Abstract
Histograms are a common technique for density estimation and they have been widely used as a tool in exploratory data analysis. Learning histograms from static and stationary data is a well known topic. Nevertheless, very few works discuss this problem when we have a continuous flow of data generated from dynamic environments. The scope of this paper is to detect changes from high-speed time-changing data streams. To address this problem, we construct histograms able to process examples once at the rate they arrive. The main goal of this work is continuously maintain a histogram consistent with the current status of the nature. We study strategies to detect changes in the distribution generating examples, and adapt the histogram to the most recent data by forgetting outdated data. We use the Partition Incremental Discretization algorithm that was designed to learn histograms from high-speed data streams. We present a method to detect whenever a change in the distribution generating examples occurs. The base idea consists of monitoring distributions from two different time windows: the reference window, reflecting the distribution observed in the past; and the current window which receives the most recent data. The current window is cumulative and can have a fixed or an adaptive step depending on the distance between distributions. We compared both distributions using Kullback-Leibler divergence, defining a threshold for change detection decision based on the asymmetry of this measure. We evaluated our algorithm with controlled artificial data sets and compare the proposed approach with nonparametric tests. We also present results with real word data sets from industrial and medical domains. Those results suggest that an adaptive window's step exhibit high probability in change detection and faster detection rates, with few false positives alarms.

Abstract
Accurately predicting values for dynamic data streams is a challenging task in decision and expert systems, due to high data flow rates, limited storage and a requirement to quickly adapt a model to new data. We propose an approach for correcting predictions for data streams which is based on a reliability estimate for individual regression predictions. In our work, we implement the proposed technique and test it on a real-world problem: prediction of the electricity load for a selected European geographical region. For predicting the electricity load values we implement two regression models: the neural network and the k nearest neighbors algorithm. The results show that our method performs better than the referential method (i.e. the Kalman filter), significantly improving the original streaming predictions to more accurate values. © 2011 Springer-Verlag Berlin Heidelberg.

Abstract
Most of the work in machine learning assume that examples are generated at random according to some stationary probability distribution. In this work we study the problem of learning when the distribution that generate the examples changes over time. We present a method for detection of changes in the probability distribution of examples. The idea behind the drift detection method is to control the online error-rate of the algorithm. The training examples are presented in sequence. When a new training example is available, it is classified using the actual model. Statistical theory guarantees that while the distribution is stationary, the error will decrease. When the distribution changes, the error will increase. The method controls the trace of the online error of the algorithm. For the actual context we define a warning level, and a drift level. A new context is declared, if in a sequence of examples, the error increases reaching the warning level at example k(w), and the drift level at example k(d). This is an indication of a change in the distribution of the examples. The algorithm learns a new model using only the examples since k(w). The method was tested with a set of eight artificial datasets and a real world dataset. We used three learning algorithms: a perceptron, a neural network and a decision tree. The experimental results show a good performance detecting drift and with learning the new concept. We also observe that the method is independent of the learning algorithm.

Abstract
This paper presents a system for induction of forest of functional trees from data streams able to detect concept drift. The Ultra Fast Forest of Trees (UFFT) is an incremental algorithm, that works online, processing each example in constant time, and performing a single scan over the training examples. It uses analytical techniques to choose the splitting criteria, and the information gain to estimate the merit of each possible splitting-test. For multi-class problems the algorithm grows a binary tree for each possible pair of classes, leading to a forest of trees. Decision nodes and leaves contain naive-Bayes classifiers playing different roles during the induction process. Naive-Bayes in leaves are used to classify test examples, naive-Bayes in inner nodes can be used as multivariate splitting-tests if chosen by the splitting criteria, and used to detect drift in the distribution of the examples that traverse the node. When a drift is detected, all the sub-tree rooted at that node will be pruned. The use of naive-Bayes classifiers at leaves to classify test examples, the use of splitting-tests based on the outcome of naive-Bayes, and the use of naive-Bayes classifiers at decision nodes to detect drift are directly obtained from the sufficient statistics required to compute the splitting criteria, without no additional computations. This aspect is a main advantage in the context of high-speed data streams. This methodology was tested with artificial and real-world data sets. The experimental results show a very good performance in comparison to a batch decision tree learner, and high capacity to detect and react to drift. Copyright 2005 ACM.

Abstract
Learning from data streams is a research area of increasing importance. Nowadays, several stream learning algorithms have been developed. Most of them learn decision models that continuously evolve over time, run in resource-aware environments, detect and react to changes in the environment generating data. One important issue, not yet conveniently addressed, is the design of experimental work to evaluate and compare decision models that evolve over time. There are no golden standards for assessing performance in non-stationary environments. This paper proposes a general framework for assessing predictive stream learning algorithms. We defend the use of Predictive Sequential methods for error estimate - the prequential error. The prequential error allows us to monitor the evolution of the performance of models that evolve over time. Nevertheless, it is known to be a pessimistic estimator in comparison to holdout estimates. To obtain more reliable estimators we need some forgetting mechanism. Two viable alternatives are: sliding windows and fading factors. We observe that the prequential error converges to an holdout estimator when estimated over a sliding window or using fading factors. We present illustrative examples of the use of prequential error estimators, using fading factors, for the tasks of: i) assessing performance of a learning algorithm; ii) comparing learning algorithms; iii) hypothesis testing using McNemar test; and iv) change detection using Page-Hinkley test. In these tasks, the prequential error estimated using fading factors provide reliable estimators. In comparison to sliding windows, fading factors are faster and memory-less, a requirement for streaming applications. This paper is a contribution to a discussion in the good-practices on performance assessment when learning dynamic models that evolve over time.

Abstract
Several predictive systems are nowadays vital for operations and decision support. The quality of these systems is most of the time defined by their average accuracy which has low or no information at all about the estimated error of each individual prediction. In many sensitive applications, users should be allowed to associate a measure of reliability to each prediction. In the case of batch systems, reliability measures have already been defined, mostly empirical measures as the estimation using the local sensitivity analysis. However, with the advent of data streams, these reliability estimates should also be computed online, based only on available data and current model's state. In this paper we define empirical measures to perform online estimation of reliability of individual predictions when made in the context of online learning systems. We present preliminary results and evaluate the estimators in two different problems. © 2008 IEEE.