linfit modules

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linfit modules

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

Applicable parts are: Optimiz "numeric" XY f φ β βlinfit X XY 1 col : Y XY 2 col : n XY rows : k x β φ length 1 - : vx raw initial vector fit i 1 X length range vx i el X i el β f eval : for M-> Cholesky Gradient Matrix j 1 k range i 1 n range M i j el X i el β φ j 1 + el eval : for for Cholesky solver [LS] Φ M transpose M * : d M transpose Y vx - (* : Δβ Φ 1 -^eval d * : b β Δβ + : 131 line :

Vector gradient ... Optimiz Symbolic x β f n φ ct 1 : i 1 n range PD ct el x β f (β ct el diff : stack f(x,β) & PD V 1 el x β f : V i 1 + el PD i el : 21 line ct ct 1 + : 41 line for V 31 line :

set N = even/odd to escape divide by 0 maybe other singularities.

L H N xd U 0 : dx H L - N / : i 1 N 1 + range U i el L dx - (dx i * + : for U 41 line :

ksmooth # plotG

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

σ 0.3708158988058244 :

The Gaussian ksmooth

d K 12 π * sqrt σ * / d 2^2 σ 2^* / - exp * :

j X Y Ksmooth μ j 11 line : X i el X μ el - bw / K Y i el * (i 1 X rows sum X i el X μ el - bw / K (i 1 X rows sum / 21 line :

Gaussian Ksmooth Kernel

x K

W j 1 X rows range s j el j X Y Ksmooth eval : for s 21 line :

The "corn Thermies" project

XY 02116121212209932120421135209862067720718207092056102083112090122072132064142081152074162054172056182058192061202 mat :

X XY 1 col : Y XY 2 col : data X Y . 14 red plotG : 31 line

A vector function of inventions x β f β 1 el β 2 el x * + β 3 el 1 0.05 x * + / + :

β 11131 mat :

in φ(x,β,f,n) -> set 'n' to last index β # el ≡

x β φ x β f 3 φ :

β XY f φ β βlinfit :

β 1910.116531 2.162484 213.288673 31 mat

L t 021 0.05 range : i 1 t rows range u i el t i el : v i el t i el β f eval : 21 line for u v augment 31 line :

"linfit" through the regional "Corn Thermies" from week 0 => 19 week [0] is the week corn crosses appear, "20 weeks thermies" is the laps weeks to harvest corn . "linfit" is a good candidate pass-through data. The function is smooth/continuous, can be derived/integrated.

L data XY 3 1 sys

x D x β f x diff :

x β f x 019 int 39531

x D

The model function is inspired from the 'J' reduced form of the Thiele continued fraction. By inspection, this form of shape will pass through the samples. The α4 is user from trial/error up to the fit is accepted.

x α Φ α 1 el α 2 el x * + α 3 el 1 α 4 el x * + / + :

x α Φ x int maple x α 2 el x * 2 α 1 el * + (* α 4 el * 2 α 3 el * 1 α 4 el x * + ln * + 2 α 4 el * /

The Gaussian ksmooth

Prepare and code the Smath project

X XY 1 col : Y XY 2 col : Yo XY 2 col : 31 line

bw 6 : bandwidth ≡ XW X W augment : vx X : vy XW 2 col : 41 line

XW vx vy o 7 red plotG : XY X Yo + 7 black plotG : 21 line

===== Finish/collect the project =====

spline join/mesh the smoothed data set Cfit t 019 0.1 range : i 1 t rows range v i el vx vy t i el cinterp eval : for t v augment 31 line :

collect plot plot Cfit XY x β f XW 41 sys :

XY=>raw data, XW=>smoothed data, Cfit=> meshed spline export

plot

From a rational fraction that best pass via the XW ksmooth x J 2092.1 467.27 x 12.789 - 51.703 x 10.398 - 55.411 x 5.5698 - 72.089 x 12.433 - / + / - / + / - :

range/mesh trend smooth J t 019 0.1 range : i 1 t rows range v i el t i el J eval : for t v augment 31 line :

Trend function rooming around the smoothed samples

J XY XW 3 1 sys

data Σ Finite differences Cumul Integration vx data 1 col : vy data 2 col : Y 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : Y i el Y i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for vx Y augment 61 line :

ΣXY XY Σ :

u ΣXY 1 col : v ΣXY 2 col : ptsΣ u v . 14 red plotG : 31 line

n J Σ rows :

n 191

J Σ ptsΣ 2 1 sys

Thermies 'linfit model' x β f β 1 el β 2 el x * + β 3 el 1 0.05 x * + / + :

x Thermies x β f :

Accumulated Thermies [1, 2, 3]

1. over the smoothed samples J Σ n row 193953512 mat

2. over the raw samples ΣXY 20 row 193953612 mat

3. over the "linfit" pass x Thermies x 019 int 39531

Relative deviation between
uncertain extrema. 13953539531 / - 0.0001 -

Observations: the 'finite differences' is known to be slightly excessive. The ksmooth trend is remarkably faithful, as well as the monotonous seasonal thermal decay. The relative deviation is not technically accountable wrt 'uncertainty' of the samples [± ?].

Another "pass through" function

x Decay 2053.7374 70.0025 x - 8.0493 / exp * + :

x Decay x 019 int 39531

expDecay from "genfit" ... @ user preference

x Thermies x Decay XY 3 1 sys

0 Decay 19 Decay 21 mat 2123.7 2060.3 21 mat

x Decay yo A x - t1 / exp * + :

Compare: linfit <=> Decay

Indefinite integral 'linfit' x ∫linfit x β 2 el x * 2 β 1 el * + (* 40 β 3 el * 20 x + ln * + 2 / :

Indfinite integral 'Decay' x ∫Decay yo x * A t1 * x t1 / - exp * - :

β 1910.116531 2.162484 213.288673 31 mat :

yo 2053.7374 : A 70.0025 : t1 8.0493 : 31 line

x Δ 1 x β f x Decay / - :

Mutual deviation: not so negligible !

0 x ≤ x 19 ≤ & x Δ out range if 0 2 1 sys

Indefinite integral curve 'linfit'

0 x ≤ ( x 19 ≤ ( & x ∫linfit 0 ∫linfit - out range if 0 2 1 sys

Conclusion

week 19 :

expDecay is to be rejected.
It under-evaluates "Thermies" week ∫Decay week ∫linfit 0 ∫linfit - 21 mat 389683953121 mat

relative error % 'linfit' <=> Decay, much
bigger relative error @ lower week # 1 week ∫linfit 0 ∫linfit - week ∫Decay / - (100 * 1.4 -

jmGiraud: 2016_07_07