Collected data for fitting session

0139ef5c-23fb-4986-b53c-9a06e8ade4cf 80 4 5 decimal

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

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1.643999934 83.24780169 1.646399975 83.3060776 1.651199937 83.36993231 1.653599977 83.43353573 1.656000018 83.49919979 1.660795569 83.56348171 1.663200021 83.62125502 1.665599942 83.6876981 1.667999983 83.75079893 1.670400023 83.80982873 1.675195575 83.87690005 1.677595615 83.94625818 1.679999948 83.99686952 1.682399988 84.05685425 1.687195659 84.11822112 1.68959558 84.17976389 1.692000031 84.23703461 1.694399953 84.30018569 1.696799994 84.3626834 1.701599956 84.4271161 1.703999996 84.49157393 1.708795667 84.55751442 1.708799958 84.61071411 1.713600039 84.67122657 1.715999961 84.72997995 1.720795631 84.78966312 1.723199964 84.8512059 1.725600004 84.9048328 1.730399966 84.96122398 1.732800007 85.01615763 1.735199928 85.07481049 1.739995599 85.13391568 1.742399931 85.19050789 1.744799972 85.24797965 1.747200012 85.31233696 1.751999974 85.36611464 1.754400015 85.42094777 1.756799936 85.47743947 1.761600018 85.54038951 1.763999939 85.59095059 1.76639998 85.64352204 1.76880002 85.7061454 1.771199942 85.76276274 1.776000023 85.81556036 1.778399944 85.8689611 1.780799985 85.92108022 1.785595655 85.97239518 1.787995577 86.01918679 1.790400028 86.07221058 1.79279995 86.12646573 1.79519999 86.17946439 1.799999952 86.23077935 1.802399993 86.27902849 1.804800034 86.33768134 1.809599996 86.39196162 1.812000036 86.45028779 1.816795588 86.50293463 1.819200039 86.55781802 1.82159996 86.60835397 1.826399922 86.65295929 1.828799963 86.70502815 1.833595634 86.76136906 1.835999966 86.81446824 1.838400006 86.85839506 1.843195558 86.91900804 1.845600009 86.97670596 1.84799993 87.02771937 1.852800012 87.07327962 1.855199933 87.1291682 1.857599974 87.18073447 1.862399936 87.22750095 1.864799976 87.27906721 1.867200017 87.33364904 1.869599938 87.38194844 1.87440002 87.43152945 1.876799941 87.47349615 1.879199982 87.52566552 1.881600022 87.57703075 1.886399984 87.63347219 1.888800025 87.67569018 1.893595576 87.73183006 1.896000028 87.78771864 1.898399949 87.83400766 1.90319562 87.88022128 1.905599952 87.92598257 1.907999992 87.98187115 1.912799954 88.03434209 1.915199995 88.08548114 1.919995666 88.13247379 1.922399998 88.18336155 1.927195668 88.22902232 1.9296 88.26617667 1.931999922 88.30937472 1.934399962 88.35714639 1.939199924 88.40220405 1.941599965 88.44316555 1.946395636 88.49088696 1.948799968 88.53800526 1.951200008 88.59012438 1.95359993 88.63701651 1.958400011 88.68069203 1.960799932 88.7218797 1.963199973 88.76907339 1.967999935 88.81981037 1.970399976 88.86587321 1.972800016 88.9055531 1.975199938 88.95583775 1.980000019 89.00230267 1.98239994 89.04910684 1.984799981 89.08790719 1.989599943 89.13105499 1.991999984 89.17695449 1.994400024 89.21912222 1.999199986 89.25594988 2.001600027 89.29818044 2.00639987 89.34258472 2.00880003 89.38574508 2.013595581 89.43032527 2.016000032 89.47400079 2.020799875 89.50661922 2.023200035 89.54487928 2.027999878 89.58800194 2.030400038 89.63861327 2.032799959 89.67598122 2.035199881 89.71789766 2.039999962 89.76129675 2.044795513 89.80320062 2.047199965 89.84622277 2.049599886 89.88446398 2.054395676 89.92328945 2.056799889 89.96736705 2.059200048 90.00722285 2.06159997 90.05662795 2.066400051 90.09227451 2.068799973 90.13138898 2.071199894 90.16381893 2.075995684 90.21111314 2.078399897 90.25642209 2.083195686 90.29175453 2.085599899 90.32466195 2.088000059 90.36328639 2.092799902 90.4030291 2.095200062 90.44618946 2.099995613 90.48750278 2.104795694 90.52454404 2.107199907 90.56164813 2.109600067 90.59698057 2.114395618 90.63914831 2.11680007 90.6757121 2.121595621 90.70835566 2.124000072 90.742507 2.126399994 90.77897028 2.131200075 90.82415358 2.135999918 90.86462506 2.138400078 90.89948003 2.140799999 90.9348376 2.143199921 90.9592386 2.148000002 91.00317798 2.150399923 91.03755549 2.152800083 91.07156862 2.157595634 91.0979423 2.160000086 91.13269675 2.162400007 91.16281477 2.167195559 91.1958227 2.16960001 91.23608058 2.171999931 91.27348622 2.176800013 91.30606695 2.179199934 91.34001726 2.183995724 91.37796319 2.186399937 91.41227788 2.191195726 91.4509777 2.193599939 91.48429976 2.198395729 91.50997609 2.200799942 91.5378073 2.203199863 91.56351504 2.207999945 91.59865901 2.212795496 91.63445635 2.215199947 91.67383468 2.220000029 91.70020836 2.22239995 91.72130479 2.227195501 91.7526416 2.229599953 91.77788444 2.231999874 91.80576591 2.236799955 91.8398293 2.239199877 91.86936933 2.243999958 91.90015328 2.246399879 91.92445376 2.248800039 91.95369223 2.253599882 91.97789219 2.256000042 92.0080102 2.260799885 92.04108096 2.263200045 92.07617467 2.267995596 92.1049985 2.270400047 92.13445058 2.272799969 92.16583765 2.27760005 92.19633261 2.279999971 92.23146401 2.284800053 92.26757547 2.289595604 92.29898139 2.292000055 92.32630372 2.296795607 92.35792952 2.299200058 92.3752502 2.303999901 92.40479023 2.306400061 92.43405384 2.311199903 92.46299075 2.313600063 92.49768238 2.318399906 92.52553872 2.320800066 92.55231448 2.325599909 92.57769553 2.328000069 92.60459694 2.332799911 92.6365243 2.335200071 92.67168083 2.339999914 92.70516623 2.342400074 92.73380159 2.347199917 92.76165793 2.349600077 92.78397943 2.351999998 92.80626325 2.356800079 92.83254897 2.359200001 92.86600925 2.363999844 92.89980877 2.366400003 92.93174869 2.371200085 92.95483038 2.373600006 92.97703252 2.378399849 93.00603226 2.380800009 93.02957884 2.38560009 93.04900414 2.388000011 93.07841853 2.392800093 93.09497275 2.397599936 93.11630792 2.400000095 93.13930793 2.404799938 93.16323146 2.409600019 93.18476766 2.411999941 93.20575729 2.416800022 93.22895206 2.419199944 93.24823287 2.424000025 93.27341288 2.426399946 93.29503704 2.431200027 93.31937521 2.433599949 93.33450332 2.43840003 93.36228427 2.440799952 93.37808461 2.445600033 93.3994449 2.447999954 93.42225644 2.452800035 93.44649409 2.457595587 93.46817479 2.460000038 93.48923353 2.46239996 93.49873258 2.467200041 93.51546899 2.469599962 93.53437285 2.474400043 93.5518883 2.479199886 93.56917129 2.481600046 93.58549935 2.486399889 93.59857939 2.49119997 93.61868948 2.493599892 93.63484163 2.498399973 93.65374549 2.500799894 93.6775685 2.505599976 93.69470699 2.510400057 93.71984303 2.512799978 93.7372077 2.51760006 93.75558383 2.519999981 93.77361443 2.524800062 93.79023777 2.527199984 93.81396654 2.532000065 93.83150711 2.536795616 93.85130936 2.539200068 93.87264453 2.543995619 93.88667321 2.54640007 93.89906219 2.551195621 93.9125443 2.553600073 93.9275342 2.558395624 93.94124248 2.560800076 93.96181119 2.565599918 93.97533728 2.568000078 93.99828704 2.572799921 94.013145 2.577600002 94.03337446 2.582400084 94.05385521 2.584800005 94.06966811 2.589600086 94.08256596 2.594399929 94.09516225 2.596800089 94.11004535 2.601599932 94.12600275 2.606400013 94.13912676 2.608799934 94.15686837 2.613600016 94.16810766 2.618399858 94.18341168 2.620800018 94.20117214 2.6256001 94.21872527 2.630399942 94.22992058 2.632799864 94.24335244 2.637595654 94.2581036 2.639999866 94.27071874 2.644799948 94.28450241 2.649600029 94.28999955 2.65199995 94.3020744 2.656800032 94.31325714 2.659199953 94.32343469 2.664000034 94.33317247 2.668799877 94.3404601 2.673595667 94.34902935 2.67599988 94.35907497 2.680799961 94.36528202 2.685595512 94.3744481 2.687999964 94.38380894 2.692800045 94.39088296 2.697599888 94.39564505 2.700000048 94.40651995 2.70719552 94.41597503 2.709599972 94.42478301 2.714400053 94.43014821 2.719195604 94.43844103 2.721595526 94.43398678 2.726395607 94.43633641 2.728795528 94.44398214 2.73119998 94.4498625 2.736000061 94.44812855 2.740799904 94.44895783 2.743200064 94.45302885 2.747999907 94.44960492 2.752799988 94.45116297 2.755199909 94.4529409 2.75999999 94.45778466 2.764800072 94.46259701 2.769595623 94.47459647 2.774395704 94.47788847 2.779195547 94.47966011 2.783995628 94.47964755 2.78640008 94.48279505 2.791199923 94.48236156 2.796000004 94.48332278 2.800799847 94.48811 10452 mat :

