# Unsupervised Detection of Changes in Usage-Phases of a Mobile App
## 2 Related Work  
Many applications require the detection of significant changes in data streams such as video streams or streams of signals. The changes of interest can be detected by methods utilizing concept drift learning [9] or methods based on anomaly detection [16] in general. The goal of concept drift techniques is to mine inherent patterns from data streams by learning the underlying distribution over time [10]. In order to obtain inherent patterns, a typical procedure for drift detection is composed of data retrieval, data modeling, the calculation of test statistics, and hypothesis testing steps [10]. Although data retrieval for data streaming is also an important research topic [17], the key to drift detection is the measurement of dissimilarity. [PERSON] et al. introduced a detection method based on the error rate of a learner in which once the error rate of a model reaches a threshold (the warning level), then the training step for a new learner is started [18]. For fast training, extreme learning machine (ELM) was introduced, whose architecture has a single hidden layer, and only the connections between the hidden layer and output layer are trained [19]. [PERSON] et al. [20] conceived of a dynamic extreme learning machine improving ELM by adjusting the architecture of a model according to the classification performance of the current model.  
The distance or dissimilarity between the distribution of the past and new data can be utilized to detect changes. In [21], the relative entropy was used to compare the dissimilarity between two distributions while the bootstrap method enhanced the statistical significance. In least-squares density-difference (LSDD) estimation, a density-difference model is fit to the target density-difference function through the squared loss [22]. For unknown distributions, the difference between two distributions represented by LSDD can be estimated by the Gaussian kernel function, thus enabling change detection [23]. [PERSON] et al. improved traditional sequential monitoring methods through nonparametric charts capable of detecting arbitrary changes to the process distribution [24]. With an estimated distribution, a sequence of the likelihood of the data stream can be computed.  
The distributions of log-likelihood were compared so as to detect changes by computing the symmetric Kullback-Leibler divergence [25].  
The occurrence of changes in a data stream can be interpreted as anomalies [26]. Recently, graph based approaches have been actively researched [16]. In order to represent graphs as points in a non-Euclidean space and detect changes, adversarial training of autoencoders was employed for graph embeddings on constant-curvature manifolds, and change detection tests were performed considering embedded graphs [27]. Dissimilarity from prototype graphs was also utilized for change detection [28].  
The test for detecting changes can be interpreted as a procedure for deciding to either reject a hypothesis that a new instance is generated from the current distribution or not reject it. The Student \(t\)-test quantifies how significant the difference between two datasets is (chap10 in [29]). For sequential data, the Mann-Kendall test and the CUSUM (cumulative sum) test are suitable [30]. With multiple parameters, the sequence of log-likelihood is a good indicator for change detection. For a log-likelihood stream, \(t\)-statistics, Kolmogorov-Smirnov, or Lepage can be utilized as the test statistic [25].