# How to increase time resolution of a time series on specific days?

Suppose I have a time series which has a frequency of 1 observation every 3 days.

I'd like to have a value for each day, that is increase the time resolution of the time series on specific days. Note: I don't need to do this for all the time series.

How can I do this? At first I thought I could somehow interpolate the time series but since it is not necessarily linear (nor quadratic or polynomial) with time this trick might not work.

Let's consider a time series which is a sum of sinusoidal and cosinusoidal terms as an example. Note that my original time series (which I can't share here) might not be that smooth... Suppose the observations are every 3 days and I'd like to find the value of the time series on the 10th day.

t <- 1:30
tsv <- sin(t) + cos(2*t)
plot(t, tsv, type="l", ylab = "tsv", xlab="t")
points(tsv, pch="o", col="green")


Again, note that t is 3 days. That is, when t=10, it is the 30th day of observation.

The 10th day of observation should be between t=3 and t=4. What statistical method could be used to calculate the value of tsv for the 10th day?

Personally, as a first try I'd use:

1. Geometrical "interpolation" by taking the line that intersects the two values of tsv at t=3 and t=4. Very rough approximation that assumes linearity of the time series between 3 and 4 and then calculate the value on the 10th day.
2. Linear approximation of tsv by taking the numerical derivative of tsv in t=3 and then calculating $\Delta tsv = f'(t=3) \Delta t$

These methods do not seem very "statistical" based to me but they seem more on the math side of things. What statistical method could (or should I) use?