Topology combined machine learning for consonant recognition

Time series analysis [58] is a prevalent tool for various applied sciences. The recent surge in TDA has opened new avenues for the integration of topological methods into time series analysis [21, 59, 60]. Much literature has contributed to the theoretical foundation in this area. For example, theoretical frameworks for processing periodic time series have been proposed by Harer and Perea [61], followed by their and their collaborators' implementation in discovering periodicity in gene expressions [62]. Their article [61] studied the geometric structure of truncated Fourier series of a periodic function and its dependence on parameters in time-delay embedding (TDE), providing a solid background for TopCap. In addition to periodic time series, towards more general and complex scenarios, quasi-periodic time series have also been the subject of scholarly attention. Research in this direction has primarily concentrated on the selection of parameters for geometric space reconstruction [63] and extended to vector-valued time series [64].

In this article, a topological space is constructed from data using TDE, a technique that has been widely employed in the reconstruction of time series (see Fig. 1a and cf. Section 2 of Supplementary Information for more background). Thanks to the topological invariance of TDE, the general construction of simplicial-complex representation (see Fig. 1b) and computation of PH from point clouds (see Fig. 1d) apply to time series data, although this transformation involves subtle technical issues in practice. For instance, Emrani et al. [47] utilised TDE and PH to identify the periodic structure of dynamical systems, with applications to wheeze detection in pulmonology. They selected the embedded dimension $d$ as 2, and their delay parameter $\tau$ was determined by an autocorrelation-like (ACL) function, which provided a range for the delay between the first and second critical points of the ACL function. Pereira and de Mello [48] proposed a data clustering approach based on PD. The data were initially reconstructed by TDE, with $d=2$ and $\tau=3$, so as to obtain the corresponding PD, which was then subjected to $k$-means clustering. The delay $\tau$ was determined using the first minimum of an auto mutual information, and the embedded dimension $d$ was set to be 2 as using 3 dimensions did not significantly improve the results. Khasawneh and Munch [49] introduced a topological approach for examining the stability of a class of nonlinear stochastic delay equations. They used false nearest neighbours to determine the embedded dimension $d=3$ and chose the delay to equal the first zeros of the autocorrelation function. Subsequently, the longest persistence lifetime in PD was used as a vectorisation to quantify periodicity. Umeda [22] focused on a classification problem for volatile time series by extracting the structure of attractors, using TDA to represent transition rules of the time series. He assigned $d=3$, $\tau=1$ in his study and introduced a novel vectorisation method, which was then applied to a convolutional neural network (CNN) to achieve classification. Gidea and Katz [51] employed TDA to detect early signs prior to financial crashes. They studied multi-dimensional time series with $\tau=1$ and used persistence landscape as a vectorisation method. For speech recognition, Brown and Knudson [27] examined the structure of point clouds obtained via TDE of human speech signals. The TDE parameters were set as $d=3$, $\tau=20$, after which they examined the structure of point clouds and their corresponding PB.

Upon reviewing the relevant literature, we see that currently there is no general framework for systematically choosing $d$ and $\tau$, and researchers often have to make choices in an ad hoc fashion for practical needs. While the TDE–PH topological approach to handling time series data is not new, TopCap extracts features from high-dimensional spaces. For example, in our experiment $d=100$. It happens in some cases that in a low-dimensional space, regardless of how optimal the choice of $\tau$ is, the structure of the time series cannot be adequately captured. In contrast, given a high-dimensional space, feature extraction from data becomes simpler. Of course, operating in a high-dimensional space comes with its own cost. For example, the adjustment of $\tau$ then requires careful consideration. Nonetheless, it also offers advantages, which we will elucidate step by step in the subsequent sections.

The impetus behind TopCap lies in an observation of how PD can capture vibration patterns within time series. To begin with, our aim is to determine which sorts of information can be extracted using topological methods. As the name indicates, topological methods quantify features based on topology, which distinguishes spaces that cannot continuously deform to each other. In the context of time series, we conduct a series of experiments to scrutinize the performance of topological methods, their limitations as well as their potential.

