#general

Title

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Farzad E

05/09/2023, 1:34 PMWhen using AutoARIMA or other methods in StatsForecast we get confidence intervals (CIs) for each individual forecast. Does anyone know how we can combine these CIs to get one CI for the average of all forecasted series? I am sure we cannot average CIs because calculation of confidence interval is not a linear operation. But I have never seen how bunch of CIs can be combined to represent one CI for the average series. This is more of a math question than related to StatsForecast but I thought someone here might have thought of this before.

f

fede (nixtla) (they/them)

05/09/2023, 6:56 PMHey **@Farzad E**! The underlying assumption behind some statistical models (such as ARIMA) is that the data they are modeling follows a normal distribution. To compute CI, the mean and the variance of such normal random variable are calculated for each timestamp in the horizon. So, you could compute those parameters (the mean corresponds to the point forecasts and you can calculate the standard deviation by computing a CI with a level of

`68.27`

). After that, you can compute the parameters of the average series (using the fact that the sum of normally distributed random variables is also a normal random variable).f

Farzad E

05/09/2023, 7:30 PMf

fede (nixtla) (they/them)

05/09/2023, 9:00 PMhey **@Farzad E**! Yes, that’s the intuition behind 68.27. Regarding the distribution of the average, if you have 100 series each of one with mean

`m_i`

, and standard deviation `sigma_i`

then the distribution of the average will have mean `(m_1 + m_2 + ... + m_100)/100`

and standard deviation `(sigma^2_1 + sigma^2_2 + ... + sigma^2_100) ^ (1/2) / 100`

(the root of the sum of the variances divided by 100).🙌 1

f

Farzad E

05/09/2023, 10:47 PM🙌 1