Time weighted averages and integrals are used in cases where a time series is not evenly sampled. Time series data points are often evenly spaced, for example every 30 seconds, or every hour. But sometimes data points are recorded irregularly, for example if a value has a large change, or changes quickly. Computing an average using data that is not evenly sampled is not always useful.
For example, if you have a lot of ice cream in freezers, you need to make sure the ice cream stays within a 0-10℉ (-20 to -12℃) temperature range. The temperature in the freezer can vary if folks are opening and closing the door, but the ice cream only has a problem if the temperature is out of range for a long time. You can set your sensors in the freezer to sample every five minutes while the temperature is in range, and every 30 seconds while the temperature is out of range. If the results are generally stable, but with some quick moving transients, an average of all the data points weights the transient values too highly. A time weighted average weights each value by the duration over which it occurred based on the points around it, producing much more accurate results.
Time weighted integrals are useful when you need a time-weighted sum of irregularly sampled data. For example, if you bill your users based on irregularly sampled CPU usage, you need to find the total area under the graph of their CPU usage. You can use a time-weighted integral to find the total CPU-hours used by a user over a given time period.
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