# Feature Construction Using Persistence Landscapes for Clustering Noisy IoT Time Series
## 6 Simulation Study  
We simulated time series of length \(T=1024\) generated from \(J\) different populations. Each time series \(X_{j,t}^{(n)}\) within the \(j\)-th population contained a (population specific) signal \(Y_{j,t}\), which was a mixture of \(M_{p}\) periodic components together with additive white noise \(\epsilon_{j,t}^{(n)}\). The scenarios varied according to their signal-to-noise ratio, defined by \(\text{SNR}=\text{Var}(Y_{j,t})/\text{Var}(\epsilon_{j,t}^{(n)})\). The set of \(M_{p}\) signal frequencies and amplitudes differed between the \(J\) populations. We studied single-frequency signals and signals that were a mixture of three different frequencies. Furthermore, within each population, we slightly perturbed the signal frequencies around the selected fixed values, in order to simulate realistic situations: as a result, each of the \(n\) time series within the same population were allowed to have nearly similar, though not identical, signals. 1. Let the number of populations be \(J=2\). We selected three different values of SNR, i.e., 0.5, 1.5, and 3.0. Corresponding to each SNR value, we generated periodic time series, with frequencies and amplitudes described below: 1. Scenario 1 with \(M_{p}=3\). We selected three periods in the first population, to be \((180,90,14)\) (i.e., half-year cycle, three-month cycle, and two-week cycle), and in the second population, as \((120,30,7)\) (i.e., four-month cycle, one-month cycle, and one-week cycle). We selected the amplitudes at these frequencies, to be \(A_{j,u}=(2,1,0.5)\) in the first population, and \((1.5,1.5,0.5)\) in the second population, for \(u=1,\dots,M_{p}\); 2. Scenario 2 with \(M_{p}=3\). We selected three periods in the first population, to be \((30,14,7)\) (i.e., one-month cycle, two-week cycle, and one-week cycle), and in the second population, as \((120,30,14)\) (i.e., four-month cycle, one-month cycle, and two-week cycle). We select the amplitudes at these frequencies, to be \(A_{j,u}=(2,1,0.5)\) in the first population, and \((0.1,0.15,2.5)\) in the second population, for \(u=1,\dots,M_{p}\); 3. Scenario 3 with \(M_{p}=1\). We selected the period, as 180 in the first population, and 120 in the second population. We selected the amplitude in the first population, as \(A_{1,1}=2\), and the amplitude in the second population, as \(A_{1,1}=1.5\). With these parameters, we generated \(Y_{j,t}=\sum_{u=1}^{M_{p}}A_{j,u}\cos(2\pi t/f_{j,u})\), to represent the simulated signal in the \(j\)-th population. 2. We simulated \(N_{j}=20\) time series in each of \(J=2\) populations. The period was not fixed, but randomly varied about the fixed periods selected for each population: 1. We simulated \(M_{p}\) random periods that uniformly varied around the fixed periods selected for each population, i.e., \(\delta f_{j,u}\sim\texttt{Uniform}(a_{j,u},b_{j,u})\), where we set \(a_{j,u}=0.8f_{j,u}\) and \(b_{j,u}=1.2f_{j,u}\); 2. We simulated a time series in the \(j\)-th population as \[X_{j,t}^{(n)}=\sum_{u=1}^{M_{p}}A_{j,u}\cos(2\pi t/\delta f_{j,u})+\epsilon_{ j,u,t},\text{ where }\epsilon_{jt}^{(n)}\sim N\bigg{(}0,\frac{\text{Var}(Y_{j,t})}{\text{SNR}}\bigg{)},\] which generated \(N_{j}=20\) time series in each of the \(J=2\) populations. 3. We constructed features based on persistence landscapes in the \(N=N_{1}+N_{2}\) series, as follows: 1. based on \(\
u=1\) only; 2. Scenario 3 with \(M_{p}=3\). We selected three periods in the first population, to be \((30,14,7)\) (i.e., one-month cycle, two-week cycle, and one-week cycle), and in the second population, as \((120,30,14)\) (i.e., four-month cycle, one-month cycle, and two-week cycle). We selected the amplitudes at these frequencies, to be \(A_{j,u}=(2,1,0.5)\) in the first population, and \((0.1,0.15,2.5)\) in the second population, for \(u=1,\dots,M_{p}\); 3. Scenario 3 with \(M_{p}=1\). We selected the period, as 180 in the first population, and 120 in the second population, as \(A_{1,1}=2\), and the amplitude in the second population, as \(A_{1,1}=1.5\).