我们尝试在我们的数据集上使用平滑的z-score算法，这导致了过度敏感或不敏感(取决于参数如何调整)，几乎没有中间地带。在我们站点的交通信号中，我们观察到一个低频基线，它代表了每天的周期，即使有最好的可能参数(如下所示)，它仍然在第4天下降，特别是因为大多数数据点被认为是异常的。

在原始z-score算法的基础上，我们提出了一种通过反向滤波来解决这个问题的方法。改进后的算法及其在电视商业流量归因中的应用详见我们的团队博客。

我认为delica的Python回答器有一个bug。我不能评论他的帖子，因为我没有代表来做这件事，编辑队列已经满了，所以我可能不是第一个注意到它的人。

avgFilter[lag - 1]和stdFilter[lag - 1]在init中设置，然后在lag == i时再次设置，而不是改变[lag]值。这个结果使得第一个信号总是1。

以下是带有轻微修正的代码:

import numpy as np

class real_time_peak_detection():
    def __init__(self, array, lag, threshold, influence):
        self.y = list(array)
        self.length = len(self.y)
        self.lag = lag
        self.threshold = threshold
        self.influence = influence
        self.signals = [0] * len(self.y)
        self.filteredY = np.array(self.y).tolist()
        self.avgFilter = [0] * len(self.y)
        self.stdFilter = [0] * len(self.y)
        self.avgFilter[self.lag - 1] = np.mean(self.y[0:self.lag]).tolist()
        self.stdFilter[self.lag - 1] = np.std(self.y[0:self.lag]).tolist()

    def thresholding_algo(self, new_value):
        self.y.append(new_value)
        i = len(self.y) - 1
        self.length = len(self.y)
        if i < self.lag:
            return 0
        elif i == self.lag:
            self.signals = [0] * len(self.y)
            self.filteredY = np.array(self.y).tolist()
            self.avgFilter = [0] * len(self.y)
            self.stdFilter = [0] * len(self.y)
            self.avgFilter[self.lag] = np.mean(self.y[0:self.lag]).tolist()
            self.stdFilter[self.lag] = np.std(self.y[0:self.lag]).tolist()
            return 0

        self.signals += [0]
        self.filteredY += [0]
        self.avgFilter += [0]
        self.stdFilter += [0]

        if abs(self.y[i] - self.avgFilter[i - 1]) > self.threshold * self.stdFilter[i - 1]:
            if self.y[i] > self.avgFilter[i - 1]:
                self.signals[i] = 1
            else:
                self.signals[i] = -1

            self.filteredY[i] = self.influence * self.y[i] + (1 - self.influence) * self.filteredY[i - 1]
            self.avgFilter[i] = np.mean(self.filteredY[(i - self.lag):i])
            self.stdFilter[i] = np.std(self.filteredY[(i - self.lag):i])
        else:
            self.signals[i] = 0
            self.filteredY[i] = self.y[i]
            self.avgFilter[i] = np.mean(self.filteredY[(i - self.lag):i])
            self.stdFilter[i] = np.std(self.filteredY[(i - self.lag):i])

        return self.signals[i]

鲁棒峰值检测算法(使用z-scores)

I came up with an algorithm that works very well for these types of datasets. It is based on the principle of dispersion: if a new datapoint is a given x number of standard deviations away from some moving mean, the algorithm signals (also called z-score). The algorithm is very robust because it constructs a separate moving mean and deviation, such that signals do not corrupt the threshold. Future signals are therefore identified with approximately the same accuracy, regardless of the amount of previous signals. The algorithm takes 3 inputs: lag = the lag of the moving window, threshold = the z-score at which the algorithm signals and influence = the influence (between 0 and 1) of new signals on the mean and standard deviation. For example, a lag of 5 will use the last 5 observations to smooth the data. A threshold of 3.5 will signal if a datapoint is 3.5 standard deviations away from the moving mean. And an influence of 0.5 gives signals half of the influence that normal datapoints have. Likewise, an influence of 0 ignores signals completely for recalculating the new threshold. An influence of 0 is therefore the most robust option (but assumes stationarity); putting the influence option at 1 is least robust. For non-stationary data, the influence option should therefore be put somewhere between 0 and 1.