Given a periodic time series, its TDE target is situated on a closed curve (i.e., a loop) in a sufficiently high-dimensional Euclidean space (see Fig. 1a). Despite the satisfactory point-cloud representation of a periodic time series, it remains rare in practical measurement and observation to capture a truly periodic series. Often, we find ourselves dealing with time series that are not periodic yet exhibit certain patterns within some time segments. For instance, Fig. 1c portrays the average temperature of the United States from the year 2012 to 2022, as documented in [65]. Although the temperature does not adhere strictly to a periodic pattern, it does display a noticeable cyclical trend on an annual basis. Typically, the temperature tends to rise from January to July and fall from August to December, with each year approximately comprising one cycle of the variation pattern. One strength of topological methods is their ability to capture ``cycles''. A question then arises naturally: Can these methods also capture the cycle of temperature as well as subtle variations within and among these cycles? To be more precise, we first observe that variations occur in several ways. For instance, the amplitude (or range) of the annual temperature variation may fluctuate slightly, with the maximum and minimum annual temperatures varying from year to year. Additionally, the trend line for the annual average temperature also shows fluctuations, such as the average temperature in 2012 surpassing that of 2013. Despite each year’s temperature pattern bearing resemblance to that depicted in the left panel in Fig. 1c (representing a single cycle of temperature within a year), it may be more beneficial for prediction and response strategies to focus on the evolution of this pattern rather than its specific form. In other words, attention should be directed towards how this cycle varies over the years. This leads to several questions. How can we consistently capture these subtle changes in the pattern’s evolution, such as variations in the frequency, amplitude, and trend line of cycles? How can we describe the similarities and differences between time series that possess distinct evolutionary trajectories? In applications, these are crucial inquiries that warrant further exploration.

To address these questions, we propose three kinds of ``fundamental variations'' which are utilised for depicting the evolutionary trace of a time series. Consider a series of a periodic function $f(t_{n})=f(t_{n}+T)$, where $T$ represents a period.

1. (1)

   Variation of frequency. Denote the frequency by $F=T^{-1}$. Note that the series is not necessarily periodic in the mathematical sense. Rather, it exhibits a recurring pattern after the period $T$. For instance, the average temperature from Fig. 1c is not a periodic series, but we consider its period to be one year since it follows a specific pattern, i.e., the one displayed in the left panel of Fig. 1c. This 1-year pattern always lasts for a year as time progresses. Hence, there is no frequency variation in this example. This type of variations can be represented as $g_{1}(t_{n})=f\big{(}F(t_{n})\cdot t_{n}\big{)}$, where $F(t_{n})$ is a series representing the changing frequency. This type of variation occurs, for example, when one switches their vocal tone or when one's heartbeats experience a transition from walking mode to running mode.

2. (2)

   Variation of amplitude. The amplitudes of temperature in the years 2014 and 2015 are 42.73^∘F and 40.93^∘F, respectively. So the variation of amplitude from 2014 to 2015 is $-1.80^{\circ}$F. This can be represented by $g_{2}(t_{n})=A(t_{n})\cdot f(t_{n})$, where $A(t_{n})$ is a series of the changing amplitude. This type of variation is observed when a particle vibrates with resistance or when there is a change in the volume of a sound.

3. (3)

   Variation of average line. The average temperatures through the years 2012 and 2013 are 55.28^∘F and 52.43^∘F, respectively. The variation of average line from 2012 to 2013 is $-2.85^{\circ}$F. Let $g_{3}(t_{n})=f(t_{n})+L(t_{n})$, where $L(t_{n})$ is a series representing the variation of average line. This type of variation is observed when a stock experiences a downturn over several days or when global warming causes a year-by-year increase in temperature.

To summarize, Fig. 1e provides a visual representation of the three fundamental variations. It is important to note that these variations are not utilised to depict the pattern itself but rather to illustrate the variation within the pattern or how the time series oscillates over time. This approach offers a dynamic perspective on the evolution of the time series, capturing changes in patterns that static analyses might overlook.

Using three simulated time series corresponding to the above three fundamental types of variation (see Section 4.1 for detailed construction), we demonstrate that PD can distinguish these variations and detect how significant they are. See Fig. 2, where a smaller value of $c$ indicates a more rapid fundamental variation. Here, regardless of which value $c$ takes, each individual diagram features a prominent single point at the top and a cluster of points with relatively short duration, except when $F(t_{n})=1$ (i.e., $c=4$). In this case, the series represents a cosine function, and thus the diagram consists of a single point. Normally, one tends to overlook the points in a PD that exhibit a short duration as they are sometimes inferred as noise. However, in this example, the distribution of those points holds valuable information regarding the three fundamental variations. As shown in Fig. 2, each fundamental variation has its distinct pattern of distribution in the lower region of a diagram, which leads to refined inferences: if the points spiral along the vertical axis of lifetime, it is probably due to a variation of amplitude; if every two or four points stay close to form a ``shuttle", it probably indicates a variation of average line; otherwise the points just seem to randomly spread over, which more likely results from a variation of frequency. It is also straightforward to distinguish the values of $c$ for a specific fundamental variation, by their most significant point in the diagram: longer lifetime for the barcode of the solitary point indicates slower variation. The lower region of a diagram also gives some hints in this respect.

