# Feature Construction Using Persistence Landscapes for Clustering Noisy IoT Time Series
## 3 Related Work  
Clustering locations or climate stations based on temperature or precipitation time series have been discussed in several recent articles. A two-step cluster analysis of 449 southeastern climate stations was described in [7], to determine general climate clusters for eight southeastern states in the US, and has been employed in several follow-up analyses involving the classification of synoptic climate types. In a similar vein, ref. [8] used a hierarchical cluster analysis to demarcate climate zones in the US, based on weather variables, such as temperature and precipitation. Spatial grouping of over 1000 climate stations in the US was discussed in [10], by using a hybrid clustering approach, based on a measure of rank correlation as a metric of statistical similarity. Based on the clustering temperatures at these stations, they showed that roughly 25% of the sites accounted for nearly 80% of the spatial variability in seasonal temperatures across the country. Recent research has used persistence landscapes constructed from time series as features to cluster the time series. In this section, we review related works that focus on using persistent homology for clustering time series. We then highlight the novelty and usefulness of our approach. A framework for implementing the break detection of critical transitions on daily price cryptocurrencies, using topological data analysis (TDA), was proposed by [19]. They (i) transformed the time series into point clouds, using [PERSON]'s delay embedding, (ii) computed persistence landscapes for each point cloud window, (iii) converted to their L\({}_{1}\) norms, and (iv) used K-means clustering for these windowed time series. Persistence landscapes of all orders were employed by [20] as topological features for time series, via [PERSON]'s time-delayed embedding transformation, and principle component analysis for denoising the time series. The problem of clustering continuous-valued EEG time series was studied by [21]. They constructed weighted Fourier transforms of the time series, and constructed persistence landscapes of all orders: they used these as features for clustering the time series. They also examined the robustness of their approach to topology-preserving transformations, while being sensitive to topology-destroying transformations. The use of persistence landscapes for clustering categorical time series from a large activity-travel transportation data set was described in [22]. They first constructed Walsh-Fourier transforms of the categorical time series, and then obtained first-order persistence landscapes from the Walsh-Fourier transforms, which they used as features for clustering the time series. In this case, they argued that the first-order landscape was sufficient for accurately clustering time series with relatively simple dependence properties, as in the activity-travel transportation data. An analysis of multivariate time series using topological data analysis was proposed in [28], by converting the time series into point cloud data, calculating Wasserstein distances between the persistence diagrams, and using the \(k\)-nearest neighbors algorithm for supervised machine learning, with an application to predicting room occupancy during a time window. Time series clustering with topological-geometric mixed distance (TGMD) was discussed in [29], which jointly considered the local geometric features and global topological characteristics of time series data. The results revealed that their proposed mixed-distance-based similarity measure could lead to promising results, and to better performance than standard time series analysis techniques that consider only topological or geometrical similarity. In our approach, we constructed persistent landscapes from the second-order spectra of stationary time series, with the goal of clustering a large number of time series. To balance computing time and clustering accuracy, we developed an algorithm to select an optimal number of landscapes to be employed as features in the clustering.