Houck, James P.

Hunt, R.D.

English

Research Papers in Economics

Staff Paper Series
      Staff Paper P68-2                                                                         October  1968
USING ABSOLUTE DEVIATIONS TO COMPUTE
LINES OF BEST FIT
By
J. P. Houck and R. D. Hunt
Department of Agricultural Economics
University of Minnesota
Institute of Agriculture
St. Paul, Minnesota 55108USING ABSOLUTE DEVIATIONS TO COMPUTE
LINES OF BEST FIT
October 1968
J. P. Houck and R. D. Hunt
P 68-2USING ABSOLUTE DEVIATIONS TO COMPUTE
LINES OF BEST FIT
J. P. Houck and R. D. Hunt
There is some renewed interest among agricultural economists and others in the
old technique of using minimized absolute deviations in computing lines of best fit.
H. B. Jones and J. C. Thompson, in a recent article in Agricultural Economics Research$
discussed philosophical and practical questions raised when fitted lines using
squared and unsquared deviations are compared ~~4]. However, agricultural economics
literature~ (includingthe Jones-Thompson paper) and elementary statistics and
econometric literature seems to contain no concise description of the straight-
forward and simple procedure for calculating lines of best fit which minimize the
sum of absolute deviations from the empirical observations to the fitted line. In
some experimental work at the University of Minnesota, we find that lines fitted
by minimizing the sum of absolute deviations can be useful in certain circumstances.
Our purpose is to present, without proving, a simple step-by-stepmethod for
calculating such lines of best fit for the two variable case. An illustrative
example is appended. Those wishing to investigate the historical development and
mathematical basis for these methods are directed to the References section,
especially ~1~ [2~ [5_7 [62 [7j. The article by Kaarst is particularly
helpful.
It is our view that the minimized sum of absoluted deviations (MSAD) can be
quite useful for linear trend fitting. Computing a trend line generally does not
involve assumptions about the error term’s statistical distribution. Therefore,
unless some quadatric loss function is postulated, minimizing absolute deviations2
has as much theoretical appeal as minimizing squared deviations and rests on a simpler
conceptual base. Selection of the MSAD procedure over least squares might hinge on
the existence of one or more extreme or ‘unusual”data points which strongly influence
the position of a least squares line. The influence of these extreme points is
reduced with the MSAD approach. Moreover, an MSAD line is easy to compute with a
desk calculator, perhaps easier than simple least squares.
Consider a set of n observations on Y and X to which a linear function is to be
fitted. The fitted function is to have the form
Y = a +bX
where a and b are parameter estimates.
Suppose that, instead of an ordinary least squares line of best fit, a MSAD line is
desired.
Two versions of this line can be computed [5]. The first is the MSAD line
constrained to pass through a pre-selected point such as the means! the medians9 or
the origin. The second is the unconstrained MSAD line. Unlike the ordinary least
squares line, the unconstrained MSAD line does not, in general, pass through the point
of means.
The Constrained MSAD Line
1. Select the reference point (Yo~ Xo) through which the MSAD line is to pass.
For each of the n observations, compute
Yi = (Yi -Yo)
i = 1,2939 ....... n
xi= (xi -Xo)
2. Calculate the n ratios, yi/x.. Then rank these ratios in ascending algebraic 1
order, beginning with largest negative or the smallest positive number.3
3. Zj Sum the absolute values of xi; ~ xi
, —
4. To the value -~ x
II
x 9 add the absolute value of x from the ratio
n i i
having the lowest rank. Then add the absolute value of xi from the ratio
with the next lowest rank. Continue until the algebraic sum changes sign
from negative to positive.
5. Note the value of the ratio (yk/xk) at which the sign change occurs. This
ratio is the slope, b, of the constrained MSAD line. This line will pass
