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Universal prediction of random binary sequences in a noisy environment
 Annals of Applied Probability
, 2004
"... Let X ={(Xt,Yt)}t∈Z be a stationary time series where Xt is binary valued and Yt,thenoisy observation of Xt, is real valued. Letting P denote the probability measure governing the joint process {(Xt,Yt)}, we characterize U(l,P), the optimal asymptotic average performance of a predictor allowed to ba ..."
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Let X ={(Xt,Yt)}t∈Z be a stationary time series where Xt is binary valued and Yt,thenoisy observation of Xt, is real valued. Letting P denote the probability measure governing the joint process {(Xt,Yt)}, we characterize U(l,P), the optimal asymptotic average performance of a predictor allowed to base its prediction for Xt on Y 1,...,Y t−1, where performance is evaluated using the loss function l. It is shown that the stationarity and ergodicity of P, combined with an additional “conditional mixing ” condition, suffice to establish U(l,P) as the fundamental limit for the almost sure asymptotic performance. U(l,P) can thus be thought of as a generalized notion of the Shannon entropy, which can capture the sensitivity of the underlying clean sequence to noise. For the case where X ={Xt} is governed by P and Yt given by Yt = g(Xt,Nt) where g is any deterministic function and N ={Nt}, the noise, is any i.i.d. process independent of X (namely, the case where the “clean ” process X is passed through a fixed memoryless channel), it is shown that, analogously to the noiseless case, there exist universal predictors which do not depend on P yet attain U(l,P). Furthermore, it is shown that in some special cases of interest [e.g., the binary symmetric channel (BSC) and the absolute loss function], there exist twofold universal predictors which do not depend on the noise distribution either. The existence of such universal predictors is established by means of an explicit construction which builds on recent advances in the theory of prediction of individual sequences in the presence of noise. 1. Introduction. Let {(Xt,Yt)}t∈Z
Scanning and sequential decision making for multidimensional data  part I: the noiseless case
 IEEE Trans. on Inform. Theory
"... We consider the problem of sequential decision making on random fields corrupted by noise. In this scenario, the decision maker observes a noisy version of the data, yet judged with respect to the clean data. In particular, we first consider the problem of sequentially scanning and filtering noisy r ..."
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We consider the problem of sequential decision making on random fields corrupted by noise. In this scenario, the decision maker observes a noisy version of the data, yet judged with respect to the clean data. In particular, we first consider the problem of sequentially scanning and filtering noisy random fields. In this case, the sequential filter is given the freedom to choose the path over which it traverses the random field (e.g., noisy image or video sequence), thus it is natural to ask what is the best achievable performance and how sensitive this performance is to the choice of the scan. We formally define the problem of scanning and filtering, derive a bound on the best achievable performance and quantify the excess loss occurring when nonoptimal scanners are used, compared to optimal scanning and filtering. We then discuss the problem of sequential scanning and prediction of noisy random fields. This setting is a natural model for applications such as restoration and coding of noisy images. We formally define the problem of scanning and prediction of a noisy multidimensional array and relate the optimal performance to the clean scandictability defined by Merhav and Weissman. Moreover, bounds on the excess loss due to suboptimal scans are derived, and a universal prediction algorithm is suggested.
Pattern Recognition for Conditionally Independent Data
 JOURNAL OF MACHINE LEARNING RESEARCH
, 2006
"... In this work we consider the task of relaxing the i.i.d. assumption in pattern recognition (or classification) , aiming to make existing learning algorithms applicable to a wider range of tasks. Pattern recognition is guessing a discrete label of some object based on a set of given examples (pairs ..."
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Cited by 8 (2 self)
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In this work we consider the task of relaxing the i.i.d. assumption in pattern recognition (or classification) , aiming to make existing learning algorithms applicable to a wider range of tasks. Pattern recognition is guessing a discrete label of some object based on a set of given examples (pairs of objects and labels). We consider the case of deterministically defined labels. Traditionally, this task is studied under the assumption that examples are independent and identically distributed. However, it turns out that many results of pattern recognition theory carry over a weaker assumption. Namely, under
Efficient Similarity Search over Future Stream Time Series
, 2008
"... With the advance of hardware and communication technologies, stream time series is gaining everincreasing attention due to its importance in many applications such as financial data processing, network monitoring, Web clickstream analysis, sensor data mining, and anomaly detection. For all of thes ..."
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Cited by 7 (0 self)
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With the advance of hardware and communication technologies, stream time series is gaining everincreasing attention due to its importance in many applications such as financial data processing, network monitoring, Web clickstream analysis, sensor data mining, and anomaly detection. For all of these applications, an efficient and effective similarity search over stream data is essential. Because of the unique characteristics of the stream, for example, data are frequently updated and realtime response is required, the previous approaches proposed for searching through archived data may not work in the stream scenarios. Especially, in the cases where data often arrive periodically for various reasons (for example, the communication congestion or batch processing), queries on such incomplete time series or even future time series may result in inaccuracy using traditional approaches. Therefore, in this paper, we propose three approaches, polynomial, Discrete Fourier Transform (DFT), and probabilistic, to predict the unknown values that have not arrived at the system and answer similarity queries based on the predicted data. We also apply efficient indexes, that is, a multidimensional hash index and a B þtree, to facilitate the prediction and similarity search on future time series, respectively. Extensive experiments demonstrate the efficiency and effectiveness of our methods for prediction and answering queries.