In this simulated example, we demonstrated how PD could be utilised as a uniform means to distinguish three fundamental variations of the cosine series and their respective rates of change. However, it is important to note that in general scenarios, identifying the fundamental variations in a time series using topological methods may encounter significant challenges. Although topological methods are indeed capable of capturing this information, vectorising this information for subsequent utilisation remains a complex task at this stage. Having recognised the potential of topological methods, we resort to an alternative algorithm for handling time series. Specifically, despite the difficulty in vectorising PD to measure each fundamental variation, we have developed a simplified algorithm to measure the vibration of time series as a whole. This approach provides a comprehensive understanding of the overall behaviour of a time series, bypassing the need for complex vectorisation.

Utilising datasets comprising human speech, we initially employ the Montreal Forced Aligner to align natural speech into phonetic segments. Following preprocessing of these phonetic segments, TDE is conducted with dimension parameter $d=100$ and delay parameter $\tau$ set to equal $6T/d$, where $T$ approximates the (minimal) period of the time series. Following additional refinement procedures, PDs are computed for these segments and are then vectorised based on the maximal persistence and its corresponding birth time. The comprehensive procedural framework is expounded in Sections 4.2 and 4.3, while the corresponding workflow is shown in Fig. 3e. In the applications of TDE, the dimension parameter $d$ is usually determined through some algorithms designed to identify the minimal appropriate dimension [45, 66]. The delay parameter $\tau$ is determined by autocorrelation function with no specific rule, but in many cases $\tau=nT/d$ for some positive integer $n$. In our pursuit of enhanced extraction of topological features, a relatively high dimension is chosen (see Section 3.1 for more discussion on dimension in TDE). Given this higher dimension, the usual case of $\tau=T/d$ with $n=1$ may prove excessively diminutive, particularly in light of the time series only taking values in discrete time steps. Consequently, in TopCap we adopt an adjusted parametrisation for $\tau=nT/d$ with a relatively large value $n=6$.

We input the pair of birth time and maximal persistence from 1-dimensional PD for each sound record to multiple traditional classification algorithms: Tree, Discriminant, Logistic Regression, Naive Bayes, Support Vector Machine, $k$-Nearest Neighbours, Kernel, Ensemble, and Neural Network. We use the application of the MATLAB (R2022b) Classification Learner, with 5-fold cross-validation, and set aside 30% records as test data. This application performs machine learning algorithms in an automatic way. There are a total of 1016 records, with 712 training samples and 304 test samples. Among them, 694 records are voiced consonants and the remaining are voiceless consonants. The models we choose in this application are Optimizable Tree, Optimizable Discriminant, Efficient Logistic Regression, Optimizable Naive Bayes, Optimizable SVM, Optimizable KNN, Kernel, Optimizable Ensemble, and Optimizable Neural Network. The results are shown in Fig. 3a–d.

The receiver operating characteristic (ROC) curve, area under the curve (AUC), and accuracy metrics collectively demonstrate the efficacy of these topological features as inputs for a variety of machine learning algorithms. Each of these algorithms achieves an accuracy of higher than 96%. The ROC and AUC together depict the high performance of our classification model across all classification thresholds. The 2D histograms depicted in Fig. 3c–d collectively illustrate the distinct distributions of voiced and voiceless consonants. Voiced consonants tend to exhibit a relatively higher birth time and lifetime, which provides an explanation for the high performance of these algorithms. Despite the intricate structure that a PD may present, appropriately extracted topological features enable traditional machine learning algorithms to separate complex data effectively. This highlights the potential of TDA in enhancing the performance of machine learning models. In summary, the most significant distinction between voiced and voiceless consonants is that the former exhibit higher maximal persistence. This can scarcely be detected in lower dimensions regardless of how we tune the delay parameter $\tau$ for TDE. See Fig. S2 of Supplementary Information for recognition of vowels as well as consonants in terms of their ``shapes".

