The most popular tools for stream data mining are based on decision trees. In previous 15 years, all designed methods, headed by the very fast decision tree algorithm, relayed on Hoeffding's inequality and hundreds of researchers followed this scheme. Recently, we have demonstrated that although the Hoeffding decision trees are an effective tool for dealing with stream data, they are a purely heuristic procedure; for example, classical decision trees such as ID3 or CART cannot be adopted to data stream mining using Hoeffding's inequality. Therefore, there is an urgent need to develop new algorithms, which are both mathematically justified and characterized by good performance. In this paper, we address this problem by developing a family of new splitting criteria for classification in stationary data streams and investigating their probabilistic properties. The new criteria, derived using appropriate statistical tools, are based on the misclassification error and the Gini index impurity measures. The general division of splitting criteria into two types is proposed. Attributes chosen based on type-I splitting criteria guarantee, with high probability, the highest expected value of split measure. Type-II criteria ensure that the chosen attribute is the same, with high probability, as it would be chosen based on the whole infinite data stream. Moreover, in this paper, two hybrid splitting criteria are proposed, which are the combinations of single criteria based on the misclassification error and Gini index.

In this paper, we propose a recursive variant of the Parzen kernel density estimator (KDE) to track changes of dynamic density over data streams in a nonstationary environment. In stationary environments, well-established traditional KDE techniques have nice asymptotic properties. Their existing extensions to deal with stream data are mostly based on various heuristic concepts (losing convergence properties). In this paper, we study recursive KDEs, called recursive concept drift tracking KDEs, and prove their weak (in probability) and strong (with probability one) convergence, resulting in perfect tracking properties as the sample size approaches infinity. In three theorems and subsequent examples, we show how to choose the bandwidth and learning rate of a recursive KDE in order to ensure weak and strong convergence. The simulation results illustrate the effectiveness of our algorithm both for density estimation and classification over time-varying stream data.

In this brief, we consider the problem of descriptors construction for the task of content-based image retrieval using deep neural networks. The idea of neural codes, based on fully connected layers’ activations, is extended by incorporating the information contained in convolutional layers. It is known that the total number of neurons in the convolutional part of the network is large and the majority of them have little influence on the final classification decision. Therefore, in this brief, we propose a novel algorithm that allows us to extract the most significant neuron activations and utilize this information to construct effective descriptors. The descriptors consisting of values taken from both the fully connected and convolutional layers perfectly represent the whole image content. The images retrieved using these descriptors match semantically very well to the query image, and also, they are similar in other secondary image characteristics, such as background, textures, or color distribution. These features of the proposed descriptors are verified experimentally based on the IMAGENET1M dataset using the VGG16 neural network. For comparison, we also test the proposed approach on the ResNet50 network.