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![Hazard function
h(t) = 𝖯[T= t | T ≥ t]
Probability failure will happen at time t given that we know
it survived up to time t.
Can be interpreted as a measure of the hazard risk, aka.
the probability that failure would happen right now.](https://image.slidesharecdn.com/deeptime-to-failure-181123095551/75/Deep-time-to-failure-predicting-failures-churns-and-customer-lifetime-with-RNN-by-Gianmario-Spacagna-Chief-Scientist-at-Cubeyou-AI-15-2048.jpg)
![Cumulative Hazard function
Λ(t)=∫[0 to t] h(z) dz
Represents the integral of the hazard-rate.](https://image.slidesharecdn.com/deeptime-to-failure-181123095551/75/Deep-time-to-failure-predicting-failures-churns-and-customer-lifetime-with-RNN-by-Gianmario-Spacagna-Chief-Scientist-at-Cubeyou-AI-16-2048.jpg)
Uploaded byData Science Milan
Deep time-to-failure: predicting failures, churns and customer lifetime with RNN by Gianmario Spacagna, Chief Scientist at Cubeyou AI
1. The document discusses using deep learning models like recurrent neural networks to predict time-to-failure events from time series data. It specifically focuses on a technique called Deep Time-to-Failure which extends a Weibull Time-to-Event Recurrent Neural Network to predict a single failure event. 2. As a case study, the technique is applied to predict failure times of NASA jet engines using sensor data as inputs. The model is trained on historical sequences of data to learn the distribution of time-to-failure and can provide probabilistic predictions and confidence intervals. 3. Key aspects of the Deep Time-to-Failure approach include using censored and uncensored training data, consuming raw time series as input
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Similar to Deep time-to-failure: predicting failures, churns and customer lifetime with RNN by Gianmario Spacagna, Chief Scientist at Cubeyou AI
Deep time-to-failure: predicting failures, churns and customer lifetime with RNN by Gianmario Spacagna, Chief Scientist at Cubeyou AI
- 1. Deep Time-to-Failure Gianmario Spacagna IBM#PartyCloud - Data Science Milan meetup 20 September 2018 @ Spirit of Milan
- 2. Companies main assets MachineriesCustomers
- 3.
- 4. Data availability (machineries) ▫Historical time-series of machines sensors telemetry ▫ Registered failures ▫ Active machines real-time measurements
- 5.
- 6. Data availability (customers) ▫Historical sequence of customers behaviours ▫ Stopped subscriptions ▫ Active customers daily activities
- 7.
- 8. Time-to-failure events ▫ Patientsexpected life ▫ Political leadership duration ▫ Product wearing out ▫ Financial default ▫ Employees quitting
- 9. Data scarcity ▫ Sensorsfailure (not the machine or user device) ▫ Loss of connectivity ▫ Errors in data collection ▫ IT blackouts ▫ Dropouts ▫ Study termination ▫ Active machine/users not “failed” yet
- 10. First part Traditional SurvivalAnalysis
- 11. Goals 1. Build amodel that can estimate the probability distribution of remaining time-to-failure 2. Train the model by exploiting censored historical data.
- 12.
- 13. Survival Function S(t)=Pr(T>t) T =failure event time Probability that the time-to-failure is greater than t
- 14. Kaplan-Meier estimator d_i: eventshappened at time t_i n_i: known survived individuals at time t_i
- 15. Hazard function h(t) =𝖯[T= t | T ≥ t] Probability failure will happen at time t given that we know it survived up to time t. Can be interpreted as a measure of the hazard risk, aka. the probability that failure would happen right now.
- 16. Cumulative Hazard function Λ(t)=∫[0to t] h(z) dz Represents the integral of the hazard-rate.
- 17. Nelson-Aalen estimator Λ̂ (t)=∑ti≤t d_i n_i d_i: events happened at time t_i n_i: known survived individuals at time t_i
- 18. Survival Regression We havecovariates X that we would like to use to map each individual to his own survival/hazard function. Features: ● age ● gender ● weight ● is a smoker? ● weekly sport time? Model ● Survival function ● Hazard function
- 19. Cox’s Proportional Hazardmodel the log-hazard of an individual is a linear function of their static covariates and a population-level baseline hazard that changes over time
- 20. Aalen’s Additive model Thehazard rate is a linear function of multiple baselines weighted by their corresponding covariates
- 21. Cox’s Time VaryingProportional Hazard model If covariates change over time X(t)
- 22. Survival Regression Limitations ▫ TraditionalSurvival regression models can only handle categorical or static numerical attributes ▫ Cox’s Time Varying Proportional Hazard model cannot perform predictions because it would require to know the future values of covariates ▫ The models do not take into account the sequence of measurements but only the current statuses ▫ Non-parametric models are hard to generalize
- 23. Second part Embrace theWeibull and Deep Learning euphoria!
