64fc8b3e-4c60-4aaf-b941-90a2b520a557 34 4 5 fraction

&[DATE] &[TIME] - &[FILENAME]

T P char size color plotG n T length : plot T 1 el P 1 el char size color augment : i 2 n range plot plot T i el P i el char size color augment stack : for plot 41 line :

data 1227311420536658787810691381018611234122681335214396154891654517643186721975020819218542287523924249382594826968279752898229995301000302 mat :

Project note: The pure 'Logistic' model is insufficient to fit this data set. We add a bias term β4 to take care of the datum low region. The choice of the model is fundamental in data fitting. From the 'savant papers', statistical tests are generally irrelevant to appreciate the goodness of a fit. If you have a good model, you have 'the model'. Residuals are interesting to highlight the region(s) where the smpled data deviate from the model and as such may indicate something about the experiments [seasonal sales, population growth ...]. In a fitting session, it is most important to recognise and eliminate outliers.

X data 1 col :

Y data 2 col :

x β f β 1 el 1 β 2 el β 3 el x * - exp + / β 4 el + :

x β φ x β f x β f β 1 el diff x β f β 2 el diff x β f β 3 el diff x β f β 4 el diff 51 mat :

10005 0.3 041 mat

=>=> initial β

β 1030 4.27 0.28 15.8 - 41 mat :

L H N xd U 0 : dx H L - N / : i 1 N 1 + range U i el L dx - (dx i * + : for U 21 line 31 line :

n X length :

n 30

k x β φ length :

k 5

j 1 k range i 1 n range M i j el X i el β φ j 1 + el : for for

i 1 Y length range vx i el X i el β f : for

L X 1 el :

H X X length el :

N 29 :

U L H N xd :

Φ M transpose M * :

d M transpose Y vx - (* :

Δβ Φ 1 -^eval d * :

β β Δβ + :

<= ... the active β

x fit x β f eval :

FIT i 1 U rows range FIT i el U i el fit eval : for FIT 21 line :

β 1031.1298 4.2853 0.2817 15.8683 - 41 mat

MyFit U FIT augment :

init next fit 1000 4.5 0.3 041 mat 1030 4.27 0.28 15.8 - 41 mat 1031 4.2853 0.2817 15.87 - 41 mat 23 mat

data X Y o 8 blue plotG :

Title 15125 Model fit in Logistic 12 black 15 mat :

data MyFit Y 30 el Y 1 el Title 5 1 sys

MyFit 1 2.61173 2 8.48228 3 16.15875 4 26.15419 5 39.09752 6 55.73891 7 76.9396 8 103.63542 9 136.7611 10 177.12511 11 225.23407 12 281.08655 13 343.98411 14 412.43203 15 484.20056 16 556.57528 17 626.74962 18 692.24603 19 751.23737 20 802.68943 21 846.32288 22 882.45309 23 911.78366 24 935.21381 25 953.69087 26 968.11438 27 979.28442 28 987.8817 29 994.46746 30 999.494 302 mat

The LGC fitting algorithm is not as flexible wrt the initials than Levenberg-Marquardt though LM often fails. In this example from MapleSoft, Smath does an impecable job in much less coding.

Good practice is to initialise the model on a separate worksheet. In fact, you could proceed in manual mode only. Sometimes it's the only possible approach, despite the savant stuff. Generally, if the model crosses the data set once, a good final fit results. That assertion is not an absolute warranty.

x β f β 1 el 1 β 2 el β 3 el x * - exp + / β 4 el + :

β 10005 0.33 014 mat transpose :

x init x β f eval :

F i 1 U rows range F i el U i el init : for F 21 line :

Initial U F augment :

data X Y o 8 blue plotG :

Title 10500 Initialise Logistic 12 black 15 mat :

The initial fit of Logistic

data Initial Y 30 el Y 1 el Title 5 1 sys