12eed524-3561-4913-9a98-1b51402d502d 131 Javier Ochoa 4 5 false false 0 0 decimal

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Regression equations forced through a coordinate point. Leonard A. Jonas, Eric B. Sansone and John C. Conion. Source: Chemical Engineering, May 27, 1986. Linear, exponential, logarithmic and power functions can be forced through a coordinate point by means of the following Table, which is presented whith the ones for a linear equation and the results were compared. The data used in these examples is in the Table 1 below.

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M x y 1 1.5 2 2.9 3 4.7 4 6.4 52 mat :

M M

M
x	y
1	1.5
2	2.9
3	4.7
4	6.4
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Source : Chemical Engineering, May 27, 1986. Regression equations forced through a coordinate point, Leonard A. Jonas, Eric B. Sansone and John C. Conlon. Linear, exponential, logarithmic and power functions can be forced through a coordinate point by means of the equations presented in the Table below.

x M 25 range 1 el :

y M 25 range 2 el :

n x length :

n 4

After doing the calculations the equation for a linear regression polynomial is presented, also with the correspondiing parameters for n data points. All the equations are from Hoel, P. G., "Introduction to Mathematical Statistics", John Wiley, NY, 1947, pp. 74 - 84 .

Case 1: linear forced through coordinate set. Let's calculate the linear regression first.

sumx_2 x k el k 1 n sum (2^:

sumx2 x k el 2^k 1 n sum :

sumx x k el k 1 n sum :

sumx2 30

sumx 10

sumx_2 100

sumy y k el k 1 n sum :

sumxy x k el y k el * k 1 n sum :

sumy2 y k el 2^k 1 n sum :

sumy 15.5

sumxy 47

sumy2 73.71

sumy_2 y k el k 1 n sum (2^:

sumy_2 240.25

blinear n sumxy * sumx sumy * - (n sumx2 * sumx_2 - (/ :

blinear 1.65

blinearf sumxy sumx2 / :

blinearf 1.5667

alinear sumy sumx2 * sumx sumxy * - (n sumx2 * sumx_2 - (/ :

alinear 0.25 -

rlinear n sumxy * sumx sumy * - (n sumx2 * sumx_2 - (12 /^n sumy2 * sumy_2 - (12 /^* / :

rlinear 0.9987

rlinearf sumxy sumx2 sumy2 * (12 /^/ :

rlinearf 0.9995

x plotlinear alinear blinear x * + :

x plotlinearf blinearf x * :

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

i 1 n range yc0 i el blinearf x i el * : for

plotl1 x y . 20 Blue plotG :

x plotlinear x plotlinearf plotl1 3 1 sys

The figure show the results of the calculations and the data presented in Case 1. The unbiased estimator of the true variance is given by the residual mean square with n - 1 degrees of freedom, denoted as standard error of estimate. In the linear fit with a constant, they are n - 2 degrees of freedom.

stdlinear y k el yc0 k el k 1 n sum n / - (k 1 n sum 2^n 1 - / sqrt :

stdlinear 0.0962

i 1 n range yc1 i el alinear blinear x i el * + : for

stdlin y k el yc1 k el k 1 n sum n / - (k 1 n sum 2^n 2 - / sqrt :

yc1 1.4 3.05 4.7 6.35 41 mat

stdlin 0

Case 2. Linear regression for the Exponential data force through (0 , 1) and also the linear equation ln y = a + b x .

sumlny y k el ln k 1 n sum :

sumxlny x k el y k el ln * k 1 n sum :

sumlny 4.874

sumxlny 14.6028

sumlny2 y k el ln (2^k 1 n sum :

sumlny_2 y k el ln k 1 n sum (2^:

sumlny2 7.1388

sumlny_2 23.7562

bexpf sumxlny (sumx2 / :

rexpf sumxlny sumx2 (12 /^sumlny2 (12 /^* / :

bexpf 0.4868

rexpf 0.9978

bexp n sumxlny * sumx sumlny * - (n sumx2 * sumx_2 - (/ :

aexp sumlny sumx2 * sumx sumxlny * - (n sumx2 * sumx_2 - (/ :