Function Utilities

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

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

Error stem plot for QuickPlot unicolor data Error stemX 1 data rows range : i 1 stemX rows range stem i el data i 1 el data i 2 el data i 1 el 022 mat : for i 1 stem rows 1 - range stem i el stem i el stem i 1 + (1 el 1 el 012 mat stack : for V stem 1 el : j 2 stem rows 1 - range V V stem j el stack : for V V data data rows row stack : 71 line :

Applicable parts are: Optimiz "numeric" XY f φ β Minimize 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 GradientMatrix j 1 k range i 1 n range M i j el X i el β φ j 1 + el eval : for for CholeskySolver LeastSquares Φ M transpose M * : d M transpose Y vx - (* : Δβ Φ 1 -^eval d * : b β Δβ + : 131 line :

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 :

Module xd(L,H,N) is a discretiser for use in-situ [user spec's]: range [L,H], density [N].

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 :

data list ListCollect collects index list select list : ct 1 : i select a ct el data i 1 el : b ct el data i 2 el : ct ct 1 + : 31 line for a b augment 51 line :

X XY 1 col : Y XY 2 col : n XY rows : 31 line

XY rows 1045

The collected data set is not usable. It is a noisy freak collection Deriv(XY). The rational fraction is a clean candidate. Here, we offer a rational fit demo. Rational fraction is exceptional fit applied to thermocouples types [J, T]

num order
denom order p.n 4 : p.d 3 : 21 sys

Total order of rational fit k 1 p.n + p.d + : 8

algo intervals a b N intervals a b a b a - (N / + range :

support 110457 intervals 0 round vectorize transpose : 1150299448598747896104518 mat

"Normalized" rational fractions have
leading "1" in denom.  Reputed stable
numerically. Most if not all computer
approximations are  "Normalized".
Generally, Num_order is Denom_order+1.
Some fit were found not "Normalized.
Ref: Hart et al.[Computer Approximations] x β f β 1 el β j 1 + el x j^* j 1 p.n sum + 1 β j 1 + p.n + el x j^* j 1 p.d sum + / maple :

THANKS & CREDIT to Jef rev.3 for helping the auto-init. Slightly recast in this demo.

x β f β 1 el β 2 el x * + β 3 el x 2^* + β 4 el x 3^* + β 5 el x 4^* + 1 β 6 el x * + β 7 el x 2^* + β 8 el x 3^* + /

iter 3 : ε 106 -^: 21 sys

recast the system x β φ x β f k φ :

βinit ct 1 : j support X# ct el XY j 1 el : Y# ct el XY j 2 el : 12 mat ct ct 1 + : 21 line for M k Ones : i 1 p.n range M M X# i^vectorize augment : for i 1 p.d range M M X# i^Y# * vectorize - augment : for M 1 -^Y# * 61 line :

β β 1 el βinit : iterate ConjugateGradient i 1 : Δβ'.max 1 : i iter ≤ (Δβ'.max ε > (& β i 1 + el XY f φ β i el Minimize : i i 1 + : n 1 k range Δβ' n el β i el n el β i 1 - el n el - β i 1 - el n el / 1 try abs : for Δβ'.max Δβ' max : 41 line while β β i el i Δβ'.max stack : 61 line eval :

Absolute residuals ... fit deviation δ i 1 XY rows range Δ i el Y i el X i el β f - 2 round eval : for X Y Δ augment 21 line :

New i 1 XY rows range U i el X i el β f eval : for X U augment 21 line :

