufb01ve of the eight smoothing\n\n8\n\nMar1995Nov1996Jul1998Mar2000Nov2001Jul2003Mar2005Nov2006Jul2008Mar2010Nov2011Aug2013012345Valueof\u03b8(t)kTimestepsSparsityofvector\u03b8k1%83%MalaysiaThailandSingaporeIndonesiaPhilippinesUSAJapan0.000.250.500.751.00Valueof\u03c6(m)kimSenders(m=1)MalaysiaSingaporeIndonesiaThailandJapanMyanmarPhilippinesReceivers(m=2)ConsultIntendStatementCoop(Dip)CoerceAppealDisapproveActiontypes(m=3)So.SudanSudanUSAChinaEthiopiaEgyptIndiaSenders(m=1)So.SudanSudanEthiopiaChinaUSAKenyaIsraelReceivers(m=2)ConsultStatementCoop(Dip)IntendDisapproveFightAppealActiontypes(m=3)060012001800time(20msintervals)0246Valueof\u03b8(t)kSparsityofvector\u03b8k83%79%77%76%75%73%72%71%69%69%060012001800time(20msintervals)1255075100componentk\u0001(\u03b8)0=0060012001800time(20msintervals)\u0001(\u03b8)0=102571012Valueof\u03b8(t)k\fstudies, the sparse variant of the PRGDS obtains a higher information gain than the PGDS; in the\nremaining three smoothing studies, there is no discernible difference between the models. For fore-\ncasting, we \ufb01nd the converse relationship. In four of the eight forecasting studies, the PGDS obtains\na higher information gain than the PGDS; in the remaining forecasting studies, there is no discernible\ndifference. In all studies, the sparse variant of the PRGDS obtains better smoothing and forecasting\nperformance than the non-sparse variant (i.e., \u0001(\u03b8)\n0 = 1). We conjecture that the better performance\nof the sparse variant can be explained by the form of the marginal expectation of \u03b8(t) (see Eq. (8)).\n0 > 0 this expectation includes an additive term that grows as more time steps are forecast.\nWhen \u0001(\u03b8)\n0 = 0, this term disappears and the expectation matches that of the PGDS (see Eq. (10)).\nWhen \u0001(\u03b8)\nQualitative analysis. We also performed a qualitative comparison of the latent structures inferred\nby the different models and found that the sparse variant of the PRGDS inferred some components\nthat the other models did not. Speci\ufb01cally, the sparse variant of the PRGDS is uniquely capable of\ninferring bursty latent structures that are highly localized in time; we visualize examples in Fig. 5. To\ncompare the latent structures inferred by the PGDS and the PRGDS, we aligned the models\u2019 inferred\ncomponents using the Hungarian bipartite matching algorithm [64] applied to the models\u2019 continuous\ngamma latent states. The kth component\u2019s activation vector \u03b8k = (\u03b8(1)\nk ) constitutes a\nsignature of that component\u2019s activity; these signatures are suf\ufb01ciently unique to facilitate alignment.\nIn the supplementary material, we provide four components that are well aligned across the models.\nIn Fig. 5(a), we visualize two components inferred by the sparse variant of the PRGDS; one of these\ncomponents (blue) was also inferred by the other models, while the other component (red) was not.\n\nk , . . . , \u03b8(T )\n\n6 Conclusion\n\nWe presented the Poisson-randomized gamma dynamical system (PRGDS), a tractable, expressive,\nand ef\ufb01cient model for sequentially observed count tensors. The PRGDS is based on a new modeling\nmotif, an alternating chain of discrete Poisson and continuous gamma latent states that yields\nclosed-form complete conditionals for all variables. We found that a sparse variant of the PRGDS,\nwhich allows the continuous gamma latent states to take values of exactly zero, often obtains better\npredictive performance than other models and infers latent structures that are highly localized in time.