其工作原理如下:

伪代码

# Let y be a vector of timeseries data of at least length lag+2
# Let mean() be a function that calculates the mean
# Let std() be a function that calculates the standard deviaton
# Let absolute() be the absolute value function

# Settings (these are examples: choose what is best for your data!)
set lag to 5;          # average and std. are based on past 5 observations
set threshold to 3.5;  # signal when data point is 3.5 std. away from average
set influence to 0.5;  # between 0 (no influence) and 1 (full influence)

# Initialize variables
set signals to vector 0,...,0 of length of y;   # Initialize signal results
set filteredY to y(1),...,y(lag)                # Initialize filtered series
set avgFilter to null;                          # Initialize average filter
set stdFilter to null;                          # Initialize std. filter
set avgFilter(lag) to mean(y(1),...,y(lag));    # Initialize first value average
set stdFilter(lag) to std(y(1),...,y(lag));     # Initialize first value std.

for i=lag+1,...,t do
  if absolute(y(i) - avgFilter(i-1)) > threshold*stdFilter(i-1) then
    if y(i) > avgFilter(i-1) then
      set signals(i) to +1;                     # Positive signal
    else
      set signals(i) to -1;                     # Negative signal
    end
    set filteredY(i) to influence*y(i) + (1-influence)*filteredY(i-1);
  else
    set signals(i) to 0;                        # No signal
    set filteredY(i) to y(i);
  end
  set avgFilter(i) to mean(filteredY(i-lag+1),...,filteredY(i));
  set stdFilter(i) to std(filteredY(i-lag+1),...,filteredY(i));
end

下面是为数据选择良好参数的经验法则。

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Demo

这个演示的Matlab代码可以在这里找到。要使用演示，只需运行它并单击上面的图表自己创建一个时间序列。算法在绘制滞后观测数后开始工作。

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结果

对于原始问题，当使用以下设置时，该算法将给出以下输出:滞后= 30，阈值= 5，影响= 0:

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在不同编程语言中的实现:

Matlab (me) R (me) Golang (Xeoncross) Golang [efficient version] (Micah Parks) Python (R Kiselev) Python [efficient version] (delica) Swift (me) Groovy (JoshuaCWebDeveloper) C++ [interactive parameters] (Jason C) C++ (Animesh Pandey) Rust (swizard) Scala (Mike Roberts) Kotlin (leoderprofi) Ruby (Kimmo Lehto) Fortran [for resonance detection] (THo) Julia (Matt Camp) C# (Ocean Airdrop) C (DavidC) Java (takanuva15) JavaScript (Dirk Lüsebrink) TypeScript (Jerry Gamble) Perl (Alen) PHP (radhoo) PHP (gtjamesa) Dart (Sga)

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配置算法的经验法则

lag: the lag parameter determines how much your data will be smoothed and how adaptive the algorithm is to changes in the long-term average of the data. The more stationary your data is, the more lags you should include (this should improve the robustness of the algorithm). If your data contains time-varying trends, you should consider how quickly you want the algorithm to adapt to these trends. I.e., if you put lag at 10, it takes 10 'periods' before the algorithm's treshold is adjusted to any systematic changes in the long-term average. So choose the lag parameter based on the trending behavior of your data and how adaptive you want the algorithm to be.

influence: this parameter determines the influence of signals on the algorithm's detection threshold. If put at 0, signals have no influence on the threshold, such that future signals are detected based on a threshold that is calculated with a mean and standard deviation that is not influenced by past signals. If put at 0.5, signals have half the influence of normal data points. Another way to think about this is that if you put the influence at 0, you implicitly assume stationarity (i.e. no matter how many signals there are, you always expect the time series to return to the same average over the long term). If this is not the case, you should put the influence parameter somewhere between 0 and 1, depending on the extent to which signals can systematically influence the time-varying trend of the data. E.g., if signals lead to a structural break of the long-term average of the time series, the influence parameter should be put high (close to 1) so the threshold can react to structural breaks quickly.

threshold: the threshold parameter is the number of standard deviations from the moving mean above which the algorithm will classify a new datapoint as being a signal. For example, if a new datapoint is 4.0 standard deviations above the moving mean and the threshold parameter is set as 3.5, the algorithm will identify the datapoint as a signal. This parameter should be set based on how many signals you expect. For example, if your data is normally distributed, a threshold (or: z-score) of 3.5 corresponds to a signaling probability of 0.00047 (from this table), which implies that you expect a signal once every 2128 datapoints (1/0.00047). The threshold therefore directly influences how sensitive the algorithm is and thereby also determines how often the algorithm signals. Examine your own data and choose a sensible threshold that makes the algorithm signal when you want it to (some trial-and-error might be needed here to get to a good threshold for your purpose).