A PD for 1-dimensional PH encodes much more information beyond the birth time and lifetime of the maximal persistence point. The three fundamental variations examined in Section 2.1 also manifest themselves in certain regions of the PD, which can in turn be vectorised. To capture these variations, we perform an experiment with two records of the vowel [A]. Specifically, we demonstrate the fundamental variations by comparing the PDs of (a) the record of [A] relatively unstable with respect to the fundamental variations and (b) the other record of the same vowel that is relatively stable. To better illustrate the results, we crop each record into 4 overlapping intervals, each starting from time 0 and ending at 600, 800, 1000, 1200, respectively. When adding a new segment of 200 units into the original sample each time, the amplitude and frequency of the series altered more drastically in case (a). A more rapid changing rate may lead to more points distributed in the lower region of the diagram. The outcomes are presented in Fig. 4. In particular, the plots in Fig. 4c show that the spectral frequency of (a) indeed varies faster than (b).

We should also mention that the 1-dimensional PD here serves as a profile for the combined effect of the fundamental variations. Currently, it is unclear how the points in the lower region change in response to a specific fundamental variation.

In the TDE plus PD approach, the determination of dimension in TDE can be complex. However, it plays a pivotal role in the extraction of maximal persistence. It is observed that a larger dimension can significantly enhance the theoretical maximal persistence of a time series, which will be discussed in this section. In TopCap construction, the dimension of TDE is set at 100, indicating a relatively large dimension for the experiment. The dimension may fluctuate based on factors such as the length of the time series, as the dimension cannot exceed the length of the time series as it would render the resulting point cloud pointless; the periodicity of the time series, as the window size should be compatible with the approximate period of the time series; among other considerations.

According to Perea and Harer [61, Proposition 5.1], there will be no information loss for trigonometric polynomials if and only if the dimension of TDE exceeds twice the maximal frequency. In this context, no data loss implies that the original time series can be reconstructed from the embedded point cloud. In general, for a periodic function, a higher dimension of TDE can yield a more precise approximation by trigonometric polynomials. Although there are no absolutely periodic functions in real data, each time series exhibits its own ``pattern of vibration'', as discussed in the Section 2.1, a higher dimension may be employed to capture a more accurate vibration pattern in time series. Furthermore, an increased embedded dimension may result in reduced computation time for PD. For instance, computation times for a voiced consonant [N] (see Fig. 5a) are 0.2671, 0.2473, and 0.2375 seconds, corresponding to embedded dimensions 10, 100, and 1000, respectively. This is attributed to the potential influence of a larger dimension on the number of points in a point cloud. While the reduction in computation time may not be considered substantial compared to the impact of skip (see Fig. 5d), it may become significant when handling large datasets. Most importantly, an increased embedded dimension can yield benefits such as enhanced maximal persistence (which serves as a motivation for a larger dimension) and a smoother shape of resulting point clouds obtained through TDE, which makes TDE more reasonable in common sense. Typically, for most algorithms, a lower dimension is preferred due to factors such as the curse of dimensionality and computation cost. However, in the TDE plus PD approach, we opt for a higher dimension.

However, the embedded dimension cannot be arbitrarily large. As illustrated in Fig. 5c, when the embedded dimension escalates to 1280, it becomes unfeasible to capture a significant maximal persistence in the phonetic time series. This is attributed to the dehiscence of the point clouds, rendering them incapable of capturing a proper maximal persistence. When the embedded dimension reaches 1290, an empty 1-dimensional barcode is procured due to the insufficiency of points necessary to form even one cycle. In this scenario, the dimension of TDE is related to the length of the time series.

Using a sound record of the voiced consonant [N] as an exemplar, we delineate the correlation between maximal persistence and embedded dimension in Fig. 5. As depicted in Fig. 5b, maximal persistence tends to escalate precipitously in a nonlinear fashion with the increase in dimension, signifying that a more substantial maximal persistence is captured in higher-dimensional TDE. Notably, two severe reductions in maximal persistence are observed, corresponding to embedded dimensions 600 and 1190. When $d=600$, this time series can theoretically attain its maximal persistence when $\tau=2$. However, given the length of the series is 1337 and the window size is $d\cdot\tau=1200$, with the skip set at 5, only 28 points are in the resulting point clouds for persistence diagram computation. The scarcity of points in the point clouds fails to represent the original series adequately, leading to a decrease in maximal persistence, the same way as when dimension equals 610, 620. Also, a similar phenomenon occurs when the dimension reaches 1190. The principal component analysis (PCA) for dimension 1280 is shown in Fig. 5c. In this scenario, the hypothetical cycle fails to form as there is a dehiscence in the point cloud, resulting in these relatively catastrophic decreases in maximal persistence. When $d=630$, this series can achieve its maximal persistence when $\tau=1$, resulting in a window size of $d\cdot\tau=630$. There are 142 points in the point cloud for the persistence diagram if skip equals 5, ensuring that the maximal persistence is reached again without any dehiscence and improper presentation in the point clouds. The embedded dimension also contributes significantly to the geometric property of time-delay embedding. As the shape becomes smoother in higher dimensions, it results in a more reasonable structure of the point clouds.

