Table 2 Comparison of data-driven stochastic model

Life prediction based on probability theory [32, 34]

Simple and easy to analyze;

Too much accumulated experience data required;

The initial design estimation;

Few parameters required and easy to get;

Only a general estimation without regard to the specific cases of working loads and failure;

Less important, small-sized, mass-produced components and parts.

Able to obtain confidence limits.

Inaccuracy prediction in terms of the individual.

Life prediction based on ARMA or ARIMA [81, 83]

Simple to model, methods mature, and widely used.

Unable to be used for long-term prediction;

Suitable for linear-time, invariant systems when the performance of its characteristics are smoothly changing.

No need for historical data or fully understanding the fault mechanism;

Unable to synthesize prior knowledge;

High efficiency in computing and allowed to be used online.

Sensitive to noise and the initial state; poor effects on the dynamic process of non-stationary.

Life prediction based on Logistic Regression (LR) [84,85,86]

No need to set the critical threshold;

Not feasible unless normal feature domain description and unpredictable behavior are both available;

Service performance evaluation for degradation failure equipment like machine tools.

Able to obtain the time dynamic characteristics without making too much assumptions for equipment failure process and the distribution function.

Unable to do real-time prediction with offline data.

Life prediction based on proportional hazards [87, 88]

Variety of status information are integrated into account;

Complex computation of parameter estimation involving numerical integration;

Risk analysis of the joint characteristic variables;

The characteristic index able to change with time.

Assumption that risks change with variables proportionally.

Analysis of data distribution, residual distribution unknown or censored data.

No need for fault history data;

Life prediction based on Neural Network [32]

Strong self-learning ability, non-linear fitting ability and good robustness;

No standard method to determine the structure of the neural network;

Pattern recognition or some certain signal processing.

Excellent memory ability and nonlinear mapping skill.

Sufficient sample data required for the parameter calibration;

Uncertainty of structure and weight.

Life prediction based on Bayesian Networks [89, 90]

Able to deal with uncertain problems.

Unable to model the unknown fault;

Used to solve uncertainty and relevance fault of complex equipment;

Able to avoid data over-fitting.

Difficulty in computing for the unknown network;

Widely used in the intelligent systems of computer intelligence science, industrial control, medical diagnostics and other fields.

Effective multi-source information fusion and expression;

Relies on reliability of the prior knowledge;

Able to visualize the dependency links among each pair of variables.

Results are sensitive to the selection of prior distribution.

Life prediction based on Support Vector Machine (SVM) [50, 91]

Concrete realization of the structural risk minimization criterion

Not suitable for large-scale data processing;

Small sample, nonlinear and high dimensional mode recognition, such as predicting porosity and clay content in oil well logging.

Simple structure, good promotion performance and fast learning;

Theoretical defects in Kernel function for nonlinear classification problems;

Only one minimal in optimization solution.

Slow solution speed.

Life prediction based on Markov Model /Hidden Markov Model (HMM) [60, 61, 92, 93]

Able to model different stages of degradation;

Unable to model the unknown fault;

Unable to directly observe the state, but able to observe the vector sequence;

No need for prior knowledge of fault mechanism, able to process incomplete data;

Complex hidden semi-Markov model is required if the fault time is not exponentially distributed;

Widely used in pattern recognition, speech recognition, behavior recognition, character recognition and fault diagnosis and other fields.

Predictions with confidence intervals.

The model calculation increases with the number of states.

State-space prediction based on Kalman Filter [93,94,95]

Able to do the linear unbiased minimum mean square error estimation;

State equation and measurement equation need to be defined;

Real-time online prediction for linear Gaussian system.

Prediction accuracy does not change with the prediction time interval; good robustness;

Noise level could influence the performance and stability of the algorithm;

Able to estimate the current state and also predict the future state.

Only works with linear system and Gaussian noise.

State-space prediction based on particle filter [93, 96,97,98]

By finding a random sample to carry out the approximate calculation for the probability density function to obtain the state minimum variance distribution;

Large amount of sampling are required to avoid degradation;

Real-time online prediction of strong-nonlinear, non-Gaussian noise system, such as radar tracking and robot localization.

High accuracy.

Computation is more complex than Kalman filtering;

High-dimensional system and increasing particles complex computation.