XY New 2 1 sys

δ 1 row 0.0048 0.1486 0.39 - 13 mat

Collected data for fitting session

XY 0.004795611 0.148570221 0.004799981 0.267785793 0.007199981 0.496265553 0.009599981 0.724745313 0.011999981 0.960764049 0.014399981 1.18914329 0.016799981 1.415612669 0.01919998 1.628914055 0.021599982 1.862520286 0.023999963 2.086979284 0.026395611 2.205491223 0.026399981 2.317368906 0.028795611 2.424783308 0.028799981 2.536459952 0.031199981 2.642507531 0.03359998 2.865257706 0.03599561 2.980553036 0.03599998 3.096551999 0.038399979 3.209233834 0.040799983 3.435200618 0.043195613 3.548083491 0.043199982 3.659961173 0.045599964 3.771637818 0.045599982 3.891557023 0.047999982 4.000117578 0.050395612 4.11199526 0.050399981 4.223671905 0.052799981 4.34047502 0.055199981 4.564129866 0.05759998 4.790398208 0.05999998 4.898556686 0.06239998 5.124623989 0.064799964 5.230671568 0.064799979 5.342750289 0.067199983 5.456336795 0.069599979 5.678885932 0.071999982 5.907868287 0.074399978 6.017434032 0.076795608 6.132427804 0.076799981 6.252347009 0.079199962 6.37065791 0.081595615 6.482032998 0.081599981 6.594011592 0.083999984 6.708301731 0.086395614 6.822993947 0.08639998 6.933363844 0.088799961 7.047050869 0.088799983 7.160637376 0.091199979 7.273620768 0.093599983 7.390825959 0.095995612 7.505116098 0.095999978 7.619908833 0.098399982 7.733193782 0.100799963 7.84758444 0.100799978 7.95252631 0.103199981 8.056865065 0.105599962 8.171054686 0.105599985 8.279112645 0.10799998 8.389784099 0.11039561 8.499249325 0.110399984 8.615348807 0.11279998 8.728432718 0.11519561 8.844130124 0.115199983 8.957314554 0.117599979 9.073112479 0.119999982 9.29837563 0.122399978 9.408745527 0.124799982 9.526252275 0.127199978 9.749706084 0.129599988 9.861985842 0.131999984 10.08664588 0.134399965 10.19812149 0.13439998 10.30919502 0.136799976 10.4269028 0.139199987 10.64663741 0.141595617 10.75861561 0.141599983 10.87169952 0.143999979 10.97664139 0.146399975 11.19577288 0.148795605 11.30885679 0.148799986 11.41470333 0.151199967 11.52487219 0.151199982 11.63051769 0.153599977 11.7438741 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1.905599952 87.92598257 1.907999992 87.98187115 1.912799954 88.03434209 1.915199995 88.08548114 1.919995666 88.13247379 1.922399998 88.18336155 1.927195668 88.22902232 1.9296 88.26617667 1.931999922 88.30937472 1.934399962 88.35714639 1.939199924 88.40220405 1.941599965 88.44316555 1.946395636 88.49088696 1.948799968 88.53800526 1.951200008 88.59012438 1.95359993 88.63701651 1.958400011 88.68069203 1.960799932 88.7218797 1.963199973 88.76907339 1.967999935 88.81981037 1.970399976 88.86587321 1.972800016 88.9055531 1.975199938 88.95583775 1.980000019 89.00230267 1.98239994 89.04910684 1.984799981 89.08790719 1.989599943 89.13105499 1.991999984 89.17695449 1.994400024 89.21912222 1.999199986 89.25594988 2.001600027 89.29818044 2.00639987 89.34258472 2.00880003 89.38574508 2.013595581 89.43032527 2.016000032 89.47400079 2.020799875 89.50661922 2.023200035 89.54487928 2.027999878 89.58800194 2.030400038 89.63861327 2.032799959 89.67598122 2.035199881 89.71789766 2.039999962 89.76129675 2.044795513 89.80320062 2.047199965 89.84622277 2.049599886 89.88446398 2.054395676 89.92328945 2.056799889 89.96736705 2.059200048 90.00722285 2.06159997 90.05662795 2.066400051 90.09227451 2.068799973 90.13138898 2.071199894 90.16381893 2.075995684 90.21111314 2.078399897 90.25642209 2.083195686 90.29175453 2.085599899 90.32466195 2.088000059 90.36328639 2.092799902 90.4030291 2.095200062 90.44618946 2.099995613 90.48750278 2.104795694 90.52454404 2.107199907 90.56164813 2.109600067 90.59698057 2.114395618 90.63914831 2.11680007 90.6757121 2.121595621 90.70835566 2.124000072 90.742507 2.126399994 90.77897028 2.131200075 90.82415358 2.135999918 90.86462506 2.138400078 90.89948003 2.140799999 90.9348376 2.143199921 90.9592386 2.148000002 91.00317798 2.150399923 91.03755549 2.152800083 91.07156862 2.157595634 91.0979423 2.160000086 91.13269675 2.162400007 91.16281477 2.167195559 91.1958227 2.16960001 91.23608058 2.171999931 91.27348622 2.176800013 91.30606695 2.179199934 91.34001726 2.183995724 91.37796319 2.186399937 91.41227788 2.191195726 91.4509777 2.193599939 91.48429976 2.198395729 91.50997609 2.200799942 91.5378073 2.203199863 91.56351504 2.207999945 91.59865901 2.212795496 91.63445635 2.215199947 91.67383468 2.220000029 91.70020836 2.22239995 91.72130479 2.227195501 91.7526416 2.229599953 91.77788444 2.231999874 91.80576591 2.236799955 91.8398293 2.239199877 91.86936933 2.243999958 91.90015328 2.246399879 91.92445376 2.248800039 91.95369223 2.253599882 91.97789219 2.256000042 92.0080102 2.260799885 92.04108096 2.263200045 92.07617467 2.267995596 92.1049985 2.270400047 92.13445058 2.272799969 92.16583765 2.27760005 92.19633261 2.279999971 92.23146401 2.284800053 92.26757547 2.289595604 92.29898139 2.292000055 92.32630372 2.296795607 92.35792952 2.299200058 92.3752502 2.303999901 92.40479023 2.306400061 92.43405384 2.311199903 92.46299075 2.313600063 92.49768238 2.318399906 92.52553872 2.320800066 92.55231448 2.325599909 92.57769553 2.328000069 92.60459694 2.332799911 92.6365243 2.335200071 92.67168083 2.339999914 92.70516623 2.342400074 92.73380159 2.347199917 92.76165793 2.349600077 92.78397943 2.351999998 92.80626325 2.356800079 92.83254897 2.359200001 92.86600925 2.363999844 92.89980877 2.366400003 92.93174869 2.371200085 92.95483038 2.373600006 92.97703252 2.378399849 93.00603226 2.380800009 93.02957884 2.38560009 93.04900414 2.388000011 93.07841853 2.392800093 93.09497275 2.397599936 93.11630792 2.400000095 93.13930793 2.404799938 93.16323146 2.409600019 93.18476766 2.411999941 93.20575729 2.416800022 93.22895206 2.419199944 93.24823287 2.424000025 93.27341288 2.426399946 93.29503704 2.431200027 93.31937521 2.433599949 93.33450332 2.43840003 93.36228427 2.440799952 93.37808461 2.445600033 93.3994449 2.447999954 93.42225644 2.452800035 93.44649409 2.457595587 93.46817479 2.460000038 93.48923353 2.46239996 93.49873258 2.467200041 93.51546899 2.469599962 93.53437285 2.474400043 93.5518883 2.479199886 93.56917129 2.481600046 93.58549935 2.486399889 93.59857939 2.49119997 93.61868948 2.493599892 93.63484163 2.498399973 93.65374549 2.500799894 93.6775685 2.505599976 93.69470699 2.510400057 93.71984303 2.512799978 93.7372077 2.51760006 93.75558383 2.519999981 93.77361443 2.524800062 93.79023777 2.527199984 93.81396654 2.532000065 93.83150711 2.536795616 93.85130936 2.539200068 93.87264453 2.543995619 93.88667321 2.54640007 93.89906219 2.551195621 93.9125443 2.553600073 93.9275342 2.558395624 93.94124248 2.560800076 93.96181119 2.565599918 93.97533728 2.568000078 93.99828704 2.572799921 94.013145 2.577600002 94.03337446 2.582400084 94.05385521 2.584800005 94.06966811 2.589600086 94.08256596 2.594399929 94.09516225 2.596800089 94.11004535 2.601599932 94.12600275 2.606400013 94.13912676 2.608799934 94.15686837 2.613600016 94.16810766 2.618399858 94.18341168 2.620800018 94.20117214 2.6256001 94.21872527 2.630399942 94.22992058 2.632799864 94.24335244 2.637595654 94.2581036 2.639999866 94.27071874 2.644799948 94.28450241 2.649600029 94.28999955 2.65199995 94.3020744 2.656800032 94.31325714 2.659199953 94.32343469 2.664000034 94.33317247 2.668799877 94.3404601 2.673595667 94.34902935 2.67599988 94.35907497 2.680799961 94.36528202 2.685595512 94.3744481 2.687999964 94.38380894 2.692800045 94.39088296 2.697599888 94.39564505 2.700000048 94.40651995 2.70719552 94.41597503 2.709599972 94.42478301 2.714400053 94.43014821 2.719195604 94.43844103 2.721595526 94.43398678 2.726395607 94.43633641 2.728795528 94.44398214 2.73119998 94.4498625 2.736000061 94.44812855 2.740799904 94.44895783 2.743200064 94.45302885 2.747999907 94.44960492 2.752799988 94.45116297 2.755199909 94.4529409 2.75999999 94.45778466 2.764800072 94.46259701 2.769595623 94.47459647 2.774395704 94.47788847 2.779195547 94.47966011 2.783995628 94.47964755 2.78640008 94.48279505 2.791199923 94.48236156 2.796000004 94.48332278 2.800799847 94.48811 10452 mat :

data XY :