Computation time assumes a critical role when processing a substantial volume of data. In this context, the parameter skip in TDE is considered, as it significantly influences the number of points within the point clouds, thereby directly impacting the number of simplices during filtration, thus reducing computation time for PD. In this subsection, we demonstrate that an appropriate increment in the skip parameter can markedly decrease computation time. However, it is noteworthy that maximal persistence exhibits resilience to an increase in skip to a certain extent. Consequently, in this case, it is feasible to augment skip in TDE to expedite the computation of PD. For details on the complexity of computing persistent homology, readers can refer to Zomorodian and Carlsson [53, Section 4.3 Discussion] as well as Edelsbrunner et al. [67, Section 4].

Using an example of a sound record of the voiced consonant [m], we elucidate the relationship between skip, computation duration, and cardinality of the resulting point clouds obtained via TDE in Fig. 5d. Computation duration is computed every time after restarting the Jupyter Notebook, measured on Dell Precision 3581, with CPU Intel^® Core${}^{\text{TM}}$ i7-13800H CPU with basic frequency 2.50 GHz, 14 cores. Computation time means the time for executing the code: ripser(Points,maxdim=1). As depicted in Fig. 5d, a substantial reduction in computation time is observed with an increase in the skip parameter. In contrast, our algorithm's computation of maximal persistence remains stable and resilient.

In this subsection, we present a table that delineates the maximum persistence in relation to dimension, delay, and skip within the context of the TDE plus PD approach. The experiment is executed on a voiced consonant [N], which comprises 887 points as its length. Theoretically, for a periodic function, one should attain the maximum persistence of a function in the desired dimension with the condition that window size is approximate to the period. However, the phonetic time series that we typically handle deviate far from being periodic. Despite our approach to calculating the period of time series by autocorrelation function, we cannot assure that the desired delay will indeed reach the actual maximum persistence of a time series in general cases. Nevertheless, the desired delay usually yields relatively good maximum persistence. For instance, as illustrated in Table 1, when the dimension is 10, the desired delay is 40. This corresponds to a maximum persistence of 0.1290, which is marginally lower than the maximum persistence achieved at a delay of 60, yielding a maximum persistence of 0.1315. However, as the dimension escalates, the point clouds resulting from TDE become more reasonable. It becomes increasingly probable that at the desired delay, one can attain maximum persistence of the time series. For example, when the dimension is either 50 or 100, the maximum persistence of the time series is achieved at the desired delay. This provides additional justification for preferring higher dimensions. This table further illustrates that an augmentation in dimension might lead to a more substantial enhancement in the maximal persistence of a time series than tuning delay.

There are three kinds of fundamental variations mentioned in Section 2.1. In order to substantiate our argument, let

Note that when $t=0$, $F(t_{n})=c/4$; when $t=7\pi$, $F(t_{n})=1$. In fact, $F(t_{n})$ is a series of line section connecting $(0,c/4)$ and $(7\pi,1)$. The frequency of $g_{1}(t_{n})$ changes more slowly as $c$ increases. In particularly, when $c=4$, $F(t_{n})=1$, so $g_{1}(t_{n})=f\big{(}F(t_{n})\cdot t_{n}\big{)}=f(t_{n})=\cos(t_{n})$, which is a periodic function. Performing TDE to $g_{1}(t_{n})$ with dimension 3, delay 100 and skip 10. Then, compute its 1-dimensional PD. The underlying space of this point cloud is 3-dimensional Euclidean space. See Fig. 2a for the result. Replace $F(t_{n})$ by $A(t_{n})$ and $L(t_{n})$, we can obtain the similar diagram in Fig. 2b and Fig. 2c.

