# Supervised Dynamic Correlated Topic Model for Classifying Categorical Time Series
## 1 Introduction  
Data on multiple time series or sequences are ubiquitous in various domains, and their classification finds applications in numerous areas, such as human motion classification [1], earthquake prediction [2], and heart attack detection [3]. The classification of multiple real-valued time series have been well studied in the statistical literature (see the detailed review in [4]). However, for analyzing categorical time series, most statistical methods are primarily focused on examining a single time series. A few methods include the Markov chain model [5], the link function approach [6], and the likelihood-based method [7]. In the computer science literature, a number of sequence classification methods have been developed that are black-box in nature and may be difficult to interpret. These include the minimum edit distance classifier with sequence alignment [8, 9] and Markov chain-based classifiers [10]. Overall, the classification of multiple categorical time series has not received much attention. Recent work has discussed a novel approach to the classification of categorical time series in the supervised learning framework, using a spectral envelope optimal scalings [11, 12]. We present an alternative approach for classifying multiple categorical time series in the topic modeling framework. Topic modeling algorithms generally examine a set of documents referred to as a corpus in the natural language processing (NLP) domain. Often, the sets of words observed in documents appear to represent a coherent theme or topic. Topic models analyze the words in the documents in order to uncover themes that run through them. Many topicmodeling algorithms have been developed over time, including non-negative matrix factorization [13], latent Dirichlet allocation (LDA) [14], and structural topic models [15]. Topic modeling in the dynamic framework has been extensively studied to model the dynamic evolution of document collections over time [16; 17]. However, the dynamic evolution of words over time within each document has not been addressed. The topic modeling literature typically assumes that words are interchangeable. We relax this assumption to model word time series (documents). We build a family of probabilistic dynamic topic models by extending the correlated topic model (CTM) in a supervised framework and modeling the dynamic evolution of the underlying latent stochastic process that reflects the thematic structure of the time series collection, and we classify the documents. Topic models like latent Dirichlet allocation, correlated topic models, and dynamic topic models are unsupervised, as only documents are used to identify topics by maximizing the likelihood (or the posterior probability) of the collection. In such modeling frameworks, we hope that the topics will be useful for categorization and are useful when no response is available and we want to infer useful information from the observed documents. However, when the main goal is prediction, a supervised topic modeling framework is beneficial, as jointly modeling the documents and the responses can identify latent topics that can predict the response variables for future unlabeled documents. The sDCTM framework is attractive because it provides a supervised dynamic topic modeling framework to (i) estimate the evolution of the latent process that captures the dynamic evolution of words in the document; and (ii) classify the time series. We apply the sDCTM method for promoter sequence identification in _E. coli_ DNA sub-sequences. A promoter is a region of DNA where RNA polymerase begins to transcribe a gene, and it is usually located upstream or at the \(5^{\prime}\) end of the transcription initiation site in DNA. DNA promoters are proven to be the primary cause of numerous human diseases, including diabetes [18] and [PERSON]'s disease [19]. Thus, the identification of DNA sequences containing promoters has gained significant attention from researchers in the field of bioinformatics.