---

警告:上面的代码每次运行时都会遍历所有的数据点。在实现这段代码时，请确保将信号的计算拆分为一个单独的函数(没有循环)。然后当一个新的数据点到达时，更新filteredY, avgFilter和stdFilter一次。不要每次有新的数据点时都重新计算所有数据的信号(就像上面的例子一样)，这在实时应用程序中是非常低效和缓慢的。

其他修改算法的方法(为了潜在的改进)有:

使用中位数而不是平均值 使用稳健的尺度测量，如中位数绝对偏差(MAD)，而不是标准偏差 使用信号裕度，这样信号就不会频繁切换 更改影响参数的工作方式 区别对待上下信号(不对称处理) 创建一个单独的影响参数的平均值和标准(在这个Swift翻译)

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(已知)学术引用此StackOverflow的答案:

Cai, Y., Wang, X., Joos, G., & Kamwa, I. (2022). An Online Data-Driven Method to Locate Forced Oscillation Sources from Power Plants Based on Sparse Identification of Nonlinear Dynamics (SINDy). IEEE Transactions on Power Systems. Yang, S., Yim, J., Kim, J., & Shin, H. V. (2022). CatchLive: Real-time Summarization of Live Streams with Stream Content and Interaction Data. CHI Conference on Human Factors in Computing Systems, 1-20. Feng, D., Tan, Z., Engwirda, D., Liao, C., Xu, D., Bisht, G., ... & Leung, R. (2022). Investigating coastal backwater effects and flooding in the coastal zone using a global river transport model on an unstructured mesh. Hydrology and Earth System Sciences Discussions, 1-31 [preprint]. Link, J., Perst, T., Stoeve, M., & Eskofier, B. M. (2022). Wearable sensors for activity recognition in ultimate frisbee using convolutional neural networks and transfer learning. Sensors, 22(7), 2560. Romeiro, J. M. N., Torres, F. T. P., & Pirotti, F. (2021). Evaluation of Effect of Prescribed Fires Using Spectral Indices and SAR Data. Bollettino della società italiana di fotogrammetria e topografia, (2), 36-56. Moore, J., Goffin, P., Wiese, J., & Meyer, M. (2021). An Interview Method for Engaging Personal Data. Proceedings of the ACM on Interactive, Mobile, Wearable and Ubiquitous Technologies, 5(4), 1-28. Rykov, Y., Thach, T. Q., Bojic, I., Christopoulos, G., & Car, J. (2021). Digital Biomarkers for Depression Screening With Wearable Devices: Cross-sectional Study With Machine Learning Modeling. JMIR mHealth and uHealth, 9(10), e24872. Hong, Y., Xin, Y., Martin, H., Bucher, D., & Raubal, M. (2021). A Clustering-Based Framework for Individual Travel Behaviour Change Detection. In 11th International Conference on Geographic Information Science (GIScience 2021)-Part II. Grammenos, A., Kalyvianaki, E., & Pietzuch, P. (2021). Pronto: Federated Task Scheduling. arXiv preprint arXiv:2104.13429. Courtial, N. (2020). Fusion d’images multimodales pour l’assistance de procédures d’électrophysiologie cardiaque. Doctoral dissertation, Université Rennes. Beckman, W. F., Jiménez, M. Á. L., Moerland, P. D., Westerhoff, H. V., & Verschure, P. J. (2020). 4sUDRB-sequencing for genome-wide transcription bursting quantification in breast cancer cells. bioRxiv. Olkhovskiy, M., Müllerová, E., & Martínek, P. (2020). Impulse signals classification using one dimensional convolutional neural network. Journal of Electrical Engineering, 71(6), 397-405. Gao, S., & Calderon, D. P. (2020). Robust alternative to the righting reflex to assess arousal in rodents. Scientific reports, 10(1), 1-11. Chen, G. & Dong, W. (2020). Reactive Jamming and Attack Mitigation over Cross-Technology Communication Links. ACM Transactions on Sensor Networks, 17(1). Takahashi, R., Fukumoto, M., Han, C., Sasatani, T., Narusue, Y., & Kawahara, Y. (2020). TelemetRing: A Batteryless and Wireless Ring-shaped Keyboard using Passive Inductive Telemetry. In Proceedings of the 33rd Annual ACM Symposium on User Interface Software and Technology (pp. 1161-1168). Negus, M. J., Moore, M. R., Oliver, J. M., Cimpeanu, R. (2020). Droplet impact onto a spring-supported plate: analysis and simulations. Journal of Engineering Mathematics, 128(3). Yin, C. (2020). Dinucleotide repeats in coronavirus SARS-CoV-2 genome: evolutionary implications. ArXiv e-print, accessible from: https://arxiv.org/pdf/2006.00280.pdf Esnaola-Gonzalez, I., Gómez-Omella, M., Ferreiro, S., Fernandez, I., Lázaro, I., & García, E. (2020). An IoT Platform Towards the Enhancement of Poultry Production Chains. Sensors, 20(6), 1549. Gao, S., & Calderon, D. P. (2020). Continuous regimens of cortico-motor integration calibrate levels of arousal during emergence from anesthesia. bioRxiv. Cloud, B., Tarien, B., Liu, A., Shedd, T., Lin, X., Hubbard, M., ... & Moore, J. K. (2019). Adaptive