# Application of a Piecewise Logistic Function to Extract Seasonality Metrics

I am working with 16-day MODIS EVI (satellite) data and I want to fit a Piecewise Logistic Function through my 23 EVI data values. The following formula is for the Piecewise Logistic Function:

y(t)= [c / (1 + e^(a+bt))] + EVIb

where: t is the day of year
a is vegetation growth time
b is rate of leaf development
c is the amplitude
bEVI is the background EVI value [constant]

For some reason there is very little clarification about each component of the equation in most papers. I am assuming that the bEVI value is a constant but am supposing that one or two other components are constants too?

I have little experience with this function, but from the literature I have read, is applied separately to the left and then subsequently to the right of this bell-shaped curve - split at the maximum y value. I am hoping someone here with a stronger mathematical background and experience using the piecewise logistic function would be able to advise me on which component of the equation above should be a constant and which should be a variable.

I tried embedding the original journal paper that I have on my OneDrive but can't yet seem to figure out how to do it. I have, however, included the link here "Monitoring Vegetation Phenology using MODIS (Zhang et al., 2003)". You may require access to Science Direct but please do let me know if you have any problems with that. The piecewise logistic regression is mentioned on page 472, section 2.

I contacted the original author regarding the equation and he clarified that the function if applied separately for the left and right side of the curve, split at the maximum point, which could most likely be somewhere in the middle of the curve.

The series of days of the year are as follows:
1
17
33
49
65
81
97
113
129
145
161
177
193
209
225
241
257
273
289
305
321
337
353

Here is an example EVI dataset I currently have for one year of 23 data points:
0.425
0.3899
0.2417
0.4724
0.4648
0.226
0.245
0.2509
0.2741
0.3711
0.3914
0.3985
0.4433
0.397
0.3329
0.3578
0.2741
0.2912
0.3053
0.2995
0.0959
0.3287
-0.3

• I believe you mean t, not y, is the date. Presumably y is an EVI. A piecewise logistic regression as you have described it requires six independent parameters: two growth rates, two growth rate offsets, one amplitude, and a break time. For just 23 values that's overfitting. Repeating it for each of many cells will be a gross overfitting, especially in light of the large variability exhibited in your example dataset. These considerations suggest being more parsimonious and fitting a model with fewer parameters--preferably just one or two. Or should a, b, c be constant over the image? – whuber Jul 3 '15 at 17:14
• Hi @whuber, thanks for that. I have made edits as above. I am currently trying to work out how the equation is meant to be applied. I understand from my recent email with the author that the function is applied once to the left of the curve, and then again to the right of the curve - split at the maximum EVI value. What I am uncertain about is how to apply it. Should I be applying the equation only once to each side of the curve (L then R), or do I apply the equation multiple times for every pair of points on each side of the curve (L then R)? – Xu Teo Jul 4 '15 at 12:29