Savitsky-Golay smoothing noisy data set

SG_PW(data,order,window)

Savitsky-Golay nth order  polysmooth window ... a primer example => sanity working data order window SG_PW O 1 : comply Smath ORIGIN 1 ≡ i window - window range t i window 1 + / : u O i + window + el t : w O i + window + el 1 t 2^- : 31 line for i 1 u rows range j 1 order range idx i el i i / : v i j el u i el (j^: X idx v augment (: 31 line for for i 1 w rows range j 1 X cols range WX i j el w i el X i j el * : for for c WX X transpose (WX * (1 -^O col * : Y data 2 col : n data rows : i O window + (O n + 1 - window - (range z Y i window - (i window + (O O submatrix : S i window - el c z * : 21 line for v window n 1 - window - (range : v S augment 101 line :

smooth data 110 SG_PW :

data rows 1045

j 1 data rows range : xy j data 2 col augment : 21 sys

smooth rows 1025

Truncated of window @ both ends

smooth xy 2 1 sys

sanity the companionship
remenber ► 'xy' is noisy smooth smooth rows row xy xy rows 11 - row 21 mat 1034 94.4603 12 mat 1034 94.4512 12 mat 21 mat

data

data 0 69.487 1 69.471 2 69.494 3 69.475 4 69.472 5 69.463 6 69.461 7 69.462 8 69.476 9 69.461 10 69.464 11 69.463 12 69.468 13 69.478 14 69.471 15 69.465 16 69.455 17 69.439 18 69.457 19 69.443 20 69.467 21 69.442 22 69.434 23 69.449 24 69.438 25 69.457 26 69.425 27 69.443 28 69.431 29 69.446 30 69.433 31 69.434 32 69.436 33 69.440 34 69.439 35 69.424 36 69.442 37 69.427 38 69.426 39 69.422 40 69.446 41 69.410 42 69.445 43 69.425 44 69.429 45 69.433 46 69.414 47 69.429 48 69.414 49 69.427 502 mat :

smooth data 110 SG_PW :

data rows 50

smooth rows 30

data smooth 2 1 sys data [709]

data 69.487 69.471 69.494 69.475 69.472 69.463 69.461 69.462 69.476 69.461 69.464 69.463 69.468 69.478 69.471 69.465 69.455 69.439 69.457 69.443 69.467 69.442 69.434 69.449 69.438 69.457 69.425 69.443 69.431 69.446 69.433 69.434 69.436 69.44 69.439 69.424 69.442 69.427 69.426 69.422 69.446 69.41 69.445 69.425 69.429 69.433 69.414 69.429 69.414 69.427 69.416 69.397 69.411 69.419 69.42 69.434 69.42 69.419 69.422 69.423 69.441 69.423 69.412 69.41 69.401 69.446 69.389 69.413 69.419 69.424 69.425 69.41 69.412 69.414 69.423 69.399 69.427 69.397 69.408 69.416 69.412 69.416 69.411 69.431 69.397 69.411 69.414 69.416 69.411 69.43 69.401 69.415 69.43 69.397 69.415 69.401 69.415 69.411 69.416 69.415 69.411 69.415 69.396 69.395 69.416 69.415 69.416 69.411 69.431 69.397 69.431 69.411 69.416 69.412 69.416 69.412 69.413 69.411 69.392 69.432 69.411 69.428 69.409 69.428 69.412 69.414 69.424 69.414 69.412 69.428 69.409 69.419 69.423 69.415 69.409 69.414 69.424 69.421 69.425 69.41 69.424 69.406 69.425 69.442 69.407 69.422 69.436 69.407 69.427 69.417 69.404 69.422 69.409 69.433 69.423 69.403 69.399 69.418 69.42 69.415 69.425 69.415 69.419 69.421 69.43 69.432 69.416 69.433 69.412 69.398 69.427 69.432 69.427 69.434 69.412 69.408 69.425 69.43 69.429 69.43 69.441 69.427 69.441 69.427 69.424 69.437 69.443 69.408 69.418 69.444 69.423 69.439 69.438 69.424 69.434 69.439 69.425 69.435 69.44 69.437 69.436 69.419 69.429 69.435 69.429 69.411 69.465 69.431 69.441 69.426 69.427 69.438 69.427 69.439 69.424 69.439 69.456 69.438 69.422 69.433 69.439 69.449 69.436 69.465 69.466 69.443 69.463 69.445 69.458 69.464 69.479 69.46 69.441 69.475 69.461 69.456 69.461 69.441 69.475 69.437 69.434 69.46 69.471 69.461 69.467 69.466 69.462 69.467 69.484 69.465 69.464 69.462 69.473 69.462 69.472 69.474 69.469 69.471 69.485 69.457 69.471 69.486 69.488 69.484 69.468 69.482 69.476 69.467 69.477 69.492 69.463 69.478 69.474 69.49 69.475 69.476 69.49 69.475 69.482 69.483 69.488 69.484 69.485 69.486 69.496 69.482 69.486 69.482 69.494 69.48 69.495 69.49 69.496 69.477 69.507 69.49 69.504 69.489 69.49 69.496 69.503 69.519 69.511 69.502 69.528 69.489 69.51 69.505 69.506 69.508 69.504 69.514 69.516 69.497 69.496 69.508 69.515 69.526 69.508 69.519 69.534 69.505 69.522 69.527 69.53 69.515 69.526 69.527 69.528 69.523 69.54 69.521 69.537 69.533 69.535 69.536 69.532 69.529 69.541 69.542 69.538 69.545 69.546 69.547 69.564 69.55 69.563 69.537 69.554 69.535 69.572 69.547 69.56 69.556 69.578 69.543 69.577 69.563 69.55 69.586 69.559 69.591 69.572 69.574 69.585 69.583 69.584 69.595 69.578 69.589 69.601 69.584 69.6 69.597 69.593 69.61 69.587 69.62 69.597 69.615 69.606 69.624 69.605 69.621 69.618 69.611 69.643 69.606 69.63 69.633 69.629 69.641 69.64 69.632 69.663 69.63 69.648 69.649 69.653 69.655 69.657 69.639 69.677 69.664 69.667 69.675 69.673 69.675 69.669 69.681 69.683 69.676 69.691 69.69 69.692 69.711 69.697 69.699 69.681 69.718 69.696 69.708 69.724 69.688 69.714 69.715 69.698 69.719 69.706 69.724 69.733 69.721 69.716 69.748 69.729 69.745 69.725 69.728 69.737 69.74 69.735 69.72 69.736 69.735 69.737 69.735 69.739 69.734 69.739 69.738 69.727 69.731 69.729 69.726 69.723 69.711 69.724 69.728 69.714 69.725 69.692 69.704 69.699 69.696 69.692 69.688 69.674 69.684 69.672 69.667 69.657 69.662 69.652 69.645 69.642 69.631 69.639 69.618 69.595 69.622 69.581 69.593 69.584 69.567 69.558 69.569 69.546 69.532 69.516 69.517 69.492 69.473 69.444 69.409 69.401 69.381 69.393 69.369 69.365 69.362 69.319 69.351 69.308 69.321 69.311 69.295 69.286 69.284 69.258 69.28 69.275 69.233 69.28 69.215 69.26 69.228 69.218 69.229 69.185 69.225 69.191 69.202 69.192 69.18 69.197 69.181 69.187 69.178 69.167 69.179 69.184 69.152 69.189 69.158 69.162 69.166 69.156 69.181 69.144 69.184 69.144 69.18 69.159 69.166 69.159 69.181 69.142 69.167 69.168 69.168 69.179 69.154 69.187 69.163 69.184 69.166 69.178 69.189 69.175 69.177 69.184 69.186 69.178 69.199 69.176 69.204 69.201 69.198 69.211 69.211 69.207 69.195 69.206 69.214 69.212 69.214 69.211 69.219 69.231 69.223 69.216 69.238 69.235 69.223 69.234 69.237 69.241 69.242 69.249 69.247 69.249 69.251 69.238 69.256 69.263 69.261 69.263 69.255 69.287 69.25 69.273 69.277 69.278 69.28 69.283 69.279 69.296 69.274 69.302 69.298 69.295 69.297 69.308 69.291 69.308 69.299 69.316 69.298 69.311 69.321 69.309 69.305 69.332 69.314 69.321 69.326 69.325 69.321 69.338 69.314 69.351 69.312 69.324 69.342 69.337 69.34 69.332 69.363 69.344 69.349 69.347 69.339 69.37 69.351 69.352 69.348 69.366 69.345 69.367 69.343 69.376 69.373 69.352 69.35 69.383 69.359 69.379 69.365 69.36 69.372 69.354 69.389 69.37 69.372 69.383 69.378 69.384 69.365 69.395 69.377 69.379 69.389 69.376 69.386 69.402 69.369 69.403 69.384 69.386 69.391 69.394 69.393 69.398 69.395 69.396 69.387 69.408 69.388 69.405 69.4 69.401 69.402 69.398 69.413 69.399 69.4 69.41 69.411 69.396 69.407 69.408 69.393 69.413 69.413 69.405 69.416 69.411 69.387 69.412 7091 mat :

