# Feature Construction Using Persistence Landscapes for Clustering Noisy IoT Time Series
###### Abstract
While persistent homology is a central tool in TDA, for summarizing geometric and topological information in data using a persistence diagram (or a bar code), it is cumbersome to construct useful statistical quantities using metrics such as the Wasserstein distance. Persistence landscapes enable us to map persistent diagrams to a Hilbert space, thereby making it easier to apply tools from statistics and machine learning. Recent research has explored the use of persistence landscapes as features to either cluster or classify time series [18; 19; 20]. Persistence landscapes of _all orders_ using weighted Fourier transforms of continuous-valued EEG time series were constructed by [21], and used as features for clustering the time series, using randomness testing to examine the robustness of the approach to topology-preserving transformations, while being sensitive to topology-destroying transformations. _First-order_ persistence landscapes, constructed from Walsh-Fourier transforms of categorical time series from a large activity-travel transportation data set, were used by [22] to create features for clustering, arguing that the first-order landscape was sufficient for accurately clustering time series with relatively simple dependence properties. Several aspects of using persistence homology in time series analysis have been discussed in-depth in [23]. It is well-known that lower-order persistence landscapes contain more important topological features than higher-order landscapes, which are closer to zero [17]. In many situations, it may be unnecessary and computationally prohibitive to use persistence landscapes of all orders to elicit useful features of time series. Selecting the order of the persistence landscapes to serve as features requires a delicate balance between missing important signals and introducing too much noise. Existing research has not addressed the problem of data-based selection of the _order_ of persistence landscapes that is sufficient to yield accurate clustering: thus, the focus of this article was to address the problem of deciding the order of persistence landscapes in a time series clustering scenario, and to study this question in the context of noisy, periodic stationary time series, using the smoothed second-order spectrum to construct persistence landscapes. The solution was an algorithm which automatically selected the optimal order of persistence landscapes in a sequential way. These features were then used in clustering the time series. The computational gain from the algorithm was demonstrated through extensive simulation studies, which showed a speed-up of approximately 13 times. We then illustrate our approach, using long temperature streams from from different US locations, and show that features constructed from the selected orders of persistence landscapes produced meaningful clusters of the locations, which may be useful for climate scientists in a comparative study of temperature patterns over time in several locations. The format of the paper is as follows. Section 2 provides the background of persistence homology for time series, and specifically describes the construction of persistent landscapes. Section 3 discusses literature that is closely related to our research. Section 4 then presents the motivation for our research, and highlights our contribution. Section 5 describes the construction and implementation of our proposed algorithm for feature construction, which is used for time series clustering. Section 6 presents a simulation study, to evaluate the performance of our clustering method. Section 7 clusters daily maximum temperatures across various locations in the United States. Section 8 summarizes the paper.