plotG

dad98d66-3bca-4edf-b428-e04317c82362 81 3 5 decimal

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

plotG

data char size clr plotG k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 line :

1. The crappy Akima

t0 1 time :

Data 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 :

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 :

X Data 1 col : Y Data 2 col : 21 line

L 0 : H 30 : N 500 : 13 mat

U L H N xd :

x F X Y x ainterp :

pts Data . 12 blue plotG :

pts x F 2 1 sys

x D x F x diff :

In Smath: linterp, ainterp, cinterp are NOT scalar functions. So, they can't be sub-used analytically. The finite differences catches the wiggling of "ainterp", thus not smooth. The 2nd deriv' out of use.

A i 1 U rows range F i el U i el F eval : for U F augment 21 line :

Finite Differences Deriv Cumul

data Deriv data Derivative BackSubstitution X data 1 col : Y data 2 col : n X rows : deriv 1 el 0 : deriv n el 4 : i 2 n 1 - range deriv i el 0.5 Y i 1 - el Y i el - X i 1 - el X i el - / Y i el Y i 1 - el - X i el X i 1 - el - / + (* eval : for deriv 1 el deriv 2 el : deriv n el deriv deriv rows el : X deriv augment 101 line :

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 eval for vx Y augment 61 line :

Logistic

Extending the use of scalar function

β 1031.1298 4.2853 0.2817 15.8683 - 41 mat :

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

Logistic i 1 U rows range F i el U i el β f eval : for U F augment 21 line :

The finite differences derivative & cumulative integral are just good tools to populate data sets from scalar functions or w/o the scalar function if the data set is clean and supplied well populated.

D A Deriv :

"ainterp" useless ... "cinterp " acceptable in reduced range from visual inspection. alternate: cinterp(X,Y,x) ... ainterp(X,Y,x)

D Deriv Logistic Deriv Deriv 2 1 sys

"ainterp" noisy 1rst Derivative

D 0 2 1 sys

Δ Logistic 2 col A 2 col - : Noise U Δ augment : cumul D Σ : 31 line

Finite differences Cumulative Integral of "ainterp". Noisy Akima does not reflects on cumulative Σ, Integration is a very effective smoother.

cumul

Akima does not join smootly @ the mesh points, an inherent default.

Noise

1rst, 2nd, 3rd FD Derivatives from Logistic

Logistic Deriv Logistic Deriv Deriv Logistic Deriv Deriv Deriv 3 1 sys Logistic Deriv Deriv Deriv

In CAS [Computer Algebra System], a "scalar function" is scalar wrt "x". In there, 'x' is a vanishing iterator. In short: the function runs over itself for immediate use or for future sub-analytical use. The other "scalar" is a number. The scalar 'x' is the "analytical iterator".

x β D x β f (x diff :

x β Int x β D x int maple :

x β D x β Int

2. The acceptable Akima

XY 0 0.00794 0.1 0.012 0.2 0.022 0.3 0.048 0.4 0.167 0.5 1 0.6 0.167 0.7 0.048 0.8 0.022 0.9 0.012 1 0.00794 112 mat :

X XY 1 col : Y XY 2 col : support XY + 10 black plotG : 31 line

L 0 : H 1 : Npts 200 : 31 line

U L H Npts xd :

x Akima X Y x ainterp :

x Cinterp X Y x cinterp :

Akim i 1 U rows range vy i el U i el Akima eval : for U vy augment 21 line :

Cin i 1 U rows range vy i el U i el Cinterp eval : for U vy augment 21 line :

1 time t0 - s 41.4

Akim Cin support 3 1 sys

Resume ainterp(X,Y,x) Akima spline

Under the hood, Akima is noisy. Adequate to populate

a trace on canvas. Fails to sub use derivatives.

--------------------------------------------------------

Though Gaussians may look simple, they are the most

difficult data sets to fit model functions. Below, we

offer "Quadratic Holistic". This method fits in

ascending order. We suggest the descending order.

