# Feature Construction Using Persistence Landscapes for Clustering Noisy IoT Time Series
###### Abstract  
With the advancement of IoT technologies, there is a large amount of data available from wireless sensor networks (WSN), particularly for studying climate change. Clustering long and noisy time series has become an important research area for analyzing this data. This paper proposes a feature-based clustering approach using topological data analysis, which is a set of methods for finding topological structure in data. Persistence diagrams and landscapes are popular topological summaries that can be used to cluster time series. This paper presents a framework for selecting an optimal number of persistence landscapes, and using them as features in an unsupervised learning algorithm. This approach reduces computational cost while maintaining accuracy. The clustering approach was demonstrated to be accurate on simulated data, based on only four, three, and three features, respectively, selected in Scenarios 1-3. On real data, consisting of multiple long temperature streams from various US locations, our optimal feature selection method achieved approximately a 13 times speed-up in computing. el 1 (20  
are likely to continue rising, even with mitigating efforts in place ([[https://www.ipcc.ch/report/ar6/wg1/](https://www.ipcc.ch/report/ar6/wg1/)]([https://www.ipcc.ch/report/ar6/wg1/](https://www.ipcc.ch/report/ar6/wg1/)) (accessed on 1 October 2019)). NOAA observation systems collect data twice every day, from nearly 100 locations in the US. The National Weather Service (NWS) launches weather balloons, carrying instrument packages called radiosondes. Radiosonde sensors measure upper air conditions, such as atmospheric pressure, temperature and humidity, and wind speed and direction. The Automated Surface Observing Systems (ASOS) program is a joint effort by the National Weather Service (NWS), the Federal Aviation Administration (FAA), and the Department of Defense (DOD). The ASOS system serves as the primary surface weather observing network in the US, updating observations every minute ([[https://www.weather.gov/about/](https://www.weather.gov/about/)]([https://www.weather.gov/about/](https://www.weather.gov/about/)) (accessed on 1 October 2019)) ([[https://nosc.noaa.gov/OSC/](https://nosc.noaa.gov/OSC/)]([https://nosc.noaa.gov/OSC/](https://nosc.noaa.gov/OSC/)) (accessed on 1 October 2019)). When weather data are available from a large number of locations, clustering/grouping locations based on stochastic properties of the data are of considerable interest [7; 8; 9; 10]. To this end, it is useful to develop effective algorithms that construct useful features that capture the behavior of the time series: clustering then proceeds on the basis of similarity/dissimilarity metrics between the features. There is a considerable literature on feature-based time series clustering. For example [11], categorized feature representations for time series fall into into four broad types: (i) data-adaptive representations, which are useful for time series of arbitrary lengths; (ii) non-data-adaptive approaches, which are used for time series of fixed lengths; (iii) model-based methods, which are used for representing time series in a stochastic modeling framework; and (iv) data-dictated approaches, which are automatically defined, based on raw time series. In this article, we describe feature construction based on persistent homology, a concept in topological data analysis (TDA), which we used to cluster locations with similar weather patterns. Topological data analysis [12] encompasses methods for discovering interesting shape-based patterns, by combining tools from algebraic topology, computer science, and statistics, and it is becoming an increasingly useful area in many time series applications. For a review of persistent homology for time series, and a tutorial using the R software, see [13]. In particular, the review discusses ideas such as transforming time series into point clouds via Takens embedding [14], creating persistence diagrams [15; 16], and constructing persistence landscapes of all orders [17].