smartphone-based sensor fusion for estimating competitive rowing kinematic metrics. PloS one, 14(12). Ceyssens, F., Carmona, M. B., Kil, D., Deprez, M., Tooten, E., Nuttin, B., ... & Puers, R. (2019). Chronic neural recording with probes of subcellular cross-section using 0.06 mm² dissolving microneedles as insertion device. Sensors and Actuators B: Chemical, 284, pp. 369-376. Dons, E., Laeremans, M., Orjuela, J. P., Avila-Palencia, I., de Nazelle, A., Nieuwenhuijsen, M., ... & Nawrot, T. (2019). Transport Most Likely to Cause Air Pollution Peak Exposures in Everyday Life: Evidence from over 2000 Days of Personal Monitoring. Atmospheric Environment, 213, 424-432. Schaible B.J., Snook K.R., Yin J., et al. (2019). Twitter conversations and English news media reports on poliomyelitis in five different countries, January 2014 to April 2015. The Permanente Journal, 23, 18-181. Lima, B. (2019). Object Surface Exploration Using a Tactile-Enabled Robotic Fingertip (Doctoral dissertation, Université d'Ottawa/University of Ottawa). Lima, B. M. R., Ramos, L. C. S., de Oliveira, T. E. A., da Fonseca, V. P., & Petriu, E. M. (2019). Heart Rate Detection Using a Multimodal Tactile Sensor and a Z-score Based Peak Detection Algorithm. CMBES Proceedings, 42. Lima, B. M. R., de Oliveira, T. E. A., da Fonseca, V. P., Zhu, Q., Goubran, M., Groza, V. Z., & Petriu, E. M. (2019, June). Heart Rate Detection Using a Miniaturized Multimodal Tactile Sensor. In 2019 IEEE International Symposium on Medical Measurements and Applications (MeMeA) (pp. 1-6). IEEE. Ting, C., Field, R., Quach, T., Bauer, T. (2019). Generalized Boundary Detection Using Compression-based Analytics. ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Brighton, United Kingdom, pp. 3522-3526. Carrier, E. E. (2019). Exploiting compression in solving discretized linear systems. Doctoral dissertation, University of Illinois at Urbana-Champaign. Khandakar, A., Chowdhury, M. E., Ahmed, R., Dhib, A., Mohammed, M., Al-Emadi, N. A., & Michelson, D. (2019). Portable system for monitoring and controlling driver behavior and the use of a mobile phone while driving. Sensors, 19(7), 1563. Baskozos, G., Dawes, J. M., Austin, J. S., Antunes-Martins, A., McDermott, L., Clark, A. J., ... & Orengo, C. (2019). Comprehensive analysis of long noncoding RNA expression in dorsal root ganglion reveals cell-type specificity and dysregulation after nerve injury. Pain, 160(2), 463. Cloud, B., Tarien, B., Crawford, R., & Moore, J. (2018). Adaptive smartphone-based sensor fusion for estimating competitive rowing kinematic metrics. engrXiv Preprints. Zajdel, T. J. (2018). Electronic Interfaces for Bacteria-Based Biosensing. Doctoral dissertation, UC Berkeley. Perkins, P., Heber, S. (2018). Identification of Ribosome Pause Sites Using a Z-Score Based Peak Detection Algorithm. IEEE 8th International Conference on Computational Advances in Bio and Medical Sciences (ICCABS), ISBN: 978-1-5386-8520-4. Moore, J., Goffin, P., Meyer, M., Lundrigan, P., Patwari, N., Sward, K., & Wiese, J. (2018). Managing In-home Environments through Sensing, Annotating, and Visualizing Air Quality Data. Proceedings of the ACM on Interactive, Mobile, Wearable and Ubiquitous Technologies, 2(3), 128. Lo, O., Buchanan, W. J., Griffiths, P., and Macfarlane, R. (2018), Distance Measurement Methods for Improved Insider Threat Detection, Security and Communication Networks, Vol. 2018, Article ID 5906368. Apurupa, N. V., Singh, P., Chakravarthy, S., & Buduru, A. B. (2018). A critical study of power consumption patterns in Indian Apartments. Doctoral dissertation, IIIT-Delhi. Scirea, M. (2017). Affective Music Generation and its effect on player experience. Doctoral dissertation, IT University of Copenhagen, Digital Design. Scirea, M., Eklund, P., Togelius, J., & Risi, S. (2017). Primal-improv: Towards co-evolutionary musical improvisation. Computer Science and Electronic Engineering (CEEC), 2017 (pp. 172-177). IEEE. Catalbas, M. C., Cegovnik, T., Sodnik, J. and Gulten, A. (2017). Driver fatigue detection based on saccadic eye movements, 10th International Conference on Electrical and Electronics Engineering (ELECO), pp. 913-917.

