Linear Additive Markov Processes

Kumar, Ravi; Raghu, Maithra; Sarlós, Tamás; Tomkins, Andrew

doi:10.1145/3038912.3052644

Cited by 8 publications

(14 citation statements)

References 37 publications

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“…Thus, approaches employing higher-order Markov processes have a cubic space complexity, and are for this reason inefficient. To cope with this issue, more efficient parameterization approaches, such as the Linear Additive Markov Process (LAMP [28]) and the Retrospective Higher-Order Markov Process (RHOMP [44]), have been proposed. In contrast to the higher-order Markov process, the number of maintained parameters in the LAMP model and the RHOMP model grow linearly, which makes them more suitable for streaming data.…”

Section: The Candidate Change Point (Detection) Approachmentioning

confidence: 99%

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Applying temporal dependence to detect changes in streaming data

2018

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Detection of changes in streaming data is an important mining task, with a wide range of real-life applications. Numerous algorithms have been proposed to efficiently detect changes in streaming data. However, the limitation of existing algorithms is that they assume that data are generated independently. In particular, temporal dependencies of data in a stream are still not thoroughly studied. Motivated by this, in this work we propose a new efficient method to detect changes in streaming data by exploring the temporal dependencies of data in the stream. As part of this, we introduce a new statistical model called the candidate change point (CCP) model, with which the main idea is to compute the probabilities of finding change points in the stream. The computed probabilities are used to generate a distribution, which is, in turn, used in statistical hypothesis tests to determine the candidate changes. We use the CCP model to develop a new algorithm called Candidate Change Point Detector (CCPD), which detects change points in linear time, and is thus applicable for real-time applications. Our extensive experimental evaluation demonstrates the efficiency and the feasibility of our approach.

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Section: The Candidate Change Point (Detection) Approachmentioning

confidence: 99%

“…In the proposed method, we introduce an adaptive estimation for detecting changes, we use CCP heritage of the CCP model to estimate the CCP mean distribution of streaming data. This method can be compared to the linear high order of states based on Markov chain [28,44].…”

Section: Adaptive Estimation Of Data In Streamingmentioning

confidence: 99%

Applying temporal dependence to detect changes in streaming data

2018

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“…First our RHOMP model de nes a speci c form of the Additive Markov Process (AMP) [21], where the transition probability is a summation of a series of memory functions that are restricted on the next state and one history state each. Applications of the AMP include LAMP [20] (see Section 1), the gravity models [39], and some dynamical systems in physics [23,35] where the memory function is empirically estimated for the application of binary state. In addition to the AMP, recent innovations include new recovery results on mixture of Markov chains [18] (a special case of HMM), which assumes a small set of Markov chains that model various classes of latent indent; and the spacey random walk [3,4,36] as a non-Markovian stochastic process that utilizes higher-order information based on the empirical occupation of states.…”

Section: Related Workmentioning

confidence: 99%

“…[20] proposed the Linear Additive Markov Process (LAMP) that is closely related to our framework. Speci cally our RHOMP model has the same formulation as the generazlied extention GLAMP from the paper [20]. We learned about this paper as we were nalizing our submission to arXiv.…”

Section: Introductionmentioning

confidence: 99%

Retrospective Higher-Order Markov Processes for User Trails

Gleich

2017

Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining

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Users form information trails as they browse the web, checkin with a geolocation, rate items, or consume media. A common problem is to predict what a user might do next for the purposes of guidance, recommendation, or prefetching. First-order and higher-order Markov chains have been widely used methods to study such sequences of data. First-order Markov chains are easy to estimate, but lack accuracy when history ma ers. Higher-order Markov chains, in contrast, have too many parameters and su er from over ing the training data. Fi ing these parameters with regularization and smoothing only o ers mild improvements. In this paper we propose the retrospective higher-order Markov process (RHOMP) as a low-parameter model for such sequences. is model is a special case of a higher-order Markov chain where the transitions depend retrospectively on a single history state instead of an arbitrary combination of history states.ere are two immediate computational advantages: the number of parameters is linear in the order of the Markov chain and the model can be t to large state spaces. Furthermore, by providing a speci c structure to the higher-order chain, RHOMPs improve the model accuracy by efciently utilizing history states without risks of over ing the data. We demonstrate how to estimate a RHOMP from data and we demonstrate the e ectiveness of our method on various real application datasets spanning geolocation data, review sequences, and business locations. e RHOMP model uniformly outperforms higher-order Markov chains, Kneser-Ney regularization, and tensor factorizations in terms of prediction accuracy.

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“…The LAMP model, proposed by Kumar et al [7], overcomes these challenges. The LAMP can fit a measured autocorrelation structure even for sequences that exhibit long-range dependence.…”

Section: Introductionmentioning

confidence: 99%

The entropy rate of Linear Additive Markov Processes

Smart,

Roughan,

Mitchell

2024

PLoS ONE

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This work derives a theoretical value for the entropy of a Linear Additive Markov Process (LAMP), an expressive but simple model able to generate sequences with a given autocorrelation structure. Our research establishes that the theoretical entropy rate of a LAMP model is equivalent to the theoretical entropy rate of the underlying first-order Markov Chain. The LAMP model captures complex relationships and long-range dependencies in data with similar expressibility to a higher-order Markov process. While a higher-order Markov process has a polynomial parameter space, a LAMP model is characterised only by a probability distribution and the transition matrix of an underlying first-order Markov Chain. This surprising result can be explained by the information balance between the additional structure imposed by the next state distribution of the LAMP model, and the additional randomness of each new transition. Understanding the entropy of the LAMP model provides a tool to model complex dependencies in data while retaining useful theoretical results. To emphasise the practical applications, we use the LAMP model to estimate the entropy rate of the LastFM, BrightKite, Wikispeedia and Reuters-21578 datasets. We compare estimates calculated using frequency probability estimates, a first-order Markov model and the LAMP model, also considering two approaches to ensure the transition matrix is irreducible. In most cases the LAMP entropy rates are lower than those of the alternatives, suggesting that LAMP model is better at accommodating structural dependencies in the processes, achieving a more accurate estimate of the true entropy.

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Linear Additive Markov Processes

Cited by 8 publications

References 37 publications

Applying temporal dependence to detect changes in streaming data

Applying temporal dependence to detect changes in streaming data

Retrospective Higher-Order Markov Processes for User Trails

The entropy rate of Linear Additive Markov Processes

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