\nAcknowledgments We thank Saurabh Vyas, Alex Williams, and Krishna Shenoy for kindly providing us with\nthe macaque monkey motor cortex data set and their corresponding preprocessing code. SWL was supported\nby the Simons Collaboration on the Global Brain (SCGB 418011). MZ was supported by NSF IIS-1812699.\nDMB was supported by ONR N00014-17-1-2131, ONR N00014-15-1-2209, NIH 1U01MH115727-01, NSF\nCCF-1740833, DARPA SD2 FA8750-18-C-0130, IBM, 2Sigma, Amazon, NVIDIA, and the Simons Foundation.\n\nReferences\n[1] Philip A Schrodt. Event data in foreign policy analysis. Foreign Policy Analysis: Continuity\n\nand Change in Its Second Generation, 1995.\n\n[2] Gary King. Proper nouns and methodological propriety: Pooling dyads in international relations\n\ndata. International Organization, 55(2), 2001.\n\n[3] Donald P Green, Soo Yeon Kim, and David H Yoon. Dirty pool. International Organization,\n\n55(2), 2001.\n\n[4] Paul Poast. (Mis)using dyadic data to analyze multilateral events. Political Analysis, 18(4),\n\n2010.\n\n[5] Robert S Erikson, Pablo M Pinto, and Kelly T Rader. Dyadic analysis in international relations:\n\nA cautionary tale. Political Analysis, 22(4), 2014.\n\n[6] Brandon Stewart. Latent factor regressions for the social sciences. Technical report, Harvard\n\nUniversity, 2014.\n\n[7] Peter D Hoff and Michael D Ward. Modeling dependencies in international relations networks.\n\nPolitical Analysis, 12(2), 2004.\n\n[8] Peter D Hoff. Multilinear tensor regression for longitudinal relational data. Annals of Applied\n\nStatistics, 9(3), 2015.\n\n9\n\n\f[9] Aaron Schein, John Paisley, David M Blei, and Hanna M Wallach. Bayesian Poisson tensor\nfactorization for inferring multilateral relations from sparse dyadic event counts. In ACM\nSIGKDD International Conference on Knowledge Discovery and Data Mining, 2015.\n\n[10] Peter D Hoff. Equivariant and scale-free Tucker decomposition models. Bayesian Analysis, 11\n\n(3), 2016.\n\n[11] Aaron Schein, Mingyuan Zhou, David M Blei, and Hanna M Wallach. Bayesian Poisson Tucker\ndecomposition for learning the structure of international relations. In International Conference\non Machine Learning, 2016.\n\n[12] Jon Kleinberg. Bursty and hierarchical structure in streams. Data Mining and Knowledge\n\nDiscovery, 7(4), 2003.\n\n[13] Eric C Chi and Tamara G Kolda. On tensors, sparsity, and nonnegative factorizations. SIAM\n\nJournal on Matrix Analysis and Applications, 33(4), 2012.\n\n[14] Tsuyoshi Kunihama and David B Dunson. Bayesian modeling of temporal dependence in large\n\nsparse contingency tables. Journal of the American Statistical Association, 108(504), 2013.\n\n[15] John Canny. GaP: A factor model for discrete data. In ACM SIGIR Conference on Research\n\nand Development in Information Retrieval, 2004.\n\n[16] David B Dunson and Amy H Herring. Bayesian latent variable models for mixed discrete\n\noutcomes. Biostatistics, 6(1), 2005.\n\n[17] Michalis K Titsias. The in\ufb01nite gamma\u2013Poisson feature model.\n\nInformation Processing Systems, 2008.\n\nIn Advances in Neural\n\n[18] A Taylan Cemgil. Bayesian inference for nonnegative matrix factorisation models. Computa-\n\ntional Intelligence and Neuroscience, 2009.\n\n[19] Mingyuan Zhou, Lauren Hannah, David B Dunson, and Lawrence Carin. Beta-negative binomial\nprocess and Poisson factor analysis. In International Conference on Arti\ufb01cial Intelligence and\nStatistics, 2012.\n\n[20] Prem K Gopalan and David M Blei. Ef\ufb01cient discovery of overlapping communities in massive\n\nnetworks. Proceedings of the National Academy of Sciences, 110(36), 2013.\n\n[21] Beyza Ermis and A Taylan Cemgil. A Bayesian tensor factorization model via variational\n\ninference for link prediction. arXiv preprint arXiv:1409.8276, 2014.