We used speech files sourced from SpeechBox [68], ALLSSTAR Corpus [69], task HT1 language English L1 file, derived on 28 January, 2023. SpeechBox is a web-based system providing access to an extensive collection of digital speech corpora developed by the Speech Communication Research Group in the Department of Linguistics at Northwestern University. This section contains a total of 25 individual files, comprising 14 files from females and 11 files from males. The age range of these speakers spans from 18 to 26 years, with an average age of 19.92 years. Each file is presented in the WAV format and is accompanied by its corresponding aligned file in Textgrid format, featuring three tiers: sentences, words, and phones. Collectively, these 25 speech files amount to a total duration of 41.21 minutes. The speech file contains each individual reading the same sentences consecutively for a duration ranging from 80 to 120 seconds, contingent upon each person's pace. The original WAV file has a sampling frequency of 22050 and comprises only one channel. Since Montreal Forced Aligner [70] is trained in a sampling frequency of 16000, we have opted to adjust the sampling frequency of the WAV files accordingly. Extracted the words tier from Textgrid and aligned words into phones using english_mfa dictionary and acoustic model (mfa version 2.0.6). Thus, we obtained corresponding phonetic data from these speech files. Then we used voiced and voiceless consonants in those segments as our dataset. Voiced consonants are consonants that vocal cords vibrate in the throat during articulation, and voiceless consonants on the contrary.

Using Praat [71], we extracted voiced consonants [N], [m], [n], [j], [l], [v], and [Z]; for voiceless consonants, we selected [f], [k], [8], [t], [s], and [tS]. These phones were then read as time series. Our selection was limited to these voiced and voiceless consonants, as we aimed to balance the ratio of voiced and voiceless consonant records in these speech files. Additionally, some consonants, such as [d] and [h], were difficult to classify using this method.

Prior to the extraction of topological features from a time series, we shall first imbue this 1-dimensional time series with a topological structure through TDE. It is noteworthy that this technique is also applicable to multidimensional time series. The underlying space throughout this article will always be Euclidean space. By establishing the topological structure, or more precisely, the distance matrix, we could subsequently calculate PD. We elaborate on the following main steps.

(1) Data cleaning. This involved eliminating the initial and final sections of the time series until the first point with an amplitude exceeding 0.03 was encountered. This approach was aimed at mitigating the impact of environmental noise at the beginning and end of a phone. Any resulting series with fewer than 500 points will be disregarded, as such series were considered insufficiently long or contained excessive environmental noise.

(2) Parameter selection for TDE. Here, we selected suitable parameters for TDE to capture the theoretical maximal persistence of a given time series. The dimension of the embedding was fixed to be 100. The principle for determining the appropriate dimension is that we want to choose the embedded dimension to be large for a time series of limited length. As discussed in the Section 3.1 and cf. Section 2 of Supplementary Information, a higher dimension results in a more accurate approximation. This approach also aimed to enhance computational efficiency and the occurrence of more prominent maximal persistence. Nonetheless, it is imperative to exercise caution when selecting the dimension, as excessively large dimensions may lead to empty point clouds and other uncontrollable factors. With a proper dimension, we then compute the delay of the embedding. According to Perea and Harer [61], in the case of a periodic function, the delay $\tau$ can be expressed as

where $T$ denotes the period, $d$ represents the dimension of the embedding, n is some positive integer. Under these conditions, we could obtain the largest maximal persistence. In the later passage, we shall call it reaches the maximal persistence of the time series. The time series under consideration in our case was far from periodic, so we used the first peak of the autocorrelation function to represent the (minimal) period and set $n=6$, thus obtaining a relatively proper delay. The common choice of $\tau$ is to let window size equal the (minimal) period. However, in the discrete time series case, one probably obtains $\tau=0$, or $\tau=1$ in this way since the dimension of TDE is too large. So, one strategy is increasing $n$ to get a relatively reasonable $\tau$. The performance of delay obtained in this way is presented in the Section 3.1. Then $\tau$ was rounded to the nearest integer. If $\tau$ was rounded to 0, retake it as 1. It was common for the case that $\tau\cdot d$ exceeded the number of points in the series, resulting in an empty embedding. In this case, we adopted to take $\tau=|S|/d$, where $|S|$ denotes the number of points, i.e., the point capacity in the time series, then rounded it downwards. This enabled us to obtain the appropriate delay for each time series, thereby facilitating the attainment of maximal persistence for this specified dimension. Lastly, took the skip=5. We chose this skip mainly to reach an adorable computation time. The impact of the skip parameter in TDE on maximal persistence and computation time is expounded upon in the Section 3.2 for those interested. Once the parameter has been established, the time series can be transformed into point clouds. If there were less than 40 points in this point cloud, we would exclude this time series from further analysis, considering that there are too few points to represent the original structure of the time series. The problem of too few points is also presented in the Section 3.1.

