# Is there a safe and simple way to estimate a standard deviation for a next subset?

In case I receive only standard deviation from a sensor of a value $$v$$ (that is btw normally distributed) each 4th minute but need to provide a standard deviation $$\sigma$$ for each 15 minutes is there a safe way to do it.

There are two things that came into my mind:

1) One and safe way is to get the mean, generate possible values using standard deviation of the 4 minute interval for the 15 minutes period (15*60 values). Calculate the $$\sigma$$ for this period

2) Alternatively one can naively estimate the value of $$\sigma$$ of the next time interval based on two previous values. For example, use and standard deviations to estimate

In case the standard deviation is increasing/descreasing in previous cases and it will increase/descreasing in the next time interval on absolute value -

The first method can be time-consuming/computationally-consuming comparing to the second method. Though the second method may suffer on precision.

Edit 16.04: Since I'm limited in the amount of data i preferably would use only the last standard deviation and no mean data

Edit 23.04: There is one more way that bring me to the result very close to 1st way of problem solving.

Let say $$\sigma_i$$ is based on $$n$$ observations while $$\sigma_{i+1}$$ is based on $$k$$ observations and $$k > n$$. Then $$\sigma^2_{i+1} = \frac{(n-1) * \sigma^2_i * \frac{k}{n}}{k-1}$$

The benefit in this case that you are not dealing with a mean value. I suppose that this solution works well only with normally distributed values.

• Is the number of values you get per 4-minute interval constant or does it vary? Apr 15 '19 at 14:42
• They can vary but not significantly
– zina
Apr 15 '19 at 15:17

Basically, the two methods you are proposing are the same.

The first one is computationally more consuming, but they are the same.

In the first method you are calculating $$\sigma$$ generating possible values of random variable with already the same $$\sigma$$ you have had historically. This is the same as calculating $$\sigma$$ with all the historical data you have.

In the second method you are doing a stimation with limited data, this is the correct way unless you have sufficient amount of data to estimate a GARCH model.

A GARCH model is a statistical model for time series data that describes the variance of the current error term or innovation as a function of the actual sizes of the previous time periods' error terms.

Meaning: $$\sigma_t^2 = w+\alpha_1\epsilon_{t-1}^2+...+\alpha_q\epsilon_{t-q}^2+\beta_1\sigma_{t-1}^2+...+\beta_p\epsilon_{t-p}^2$$

This model requires a sufficient amount of data and knowledge on time series analysis, of the two options you posted, I would use the second with as much data as possible.

• Juan, thanks for the quick answer. I will try the model and get back to you. Need to check then what coefficients to use...
– zina
Apr 15 '19 at 15:21