FSSA Detects Decade Long Oscillation that MSSA Cannot

The following plot displays the kernel density estimation functional time series results of the NDVI data.

We run FSSA on the NDVI data and obtain the following plots that suggest we should have three groups for reconstruction. The first group used in reconstruction uses only the first component, the second group should use components two and three, and a fourth group that only uses reconstruction four.

We see a periodicity here of the 16 day periods between capturing images from the NDVI data

Here is the reconstruction using only the first components which gives us trend

Here we see oscillatory behavior in reconstruction that used components 2 and 3 which captures the 16 day periodicity

This is the interesting component that captures roughly decade long oscillations in the data.

This fourth interesting component is telling us that there is a decade long oscillation in the distribution of pixel intensities which indicates a change in vegetation. We see that if time is greater than about 200 days, then the probability we observe a pixel that is indicative of vegetative land cover (pixel value is between 0.5 and 1) is much less than the probability density that we observe non-vegetative land cover (pixel value between 0 and 0.5). Season is thought of to be the most interesting player in vegetative land cover changes (Lambin 1999), but our analysis shows that there might be more destruction of vegetative land cover in the Jambi region and while short term seasonality is at play, we detect that there is more long term destruction and loss of vegetative land cover. We also find that for large lag parameter L, MSSA can also detect this fourth component.

Trend

Oscillatory behavior

MSSA reconstruction with component 4 roughly approximates the same result that FSSA found.

The reason why MSSA is able to approximate the results of FSSA is because for large lag, we are stacking functional time series components on top of one another evaluated at points. In other words, the first column in the trajectory matrix will hold an approximation of day one in the first L (in this case L=200) entries and then next L entries in the first column will hold an approximation of day 16 (due to 16 day cycles) and so forth. So, for large L, MSSA is approximating FSSA but since the data is inherently functional, rather than approximating FSSA results with large L, the user should just use FSSA not MSSA.

FSSA is able to detect significant change in vegetation of the Jambi region over the period of a decade which gives clues into possible destruction of forested area and that seasonality is not only the most interesting factor in vegetative land cover as stated in Lambin 1999. We also found that MSSA is approximating FSSA results makes FSSA the algorithm of choice for this problem.