(3) Computing PD. Using Ripser [72, 73], we could compute the PD of the point clouds in a fast and efficient way. Then we extracted the maximal persistence from the 1-dimensional PD, using its birth time and lifetime as two features of this time series. The process of vectorising a PD presents a challenge due to the indeterminate (and potentially large) number of tuples in the barcode, coupled with the ambiguous information they contain. This ambiguity arises from our lack of knowledge about the types of information that can be derived from different parts of the PD. Here we only extracted the maximal persistence and corresponding birth time. This decision was informed by our prior selection of an appropriate parameter, which ensured that the PD reached its maximum. See Fig. 3e for the flow chart of this section.

Department of Mathematics, Southern University of Science and Technology, Shenzhen, China
Pingyao Feng, Siheng Yi, Qingrui Qu, Zhiwang Yu & Yifei Zhu

Y.Z. planned the project. P.F. and S.Y. constructed the theoretical framework. P.F. designed the sample, built the algorithms, and analysed the data. S.Y. assisted with the algorithms. P.F., S.Y., Q.Q., Z.Y., and Y.Z. wrote the paper and contributed to the discussion.

Correspondence to Pingyao Feng or Yifei Zhu.

The evolution of TDA is relatively nascent when juxtaposed with other enduring fields, and its applications are still somewhat delimited. The genesis of the concept of invariants of filtered complexes can be traced back to Barannikov in 1994 \citeSSbarannikov, which are nowadays referred to as PD/PB (persistence diagram / barcode). These invariants were conceived with the objective of quantifying some specific critical point within some ambit of an extension of function. In 1999, Robins \citeSSHistory1 pioneered the concept of ``persistent Betti numbers'' of inverse systems and underscored their stability in Hausdorff distance. The modern incarnation of persistent homology was established in the first decade of the 21st century. Zomorodian, under the tutelage of Edelsbrunner, submitted his doctoral thesis in 2001 \citeSSZomorodian2001ComputingAC, wherein he employed persistence to distinguish between topological noise and inherent features of a space. After that, the term ``persistent homology group'' first appeared in the work by Edelsbrunner et al. \citeSSedel_persistent_diagram in 2002. This seminal work formalised topological methodologies to chronicle the evolution of an expanding complex originating from a point set in $\mathbb{R}^{3}$, a process they termed as topological simplification. The expansion process is recognised as filtration. They classified topological modifications based on the lifetime of topological features during filtration and proposed an algorithm to compute this simplification process. Subsequently, in 2005, Carlsson et al. \citeSSpersistent_shape applied persistent homology to generate a barcode as a shape descriptor. Their methodology was able to distinguish between shapes with varying degrees of ``sharp'' features, such as corners. In the same year, Zomorodian and Carlsson \citeSSPH1 presented an algebraic interpretation of persistent homology and developed a natural algorithm for computing persistent homology of spaces in any dimension over any field. Cohen-Steiner et al. \citeSSBarcode3 considered the stability property of persistence algorithm. Robustness is measured by the bottleneck distance between persistence diagrams. In 2008, Carlsson, Singh and Sexton founded Ayasdi, a company that combines mathematics and finance to truly put theory into practice. The inception of TDA may be complex, as it originates from some pure mathematical fields such as Morse theory and PH. However, the underlying principle remains steadfast: to identify topological features that can quantify the shape of the data to certain degrees, which is robust against noise and perturbations.

An abundance of materials is available that offer a thorough understanding of TDA for both specialists and non-specialists. In 2009, Carlsson \citeSScarlsson_topology_2009 wrote an extensive survey on the applications of geometry and topology to the analysis of various types of data. This work introduced topics such as the characteristics of topological methods, persistence, and clusters. More recent work by Carlsson and Vejdemo-Johansson \citeSScarlsson_TDA_application discussed practical case studies of topological methods, such as their applications to image data and time series. For non-specialists seeking to delve into TDA, the introductory article by Chazal and Michel \citeSSintroDS may be more readable. It provides explicit explanations and hands-on guidance on both the theoretical and practical aspects of TDA. Several software tools assist researchers in building case studies on data. The GUDHI library \citeSSGUDHI, an open-source C++ library with a Python interface, includes a comprehensive set of tools involving different complexes and vectorisation tools. Ripser \citeSSBauer2021Ripser, also a C++ library with a Python binding, surpasses GUDHI in terms of computing Vietoris–Rips PD/PB, especially when high-dimensional cases or large quantities of PD/PB are required. TTK \citeSSTTK is both a library and software designed for topological analysis with a focus on scientific visualisation. Other standard libraries include Dionysus²²2https://mrzv.org/software/dionysus2, PHAT³³3https://bitbucket.org/phat-code/phat, DIPHA⁴⁴4https://github.com/DIPHA/dipha, and Giotto⁵⁵5https://giotto-ai.github.io/gtda-docs/0.4.0. Additionally, an R interface named TDA \citeSSfasy2015introduction is available for the libraries GUDHI, Dionysus, and PHAT.