bexp 0.4835

aexp 0.0097

rexp n sumxlny * sumx sumlny * - (n sumx2 * sumx_2 - (12 /^n sumlny2 * sumlny_2 - (12 /^* / :

rexp 0.9871

x plotexp aexp bexp x * + :

x plotexpf bexpf x * :

k 1 n range yy k el y k el ln : for

yy 0.4055 1.0647 1.5476 1.8563 41 mat

plot2 x yy + 20 Green plotG :

i 1 n range yc i el e bexpf x i el * (^(: for

x plotexp x plotexpf plot2 3 1 sys

The figure show the results of the calculations and the data presented in Case 2. The standard error was calculated with the following formula:

stdexpf y k el yc k el k 1 n sum n / - (k 1 n sum 2^n 1 - / sqrt :

i 1 n range yc2 i el e aexp bexp x i el * + (^eval : for

stdexpf 0.0516

stdexp y k el yc2 k el k 1 n sum n / - (k 1 n sum 2^n 2 - / sqrt :

stdexp 0.0607

Case 3. Linear regression for the logarithmic data force through (1 , 1) and also the linear equation y = a + b ln x .

sumlnx x k el ln k 1 n sum :

sumylnx y k el x k el ln * k 1 n sum :

sumylnx 16.0459

sumlnx 3.1781

sumlnx2 x k el ln (2^k 1 n sum :

sumlnx_2 x k el ln k 1 n sum (2^:

sumlnx2 3.6092

sumlnx_2 10.1

blnf sumylnx sumlnx2 / :

blnf 4.4458

rlnf sumylnx sumlnx2 sumy2 * (12 /^/ :

rlnf 0.9838

bln n sumylnx * sumlnx sumy * - (n sumlnx2 * sumlnx_2 - (/ :

bln 3.4412

aln sumy sumlnx2 * sumlnx sumylnx * - (n sumlnx2 * sumlnx_2 - (/ :

aln 1.141

rln n sumylnx * sumlnx sumy * - (n sumlnx2 * sumlnx_2 - (12 /^n sumy2 * sumy_2 - (12 /^* / :

rln 0.9699

k 1 n range xx k el x k el ln : for

xx plotln aln bln xx * + :

xx plotlnf blnf xx * :

xx 0 0.6931 1.0986 1.3863 41 mat

plot3 xx y o 20 Magenta plotG :

i 1 n range yc3 i el blnf x i el ln * : for

x plotln x plotlnf plot3 3 1 sys

The figure show the results of the calculations and the data presented in Case 3. The standard error was calculated with the following formula:

stdlnf y k el yc3 k el k 1 n sum n / - (k 1 n sum 2^n 1 - / sqrt :

stdlnf 0.7915

i 1 n range yc4 i el aln bln x i el ln * + : for

stdln y k el yc4 k el k 1 n sum n / - (k 1 n sum 2^n 2 - / sqrt :

stdln 2.2168 1014 -^*

Case 4. Linear regression for the power function and data force through (0 , 1) and also the linear equation ln y = a + b ln x .

sumlnxlny x k el ln y k el ln * (k 1 n sum :

sumlnxlny 5.0115

bpowf sumlnxlny sumlnx2 / :

rpowf sumlnxlny sumlnx2 sumlny2 * (12 /^/ :

bpowf 1.3885

rpowf 0.9873

bpow n sumlnxlny * sumlnx sumlny * - (n sumlnx2 * sumlnx_2 - (/ :

bpow 1.0506

apow sumlny sumlnx2 * sumlnx sumlnxlny * - (n sumlnx2 * sumlnx_2 - (/ :

apow 0.3838

rpow n sumlnxlny * sumlnx sumlny * - (n sumlnx2 * sumlnx_2 - (12 /^n sumlny2 * sumlny_2 - (12 /^* / :

rpow 0.9987

xx plotpow apow bpow xx * + :

xx plotpowf bpowf xx * :

plot4 xx yy o 20 Black plotG :

i 1 n range yc3 i el e xx i el plotpowf (^: for

x plotpow x plotpowf plot4 3 1 sys

The figure show the results of the calculations and the data presented in Case 4. The standard error was calculated with the following formula:

stdpowf y k el yc3 k el k 1 n sum n / - (k 1 n sum 2^n 1 - / sqrt :

stdpowf 0.2482

i 1 n range yc4 i el e xx i el plotpow (^: for

stdpow y k el yc4 k el k 1 n sum n / - k 1 n sum (2^n 2 - / sqrt :

stdpow 0.0272

The results of the four Cases may be grouped in a Table using the followin matrix:

