Topological time-series analysis with delay-variant embedding

Identifying the qualitative changes in time-series data provides insights into the dynamics associated with such data. Such qualitative changes can be detected through topological approaches, which first embed the data into a high-dimensional space using a time-delay parameter and subsequently extract topological features describing the shape of the data from the embedded points. However, the essential topological features that are extracted using a single time delay are considered to be insufficient for evaluating the aforementioned qualitative changes, even when a well-selected time delay is used. We therefore propose a delay-variant embedding method that constructs the extended topological features by considering the time delay as a variable parameter instead of considering it as a single fixed value. This delay-variant embedding method reveals multiple-timescale patterns in a time series by allowing the observation of the variations in topological features, with the time delay serving as an additional dimension in the topological feature space. We theoretically prove that the constructed topological features are robust when the time series is perturbed by noise. Furthermore, we combine these features with the kernel technique in machine learning algorithms to classify the general time-series data. We demonstrate the effectiveness of our method for classifying the synthetic noisy biological and real time-series data. Our method outperforms a method that is based on a single time delay and, surprisingly, achieves the highest classification accuracy on an average among the standard time-series analysis techniques.

Figure 1

Time series $x\left(t\right)$ is embedded in an $m$-dimensional space (in this illustration, $m=3$) through delay embedding (left panel). The embedded points at different values of time delay $\tau$ have different geometric features, such as clusters and loops (middle panel). We extract the topological features such as the emergence (birth scale) and disappearance (death scale) of these geometric features. The birth and death scales at each $\tau$ are represented as points in a two-dimensional persistence diagram. By observing the manner in which the topological features vary with $\tau$ serving as an additional dimension, we obtain the three-dimensional persistence diagram (right panel), which can be considered to be a typical feature of the time series.

Figure 2

(a) Bottleneck distance between two diagrams $D_1^{\left(3\right)}$ (green circles) and $D_2^{\left(3\right)}$ (red squares). The $\tau$ axis is rescaled with the positive coefficient $\xi$ to adjust the scale difference between the pointwise distance and time. A green (red) point is matched to either a red (green) point or its projection on the diagonal plane, ${\mathcal{W}}^{\left(3\right)}$. The distance between two matched points is the longest edge of their differences along any coordinate dimension. The bottleneck distance is defined as the longest distance between two matched points that is minimal compared to all other matchings. (b) Kernel between $D_1^{\left(3\right)}$ (green circles) and $D_2^{\left(3\right)}$ (red squares). The faint red points are symmetric to the dark red points with respect to ${\mathcal{W}}^{\left(3\right)}$. Points near ${\mathcal{W}}^{\left(3\right)}$, i.e., ${\mathit{p}}_1,{\mathit{q}}_2$, exhibit less influence, whereas the pairs of points far away from ${\mathcal{W}}^{\left(3\right)}$, i.e., $({\mathit{p}}_2,{\mathit{q}}_1),({\mathit{p}}_2,{\mathit{q}}_3),({\mathit{p}}_3,{\mathit{q}}_1),({\mathit{p}}_3,{\mathit{q}}_3)$, make essential contributions to the kernel.

Figure 3

(a, d) Examples of two time series generated from the (a) nonoscillatory and (d) oscillatory regimes of the Hes1 model measured after every 16 min. (b, e) Two-dimensional persistence diagrams of the loop patterns calculated with embedding dimension $m=3$ and a single value of $\tau$ selected according to Ref. [11] for the (b) nonoscillatory ($\tau =4$) and (e) oscillatory ($\tau =2$) time series. (c, f) Three-dimensional persistence diagrams of loop patterns calculated with $m=3$ and 10 values of $\tau$ given by ${\tau }_1=1,{\tau }_2=2,...,{\tau }_{10}=10$ for the (c) nonoscillatory and (f) oscillatory time series.

Figure 4

Average classification accuracies of the delay-variant (circles) and single-delay (squares) methods over 100 random splits of the Hes1-model data at different (a, c, e, g) embedding dimensions $m$ and different (b, d, f, h) effective embedding dimensions $M$. The time-series data were obtained at different measurement intervals (a, b) $\nu =8$, (c, d) $\nu =16$, (e, f) $\nu =32$, and (g, h) $\nu =64$. The insets in (a) ($\nu =8$) and (c) ($\nu =16$) highlight the accuracy over the embedding dimension $\mathrm{m}=6–10$.

Figure 5

Average classification accuracies of the delay-variant (circles) and single-delay (squares) methods over 100 random splits of the Hes1-model data at different embedding dimensions $m$. The time-series data were obtained by adding the Gaussian white noise with variance (a) ${\sigma }_n^2=0.1$, (b) ${\sigma }_n^2=0.3$, (c) ${\sigma }_n^2=0.5$, and (d) ${\sigma }_n^2=0.7$ to the time series generated with $\nu =16$ min. The inset in panel (a) (${\sigma }_n^2=0.1$) depicts the accuracy over the embedding dimension $m=6–10$.

Figure 6

(a) Time series generated from the frequency-modulated model. (b) Three-dimensional persistence diagram of the loop patterns calculated with $m=6$ and 20 values of $\tau$ given by ${\tau }_1=1,{\tau }_2=2,...,{\tau }_{20}=20$. The multiple-timescale components in the time series corresponding to the limit-cycle oscillations of the dynamics are revealed from the points in the diagram over different values of $\tau$.

Figure 7

Projection of the features of synthetic frequency-modulated time series from kernel space to principal components for (a) the delay-variant method with embedding dimension $m=6$, (b) the single-delay method with $m=6$ and $\tau =1,2,...,20$, and (c) the single-delay method with $m=120$ (for performing comparison with the delay-variant method at the same effective embedding dimension $M=120$) and $\tau =1,2,3,4$. Different colors represent the data generated using different values of the carrier frequency $f_c$.

Figure 8

We consider $L$-dimensional hyperspheres with radius $\varepsilon$ centered at each point. From (a) to (f), we increase the radius $\varepsilon$ gradually. By increasing $\varepsilon$, holes appear and disappear in the region. The first one-dimensional hole (blue loop) appears at (c), while the second (red loop) appears at (d). Then at (e), the first hole disappears and finally the second one disappears at (f). (g) An exemplary two-dimensional persistence diagram calculated by considering the appearance and disappearance of loops from a filtration of the Vietoris-Rips complex. The radius $\varepsilon$ takes the discrete values from the set {0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6}. There are two persistence pairs (0.2, 0.4) and (0.3, 0.6), which represent the appearance and disappearance of the blue loop and the red loop, respectively. These persistence pairs are displayed as a blue circle and a red triangle in the Cartesian plane, respectively. The collection of all the persistence pairs in the filtration is a two-dimensional persistence diagram.