09f2a581-a5bd-45bc-9216-4868b9d77380 71 Javier Ochoa 4 5 false false 0 0 decimal

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This is the same problem solved used the linfit.pdf as a guide.

3 appVersion 1.0.8348

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 and standard errors where compared at the end of each case.

x M 25 range 1 el :

y M 25 range 2 el :

N Mathematical Function Equation Typical application Linear y=b*x D’Arcy Law pressure drop vs. Linear flow velocity Exponential ln(y)=-b*x First-order decrease in concentration (or radiation) with time Logarithmic y=b*ln(x) Same as exponential but and independent variable transposed. Power ln(y)=b*ln(x) Pressure drop vs. Flowrate in a capillary or oriffice. 53 mat :

N M N y x

N
Mathematical Function	Equation	Typical application
Linear	y=b*x	D’Arcy Law pressure drop vs. Linear flow velocity
Exponential	ln(y)=-b*x	First-order decrease in concentration (or radiation) with time
Logarithmic	y=b*ln(x)	Same as exponential but and independent variable transposed.
Power	ln(y)=b*ln(x)	Pressure drop vs. Flowrate in a capillary or oriffice.
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Case 1: linear forced through coordinate set. Then, we will caculate the linear regression .

x y b x * ≡

x φ x 11 mat :

n x length :

n 4

i 1 n range X i 1 el x i el φ : for

Φ X transpose X * :

d X transpose y * :

blf Φ 1 -^eval d * :

blf 1.5667 11 mat

i 1 y length range yc i el x i el ycalc : for

stdf 1 n / y yc - (2^* sqrt :

x ycalc blf x φ * :

stdf 0.1384

x φ 1 x 21 mat :

y b 1 el b 2 el x * + ≡

k x φ length :

k 2

j 1 k range i 1 n range X i j el x i el φ j el : for for

Φ X transpose X * :

d X transpose y * :

blin Φ 1 -^eval d * :

blin 0.25 - 1.65 21 mat

x ycalc1 blin x φ * :

i 1 y length range yc1 i el x i el ycalc1 : for

std 1 n k - (/ y yc1 - (2^* sqrt :

std 0.1323

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

y ln b x * ≡

x φ x :

n x length :

n 4

i 1 n range u i el y i el ln : for

i 1 n range X2 i 1 el x i el φ : for

Φ X2 transpose X2 * :

d X2 transpose u * :

bexpf Φ 1 -^eval d * :

bexpf 0.4868

i 1 y length range yc i el x i el ycalc2 : for

x ycalc2 e bexpf x φ *^eval :

std2f 1 n / y yc - (2^* sqrt :

std2f 0.3885

y ln b 1 el b 2 el x * + ≡

x φ 1 x 21 mat :

k x φ length :

k 2

j 1 k range i 1 n range X3 i j el x i el φ j el : for for

Φ X3 transpose X3 * :

d X3 transpose u * :

bexp Φ 1 -^eval d * :

bexp 0.0097 0.4835 21 mat

i 1 y length range yc3 i el x i el ycalc3 : for

std2 1 n k - (/ y yc3 - (2^* sqrt :

x ycalc3 e bexp x φ *^eval :

std2 0.5364

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

y b x ln * ≡

x φ x ln :

n x length :

n 4

i 1 n range X4 i 1 el x i el φ : for

X4 0 0.6931 1.0986 1.3863 41 mat

Φ X4 transpose X4 * :

d X4 transpose y * :

blogf Φ 1 -^eval d * :

blogf 4.4458

i 1 y length range yc i el x i el ycalc2 : for

std3f 1 n / y yc - (2^* sqrt :

ycalc4 blogf x φ * :

std3f 0.3885

y b 1 el b 2 el x ln * + ≡

x φ 1 x ln 21 mat :

k x φ length :

k 2

j 1 k range i 1 n range X5 i j el x i el φ j el : for for

Φ X5 transpose X5 * :

d X5 transpose y * :

blog Φ 1 -^eval d * :

blog 1.141 3.4412 21 mat

x ycalc5 blog x φ * :

std3 1 n k - (/ y yc - (2^* sqrt :

i 1 y length range yc i el x i el ycalc5 : for

std3 0.5495

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 .

y ln b x ln * ≡

x φ x ln :

n x length :

n 4

i 1 n range u i el y i el ln : for

i 1 n range X6 i 1 el x i el φ : for

Φ X6 transpose X6 * :

d X6 transpose u * :

bpowf Φ 1 -^eval d * :

bpowf 1.3885

i 1 y length range yc i el x i el ycalc6 : for

std4f 1 n / y yc - (2^* sqrt :

x ycalc6 e bpowf x φ *^eval :

std4f 0.3697

y ln b 1 el b 2 el x ln * + ≡

x φ 1 x ln 21 mat :

k x φ length :

k 2

j 1 k range i 1 n range X7 i j el x i el φ j el : for for

Φ X7 transpose X7 * :

d X7 transpose u * :

bpow Φ 1 -^eval d * :

bpow 0.3838 1.0506 21 mat

x ycalc7 e bpow x φ *^eval :

i 1 y length range yc i el x i el ycalc7 : for

std4 1 n k - (/ y yc - (2^* sqrt :

std4 0.1288

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 blf bexpf blogf bpowf Standard Error stdf std2f std3f std4f a Unforced blin 1 el bexp 1 el blog 1 el bpow 1 el b Unforced blin 2 el bexp 2 el blog 2 el bpow 2 el Standard Error std std2 std3 std4 65 mat : Curve Type Linear Exponential Logarithmic b forced 1.5667 11 mat 0.4868 4.4458 Standard Error 0.1384 0.3885 0.3885 a Unforced 0.25 - 0.0097 1.141 b Unforced 1.65 0.4835 3.4412 Standard Error 0.1323 0.5364 0.5495 64 mat

U std4 ycalc7 ycalc6 std4f bpowf std3 ycalc5 ycalc4 std3f blogf ycalc3 std2 std2f ycalc2 bexpf std ycalc1 ycalc stdf blf U std4 ycalc7 bpow ycalc6 std4f bpowf X6 std3 ycalc5 blog ycalc4 std3f blogf X4 ycalc3 std2 yc3 bexp std2f ycalc2 bexpf X2 u std yc1 ycalc1 blin ycalc stdf yc blf d φ y x std4 bpowf std3 ycalc5 ycalc4 std3f blogf std2f bexpf std ycalc1 ycalc

U
Curve Type	Linear	Exponential	Logarithmic	Power
b forced	mat(1.57,1,1)	0.49	4.45	1.39
Standard Error	0.14	0.39	0.39	0.37
a Unforced	-0.25	0.01	1.14	0.38
b Unforced	1.65	0.48	3.44	1.05
Standard Error	0.13	0.54	0.55	0.13
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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 stdf x k el (2^k 1 n sum / : 0.0046

sbexp std2f x k el ln 2^k 1 n sum / : 0.1077

sbln std3f x k el (2^k 1 n sum / : 0.013

sbpow std4f x k el ln (2^k 1 n sum / : 0.1024

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