U Curve Type Linear Exponential Logarithmic Power b Forced blinearf bexpf blnf bpowf Standard Error stdlinear stdexpf stdlnf stdpowf R coefficient rlinearf rexpf rlnf rpowf a Unforced alinear aexp aln apow b Unforced blinear bexp bln bpow Standard Error stdlin stdexp stdln stdpow R Coefficient rlinear rexp rln rpow 85 mat : Curve Type Linear Exponential Logarithmic b Forced 1.5667 0.4868 4.4458 Standard Error 0.0962 0.0516 0.7915 R coefficient 0.9995 0.9978 0.9838 a Unforced 0.25 - 0.0097 1.141 b Unforced 1.65 0.4835 3.4412 Standard Error 0 0.0607 2.2168 1014 -^* R Coefficient 0.9987 0.9871 0.9699 84 mat

U stdpow stdpowf plot4 plotpowf plotpow rpow apow bpow rpowf bpowf sumlnxlny stdln stdlnf plot3 plotlnf plotln rln aln bln rlnf blnf sumlnx_2 sumlnx2 sumylnx sumlnx stdexp stdexpf plot2 plotexpf plotexp rexp aexp bexp rexpf bexpf sumlny_2 sumlny2 sumxlny sumlny stdlinear plotG plotlinearf plotlinear rlinearf rlinear alinear blinearf blinear sumy_2 sumy2 sumy U stdpow stdpowf plot4 plotpowf plotpow rpow apow bpow rpowf bpowf sumlnxlny stdln yc4 stdlnf yc3 plot3 plotlnf plotln xx rln aln bln rlnf blnf sumlnx_2 sumlnx2 sumylnx sumlnx stdexp yc2 stdexpf yc plot2 yy plotexpf plotexp rexp aexp bexp rexpf bexpf sumlny_2 sumlny2 sumxlny sumlny yc1 stdlinear plotl1 yc0 plotG plotlinearf plotlinear rlinearf rlinear alinear blinearf blinear sumy_2 sumy2 sumy y x stdpow stdpowf plotpowf plotpow rpow apow bpow rpowf bpowf sumlnxlny stdln stdlnf plotlnf plotln rln aln bln rlnf blnf sumlnx_2 sumlnx2 sumylnx sumlnx stdexp stdexpf plotexpf plotexp rexp aexp bexp rexpf bexpf sumlny_2 sumlny2 sumxlny sumlny stdlinear plotlinearf plotlinear rlinearf rlinear alinear blinearf blinear sumy_2 sumy2 sumy

U
Curve Type	Linear	Exponential	Logarithmic	Power
b Forced	1.567	0.487	4.446	1.389
Standard Error	0.096	0.052	0.792	0.248
R coefficient	0.999	0.998	0.984	0.987
a Unforced	-0.250	0.010	1.141	0.384
b Unforced	1.650	0.484	3.441	1.051
Standard Error	0.000	0.061	0.000	0.027
R Coefficient	0.999	0.987	0.970	0.999
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Althoug the correlation coefficients are good, as the authors claimed, significant differences in the Standard Error coefficient for the estimates were found in all the Cases, which seems to be a better to decide whether the method is good enough. The authors also calculated the standard error for the slope, on each forced regression , which gave them: 0.02919, 0.01849, 0.4615 and 0.12895 for the Linear, Exponential, Logarithmic and Power forced regressions, respectively.

The estimate of the Standard Error for the forced slope on each regression can be calculated by the following formulas:

sbl stdlinear x k el 2^k 1 n sum / : 0.0032

sbexp stdexpf x k el ln 2^k 1 n sum / : 0.0143

sbln stdlnf x k el (2^k 1 n sum / : 0.0264

sbpow stdpowf x k el ln (2^k 1 n sum / : 0.0688

The values follow the trend found by the authors, who made the calculations with a Texas Instruments 59 calculator by this time.