The recent proliferation of TDA has established it as an effective instrument in numerous studies. Owing to the characteristics of topological methods \citeSScarlsson_topology_2009, a multitude of applications have been discovered, particularly in the realm of recognition. In the field of biomedicine, Nicolau et al. \citeSSBio4 utilised the topological methods Mapper \citeSSMapper to analyse transcriptional data related to breast cancer. This method is used due to its high performance in shape recognition in high dimensions. The book authored by Rabadán and Blumberg \citeSSBio5 provides an introduction to TDA techniques and their specific applications in biology, encompassing topics such as evolutionary processes and cancer genomics. In signal processing, Emrani et al. \citeSSSP1 introduced a topological approach for the analysis of breathing sound signals for the detection of wheezing, which can distinguish abnormal wheeze signals from normal breathing signals due to the periodic patterns within wheezing. Robinson \citeSStopo_sig offers a systematic exploration of the intersection between topology and signal processing. In the context of deep learning, Bae et al. \citeSSDL1 proposed a PH-based deep residual learning algorithm for image restoration tasks. Hofer et al. \citeSSDL2 incorporated topological signatures into deep neural networks to learn unusual structures that are typically challenging for most machine learning techniques. Love et al. \citeSSDL3 extracted the topological features of images and videos by TDA, and input these features into the kernel of the convolutional layer. In their case, manifolds with relationships to the natural image space are used to parameterize image filters, which also parameterize slices in layers of neural networks. For networks, an early application of PH on sensor networks is presented in the work by de Silva and Ghrist \citeSSsensor_Silva. They applied topological methods to graphs representing the distance estimation between nodes and a proximity sensor. Subsequently, Horak et al. \citeSSPH_CN discussed PH in different networks, observing that persistent topological attributes are related to the robustness of networks and reflect deficiencies in certain connectivity properties of networks. Additionally, Jonsson’s book \citeSSsim_com_graph provides insights on how to construct a simplicial complex from a graph. Given the potential of topological methods, researchers from various fields may be inspired to incorporate these techniques into their studies in future possible cases to see if this leads to unexpected and exciting discoveries.

When executing PH, one typically obtains PD/PB, which is a set of intervals on the extended line. Indeed, PD/PB can be considered a form of vectorisation of original data. However, they may not be sufficiently accessible for further applications, such as integration into machine learning algorithms for future model development. Since the intervals exist on the extended line, some may involve $+\infty$ as their termination point, which could pose challenges for certain algorithms when encountering infinity. This issue can be mitigated by something like setting a threshold for the maximal lifetime, which is a relatively straightforward solution. However, there are more intrinsic challenges embedded in the vectorisation of PD/PB that are not easily resolved and may pose difficulties for researchers attempting to leverage this powerful tool. For example, the number of intervals in PD/PB is not fixed; sometimes, there may be 10, and other times there may be 100. Moreover, PD is too sparse to put into machine learning algorithms. Researchers might extract the top five longest intervals from the set as a method of vectorisation, or remove intervals with a length less than a certain threshold from the set, or implement the distance functions and kernel methods of PD/PB to achieve vectorisation. In this article, vectorisation in TopCap is relatively simple, as we extract the maximal persistence and its corresponding birth time as two topological features to feed into machine learning algorithms.

There is no definitive rule to determine that one method of vectorisation is superior to another, as the performance of vectorisation methods largely depends on the data and how it is transformed into a topological space. Nevertheless, there are a great many creative methods for vectorising PH. Persistence Landscapes (PL) \citeSSVec3, developed by Bubenik, is one popular method. Bubenik’s work introduces both theoretical and experimental aspects of PL in a statistical manner. Generally speaking, PL maps PD into a function space that is stable and invertible \citeSSPLProperty. A toolbox \citeSSPLTool is also available for implementing PL. Persistence Image (PI) \citeSSVec4, another vectorisation method developed by Adams et al., stably maps PD to a finite-dimensional vector representation depending on resolution, weight function, and distribution of points in PD. For additional vectorisation methods, one might consider the article by Ali et al. \citeSSVec2, which presents 13 ways to vectorise PD.