- 24. Weibull Distribution ⍺ =ƛ: scale parameter β = k: shape parameter
- 25. Universal PDF forscientists and engineers Source: https://ragulpr.github.io/2016/12/22/WTTE-RNN-Hackless-churn-modeling/#embrace-the-weibull-euphoria
- 26. Weibull Time-to-Event RecurrentNeural Net “An algorithm & philosophy about predicting when things will happen.” Egil Martinsson https://github.com/ragulpr/wtte-rnn
- 27.
- 28. WTTE-RNN architecture The outputlayer consists in the Weibull parameters
- 29. Loss function u =E = 1 censored u = E = 0 uncensored Source: https://ragulpr.github.io/assets/draft_master_thesis_martinsson_egil_wtte_rnn_2016.pdf
- 30.
- 31. Deep Time-to-Failure ▫ Extensionof WTTE for failure events ▫ Only one single event to predict (the failure) ▫ Prediction is done at the end of the observed sequence instead at each step Failure event ?
- 32. Case Study: NASAjet engines degradation
- 33.
- 34. Data preparation Selected 17relevant features Values normalized between -1 and 1 Maximum lookback period = 100 Each point corresponded to the subsequence between time 0 and t up to the failure event Shorter sequences masked with a special value -99
- 35. Train/test splits train_x (20631,100, 17) train_y (20631, 2) test_x (100, 100, 17) test_y (100, 2) X contains axis 0: subsequence identifier axis 1: time (100 steps) axis 3: covariate feature Y contains axis 0 (T’): latest observation time axis 1 (E’): 1 if failure event 0 otherwise
- 36. Training with CensoredData T = failure time T'(t) = min(T, t) E'(t) = if T <= t 1 (observed) else 0 (censored) Only the full sequence observe the failure, the other subsequences always have E’ = 0
- 37.
- 38.
- 39. RNN parameters ▫ stateful= False - each subsequence is independent ▫ return_state=False - only return the output of the recurrent layer ▫ return_sequences = False - despite WTTE which is True - we modeled the problem to give one prediction at the end of each subsequence instead
- 40. Initialize alpha andbeta The parameter alpha is proportional to the mean failure time so we initialize with the mean of observed failures Beta is a measure of variance so we set max_beta_value to 100 time steps
- 41. Output layer activations Replacedsoftplus functions from original paper with: alpha neuron (a): alpha = exp(a) * init_alpha beta neuron(b): beta= sigmoid(b) * max_beta_value
- 42.
- 43.
- 44. Tunable Hyper-parameters GRU: ▫ activation='tanh' ▫recurrent_dropout=0.25 ▫ dropout=0.3 Optimizer: ▫ algorithm: adam ▫ lr=0.01 (learning_rate) ▫ clipnorm=1
- 45.
- 46. Loss function overepochs ● Validation loss < training loss ● Test data contains only full sequences ● easier to predict ● higher accuracy ● lower loss
- 47. Training biases andweights
- 48. Inspecting Final RecurrentActivations Covariates subsequences Hidden activations Output 20 GRUs 2 output nodes E Ground truth T𝛃𝛂time t x1 x2 x3 Encoding of the ith subsequence in a 20-dim vector x1_i x2_i x3_i
- 49. time t Inspecting HiddenRecurrent States x1 x2 x3 20 recurrent states Output: T, 𝛂, 𝛃 Input covariates Encoding at time t_i t_i Subsequence at time t_i Weibull parameters at time t_i t_i t_i
- 50. Debugging nan weights Canhappen if: ▫ beta value too large (e.g. max_beta > 1000) ▫ if needs to reduce beta: - downscale the time axis - use shorter sequences ▫ If TensorFlow backend set epsilon to 1e-10 ▫ Add a clip_value of 0.5 or less to the optimizer ▫ Clip log-likelihood ▫ Pre-train the output layer (transfer learning)
- 51.
- 52. Evaluating last subsequences Weibulldistributions One curve for each engine Only last subsequence evaluated Dots represents expected mode (maximum likelihood)
- 53. 𝛃 (∝ variance)Vs. 𝛂 (∝ mean)
- 54.
- 55.
- 56. Single engine ttfprobability distribution over time t One engine All of the Weibull distributions of T estimated at each timestep
- 57. Single engine Tprediction over time T: time-to-failure t: current time
- 58. Conclusions We learnt atechnique deep-ttf which is the extension of wtte for the specific case of predicting single failures. The strengths of this approach are: ▫ Consuming raw time-series or sequences ▫ Training with censored and uncensored data ▫ Probabilistic predictions with confidence intervals ▫ Can be applied to any survival regression problem
- 59. References Tutorial link: https://github.com/gm-spacagna/deep-ttf NASAdata: https://c3.nasa.gov/dashlink/resources/139/ wtte-rnn: https://github.com/ragulpr/wtte-rnn Analysis of WTTE-RNN variants that improve performance, R. Cawley et al. Recurrent Neural Networks for real-time distributed collaborative prognostics, A. S. Palau et al.