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Fundamental Relationship between Averages
Timeaverage , delayedaverage , movingaverage and delayedmovingaverage have several interesting relationships.
Relationship of TimeAverage and Movingaverage
When we set the length of kernel equal to the length of the sequence, that is , the movingaverage is equal to the timeaverage (tag=movag)
(13)
Proof:
QED
Relationship of DelayedAverage and DelayedMovingaverage
If we set the kernel length of delayedmovingaverage into , the delayedaverage is equal to the delayedmoving average (dag=demovag)
(14)
Proof:
QED
Relationship of DelayedAverage and Movingaverage and TimeAverage
From the average decomposition diagram (ADD), we can decompose the average of whole sequence into two parts of weighted averages
I call Equation (15) as the fundamental theorem of average : timeaverage is equal to the sum of lengthweighted delayedaverage and movingaverage.
Proof:
Multiply both side of equation (15) with we have
(15a)
Expanding the right hand side of equation (15a) we have
QED.
From equation (15) we can derive what I call as Shift Property of Average
(16)
Notice that the right hand side of (16) is evaluated for time while the left hand side is evaluated at time
Based on equation (15), we can also derive a ratio of difference. The ratio between the difference of moving average to delayed average and the difference of average to delayed average is equal to the ratio between the delay and the length of sequence in consideration. (relationship of movag, dag and tag)
(17)
Proof:
Taking the difference of average and delayedaverage multiplies by its own length, we have
because .
Expand the second term in the left hand side and put the right hand side to the left we get
Gather the same terms
Then,
QED
Relationship of DelayedAverage and DelayedMovingAverage and TimeAverage
We can also decompose the measurement sequence into three parts of lengthweighted averages
From the average decomposition diagram (ADD), we can decompose
(18)
Combining equation (15) and (18) we have
Removing redundant last term of both side gives
(19)
Equation (19) is the relationship of delayedaverage, time average and delayed moving average
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