CWD C:\SmathFile CurrentDirectory :

data rows 709

data n Decimate collects every 'n' points range 1 data rows n range : k 1 : i range u k el data i 1 el : k k 1 + : 31 line for u 31 line 31 line :

data data 3 Decimate :

data CWD Lea wfile 1

Lea CWD Lea rfile :

XYLea i 1 Lea rows range idx i el i : vy i el Lea i el : 21 line for idx vy augment 21 line :

Note: Lea did not explain the data set. Maybe a portion of a chirp ? Though, a window length is lost at both ends, the essential of the information is sufficiently well recovered for spectral analysis. Both ends can be smoothed with parabolic arcs.

SG_PW(data,order,window)

Savitsky-Golay nth order  polysmooth window ... adapted from Paul W. Mathcad 11 data order window SG_PW O 1 : (comply Smath ORIGIN 1 ≡ range window - window range : i window - window range t i window 1 + / : u O i + window + el t : w O i + window + el 1 t 2^- : 31 line for i 1 u rows range j 1 order range idx i el 1 : v i j el u i el (j^: X idx v augment (: 31 line for for i 1 w rows range j 1 X cols range WX i j el w i el X i j el * : for for matrix X transpose WX * (1 -^: c WX matrix O col * : Y data 2 col : n data rows : i O window + (n window - (range z Y i window - (i window + (O O submatrix : S i window - el c z * : 21 line for v O window + (O n + 1 - window - (range : v S augment 121 line eval :

order 2 : window 30 : 12 mat

smooth XYLea order window SG_PW :

data rows smooth rows 21 mat 35529521 mat

XYLea smooth window 1 + 69.6 window 1 + 69.3 2 2 mat data rows window - 69.5 data rows window - 69.2 2 2 mat 4 1 sys

Smoothing by convolving ksmooth

XY raw data [100]

XY 0.5 - 0.105 0.48 - 0.115 0.46 - 0.126 0.44 - 0.138 0.42 - 0.151 0.4 - 0.165 0.38 - 0.181 0.36 - 0.198 0.34 - 0.217 0.32 - 0.238 0.3 - 0.261 0.28 - 0.286 0.26 - 0.313 0.24 - 0.343 0.22 - 0.375 0.2 - 0.411 0.18 - 0.45 0.16 - 0.493 0.14 - 0.539 0.12 - 0.59 0.1 - 0.645 0.08 - 0.705 0.06 - 0.77 0.04 - 0.841 0.02 - 0.918 01 0.02 1.09 0.04 1.186 0.06 1.288 0.08 1.398 0.1 1.515 0.12 1.638 0.14 1.767 0.16 1.902 0.18 2.043 0.2 2.186 0.22 2.333 0.24 2.48 0.26 2.627 0.28 2.773 0.3 2.914 0.32 3.05 0.34 3.18 0.36 3.326 0.38 3.415 0.4 3.52 0.42 3.62 0.44 3.73 0.46 3.78 0.48 3.848 0.5 3.903 0.52 3.971 0.54 4.023 0.56 4.072 0.58 4.117 0.6 4.16 0.62 4.178 0.64 4.208 0.66 4.235 0.68 4.259 0.7 4.281 0.72 4.301 0.74 4.319 0.76 4.338 0.78 4.351 0.8 4.362 0.82 4.377 0.84 4.387 0.86 4.4 0.88 4.407 0.9 4.419 0.92 4.424 0.94 4.436 0.96 4.444 0.98 4.451 1 4.457 1.02 4.463 1.04 4.468 1.06 4.475 1.08 4.48 1.1 4.484 1.12 4.489 1.14 4.493 1.16 4.497 1.18 4.501 1.2 4.505 1.22 4.508 1.24 4.511 1.26 4.513 1.28 4.518 1.3 4.52 1.32 4.523 1.34 4.525 1.36 4.527 1.38 4.531 1.4 4.533 1.42 4.535 1.44 4.537 1.46 4.539 1.48 4.54 1002 mat :

ksmooth kernel

σ 0.3708158988058244 :

The Gaussian ksmooth

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

Smath recognises the silent 'bw', something
like the Mathcad "global definition". W j X Y Ksmooth X i el X j el - bw / K Y i el * (i 1 X rows sum X i el X j el - bw / K (i 1 X rows sum / : j 1 X rows range s j el j X Y Ksmooth eval : for s 31 line :

Gaussian Ksmooth Kernel

x K

X Y bw 13 mat

from below

Discrete derivative by convolution ... better than finite differences. data scale convol vx data 1 col : vy data 2 col : 12 mat f 1 - 0113 mat transpose : Dy vy f al_convr1d eval : Dy Dy 2 Dy rows 1 - 11 submatrix : Deriv vx scale Dy * augment : Deriv Deriv 2 Deriv rows 1 - 12 submatrix : 61 line :

scale XY 2 col Max XY 1 - convol 2 col Max / (- : 15.4422 -

NOT smooth for sub-use ►

XY XY scale convol

X XY 1 col : Y XY 2 col : 12 mat bw 0.125 : (bandwidth ≡ XW X W augment : 31 line

Explain Project:

We are given a data set to quality control. Though it loks good,

noisy and unfit for publishing. The Gaussian ksmooth reveals an

excellent candidate. The discrete derivative is a noise amplifier.

XW is the smoothed derivative.

You can't ksmooth this data set XY directly . . . not visible.

]]>

Sanity smoothed derivative ... XW ► smoothed deriv. Disable blue to inspect quality of red... adjust 'bw'.

