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So I have amplitudes following a normal distribution with $\mu = 0$ and $\sigma = 1$. Now a certain amplitude "live" for a fixed time and then a new one is drawn from the distribution. Due to physical reasons I now want to interpolate between these amplitudes, which lowers the standard deviation. How can I calculate this correction factor? Empirically it is ~1.22, but how can one calculate it?

Example:

enter image description here

Shall become:

enter image description here R code sample:

amplitudes_digital <- rnorm(10)
amplitudes_digital_stretched <- sapply(amplitudes_digital, rep, times=10)
# test3 <- matrix(nrow=ncol(test2)*nrow(test2), ncol=1)
amplitudes_digital_stretched2 <- stack(as.data.frame(amplitudes_digital_stretched), select=1:10)
plot(c(1:nrow(amplitudes_digital_stretched2)),
       amplitudes_digital_stretched2[,1], xlab="Time [s]", ylab="Value")

amplitudes_interpolated <- approx(x=c(0:9)*10+1,
                                  y=amplitudes_digital, method="linear", 
                                  n=nrow(amplitudes_digital_stretched2)-length(amplitudes_digital)+1)
points(amplitudes_interpolated$y, col="red")
mtext(paste("SD black points:", sd(amplitudes_digital_stretched2[,1]), "\nSD red points:",
        sd(amplitudes_interpolated$y) ) )
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  • $\begingroup$ Could you explain what the "live for a short time" really means? What specifically are you measuring or observing? That ought to play an important role in suggesting any reasonable solution. $\endgroup$
    – whuber ♦
    Commented May 29, 2020 at 21:45
  • $\begingroup$ @whuber well, I am simulating random voltage fluctuations with a standard deviation of 1 and a mean of 0 (electrical noise). A certain voltage decays after a specific time, which I called "living for a short time". To make the transitions between my dice-rolled voltages smooth, I want to do a linear interpolation as the most simple transition. However, as you can see above, this changes the standard deviation. $\endgroup$
    – Christian
    Commented May 30, 2020 at 17:33
  • $\begingroup$ I still don't see enough information to understand your data. What do the individual dots mean? How many measurements do you actually have? Do you have a physical theory to describe the voltage decay? All this information would be useful for formulating good, relevant answers. $\endgroup$
    – whuber ♦
    Commented May 30, 2020 at 19:12
  • $\begingroup$ The individual dots are discrete data points for certain times. The physical theory is what I tried to explain in my first comment: I know that it is a Gaussian distribution and I know the mean and standard deviation. I also know that the time behavior is smooth. The total amount of data points is much larger than depicted here. $\endgroup$
    – Christian
    Commented May 30, 2020 at 20:49
  • $\begingroup$ If the time behaviour is known to be smooth, why does the data show the contrary? Can you explain that disparity, because it might help us understand the dynamics of your question. $\endgroup$
    – Ben
    Commented Dec 16, 2020 at 22:32

1 Answer 1

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A suggestion, employ interpolation plus a randomization component. This is likely required to effectively address one (or more) of what I would frame as a missing data connection link.

Here is a source discussing various methods in use:

Sixty-eight papers (83%) described how they dealt with missing data in the analysis. Most of the papers excluded participants with missing data and performed a complete-case analysis (n = 54, 66%). Other papers used more sophisticated methods including multiple imputation (n = 5) or fully Bayesian modeling (n = 1). Methods known to produce biased results were also used, for example, Last Observation Carried Forward (n = 7), the missing indicator method (n = 1), and mean value substitution (n = 3). For the remaining 14 papers, the method used to handle missing data in the analysis was not stated.

Currently, you are apparently employing a bias 'Last Observation Carried Forward' methodology, where the carried forward value employed here is, more precisely, been interpolated adjusted.

A conservative approach, which elevates the variance, is exclusion (adjust the time series, excluding data gaps).

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