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I have a time series data as shown in the figure below where the X axis is the serial number of the day of the year form 1 to 365 where 1 is 1-Jan and 365 or 366 is 31-Dec. The Y axis represents the value of a quantity I am measuring on each day of the year since 2015.

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The the trend in the red portion of the graph are events due to Islamic religious festivals and clearly these events shifts forward each year by about two weeks since the lunar based Islamic calendar year is shorter than the solar based Gregorian calendar. The blue portion of the trend is the part of the year where Islamic calendar has little or no impact.

Question 1: Blue portion has its own seasonality, red portion has its own seasonality and red portion shifts forward by 2 weeks every year relative to the blue portion. What kind of time series technique is best suitable for modeling this kind of observations?

Question 2: If we are to use ARIMA, what modifications do we need?

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Might be useful to check out the FASSTER package in R. It has the capability of handling switching seasonality.

Another option could be to use dummy variables to denote which portion and which day of the portion you're in to capture seasonality and then ARIMA on the residuals.

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