XY scale convol XW scale convol 2 1 sys

x λ 1 0.5 - x ≤ (x 1.48 ≤ (& cases :

data 0 :

Y 0 :

smoothed XW scale convol eval :

data Σ Finite differences Cumulative ∫ vx data 1 col : vy data 2 col : 12 mat 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 XY 12 el + augment 51 line :

Export clean Σ smoothed Σ :

vx Σ 1 col : vy Σ 2 col 4.54 2.8281 / * : 12 mat x Σ vx vy x ainterp : 21 sys

interpolate in-situ u 1.25 :

track 22 0.75 2 u u Σ 32 mat :

Σ Σ rows row 1.46 2.8281 12 mat

The original noisy data set has been cleaned via the convolving adjustable ksmooth. At this point, we can export a more reliable Σ(smoothed).

x Σ x λ * track 2 1 sys

Export vx vy augment :

Export XY 2 1 sys

u u Σ 12 mat 1.25 4.516 12 mat

∫

data [145 pairs]

XY 0 0.152 0.006944 0.063 0.014 0.098 0.021 0.263 0.028 0.125 0.035 0.107 0.042 0 0.049 0.25 0.056 0.071 0.063 0.259 0.069 0.286 0.076 0.205 0.083 0.295 0.09 0.214 0.097 0.188 0.104 0.152 0.111 0.295 0.118 0.214 0.125 0.125 0.132 0.098 0.139 0.107 0.146 0.357 0.153 0.473 0.160 0.214 0.167 0.232 0.174 0.259 0.181 0.268 0.188 0.196 0.194 0.152 0.201 0.179 0.208 0.241 0.215 0.143 0.222 0.196 0.229 0.321 0.236 0.134 0.243 0.179 0.25 0.107 0.257 0.071 0.264 0.143 0.271 0.152 0.278 0.330 0.285 0.339 0.292 0.170 0.299 0.268 0.306 0.348 0.313 0.268 0.319 0.143 0.326 0.080 0.333 0.045 0.340 0.143 0.347 0.205 0.354 0.036 0.361 0.00893 0.368 0.027 0.375 0.063 0.382 0.045 0.389 0.116 0.396 0.107 0.403 0.259 0.410 0.339 0.417 0.143 0.424 0.036 0.431 0.134 0.438 0.223 0.444 0.268 0.451 0.286 0.458 0.196 0.465 0.223 0.472 0.170 0.479 0.259 0.486 0.393 0.493 0.295 0.5 0.268 0.507 0.134 0.514 0.330 0.521 0.402 0.528 0.366 0.535 0.259 0.542 0.366 0.549 0.321 0.556 0.366 0.563 0.438 0.569 0.527 0.576 0.455 0.583 0.455 0.590 0.527 0.597 0.509 0.604 0.509 0.611 0.643 0.618 0.527 0.625 0.366 0.632 0.375 0.639 0.375 0.646 0.375 0.653 0.286 0.660 0.402 0.667 0.482 0.674 0.536 0.681 0.313 0.688 0.313 0.694 0.241 0.701 0.509 0.708 0.563 0.715 0.500 0.722 0.464 0.729 0.500 0.736 0.491 0.743 0.518 0.750 0.259 0.757 0.330 0.764 0.411 0.771 0.402 0.778 0.384 0.785 0.500 0.792 0.446 0.799 0.304 0.806 0.438 0.813 0.554 0.819 0.321 0.826 0.366 0.833 0.286 0.84 0.527 0.847 0.446 0.854 0.527 0.861 0.571 0.868 0.589 0.875 0.527 0.882 0.688 0.889 0.500 0.896 0.482 0.903 0.563 0.910 0.688 0.917 0.696 0.924 0.634 0.931 0.777 0.938 0.732 0.944 0.598 0.951 0.643 0.958 0.705 0.965 0.813 0.972 0.670 0.979 0.857 0.986 1 0.993 0.777 1 0.732 1452 mat :

data spec pts i 1 data rows range p i el a b c spec : for v p 1 el : Unest 'p' j 2 p rows range v v p j el stack : for data v augment 51 line :

ρ 0.7125 : m 3 : μ 0 : σ 0.5625 : bw 0.1875 : 51 sys

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

r m ρ * - ρ - ρ m ρ * 14 mat transpose :

visible Lagrange iterated product x L x r 1 el - (x r 2 el - (* x r 3 el - (* x r 4 el - (* :

r 2.1375 - 0.7125 - 0.7125 2.1375 41 mat

t0 1 time :

x μ σ dnorm 12 π * σ * sqrt / 12 σ 2^* / x μ - (2^* - exp * :

x c 10 x ≤ (x 1 ≤ (& cases :

Kernel(x) exposed below x Kernel x L x μ σ dnorm * :

Smoothing Kernel

weighted moving average smoothing kernel j X Y Sup X i el X j el - bw / Kernel Y i el * (i 1 X rows sum X i el X j el - bw / Kernel (i 1 X rows sum / :

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

W W :

XW X W augment : vx X : vy XW 2 col : Δ Y W - : 41 line

a b c spec . 10 black 13 mat : data XY a b c spec pts : 21 line

x f vx vy x linterp :

data x f x c * 2 1 sys x L x μ σ dnorm 2 1 sys

1 time t0 - min 2.3

J_Fraction from Padé rational x J 1.0867 x * 0.099818 + 0.10829 x 0.88046 - 0.15161 x 0.078209 - 0.010073 x 1.8302 - 1.5240 x 0.60471 + / + / - / + / - :

data x f x c * x J 3 1 sys x Kernel r 2 el 0.5 r 2 el 0.5 - 2 2 mat r 3 el 0.5 r 3 el 0.5 - 2 2 mat 3 1 sys

x Kernel 0 ≡ x 1 - 1 solve 0.7125 - 0.7125 21 mat

x L x μ σ dnorm - 0 ≡ x 1 - 1 solve 0.6661 - 0.6661 21 mat

The Scientific Carbon α 12

plotG(vx,vy,char,size,color)

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 :

Carbon12 0.175 317.303 0.35 195.232 0.525 124.387 0.7 43.733 0.875 40.463 1.05 55.722 1.225 90.599 1.4 115.668 1.575 184.332 1.75 140.736 1.925 100.409 2.1 57.902 2.275 42.643 2.45 58.992 2.625 98.229 2.8 206.131 2.975 316.212 3.15 345.64 3.325 284.605 3.5 235.559 3.675 90.599 3.85 35.014 4.025 36.104 4.2 11.035 242 mat :

Xo Carbon12 1 col :

X Carbon12 1 col :

Yo Carbon12 2 col :

Y Carbon12 2 col :

n X rows : 24

t0 1 time :

ρ 0.525 : m 2.5 : μ 0 : σ 0.385 : 41 sys

X 2 el 0.315 : Y 9 el 230 : 21 line

r m ρ * - ρ - ρ m ρ * 14 mat transpose :

Kernel(x) designed specially for Carbon α 12

visible Lagrange iterated product x L x r 1 el - (x r 2 el - (* x r 3 el - (* x r 4 el - (* :

r 1.3125 - 0.525 - 0.525 1.3125 41 mat

x μ σ dnorm 12 π * σ * sqrt / 12 σ 2^* / x μ - (2^* - exp * :

Kernel Carbon α 12 x Kernel x L x μ σ dnorm * :

unique for that application bw 1 :

weighted moving average smoothing kernel j X Y Sup X i el X j el - bw / Kernel Y i el * (i 1 X rows sum X i el X j el - bw / Kernel (i 1 X rows sum / :

W j 1 X rows range s j el j X Y Sup eval : constrain 1rst node s 1 el 301.5 : 31 line for s 21 line :

W W eval :

XW X W augment : vx Xo : vy XW 2 col : Δ Yo W - : 41 line

t 0.175 4.2 0.185 range :

t rows 403

Cfit i 1 t rows range u i el t i el : v i el vx vy t i el cinterp eval : 21 line for u v augment 21 line :

C12 Xo Yo + 7 blue plotG : XW XW 1 col XW 2 col o 7 blak plotG : Title 1360 Carbon α 12 12 black 15 mat : 31 sys

1 time t0 - 3.4 s *

0 Cfit C12 XW Title 5 1 sys Stem construct

iii 1 Δ rows range P iii el 3 : P Δ rows el 2 : 21 line for

ii 1 Δ rows range vv ii el ii : for

i 1 vv length range YΔ 3 i * el Δ i el : for

Core component of the stem style P v replicate Mathcad 8 Pro. Treasury k 1 : i 1 v length range P i el 1 ≡ continue j 1 P i el range T k j + el v i el : for k k P i el + : 21 line if for T 41 line :

datum 0 :

u P vv replicate :

V u YΔ augment :

Δx P Xo replicate :

Residuals Δx V 2 col augment :

β 1403.4 6.8679 3.4766 4.6785 0.10541 5.9138 1.0965 2933.1 33.952 0.32404 4.1246 0.50485 14.081 131 mat :

Title 145 Cα12 SuperSmoothlet 12 black 15 mat :

Residuals on SuperSmoothlet... extensible at will.

