Hi,

Here's a breakdown of how to interpret the relevant terms and calculate the α-Lambda performance metric:

True Remaining Useful Life (RUL):

- In prognostic scenarios, you might have access to historical data or labels indicating the failure times of similar systems.
- If you have this information, you can directly calculate the true RUL for each data point in your test set as the difference between the current time step (t) and the actual failure time (t_failure).

Estimated RUL:

- Once you've obtained the state posterior probabilities for a given time step (t), you can estimate the RUL using various approaches:

1) State-Averaged Mapping:

- Assign a pre-defined RUL value to each hidden state in your HMM. This value could be based on domain knowledge or historical data.
- Calculate the weighted average of these assigned RUL values using the state posterior probabilities as weights:

estimatedRUL_t = sum(pi_t(i) * RUL_i) for all states i

- Here, pi_t(i) is the posterior probability of being in state "i" at time "t", and RUL_i is the assigned RUL value for state "i".

2) Learning-Based Mapping:

- Train a separate regression model to map the state posterior probabilities to the estimated RUL.
- This model could be a simple linear regression, a neural network, or any other suitable regression technique.
- Train the regression model on a dataset where you have both the state posterior probabilities and the corresponding true RUL values.

α-Lambda Performance Metric:

- It's calculated as follows: alpha_lambda = 1 / N * sum( |estimatedRUL_t - trueRUL_t| / trueRUL_t) for all N data points in the test set
- A lower α-Lambda value indicates better prediction accuracy. It represents the average fractional error between the estimated and true RUL, expressed as a proportion of the true RUL.

Hope it helps.