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My data is constructed from monthly mean Sea level measurements from 1950 till 2020. Univariate data.

Some of the years/ months in 1960s / 1970s are missing and I would like to impute them. We know that there are seasonal effects and other cycles (lunar cycles etc)

I was aiming to use ARIMA for imputation, but since most of my missing data is at the beginning of my series, what would be the best method to handle it? can I train the model backward?

Also, any other ideas and information regarding missing data imputations that will fit my data type and task will be really helpful .

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You can back-cast for the initial missing values. See https://otexts.com/fpp2/backcasting.html for some R code to do this using the forecast package.

You can interpolate the interior missing values using the fitted ARIMA model via the imputeTS package.

The same ARIMA model should be used for both steps to ensure consistency.

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