involving stations which have been set up recently in SPI mapping
You have to start somewhere, so as long as you have more than about 5 or 6 time-periods as a basis for computing the mean and standard deviation, I would recommend just using whatever data you have to compute an initial estimate of the mean and standard deviation of precipitation for each of your stations. You can use those values then to compute the SPI for any single time-period. As time goes on and more data are collected, the underlying or long-term means and std. deviations can be re-computed to provide more accurate measures of the "background" climate that the SPI is based on. When presenting your initial calculations, just include the proviso that the underlying time-period is very short - simply say exactly how long it is, and let people evaluate the index accordingly.
In addition to the excellent approach mentioned by Enda William O'Brien, which involves statistical adjustments based on available data, it is also possible to adopt approaches that take into account the scarcity of available data. One such approach is spatial interpolation, which is highly useful for estimating precipitation values in locations with insufficient data. By utilizing information from nearby stations with more complete records, it is possible to fill in the data gaps of recently installed stations.
Furthermore, historical data analysis from nearby stations with longer precipitation records can be employed to infer precipitation patterns in the area and fill in the data gaps of the new station, allowing for a better understanding of the climatic context in which the Standardized Precipitation Index (SPI) is being calculated.
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