\n\n[22] Aaron Schein, Hanna M Wallach, and Mingyuan Zhou. Poisson-gamma dynamical systems. In\n\nAdvances in Neural Information Processing Systems, 2016.\n\n[23] Lin Yuan and John D Kalb\ufb02eisch. On the Bessel distribution and related problems. Annals of\n\nthe Institute of Statistical Mathematics, 52(3), 2000.\n\n[24] Richard A Harshman. Foundations of the PARAFAC procedure: Models and conditions for an\n\n\u201cexplanatory\u201d multimodal factor analysis. UCLA Working Papers in Phonetics, 16, 1970.\n\n[25] Roman N Makarov and Devin Glew. Exact simulation of Bessel diffusions. Monte Carlo\n\nMethods and Applications, 16(3-4), 2010.\n\n[26] Milton Abramowitz and Irene A Stegun. Handbook of mathematical functions: with formulas,\n\ngraphs, and mathematical tables. Courier Corporation, 1965.\n\n[27] Mingyuan Zhou, Yulai Cong, and Bo Chen. Augmentable gamma belief networks. Journal of\n\nMachine Learning Research, 17(1), 2016.\n\n[28] Chengyue Gong and Win-bin Huang. Deep dynamic Poisson factorization model. In Advances\n\nin Neural Information Processing Systems, 2017.\n\n[29] Dandan Guo, Bo Chen, Hao Zhang, and Mingyuan Zhou. Deep Poisson gamma dynamical\n\nsystems. In Advances in Neural Information Processing Systems, 2018.\n\n10\n\n\f[30] Ayan Acharya, Joydeep Ghosh, and Mingyuan Zhou. Nonparametric Bayesian factor analysis\nfor dynamic count matrices. In International Conference on Arti\ufb01cial Intelligence and Statistics,\n2015.\n\n[31] Sikun Yang and Heinz Koeppl. Dependent relational gamma process models for longitudinal\n\nnetworks. In International Conference on Machine Learning, 2018.\n\n[32] Mingyuan Zhou and Lawrence Carin. Augment-and-conquer negative binomial processes. In\n\nAdvances in Neural Information Processing Systems, 2012.\n\n[33] Mingyuan Zhou, Yulai Cong, and Bo Chen. The Poisson gamma belief network. In Advances\n\nin Neural Information Processing Systems, 2015.\n\n[34] A Taylan Cemgil and Onur Dikmen. Conjugate gamma Markov random \ufb01elds for modelling\nnonstationary sources. In International Conference on Independent Component Analysis and\nSignal Separation, 2007.\n\n[35] C\u00e9dric F\u00e9votte, Jonathan Le Roux, and John R Hershey. Non-negative dynamical system with\napplication to speech and audio. In IEEE International Conference on Acoustics, Speech and\nSignal Processing, 2013.\n\n[36] Ghassen Jerfel, Mehmet Basbug, and Barbara Engelhardt. Dynamic collaborative \ufb01ltering with\ncompound poisson factorization. In International Conference on Arti\ufb01cial Intelligence and\nStatistics, 2017.\n\n[37] Rajesh Ranganath, Linpeng Tang, Laurent Charlin, and David M Blei. Deep exponential\n\nfamilies. In International Conference on Arti\ufb01cial Intelligence and Statistics, 2015.\n\n[38] Aaron Schein. Allocative Poisson Factorization for Computational Social Science. PhD thesis,\n\nUniversity of Massachusetts Amherst, 2019.\n\n[39] Robert GD Steel. Relation between Poisson and multinomial distributions. 1953.\n\n[40] Anirban Bhattacharya and David B Dunson. Simplex factor models for multivariate unordered\n\ncategorical data. Journal of the American Statistical Association, 107(497), 2012.\n\n[41] Anne C Smith and Emery N Brown. Estimating a state-space model from point process\n\nobservations. Neural Computation, 15(5), 2003.\n\n[42] Liam Paninski, Yashar Ahmadian, Daniel Gil Ferreira, Shinsuke Koyama, Kamiar Rahnama\nRad, Michael Vidne, Joshua Vogelstein, and Wei Wu. A new look at state-space models for\nneural data. Journal of Computational Neuroscience, 29(1-2), 2010.\n\n[43] Jakob H Macke, Lars Buesing, John P Cunningham, Byron M Yu, Krishna V Shenoy, and\nManeesh Sahani. Empirical models of spiking in neural populations. In Advances in Neural\nInformation Processing Systems, 2011.