Efficient Evaluation of Composite Correlations for Streaming Time Series
 IN: PROCEEDINGS OF THE 4TH INTERNATIONAL CONFERENCE ON WEBAGE INFORMATION MANAGEMENT (WAIM 2003
, 2003
"... In applications ranging from stock trading to space mission operations, it is important to monitor the correlations among multiple streaming time series efficiently in order to make timely decisions. The challenge is that both the number of streaming time series and the number of interested corr ..."
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In applications ranging from stock trading to space mission operations, it is important to monitor the correlations among multiple streaming time series efficiently in order to make timely decisions. The challenge is that both the number of streaming time series and the number of interested correlations can be large. The straightforward way of performing the evaluation by computing the correlation value for each relevant stream pair at each time position is not efficient enough in many situations. In
Universal Scanning and Sequential Decision Making for Multidimensional Data
"... Abstract — We investigate several problems in scanning of multidimensional data arrays, such as universal scanning and prediction (“scandiction”, for short), and scandiction of noisy data arrays. These problems arise in several aspects of image and video processing, such as predictive coding, filter ..."
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Abstract — We investigate several problems in scanning of multidimensional data arrays, such as universal scanning and prediction (“scandiction”, for short), and scandiction of noisy data arrays. These problems arise in several aspects of image and video processing, such as predictive coding, filtering and denoising. In predictive coding of images, for example, an image is compressed by coding the prediction error sequence resulting from scandicting it. Thus, it is natural to ask what is the optimal method to scan and predict a given image, what is the resulting minimum prediction loss, and if there exist specific scandiction schemes which are universal in some sense. More specifically, we investigate the following problems: First, given a random field, we examine whether there exists a scandiction scheme which is independent of the field’s distribution, yet asymptotically achieves the same performance as if this distribution was known. This question is answered in the affirmative for the set of all spatially stationary random fields and under mild conditions on the loss function. We then discuss the scenario where a nonoptimal scanning order is used, yet accompanied by an optimal predictor, and derive a bound on the excess loss compared to optimal scandiction. Finally, we examine the scenario where the random field is corrupted by noise, but the scanning and prediction (or filtering) scheme is judged with respect to the underlying noiseless field. I.
Nonparametric sequential prediction for stationary processes
 Ann. Probab
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ON UNIVERSAL ESTIMATES FOR BINARY RENEWAL PROCESSES
, 811
"... A binary renewal process is a stochastic process {Xn} taking values in {0,1} where the lengths of the runs of 1’s between successive zeros are independent. After observing X0,X1,...,Xn one would like to predict the future behavior, and the problem of universal estimators is to do so without any prio ..."
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A binary renewal process is a stochastic process {Xn} taking values in {0,1} where the lengths of the runs of 1’s between successive zeros are independent. After observing X0,X1,...,Xn one would like to predict the future behavior, and the problem of universal estimators is to do so without any prior knowledge of the distribution. We prove a variety of results of this type, including universal estimates for the expected time to renewal as well as estimates for the conditional distribution of the time to renewal. Some of our results require a moment condition on the time to renewal and we show by an explicit construction how some moment condition is necessary. 1. Introduction. The
Continually Evaluating SimilarityBased Pattern Queries on a Streaming Time Series*
"... In many applications, local or remote sensors send in streams of data, and the system needs to monitor the streams to discover relevant events/patterns and deliver instant reaction correspondingly. An important scenario is that the incoming stream is a continually appended time series, and the pat ..."
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In many applications, local or remote sensors send in streams of data, and the system needs to monitor the streams to discover relevant events/patterns and deliver instant reaction correspondingly. An important scenario is that the incoming stream is a continually appended time series, and the patterns are time series in a database. At each time when a new value arrives (called a time position), the system needs to find, from the database, the nearest or near neighbors of the incoming time series up to the time position. This paper attacks the problem by using Fast Fourier Transform (FFT) to efficiently find the cross correlations of time series, which yields, in a batch mode, the nearest and near neighbors of the incoming time series at many time positions. To take advantage of this batch processing in achieving fast response time, this paper uses prediction methods to predict future values. FFT is used to compute the cross correlations of the predicted series (with the values that have already arrived) and the database patterns, and to obtain predicted distances between the incoming time series at many future time positions and the database patterns. When the actual data value arrives, the prediction error together with the predicted distances is used to filter out patterns that are not possible to be the nearest or near neighbors, which provides fast responses. Experiments show that with reasonable prediction errors, the performance gain is significant. 1.
Sensor Networks: from Dependence Analysis Via Matroid Bases to Online Synthesis
, 2012
"... Consider the two related problems of sensor selection and sensor fusion. In the first, given a set of sensors, one wishes to identify a subset of the sensors, which while small in size, captures the essence of the data gathered by the sensors. In the second, one wishes to construct a fused sensor, w ..."
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Consider the two related problems of sensor selection and sensor fusion. In the first, given a set of sensors, one wishes to identify a subset of the sensors, which while small in size, captures the essence of the data gathered by the sensors. In the second, one wishes to construct a fused sensor, which utilizes the data from the sensors (possibly after discarding dependent ones) in order to create a single sensor which is more reliable than each of the individual ones. In this work, we rigorously define the dependence among sensors in terms of joint empirical measures and incremental parsing. We show that these measures adhere to a polymatroid structure, which in turn facilitates the application of efficient algorithms for sensor selection. We suggest both a random and a greedy algorithm for sensor selection. Given an independent set, we then turn to the fusion problem, and suggest a novel variant of the exponential weighting algorithm. In the suggested algorithm, one competes against an augmented set of sensors, which allows it to converge to the best