Piecewise Logistic Function [Satellite Data]

**teoxu** · 07-05-2015, 12:15 PM

Hello everyone,

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

Formula:

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where:

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

The literature provides some ambiguous definition as to which component is a constant, but I am assuming that the EVIb value is a constant but am supposing that we need possibly one or 2 others to be constant to for the function to work properly?

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 would be able to advise me on which component of the equation above should be a constant and which should be a variable. I have attached the original paper source here 2003 Monitoring vegetation phenology using MODIS.pdf, with a description of the function on page 2, section 2. I have included a series of sample data for one year below.

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 data-set 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 have tried quite for quite some time now to properly model my data using the above equation but have so far been unsuccessful and am really hoping someone would be kind enough to shed some light!
Any help at all would be greatly appreciated. Thank you!

**shg** · 07-05-2015, 12:44 PM

If you plot your data, it doesn't look remotely like the curves in the paper, even considering multiple growth phases. Got a better dataset?

EDIT:

This

y(t) = (c / 1 + exp(a + b*t)) + EVIb

should be

y(t) = c / (1 + exp(a + b*t)) + EVIb

where c is the minimum value and c + EVIb is the peak value of the segment of interest (from min to max or max to min), and a & b are the parameters of the fit.

**teoxu** · 07-05-2015, 01:05 PM

Hi shg,

Thank you so much for your reply and for that edit. I do indeed have more data - I have included a better one at the end of this post. I understand that this formula is meant to be applied separately to the left and then right of the curve - split at the maximum value. However, what am struggling to understand is what the parameters of fit, as you mentioned above, refers to precisely? Would you be kind enough to insert one example of the use of the formula using the dataset below please? I have been trying it out for quite awhile now and don't seem to be getting anywhere with it.

Really appreciate your help! =)

0.0197
0.0082
0.0502
0.1172
0.3359
0.4533
0.6716
0.668
0.70955
0.7511
0.7721
0.7875
0.8207
0.7832
0.8055
0.7608
0.7582
0.8119
0.7224
0.5318
0.46
0.0054
-0.0071

**shg** · 07-05-2015, 04:01 PM

I assigned evenly-spaced days to the measurements, then used Solver to set aa and bb for the left half by minimizing RMS error, then repeated for the right half.

Row\Col	A	B	C	D	E	F	G	H
1		aa		3.1247	D1: Input		-24.5696	G1: Input
2		bb		-0.0463	D2: Input		0.0730	G2: Input
3		cc		0.8125	D3: =MAX($B$9:$B$20) - D4		0.8153	G3: =MAX($B$20:$B$29) - G4
4		dd		0.0082	D4: =MIN($B$9:$B$20)		0.0054	G4: =MIN(B20:B29)
5		RMS Err		0.0411	D5: {=SQRT(AVERAGE((D9:D20 - B9:B20)^2))}		0.0660	G5: {=SQRT(AVERAGE((G20:G29 - B20:B29)^2))}
6
7
8	Day	Value		Left Fit			Right Fit
9	1	0.00820		0.042407	D9: =D$3 / (1 + EXP(D$1 + D$2 * ($A9 - $A$9))) + D$4
10	19	0.05020		0.082805
11	37	0.11720		0.161514
12	55	0.33590		0.291394
13	73	0.45330		0.456478
14	91	0.67160		0.608631
15	109	0.66800		0.712556
16	127	0.70955		0.769864
17	145	0.75110		0.797791
18	163	0.77210		0.810580
19	181	0.78750		0.816270
20	199	0.82070		0.818768			0.820667	G20: =G$3 / (1 + EXP(G$1 + G$2 * ($A20 - $A$9))) + G$4
21	217	0.78320					0.820578
22	235	0.80550					0.820247
23	253	0.76080					0.819017
24	271	0.75820					0.814474
25	289	0.81190					0.798015
26	307	0.72240					0.742265
27	325	0.53180					0.589476
28	343	0.46000					0.335169
29	361	0.00540					0.131299

**teoxu** · 07-06-2015, 07:15 AM

Hi shg,

Thank you so very much for that - finally understanding how to apply the formula properly!

Just one last question regarding aa and bb. What do they actually represent? I do have other sets of data I will be working through and was wondering how else I can derive them? Do they represent

Thanks so much!

**shg** · 07-06-2015, 08:17 AM

I did the parameterization differently than the article. The article said a+bx, I did a+b(x-x0). Corrected below.

Row\Col	A	B	C	D	E	F	G	H
1		aa		3.1710	D1: Input		-24.6422	G1: Input
2		bb		-0.0463	D2: Input		0.0730	G2: Input
3		cc		0.8125	D3: =MAX($B$9:$B$20) - D4		0.8153	G3: =MAX($B$20:$B$29) - G4
4		dd		0.0082	D4: =MIN($B$9:$B$20)		0.0054	G4: =MIN(B20:B29)
5		RMS Err		0.0411	D5: {=SQRT(AVERAGE((D9:D20 - B9:B20)^2))}		0.0660	G5: {=SQRT(AVERAGE((G20:G29 - B20:B29)^2))}
6
7
8	Day	Value		Left Fit			Right Fit
9	1	0.00820		0.042407	D9: =D$3 / (1 + EXP(D$1 + D$2 * $A9)) + D$4
10	19	0.05020		0.082805
11	37	0.11720		0.161514
12	55	0.33590		0.291394
13	73	0.45330		0.456478
14	91	0.67160		0.608631
15	109	0.66800		0.712556
16	127	0.70955		0.769864
17	145	0.75110		0.797791
18	163	0.77210		0.810580
19	181	0.78750		0.816270
20	199	0.82070		0.818768			0.820667	G20: =G$3 / (1 + EXP(G$1 + G$2 * $A20)) + G$4
21	217	0.78320					0.820578
22	235	0.80550					0.820247
23	253	0.76080					0.819016
24	271	0.75820					0.814473
25	289	0.81190					0.798014
26	307	0.72240					0.742263
27	325	0.53180					0.589473
28	343	0.46000					0.335168
29	361	0.00540					0.131300

I can't give you a straightforward physical interpretation of a and b. The sign of b dictates a rising or falling function, and its size controls the speed of the transition. The values of a and b together control the onset.

There's a good general discussion of the parameterized logistics function in wikipedia.

**teoxu** · 07-16-2015, 12:17 PM

Hi shg,

Thank you so much for your previous post. I have since been trying to obtain the aa and bb values but so far not gotten anywhere with getting them with solver. I'd just like to ask how you did it and what the settings in solver were that you used to obtain the values?

Thank you so much!

**shg** · 07-16-2015, 12:23 PM

The solver model is already set up in the posted workbook for the right side fit. Change the values of G1 & G2 to something else, then solve.

**shg** · 07-17-2015, 01:45 PM

If you are still struggling with this, I have a suggestion for a more intuitive parametrization that will allow you to set initial values by observation of the data. See attached.

Piecewise Logistic Function [Satellite Data]

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