\n\n[44] Rudolph E Kalman and Richard S Bucy. New results in linear \ufb01ltering and prediction theory.\n\nJournal of Basic Engineering, 83(1), 1961.\n\n[45] Zoubin Ghahramani and Sam T Roweis. Learning nonlinear dynamical systems using an EM\n\nalgorithm. In Advances in Neural Information Processing Systems, 1999.\n\n[46] John A Nelder and Robert WM Wedderburn. Generalized linear models. Journal of the Royal\n\nStatistical Society: Series A (General), 135(3), 1972.\n\n[47] Laurent Charlin, Rajesh Ranganath, James McInerney, and David M Blei. Dynamic Poisson\n\nfactorization. In ACM Conference on Recommender Systems, 2015.\n\n[48] Patrick T Brandt and Todd Sandler. A Bayesian Poisson vector autoregression model. Political\n\nAnalysis, 20(3), 2012.\n\n[49] Alan G Hawkes. Spectra of some self-exciting and mutually exciting point processes.\n\nBiometrika, 58(1), 1971.\n\n11\n\n\f[50] Aleksandr Simma and Michael I Jordan. Modeling events with cascades of poisson processes.\n\nIn Conference on Uncertainty in Arti\ufb01cial Intelligence, 2010.\n\n[51] Charles Blundell, Jeff Beck, and Katherine A Heller. Modelling reciprocating relationships\n\nwith Hawkes processes. In Advances in Neural Information Processing Systems, 2012.\n\n[52] Scott W Linderman and Ryan Adams. Discovering latent network structure in point process\n\ndata. In International Conference on Machine Learning, 2014.\n\n[53] Luc Devroye. Simulating Bessel random variables. Statistics & Probability Letters, 57(3),\n\n2002.\n\n[54] Hanna M Wallach. Structured topic models for language. PhD thesis, University of Cambridge\n\nCambridge, UK, 2008.\n\n[55] Hanna M Wallach, Iain Murray, Ruslan Salakhutdinov, and David Mimno. Evaluation methods\n\nfor topic models. In International Conference on Machine Learning, 2009.\n\n[56] Andrew Gelman, Jessica Hwang, and Aki Vehtari. Understanding predictive information criteria\n\nfor bayesian models. Statistics and Computing, 24(6), 2014.\n\n[57] NeurIPS corpus. UCI Machine Learning Repository.\n\n[58] dblp computer science bibliography. http://dblp.uni-trier.de/.\n\n[59] State of the Union Addresses (1790-2006) by United States Presidents. https://www.\n\ngutenberg.org/ebooks/5050?msg=welcome_stranger.\n\n[60] Kalev Leetaru and Philip A Schrodt. GDELT: Global data on events, location, and tone,\n\n1979\u20132012. In ISA Annual Convention, volume 2. Citeseer, 2013.\n\n[61] Elizabeth Boschee, Jennifer Lautenschlager, Sean O\u2019Brien, Steve Shellman, James Starz, and\n\nMichael Ward. ICEWS coded event data. Harvard Dataverse, 2015.\n\n[62] Saurabh Vyas, Nir Even-Chen, Sergey D Stavisky, Stephen I Ryu, Paul Nuyujukian, and\nKrishna V Shenoy. Neural population dynamics underlying motor learning transfer. Neuron, 97\n(5), 2018.\n\n[63] Alex H Williams, Tony Hyun Kim, Forea Wang, Saurabh Vyas, Stephen I Ryu, Krishna V\nShenoy, Mark Schnitzer, Tamara G Kolda, and Surya Ganguli. Unsupervised discovery of\ndemixed, low-dimensional neural dynamics across multiple timescales through tensor compo-\nnent analysis. Neuron, 98(6), 2018.\n\n[64] Harold W Kuhn. The Hungarian method for the assignment problem. Naval Research Logistics\n\nQuarterly, 2(1-2), 1955.\n\n12\n\n\f", "award": [], "sourceid": 395, "authors": [{"given_name": "Aaron", "family_name": "Schein", "institution": "UMass Amherst"}, {"given_name": "Scott", "family_name": "Linderman", "institution": "Columbia University"}, {"given_name": "Mingyuan", "family_name": "Zhou", "institution": "University of Texas at Austin"}, {"given_name": "David", "family_name": "Blei", "institution": "Columbia University"}, {"given_name": "Hanna", "family_name": "Wallach", "institution": "MSR NYC"}]}