]]>

3. The Quadratic Holistic

ε 1012 -^:

data ε 0.00794 0.1 0.012 0.2 0.022 0.3 0.048 0.4 0.167 0.5 1 0.6 0.167 0.7 0.048 0.8 0.022 0.9 0.012 1 ε - 0.00794 112 mat :

<= Holistic matrix does not take '0'

X data 1 col : Y data 2 col : 21 line

data char size clr plotG k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 line :

Tridiagonal matrix whose inverse is needed to solve for 
the quadratic coefficients, given the vector 'S' ... => A n X rows : i 13 n 1 - (* (range j 13 n 1 - (* (range T i j el 0 : for for i 1 n 1 - (range j 12 range k 13 range T 2 i * 1 - j + (3 i * 3 - k + (el X i 1 - j + el (k 1 -^: for for for i 2 n 1 - (range j 12 range k 12 range T 2 n 1 - (* i + 3 i * 8 - k + j 3 * + (el 1 - (j^- (2 X i el * (k 1 -^* : for for for T 13 el 1 : T 61 line :

Vector associated with the 
Holistic tridiagonal 'A'. S n X rows : i 13 n 1 - (* (range V i el 0 : for i 1 n 1 - (range j 01 range V 2 i 1 + (* 2 - j + el Y i j + el : for for V 41 line :

A 1 -^S * sometimes needed ≡ 11 line

C A 1 -^S * :

S is the quadratic solving vector associated with 'A' for solving the vector 'C' of the coefficients

CWD C:\SmathFile CurrentDirectory :

We construct individual segments with their respective 'α' set of coefficients. As well, we isolate 'C' by spooling to file.

C CWD C_QuadraticHolistic wfile 1

C CWD C_QuadraticHolistic rfile :

α u f u C 2 α + el u * C 1 α + el + (* C α el + :

1 X 1 el f 4 X 2 el f 7 X 3 el f 10 X 4 el f 13 X 5 el f 13 X 6 el f 61 mat 0.00794 0.012 0.022 0.048 0.167 161 mat

<= Sanity check

xy data . 14 black plotG :

x collect 1 x f 4 x f 7 x f 10 x f 13 x f 13 0.232 - x + f 10 0.443477 - x + f 7 0.758449 - x + f 4 0.86835 - x + f 11 - x + f 101 sys :

x collect xy 2 1 sys 1 x f 4 x f 7 x f 10 x f 13 x f 5 1 sys xy 2 1 sys Truncated 'C' for visible export

C 0.00794 0.0406 0 0.01388 0.0782 - 0.594 0.03036 0.243 - 1.006 0.68628 4.6158 - 8.294 9.4562 48.4654 - 63.106 63.7468 - 244.3466 229.706 - 127.34552 392.6278 - 301.106 163.18136 - 437.449 291.806 - 211.35432 498.8902 - 293.406 263.48226 - 556.3022 292.812 - 301 mat :

α u f u C α 1 + el C α 2 + el u * + (* C α el + :

1 u f 4 u f 7 u f 10 u f 13 u f 51 mat 3972030 u * + 50000 / 3475 u * 391 - 2970 u * + (* + 25000 / 75925 u * 243 - 1006 u * + (* + 25000 / 171575 u * 23079 - 41470 u * + (* + 25000 / 47281 u 242327 - 315530 u * + (* + 5000 / 51 mat

The Quadratic Holistic segments

x s1 1 x f X 1 el x ≤ (x X 2 el ≤ (& cases :

x s4 10 x f X 4 el x ≤ (x X 5 el ≤ (& cases :

x s1 x s2 x s3 x s4 x s5 xy 61 sys

x s2 4 x f X 2 el x ≤ (x X 3 el ≤ (& cases :

x s5 13 x f X 5 el x ≤ (x X 6 el ≤ (& cases :

x s3 7 x f X 3 el x ≤ (x X 4 el ≤ (& cases :

X_Y plot rejects plotting the "if" segments

1 time t0 - s 44.9

Expose the weak Akima region that can be replaced by the better Holistic segment(s). They will join smooth for a final reconstruction ... left as home work exercise.

x s1 x s2 x s3 x s4 x s5 xy Akim 7 1 sys

4. Brute force reconciliation

x Holistic 1 x f X 1 el x ≤ (x X 2 el ≤ (& 4 x f X 2 el x ≤ (x X 3 el ≤ (& 7 x f X 3 el x ≤ (x X 4 el ≤ (& 10 x f X 4 el x ≤ (x X 5 el ≤ (& 13 x f X 5 el x ≤ (x X 6 el ≤ (& cases :

x Holistic

x HolisticBest 1 x f X 1 el x ≤ (x X 2 el ≤ (& 4 x f X 2 el x ≤ (x X 3 el ≤ (& 7 x f X 3 el x ≤ (x X 4 el ≤ (& 10 x f X 4 el x ≤ (x X 5 el ≤ (& cases :

x HolisticBest

===== End of this visit =====