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Suppose you are given a set of time series, all with identical time stamps (a vector-valued time-series, if you will). These could be for example measurements of various medical metrics, such as blood pressure, heart rate etc of a single patient. For a pool of patients such time series are given. For each patient there is a certain designated time stamp, where an event happens (e.g., "patient dies"). If the task is to predict when patients die, what is is the (general) name of this data science problem?

Please notice: It is not "time series event prediction", since the fact that the event happens can't be read off the time series values (since the event is given as an additional "label" of a time point) and as far as I know event prediction refers to predicting events that can be read of the time series values (such as predicting the event "heart attack" which can be defined as the "heart rate dropping suddenly below a 20" - I'm not a doctor, so don't quote me on this number).

Could you give me references where I can find standard solutions how such a prediction, with an event given by a time label, can be made?

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Predicting the time until a given event occurs is survival analysis. In the medical field, the event is typically the patient's death, and it is predicted using medical data. For instance: how long do we expect a patient with a given medical condition and history to survive?

Classical survival analysis, especially in the medical field, does not really leverage time series information though, only snapshot data. Nevertheless, you may be able to find something by combining the two search terms.

A more recent field of study that may be of interest would be predictive maintenance. Here, we typically predict when a piece of machinery will fail, based on a time series of sensor data. The idea is to schedule maintenance before the machine breaks down, so we can arrange for maintenance, spare parts and a replacement machine more gracefully than if we wait for the machine to actually die on us.

The underlying problem is similar to survival analysis, but time series data are commonly used - as above, one typically uses data from the hundreds of sensors that modern machinery contains. (I saw an interesting presentation on predicting when electric train engines will fail, based on time series of power readings and similar data of their batteries.) One difference is that medical survival analysis typically is concerned with a single unambiguous type of "event", i.e., the patient's death, whereas predictive maintenance looks at many different possible ways a piece of machinery can break down.

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