Residuals datum Title 3 1 sys

Δ Max Δ Min 21 mat 22 27.9 - 21 mat

Padé rational fraction from Cfit, converted J_Fraction x J β 1 el β 8 el x β 2 el - β 9 el x β 3 el + β 10 el x β 4 el - β 11 el x β 5 el + β 12 el x β 6 el - β 13 el x β 7 el + / + / - / + / - / + / - :

x q 0.175 x ≤ (x 4.2 ≤ (& x J eval out of range if :

x D 0.175 x ≤ (x 4.2 ≤ (& x J (x diff eval out of range if :

D(x) takes 3 min to plot instead: plot Dinf(x,h,f)

x q 0.5 x D * 2 1 sys

Hyper fast conclusion 1000 points

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 :

range/discretise L 0.175 : H 4.2 : N 1000 : 13 mat U L H N xd : 21 sys

x fnct Q i 1 U rows range vx i el U i el : vy i el U i el fnct eval : 21 line for vx vy augment 21 line :

Ω x J Q :

Convolution Derivative  on Padé J_Fraction.
The J_Fraction is collected from Mathcad 11 Φ U Δeriv_f dx U 2 el U 1 el - : scale 1 - 2 dx * / : f 1 - 0113 mat transpose : dy Φ 2 col f al_convr1d : dy dy 2 dy rows 1 - 11 submatrix : Deriv U scale dy * augment : Deriv Deriv 2 Deriv rows 1 - 12 submatrix : 71 line :

◄ superb auto-scale by coincidence

mesh fine enough
detect noise ... h 0.001 : x f x J : 21 sys

◄ -1 point @ each end

x c 1 0.175 x ≤ (x 4.2 ≤ (& cases :

x h f Dinf 0.5 x h - f x f - x h - (x - / x f x h - f - x x h - (- / + (* :

Discretised function

C12 Ω 2 1 sys

Scalar infinitesimal derivative

0 x h f Dinf x c * 2 1 sys

Convolution derivative wrt J(x)

0 Ω U Δeriv_f 2 1 sys

At this point, you can spline both Carbon α 12 and its derivative

x J x Ω 1 el Ω rows int 1.38 106^*

x Kernel r 2 el 0.125 r 2 el 0.1255 - 2 2 mat r 3 el 0.125 r 3 el 0.125 - 2 2 mat 3 1 sys

1 time t0 - 10 s *

jmGiraud: 2015/10/06

x Kernel 0 ≡ x 1 - 1 solve 0.525 - 0.525 21 mat

x L x μ σ dnorm - 0 ≡ x 2 - 2 solve 1.313 - 1.313 21 mat

The cubic spline modeling "Curvator"

t0 1 time :

Inspired/adapted from Paul R. Lorczak [Mathsoft] Quick Polyfit & linfiCheby fail

Point(s) spec's algo data spec pts i 1 data rows range p i el a b c spec : for v p 1 el : Unest 'p' j 2 p rows range v v p j el stack : for data v augment 51 line :

Sampled data to fit Curvdata 0.192 0.217 0.224 0.07 0.256 0.185 0.288 0.434 0.32 0.689 0.352 0.822 0.385 0.705 0.417 0.359 0.449 0.418 0.481 0.711 0.513 0.94 0.545 1 0.577 0.932 0.609 0.775 0.641 0.566 0.673 0.346 0.706 0.154 0.738 0.028 0.77 0.00694 0.802 0.13 202 mat :

Curvature at
calib'n points Deriv 10 - 15 - 6.5 6.5 41 mat :

Calibration points cal 0.192 0.2174 0.4 0.52174 0.5 0.87 0.8018 0.1304 42 mat :

X cal 1 col :

Y cal 2 col :

x X :

y Y :

Linear solving the system S j 13 range algo 1 j el x j el (3^x j el (2^x j el 1 x j 1 + el (3^x j 1 + el (2^x j 1 + el 13 x j el (2^* 2 x j el * 103 x j 1 + el (2^* 2 x j 1 + el * 1044 mat (1 -^y j el y j 1 + el Deriv j el Deriv j 1 + el 41 mat * : for algo 21 line :

Cutvator in there

S 645.487 - 561.173 154.105 - 13.687 41 mat 1546.52 - 2195.302 1028.912 - 159.816 41 mat 196.537 383.778 - 242.875 49.19 - 41 mat 13 mat

The 3 intervals

t s3 S 3 el t 3^t 2^t 114 mat transpose * :

a b c spec . 14 red 13 mat :

t s2 S 2 el t 3^t 2^t 114 mat transpose * :

Data Curvdata a b c spec pts :

t s1 S 1 el t 3^t 2^t 114 mat transpose * :

a b c spec + 14 black 13 mat :

calib cal a b c spec pts :

vx Curvdata 1 col : vy Curvdata 2 col : 12 mat

Piecewise construct t spline t X 3 el ≥ t s3 t X 2 el ≥ t s2 t s1 if if :

The modeling "Curvator" passes through the support points. The remaining of the shape is achieved by the slope [Deriv] @ support points associated with the linear solver. The data points can be noisy. This demo results from a real noisy project from which a smooth fitted 20 data points were collected [Mathsoft Collaboratory].

Discretise/collect the project Spline u 0.175 0.825 0.176 range : i 1 u rows range v i el u i el spline eval : for u v augment 31 line :

The "Curvator" adjusted by hand

Spline Data calib 3 1 sys

Piecewise construct t curvator t X 3 el ≥ t s3 t X 2 el ≥ t s2 t s1 if if :

vx Curvdata 1 col : vy Curvdata 2 col : 12 mat

Discretise/collect the project Δcurv i 1 vx rows range fit i el vx i el curvator eval : for vx vy vy fit - augment 21 line :

Δcurv 0.192 0.217 0.0004 - 0.224 0.07 0.0002 - 0.256 0.185 0.0011 0.288 0.434 0.0024 0.32 0.689 0.0026 0.352 0.822 0.0006 0.385 0.705 0.0041 0.417 0.359 0.0017 0.449 0.418 0.0019 - 0.481 0.711 0.0012 - 0.513 0.94 0.0001 0.545 1 0.0002 - 0.577 0.932 0.0007 - 0.609 0.775 0.0009 - 0.641 0.566 0.0026 - 0.673 0.346 0.0033 - 0.706 0.154 0.0025 0.738 0.028 0.0012 0.77 0.0069 0.0004 - 0.802 0.13 0.0017 - 203 mat

1 time t0 - 2.3 s *

β Clear 1

NBS 125

T 6.25629 - 268.935 - 6.22919 - 259.340 - 6.19773 - 252.870 - 6.17138 - 248.595 - 6.15358 - 246.048 - 5.87302 - 218.789 - 5.75328 - 210.002 - 5.53559 - 195.802 - 5.31472 - 182.962 - 2.74070 - 78.476 - 1.43494 - 38.862 - 00 1.0679 26.87 4.2773 100 5.3414 122.37 7.0364 156.634 11.0133 231.9681 13.2188 271.442 16.0953 321.108 16.4733 327.502 18.2179 356.66 212 mat :

