Methods of modelling systems or performing predictive maintenance of systems, such as lithographic systems and associated lithographic systems
A method for determining a causal relationship between events in a plurality of parameter time series, the method including: identifying a first event associated with a parameter excursion event; identifying a second event associated with a failure event, wherein there are a plurality of events including the first events and second events; determining values of transfer entropy for pairs of the events to establish a causal relationship for each of the pairs of events; using the determined values of transfer entropy and identified causal relationships to determine a process network, wherein each of the events is a node in the process network, the edges between nodes being dependent upon the values of transfer entropy; identifying a directed cycle within the plurality of events and the causal relationships; classifying a directed cycle; and classifying one or more events having a causal relation to the classified directed cycle.
1. A method comprising:
identifying at least a first event associated with a parameter excursion event;
identifying at least a second event associated with a failure event, wherein there are a plurality of events in a plurality of parameter time series associated with an industrial process, the plurality of events comprising a plurality of the first events and second events;
determining values of transfer entropy for pairs of the events to establish a causal relationship for each of the pairs of events;
using the determined values of transfer entropy and identified causal relationships to determine a process network, wherein each of the events is a node in the process network, the edges between nodes being dependent upon the determined values of transfer entropy;
identifying one or more directed cycles within the plurality of events and the causal relationships;
classifying a directed cycle of the one or more directed cycles, based on a nominal system behavior; and
classifying one or more events having a causal relation to the classified directed cycle based on the cycle classification.
2. The method as claimed in claim 1 , wherein the classifying one or more events comprises classifying one or more of the first events as a root cause event for one of the second events.
3. The method as claimed in claim 1 , wherein a parameter excursion event is identified as one or more selected from: a spike, a step, a change in trend, a change in standard deviation, a change in standard deviation relative to trend, and/or a control limit violation in the corresponding parameter time series.
4. The method as claimed in claim 1 , wherein each subsection of the process network which comprises a node representing a parameter excursion event leading to a node representing a failure event is determined to be a predictive model.
5. The method as claimed in claim 4 , further comprising managing unattended alerts generated by one or more of the predictive models based on detection of one or more parameter excursion events, by:
obtaining a cost metric relating to a measure of a cost of attending to each alert generated;
obtaining a benefit metric relating to a measure of a benefit in attending to each alert generated as a function of time; and
performing a management action on the unattended alerts based on an evaluation of the benefit metric against the cost metric for each alert.
6. The method as claimed in claim 1 , further comprising identifying one or more failure paths from a parameter excursion event to a failure event.
7. The method as claimed in claim 6 , wherein one of the failure paths comprises a plurality of parameter excursion events, and further comprising validating an occurrence of an earlier parameter excursion event on the failure path with a subsequent parameter excursion event on the failure path.
8. The method as claimed in claim 1 , further comprising:
an initial step of determining context data relating to a context in which the industrial process is operating from the parameter time series, wherein the process is operable in at least one of a plurality of contexts at any one time; and
applying a quality weighting to the context data, the quality weighting being dependent upon a measure of the accuracy of the context data for a particular context segment, each context segment comprising a segment of one of the contexts, wherein each context is segmented temporally.
9. The method according to claim 8 , wherein the quality weighting is applied to each context segment of the context data in further dependence of the preceding and/or succeeding context segment.
10. The method according to claim 8 , wherein the quality weighting determines the degree of consideration accorded to the context data and/or an event to which the context data corresponds when determining the causal relationship between events.
11. The method according to claim 10 , wherein the quality weighting is binary such that the context data and/or the event to which the context data corresponds is weighted to be either included or not included in determining the causal relationship between events.
12. The method according to claim 8 , wherein the context comprises an operational mode or operational state of the system and/or process.
13. The method as claimed in claim 1 , further comprising determining the time of one or more of the parameter excursion events by, for the parameter time series comprising the parameter excursion event;
applying a causal filter to the parameter time series to obtain first filtered time series data;
applying an anti-causal filter to the parameter time series to obtain second filtered time series data; and
combining the first filtered time series data and the second filtered time series data to determine a corrected time for the parameter excursion event.
14. The method as claimed in claim 1 , wherein the plurality of parameter time series comprises parameter excursion time series comprising one or more of the parameter excursion events and failure parameter time series comprising one or more of the failure events, wherein the parameter excursion time series and failure time series having variation between them in matching quality of their data, and the method further comprises repeatedly matching and filtering events from each of the parameter excursion time series and failure time series to obtain matched events using successively less strict matching rules, the matching rules being defined based on the variation in the matching quality.
15. The method as claimed in claim 1 , wherein the industrial process comprises a lithographic process for applying a pattern on a substrate.
16. A non-transitory computer-readable medium comprising instructions stored therein, the instructions, upon execution by a computer system, configured to cause the computer system to at least:
identify at least a first event associated with a parameter excursion event;
identify at least a second event associated with a failure event, wherein there are a plurality of events in a plurality of parameter time series associated with an industrial process, the plurality of events comprising the first event and second event;
determine values of transfer entropy for pairs of the events to establish a causal relationship for each of the pairs of events;
use the determined values of transfer entropy and identified causal relationships to determine a process network, wherein each of the events is a node in the process network, the edges between nodes being dependent upon the determined values of transfer entropy;
identify one or more directed cycles within the plurality of events and the causal relationships;
classify a directed cycle of the one or more directed cycles, based on a nominal system behavior; and
classify one or more events having a causal relation to the classified directed cycle based on the cycle classification.
17. The computer-readable medium as claimed in claim 16 , wherein the classification of one or more events comprises classification of one or more of the first events as a root cause event for one of the second events.
18. The computer-readable medium as claimed in claim 16 , wherein a parameter excursion event is identified as one or more selected from: a spike, a step, a change in trend, a change in standard deviation, a change in standard deviation relative to trend, and/or a control limit violation in the corresponding parameter time series.
19. The computer-readable medium as claimed in claim 16 , wherein each subsection of the process network which comprises a node representing a parameter excursion event leading to a node representing a failure event is determined to be a predictive model.
20. A lithographic apparatus comprising:
an illumination optical system arranged to illuminate a pattern;
a projection optical system arranged to project an image of the pattern onto a substrate; and
the non-transitory computer-readable medium of claim 16 , wherein the plurality of events pertain to operation of a lithographic process by the lithographic apparatus.