I
SABRI ERGUN Branch of Bituminous Coal, Bureau of Mines, Pittsburgh, Pa.
Application of the Principle of Least Squares to Families of Straight lines
DATA
relating to numerous physicochemical phenomena can be presented in the form of parallel straight lines or lines having common intercepts. Methods have been developed for treating such data statistically.
considered. The coordinate which has the error, the dependent variable, is designated as Y. Theory. A family of straight lines having equal slopes can be expressed as yi = A ,
Parallel Straight lines Physicochemical phenomena such as absorption (23), crystallization (77), distillation (78), filtration (77, 79), flame stability (73),heat transfer (2, 7 4 , hydroextraction ( 7 7), reaction kinetics (3, 7, 76, 27), and others (4,72, 24, 26) are frequently expressed in the form of families of parallel straight lines. The aim of the present study has been to develop a special method of least squares whereby data obtained for such cases can be treated statistically. The principle of the method of least squares involves minimizing the sum of the squares of the residuals. In curve fitting either or both of the Y and X observations may be subject to error. Accordingly, the sum will be written explicitly for the Y residual alone, or the X residual alone, or both, depending upon the experimental conditions. I n the present treatment the case in which the error of one coordinate, the independent variable is negligible compared with the error of the other coordinate is
f SX,
(li)
where subscript i denotes any of the individual lines designated by subscripts a, b, , n; X,is the measured value of the independent variable on line i; y , is the corresponding calculated value of ordinate; B is the estimate of the common slope; and A i is the estimate of the intercept of line i. The deviation of the calculated results, y , from the observed results, Y , for any point on line i can be expressed as
...
A, = yi
- Y , = A , + SA', - Y,
(2i)
where A, is the deviation or the residual. Then, Z4, represents the sum of the residuals of all the points on line i and 24: represents the sum of the squares of the residuals. If the sum of the squares of the residuals for each line is kept a t a minimum, independent regression lines will result and a common slope will not be obtained. If, however, the sums of the squares of all residuals are simultaneously kept a t a minimum, parallel regression lines will be obtained. The
sum of the sums of squares of the residuals can be expressed as i(ZAT) a = 2AZ -k ZA:
+ .. . f2A;
(3)
T o determine the values of A's and of n
the common slope B such that Z(Z4:) a
will be minimum, the partial summations with respect to B and each A must equal zero :
+ . . . + 2A;) = 0 ( 4 ) (EA: f 2A: + . . . + 2AZ) = 0 (4a)
b a (EA: b b
2AZ
=,(ZA:
-k ZA; f . . .
b -(ZA: hA n
+ 2A: f
+ 2A:)
... f
=0
(4b)
EA:) = 0
(4n)
Evaluating the partial term by term, bZAZ,/bB = 22X,A,, bZA:/bB = 2zXb&,, etc.
(5)
b Z A : / b A , = 22A,, bZA:/bA. = 0, . . b Z A : / b A , = 0 (sa)
.
bZA:/bAb =: 0, bZA:/bA, = 2ZAb, VOL. 48, NO. 1 1
e
. . . bZA:/Ab
= 0 (5b)
NOVEMBER 1956
2063
Table I.
Determination of Energy of Activation for Conversion of Hexagonal Close-Packed Iron Carbide to Hagg Iron Carbide Separate Least Squares
Hexagonal Carbide, Fraction
Y (Loo 0
0.75
1.57 4.12
0.65
0.55
X
A
calcd.
A
1.815 1.880
...
....
1.72 3.97
+ O . 15
1.19 3.84 5.73
1.745 1.815 1.880
1.29 3.64 5.83
3.0.10 -0.20 $0.10
1.21 3.65 5.90
1-0.02 -0.19 $0.17
-0.916 $0.285 2.69 5.07 7.09
1.653 1.686 1.745 1.815 1.880
-0.810 $0.363 2.46 4.95 7.26
$0.11 $0.08 -0.23 -0.12 1-0.15
-0,727 +0.419 2.47 4.90 7.16
$0.19 + O . 13 -0.22 -0.17 $0.07
0.587 1.73 3.56 5.98
1.653 1.686 1.745 1.815
0.592 1.68 3.63 5.95
+O.Ol -0.05 1-0.07 -0.03
0 472 6.10
-0.12 -0.11 3.0.11 +O.lZ
1.41 2.61 4.34
7.07
1.653 1.686 1.745 1.815
1.39 2.52 4.55 6.97
-0.02 -0.09 $0.21 -0.10
1.37 2.51 4.56 6.99
-0.04 -0.10 $0.22 -0.08
2.06 3.44 5.24
1.653 1.686 1 745
2.16 3.28 5.30
$0. 10 -0.16 1-0.06
2.13 3.28 5.33
$0.07 -0.16 $0.09
0.15
2.88 4.30 6.18
1.653 1.686 1.745
2.98 4.15 6.24
+o.
3.01 4.15 6.20
3.0.13 -0.15 $0.02
0.05
4.20 5.47
1.653
I 686
...
4.26 5.41
1-0.06 -0.06
0.35
0.25
(1OOO/
I
(5n)
Substitution of Equations 5 to 5, in 4 to leads to
411,respectively,
+ ZX*& + . . . +
ZX,A,
n
ZX,A,
Z(ZXi4i) = 0
(6)
4
and ZAa = ZAb = ZAn = ZSi
=
0
(6a-n)
Substitution of Equation 2 in 6i and proper rearrangement lead to (BY< - B Z X , ) / N