其他工作使用的算法从这个答案

Bergamini, E. and E. Mourlon-Druol (2021). Talking about Europe: exploring 70 years of news archives. Working Paper 04/2021, Bruegel. Cox, G. (2020). Peak Detection in a Measured Signal. Online article on https://www.baeldung.com/cs/signal-peak-detection. Raimundo, D. W. (2020). SwitP: Mobile Application for Real-Time Swimming Analysis.. Semester Thesis, ETH Zürich. Bernardi, D. (2019). A feasibility study on pairing a smartwatch and a mobile device through multi-modal gestures. Master thesis, Aalto University. Lemmens, E. (2018). Outlier detection in event logs by using statistical methods, Master thesis, University of Eindhoven. Willems, P. (2017). Mood controlled affective ambiences for the elderly, Master thesis, University of Twente. Ciocirdel, G. D. and Varga, M. (2016). Election Prediction Based on Wikipedia Pageviews. Project paper, Vrije Universiteit Amsterdam.

算法的其他应用从这个答案

Avo审计dbt包。Avo公司(下一代分析治理)。 用OpenBCI系统合成语音，SarahK01。 Python包:机器学习金融实验室，基于De Prado, m.l.(2018)的工作。金融机器学习的进展。John Wiley & Sons。 Adafruit CircuitPlayground Library, Adafruit board (Adafruit Industries) 步长跟踪算法，Android App (jeeshnair) R包:《动物追踪者》(乔·钱皮恩、西娅·苏基安托)

链接到其他峰值检测算法

噪声正弦时间序列中的实时峰值检测

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如何引用该算法:

Brakel, J.P.G. van(2014)。使用z分数的鲁棒峰值检测算法。堆栈溢出。下载地址:https://stackoverflow.com/questions/22583391/peak-signal-detection-in-realtime-timeseries-data/22640362#22640362(版本:2020-11-08)。

助理 @misc{brakel2014，作者= {Brakel, J.P.G van}，标题={使用z-scores的鲁棒峰值检测算法}，url = {https://stackoverflow.com/questions/22583391/peak-signal-detection-in-realtime-timeseries-data/22640362#22640362}，语言= {en}，年份= {2014}，urldate ={2022-04-12}，期刊= {Stack Overflow}， howpublished = {https://stackoverflow.com/questions/22583391/peak-signal-detection-in-realtime-timeseries-data/22640362#22640362}}

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如果你在某个地方使用这个功能，请使用上面的参考资料来感谢我。如果你对算法有任何问题，请在下面的评论中发表，或在LinkedIn上与我联系。

