Data Revival from the Statistics

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In the previous example, we have the following results:

Suppose we only stored all the time-average data. Can we get back the real measurement data only based on the stored time-average? Yes, we can revive the measurement data only based on the time-average, provided we know the sequence data number .

Remember that we have recursive formula (3) to compute the time-average. I write it again in here for your convenient:

                           (3)

Rearrange the equation (3) for  we have

                           (5)

As before, the subscript  start at 1, therefore  is undefined and we can put any number for it. Using equation (5) we can compute back the measurement data based only on two consecutive time-averages.

To restore data from the statistics using equation (5) above, we need to know the sequence data number . Suppose we do not know the sequence data number  but we only know time average and time variance of the data, can we revive the real measurement?

Yes, we can revive the real measurement data from two consecutive time average and time variance using the following formula

                                        (6)

where,

Using previous example, we have the statistics (time average and time variance) and we can revive the data. Obviously, we use quadratic formula to get the data, thus two possible value are the results.

Note that the only first measurement data is revived using positive sign of equation (6), while the others revival measurement data are obtained using the negative sign. This rule is true for any measurement data.

See proof of the quadratic formula here

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