In this section we describe some commonly used terms in prognostics. Similar terms have been used interchangeably by different researchers and in some cases the same term has been used to represent different notions. This list is provided to reduce ambiguities that may arise by such non-standardized use. Further, some concepts on incorporating uncertainty in prognotics for decision making have been introduced. These methods are used in the new performance metrics.

- Prognostics is condition based health assessment that includes detection of failure precursors from sensor data, prediction of RUL by generating a current state estimate and using expected future operational conditions for a specific system.
- A suitable diagnostic algorithm correctly detects, identifies and isolates the system fault before it triggers a PA to predict evolution for that specific fault mode.
- If the information about future operational conditions is available it may be explicitly used in the predictions. Any prediction, otherwise, implicitly assumes current conditions would remain in the future and/or variations from current operating conditions do not affect the life of a system.
- RUL estimation is a prediction/ forecasting/ extrapolation process.
- Algorithms incorporate uncertainty representation and management methods to produce RUL distributions. Point estimates for RUL may be generated from these distributions through suitable methods when needed.
- RtF data are available that include sensor measurements, operating condition information, and EoL ground truth.
- A definition of failure threshold is available that determines the EoL for a system beyond which the system is not recommended for further use.
- In the absence of true EoL (determined experimentally) statistical (reliability) data such as MTTF (Mean Time to Failure) or MTBF (Mean Time Between Failures) may be used to define EoL with appropriate caution.

Figure 1. Illustration depicting some important prognostic time definitions and prediction concepts.

**Time Index:**The time in a prognostics application can be discrete or continuous. We will use a time index *i* instead of the actual time, e.g., *i*=10 means *t _{10}*. This takes care of cases where sampling time is not uniform. Furthermore, time indexes are invariant to time-scales.

**Time of Detection of Fault:** Let D be the time index (*t _{D}*) at which the diagnostic or fault detection algorithm detected the fault. This process will trigger the prognostics algorithm which should start making RUL predictions shortly after the fault was detected as soon as enough data has been collected. For some applications, there may not be an explicit declaration of fault detection, e.g., applications like battery health management, where prognosis is carried out on a decay process. For such applications

**Time to Start Prediction:** We will differentiate between the time when a fault is detected (*t _{D}*) and the time when the system starts predicting (

Figure 2. Features and conditions for l^{th} UUT

**Prognostics Features:** Let *f _{n}^{l}*(

**Definition 5 - Operational Conditions:** Let be an operational condition at time index i, where m = 1, 2, … , M is the condition number, and l = 1, 2, … , L is the UUT index. The operational conditions describe how the system is being operated and are sometimes referred to as the load on the system. The conditions can also be referred to as a vector Cl(i) of the lth UUT at time index i.

**Definition 6 - Health Index:** Let be a health index at time index i for UUT l = 1, 2, … , L. h can be considered a normalized aggregate of health indicators (relevant features) and operational conditions.

**Definition 7 - Ground Truth:** Ground truth, denoted by the subscript *, represents our best belief of the true value of a system variable. In the feature domain may be directly or indirectly calculated from measurements. In the health domain, is the computed health at time index i for UUT l = 1, 2, … , L after a run to failure test. This health index represents an aggregate of information provided by features and operational conditions up to time index i.

**Definition 8 – History Data:** History data, denoted by the subscript #, encapsulates all the information we know about a system a priori. Such information may be of the form of archived measurements or EOL distributions, and can refer to variables in both the feature and health domains represented by and respectively.

**Definition 9 - Point Prediction:** Let be a point prediction of a variable of interest at time index i given information up to time tj, where tj ≤ ti. for i = EOL represents the critical threshold for a given health indicator. Predictions can be made in any domain, features or health. In some cases it is useful to extrapolate features and then aggregate them to compute health and in other cases features are aggregated to a health and then extrapolated to estimate RUL.

**Definition 10 - Trajectory Prediction:** Let be the trajectory of predictions at time index i such that =

Figure 3. Illustration showing a trajectory prediction. Predictions may modify every time instant and hence the corresponding RUL estimate.

**Definition 11 - RUL:** Let *r ^{l}(i)* be the remaining useful life estimate at time index i given that the information (features and conditions) up to time index i and an expected operational profile for the future are available. RUL is computed as the difference between the predicted time of failure (where health index approaches zero) and the current time

**Definition 12 - RUL vs. time Plot:** RUL values are plotted against time to compare with RUL ground truth (represented by a straight line). As illustrated below, this visually summarizes prediction performance as it evolves through time. This plot is the foundation of prognostic metrics developed in subsequent sections.

Figure 4. Comparing RUL predictions from ground truth (t_{p} [70,240], *t*_{EOL}= 240, *t*_{EOP} > 240).

Prognostic estimates of RUL include uncertainties from various sources and often represent them as probability distributions. While, there are other methods to represent uncertainty, the discussion here is restricted to use of probability distributions for consistency and a concise description. Generally a common practice has been to compute mean and variance estimates of these distributions while assessing performance. However, in reality these distributions are rarely smooth or symmetric, thereby resulting in large errors. It is, therefore, suggested that other estimates of central tendency (location) and variance (spread) be used instead of mean and standard deviation, which are appropriate only for Normal cases. For instance, the situations were normality of the distribution cannot be established, it is preferable to rely on median as a measure of location and the quartiles or Inter Quartile Range (IQR) as a measure of spread. Here these distributions are broadly categorized into four classes and corresponding methods are suggested to compute more appropriate location and spread measures.

Table 1. Methodology to select location and spread measures for various distribution types.

For purposes of plotting and visualizing the data, use of error bars and box-plots is suggested as shown below.

Figure 5. Visual representation of distributions using boxplots and error bars.

In order to account for information from predicted distributions, it is suggested to specify an allowable error bound around the point of interest and then one could use the total probability of failure within that error bound instead of basing a decision on a single point estimate. This concept is shown below. It must be noted that this error bound may be asymmetric for prognostics, since it is often argued that an early prediction is preferred over a late prediction.

Figure 6. Concept of computing the total probability mass within desired accuracy levels.

A formal way to include this probability mass into the analytical framework is by introducing a β-criterion, where a prediction is considered inside α-bounds only if the probability mass of the corresponding distribution within the α-bounds is more than a predetermined threshold β. This parameter is also linked to the issues of uncertainty management and risk absorbing capacity of the system. A more systematic approach is summarized in the flowchart below. Table 1 above guides in identifying the class of distribution for these calculations.

Figure 7. Steps to compute a more robust point estimate from probability distributions.