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如果你的数据在一个数据库表中，这里是一个简单的z-score算法的SQL版本:

with data_with_zscore as (
    select
        date_time,
        value,
        value / (avg(value) over ()) as pct_of_mean,
        (value - avg(value) over ()) / (stdev(value) over ()) as z_score
    from {{tablename}}  where datetime > '2018-11-26' and datetime < '2018-12-03'
)


-- select all
select * from data_with_zscore 

-- select only points greater than a certain threshold
select * from data_with_zscore where z_score > abs(2)

假设你的数据来自传感器(所以算法不可能知道未来的任何事情)，

我做了这个算法，它与我在自己的项目中获得的数据非常好。

该算法有2个参数:灵敏度和窗口。

最后，只需一行代码就可以得到你的结果:

detected=data.map((a, b, c) => (a > 0) ? c[b] ** 4 * c[b - 1] ** 3 : -0).map((a, b, c) => a > Math.max(...c.slice(2)) / sensitivity).map((a, b, c) => (b > dwindow) && c.slice(b - dwindow, b).indexOf(a) == -1);

因为我是程序员而不是数学家，所以我不能更好地解释它。但我相信有人可以。

sensitivity = 20; dwindow = 4; data = [1., 1., 1., 1., 1., 1., 1., 1.1, 1., 0.8, 0.9, 1., 1.2, 0.9, 1., 1., 1.1, 1.2, 1., 1.5, 1., 3., 2., 5., 3., 2., 1., 1., 1., 0.9, 1., 1., 3., 2.6, 4., 3., 3.2, 2., 1., 1., 1., 1., 1. ]; //data = data.concat(data); //data = data.concat(data); var data1 = [{ name: 'original source', y: data }]; Plotly.newPlot('stage1', data1, { title: 'Sensor data', yaxis: { title: 'signal' } }); filtered = data.map((a, b, c) => (a > 0) ? c[b] ** 4 * c[b - 1] ** 3 : -0); var data2 = [{ name: 'filtered source', y: filtered }]; Plotly.newPlot('stage2', data2, { title: 'Filtered data<br>aₙ = aₙ⁴ * aₙ₋₁³', yaxis: { title: 'signal' } }); dwindow = 6; k = dwindow; detected = filtered.map((a, b, c) => a > Math.max(...c.slice(2)) / sensitivity).map((a, b, c) => (b > k) && c.slice(b - k, b).indexOf(a) == -1) var data3 = [{ name: 'detected peaks', y: detected }]; Plotly.newPlot('stage3', data3, { title: 'Window 6', yaxis: { title: 'signal' } }); dwindow = 10; k = dwindow; detected = filtered.map((a, b, c) => a > Math.max(...c.slice(2)) / 20).map((a, b, c) => (b > k) && c.slice(b - k, b).indexOf(a) == -1) var data4 = [{ name: 'detected peaks', y: detected }]; Plotly.newPlot('stage4', data4, { title: 'Window 10', yaxis: { title: 'signal' } }); <script src="https://cdn.jsdelivr.net/npm/plotly.js@2.16.5/dist/plotly.min.js"></script> <div id="stage1"></div> <div id="stage2"></div> <div id="stage3"></div> <div id="stage4"></div>

在计算拓扑学中，持久同调的思想导致一个有效的 -快如排序数字-解决方案。它不仅检测峰值，还以一种自然的方式量化峰值的“重要性”，使您能够选择对您重要的峰值。

算法的总结。 在一维设置(时间序列，实值信号)中，算法可以简单地描述为下图:

Think of the function graph (or its sub-level set) as a landscape and consider a decreasing water level starting at level infinity (or 1.8 in this picture). While the level decreases, at local maxima islands pop up. At local minima these islands merge together. One detail in this idea is that the island that appeared later in time is merged into the island that is older. The "persistence" of an island is its birth time minus its death time. The lengths of the blue bars depict the persistence, which is the above mentioned "significance" of a peak.

效率。 在对函数值进行排序之后，找到一个在线性时间内运行的实现并不难——实际上它是一个单一的、简单的循环。因此，这种实现在实践中应该是快速的，也很容易实现。

参考文献 一篇关于整个故事的文章和对持久同调(计算代数拓扑中的一个领域)动机的引用可以在这里找到: https://www.sthu.org/blog/13-perstopology-peakdetection/index.html