J 8.0957 - 210.002 - 7.7963 - 195.802 - 7.4807 - 182.962 - 3.7187 - 78.476 - 1.9063 - 38.862 - 00 1.3739 26.87 5.2677 100 6.4886 122.37 8.3743 156.634 12.5517 231.9681 14.7427 271.442 17.4928 321.108 17.8462 327.502 19.456 356.66 22.9259 419.58 24.3123 444.674 30.1095 548.23 34.9108 630.74 36.6933 660.37 55.669 961.93 61.716 1064.43 62.880 1084.5 232 mat :

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

data list ListCollect collects index list select list : ct 1 : i select a ct el data i 1 el : b ct el data i 2 el : ct ct 1 + : 31 line for a b augment 51 line :

XY J :

X XY 1 col : Y XY 2 col : n XY rows : 31 line

num order
denom order p.n 8 : p.d 7 : 21 sys

...k is total order of rational fit k 1 p.n + p.d + : 16

"Normalized" rational fraction have
leading "1" in denom.  Reputed stable
numerically. Most if not all computer
approximations are  "Normalized".
Generally, Num_order is Denom_order+1.
Some fit were found not "Normalized. x β f β 1 el β j 1 + el x j^* j 1 p.n sum + 1 β j 1 + p.n + el x j^* j 1 p.d sum + / maple :

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 :

support points ≡ k support 1346789101112141618202122116 mat :

x β Clear 1

β Discrete Cholesky solver slight recast from Alyles ct 1 : j support Xs ct el XY j 1 el : Ys ct el XY j 2 el : 12 mat ct ct 1 + : 21 line for Ms k Ones : i 1 p.n range Ms Ms Xs i^vectorize augment : for i 1 p.d range Ms Ms Xs i^Ys * vectorize - augment : for Ms 1 -^Ys * 81 line :

b β :

β ct 1 : j 1 b rows range u ct el b j el 15 round : ct ct 1 + : 21 line for u 31 line :

CWD C:\SmathFile CurrentDirectory :

β CWD βTypeJ wfile 1

β CWD βTypeJ rfile :

β β :

β rows 72

faster than
f(x,β) ... x f β 1 el β 2 el x * + β 3 el x 2^* + β 4 el x 3^* + β 5 el x 4^* + β 6 el x 5^* + β 7 el x 6^* + β 8 el x 7^* + β 9 el x 8^* + 1 β 10 el x * + β 11 el x 2^* + β 12 el x 3^* + β 13 el x 4^* + β 14 el x 5^* + β 15 el x 6^* + β 16 el x 7^* + / :

Absolute residuals [mutual conformity] conformity i 1 XY rows range Δconform i el Y i el X i el f - 2 round eval : for X Y Δconform augment 21 line :

conformity 8.0957 - 210.002 - 0 7.7963 - 195.802 - 0.04 7.4807 - 182.962 - 0 3.7187 - 78.476 - 0 1.9063 - 38.862 - 0000 1.3739 26.87 0 5.2677 1000 6.4886 122.37 0 8.3743 156.634 0 12.5517 231.9681 0 14.7427 271.442 0 17.4928 321.108 0 17.8462 327.502 0 19.456 356.66 0 22.9259 419.58 0 24.3123 444.674 0 30.1095 548.23 0 34.9108 630.74 0.02 36.6933 660.37 0 55.669 961.93 0 61.716 1064.43 0 62.88 1084.5 0.62 - 233 mat

FIT u 8.0957 - 65 8.05 - range : i 1 u rows range FIT i el u i el f eval : for u FIT augment 31 line :

8.0957 - 65 8.085 - range

12946832 / 0.19

1. selcted  support pts
2. collect for plot list support : xy XY list ListCollect : 21 sys

data XY . 12 black plot : support xy + 20 blue plot : x c 1 8.0957 - x ≤ (x 65 ≤ (& cases : 31 line

Rational fraction to Type 'J' T/C, NBS Monograph 125 Certified free of glitch @ XTR resolution ± 0.2 °C

data x f x c * support 3 1 sys

Support pts 'J' NBS-125 xy 8.0957 - 210.002 - 7.4807 - 182.962 - 3.7187 - 78.476 - 00 1.3739 26.87 5.2677 100 6.4886 122.37 8.3743 156.634 12.5517 231.9681 14.7427 271.442 17.8462 327.502 22.9259 419.58 30.1095 548.23 36.6933 660.37 55.669 961.93 61.716 1064.43 162 mat

__ Type 'J' Thermocouple __

-210.002 . . . 1084.5 °C

]]> 'c' normalised coefficients

'c' ...recovered from Mathcad c 19.8518394369938 3.39708430077966 0.052276519294839 8.91649190043396 103 -^* - 9.92828185332406 105 -^* 8.38053206800752 106 -^* 3.1464248690302 107 -^* - 4.25243756863902 109 -^* 2.07174891774966 1011 -^* - 0.183112912617708 3.87083881009766 103 -^* 5.08713285834413 104 -^* - 2.26745307368593 106 -^* 6.6275111583499 107 -^* 2.21691344604783 108 -^* - 2.87497042511128 1010 -^* 1.36525976453514 1012 -^* - 171 mat :

'β' ... recovered from Smath b 2.62726958173969 109 -^* - 19.8520614070612 1.44449548339689 - 0.182140077987682 - 0.023436694906595 0.000180113626196 3.3011046229056 105 -^* - 8.43291562951266 107 -^* 5.49764510078889 109 -^* - 0.060761237973477 - 0.010863420285349 - 0.001177409875546 2.39245028827081 105 -^* 2.44239684968009 106 -^* - 5.85871669914824 108 -^* 3.81957376430255 1010 -^* - 161 mat

recover 
reduced x J1084 x c 1 el c 2 el x * + c 3 el x 2^* + c 4 el x 3^* + c 5 el x 4^* + c 6 el x 5^* + c 7 el x 6^* + c 8 el x 7^* + c 9 el x 8^* + 1 c 10 el x * + c 11 el x 2^* + c 12 el x 3^* + c 13 el x 4^* + c 14 el x 5^* + c 15 el x 6^* + c 16 el x 7^* + c 17 el x 8^* + / * :

X J 1 col : Y J 2 col : 12 mat XY X Y augment : J XY . 12 black plot : 31 line

J1084 mV 8.25 - 638 - range : i 1 mV rows range °C i el mV i el J1084 eval : for mV °C augment 31 line :

Type 'J' ultimate fit to NBS 125

J J1084 2 1 sys

ΔJ1084 X Y Y X J1084 - (vectorize augment :

ΔJ1084 8.096 - 210.002 - 0.000 7.796 - 195.802 - 0.000 7.481 - 182.962 - 0.000 3.719 - 78.476 - 0.000 1.906 - 38.862 - 0.000 0.000 0.000 0.000 1.374 26.870 0.000 5.268 100.000 0.000 6.489 122.370 0.000 8.374 156.634 0.000 12.552 231.968 0.000 14.743 271.442 0.001 17.493 321.108 0.000 17.846 327.502 0.000 19.456 356.660 0.000 22.926 419.580 0.000 24.312 444.674 0.000 30.110 548.230 0.000 34.911 630.740 0.000 36.693 660.370 0.000 55.669 961.930 0.000 61.716 1064.430 0.000 62.880 1084.500 0.000 233 mat

Calibrate the 'J' mV/I converter

XmV 100 - :

XmV °C ≡

mV 1 :

°C 1 :

mV J1084 XmV - 0 ≡ mV 6 - 4 - solve mV 4.6322 -

4.6322 - J1084 °C 100 -

XmV 200 :

mV J1084 XmV - mV 812 solve mV 10.7765

10.7765 J1084 °C 200

From the "Calibrate", you can categorize all similar ranges of the 100's Type 'J' of the Industrial Plant. That does not warrant better accuracy than specified my the Manufacturer wrt their operating range. Extension wiring adds to the global ± °C of the installation.