You are trying to predict an early warning sign of a critical transition in a time series, i.e. a sign of imminent failure. The most important clue that has been suggested in the literature whether a system is getting close to a critical transition (failure) is the so-called critical slow down (CSD). There are three main types of signals have been suggested: (i) increased lag-1 autocorrelation compared to the previous one, (ii) increased variance, (iii) flickering. A simple mathematical relationship demonstrates that as lag-1 autocorrelation tends to 1 the variance of the time series tends to infinity, i.e. it is sharply increasing as a sign of incoming failure.

Flickering happens if stochastic intrinsic forces are strong enough to move the system back and forth between two or more relatively stable states. Statistically flickering can be observed as frequency distribution skewness and/or bi-modality that reflects the jumping between two alternative states, i.e. incoming failure.

At the same time, there is no 100% reliable fool-proof indicators of failure based on a time series. Some warning signs could be quite strong but no critical transition will happen (false positive), or no above mentioned signs but the failure happens (false negative). In fact, without knowing details of your system performance your system might have a characteristic subset of the above early warning signals. This individuality is based on the intrinsic random forces originating from the stochastic nature of the system that lead to some non-trivial effects like stochastic and coherence resonance.

Nonetheless, I suggest to test the signs of increasing lag-1 autocorrelation and/or the variance of a time series before plunging into more complicated methods.

@Janusz Milek: The description of the process is: It is a problem of hydraulic pulse test being done on several equipments. We have the amplitude ( which I got from other two columns: max and min value of a wave, though it is nowhere defined with its time period or frequency. It doesn't have any labels so clearly it's not a supervised learning problem. We can plot the graphs and check for abnormal jump in the amplitude. It shows that crack has occurred on an equipment under test. Like this, we need to predict when can we observe the crack on the equipment. The time here is the time when the row was generated. Ideally the test machine generates a row after each 54-55 seconds but it may vary too.

@Alexander Kolker: Thank you very much for the suggestions.

@Janusz Milek: Thanks and yes, even the equipment type varies in my case and sadly, I have been given this much of information only.

You can get the frequencies (notabene: not frequency) by applying Fourier transformation. Choose a number of samples as input, a power of 2, eg. 1024. Apply FFT. You obtain an array of complex numbers of this size as result. Use the lengths of these complex numbers (magnitudes) for further considerations. The entry with the greatest magnitude is the main frequency of your motor, but you will also have to observe all other frequencies as well.

Here's where machine learning fun comes in. The magnitudes of most frequencies will change over time, there are infinite possibilities to define clusters, states, variances, abnormalities.

https://www.nayuki.io/page/free-small-fft-in-multiple-languages

https://rosettacode.org/wiki/Fast_Fourier_transform

https://stackoverflow.com/questions/4364823/how-do-i-obtain-the-frequencies-of-each-value-in-an-fft

Regards,

Joachim

Thanks Joachim Pimiskern ! I have tried using FFT but the problem with my data is that, it is non-uniformly sampled and non-stationary. So would it be fine to use FFT? Also, I do not know the time period/frequency of the waves, so could you please give me some hints to proceed in this direction. I am new to frequency domain analysis and would request you for some help/hints.

You could compute equally spaced samples by interpolation.

Regards,

Joachim

Thanks a lot for your help Alexander Kolker ! Could you please give some more ideas/names of more complicated algorithms/methods to proceed for this problem (as you have mentioned in the last line of your answer).

Ujjwal Anand Most of the work on early warning signals for critical transitions has so far been done on relatively simple low dimensional mathematical models of dynamic systems with low gaussian noise. Therefore, the question still remains open of whether highly complex real systems will consistently show the theoretically expected results.

The theoretical foundation of forecasting of the critical transition is the mathematical theory of dynamic systems. This is a complex area of applied mathematics.

I suggest the following references for more details:

Lyapunov, A. M. The general problem of the stability of motion. Int. J. Control 55, 531–534 (1992).

Scheffer, M. et al. Anticipating critical transitions. Science 338, 344–348, (2012 )

Barzel, B., Liu, Y.-Y. & Barabási, A.-L. Constructing minimal models for complex system dynamics. Nature Commun. 6, 7186 (2015).

as well as

Gao et al. Universal resilience patterns in complex networks. NATURE | VOL 530 | 18 FEBRUARY 2016

Thank you.

Hi Ujjwal.

I see that you have some time series data and you aim to predict the failures. So I feel there are 2 tasks associated with it - 1st forecast the time series and 2nd predict if that value is a possible failure. Probably you can try doing some Time series forecasting and then check those predicted values for anomalies (as you mentioned that the failure is generally when the value goes out of range). You will have multiple algorithms to test and see which one fits best with your data. You can refer to this blog I found for an example approach.

https://towardsdatascience.com/anomaly-detection-with-time-series-forecasting-c34c6d04b24a

Hi Ujwal,

This blog can help you to identify an approach to address the problem.

https://medium.com/bigdatarepublic/machine-learning-for-predictive-maintenance-where-to-start-5f3b7586acfb .

From the information you have provided, I guess I can recommend one approach. If you are comfortable using python, here is one Machine Learning solution using Long Short Term Memory.

https://www.curiousily.com/posts/anomaly-detection-in-time-series-with-lstms-using-keras-in-python/

If your time domain data does not give good result, only then try with frequency domain data. Since it is non-uniformly sampled, I recommend Time-Domain for your case.

If you have any interesting developments, kindly update.

Correct me if I am wrong, but in anomaly detection/predictive maintenance we observe the abnormal behaviour only after it actually occurs but my intention is to predict the time when we may observe this abnormality (ahead of time, possibly asap). Thanks!