through both the pre-selected point and the point at which the ratio
(xk/yk) is computed. Having calculated b:
a=Y - bXo
o
The Unconstrained MSAD Line
Finding the unconstrained MSAD line of best fit involves a simple iterative
procedure based on the method described above for the constrained line. The procedure
is:
1. Compute a constrained MSAD line, but select one of the data points as the
initial reference point (Fewer iterations are needed if the selected point
is on or near a freehand line of best fit.)
2. This constrained line will pass through at least one other data point in the
sample. Select this point as the next reference point, and compute another
MSAD line.
3. This second line will pass through still another data point. Using this point
as the reference, re-compute.
4. Continue until the fitted line reflects back through a data point already
used. This line is then the unconstrained MSAD line of best fit. By
judicious selection of the initial reference point on or near a freehand4
trend line, only one or two iterations usually will be needed.
Goodness of Fit
Measuring the goodness of fit of an MSAD line is not as straightforward
with the least squares technique. This is because the total sum of absolute
deviations of
unambiguously
not accounted
Yi about a point,
into that portion
for by the fitted
as
the sample mean in least squares.
say the mean or medianj cannot be partitioned
accounted for by the fitted line and that portion
line, as can be done with squared deviations about
a measure of how well an MSAD line
f=l-
Where?i are values of Y along the
N
and Y is the median value of Y.
The median is selected as the
However, we suggest the following coefficient as
fits the sample observations on Y,.
~ y~i~~lyfii
fitted MSAD line associated with the sample X. 1
reference point since, for any given set of
numbers, the sum of the absolute deviations is smaller when measured from the median
than from any other number ~8_7. The coefficient, f, can range between zero and
-+-1.0.If there is no systematic association between X and Y in the data, then the
minimizing of absolute deviations will yield ?i =
&
Y and f will be zero. On the
other hand, if the fit is perfect and ?i = Yi, then f will be +1.0. Intermediate
values of f will then indicate how well the MSAD line fits the data relative to a
goodness
scale of zero to +1.0. As a criterion for judging/of fit, the coefficient f
probably should not be compared with the least squares r2 for the same set of data.
Its use should be confined to comparisons of MSAD lines of best fit. For example?
the coefficient f can be used to compare constrained ~ unconstrained MSAD lines
or constrained MSAD lines fitted through va:riouspreselected points.5
Concluding Comments
Tables 1 and 2 contain an illustrative example of the calculation of both the
constrained and the unconstrained MSAD line based on a single set of hypothetical
data. The f coefficient is computed for both. Note that f is slightly larger with
the unconstrained line. Figure 1 illustrates the data and the two computed lines.
Extending these methods beyond two variables is not easy, though it has been
done mathematically ~3]~7~. But perhaps the step-by-step sequences presented
here will be useful to agricultural economists and other researchers who can
visualize using MSAD lines of best fit in their work.6
Table 1: Calculation of constrained MSAD line of best fit through the
data means.
x Y x Y Ranking of
(X-Y) (Y-Y)
y/x y/x
1
2
3
4
5
6
7
8
9
10
11
10
7
11
4
6
3
1
5
2
3
1
-5
-4
-3
-2
-1
0
1
2
3
4
5
5.18
2.18
6.18
- .82
1.18
-1.82
-3.82
.18
-2.82
-1.82
-3.82
-1.036
- .545
-2.060
- .410
-1.180
---
-3.820
.090
- .940
- .455
- .764
4
7
2
9
3
11
1
10
5
8
6
?= 4.82 Y = 4.0
= -15 (here the subscript on x
= +1 denotes ranking) i
-14
-— +-?
-11
— — +1
-10
— — +5 ——
“5
— +3 ——
-2~
=
--H--l
Note sign change at rank 6
Then: b = (y6/X6) = -.764
a =~-b? = 9.402
f = 1 - (17.238 / 29.000) = .4067
Table 2: Calculation of unconstrained MSAD line of best fit selecting
X = 5, Y = 6 as the first reference point.
(First iteration)
x Y
{X:5) [Y~6)
y/x Ranking
1
2
3
4
5
6
7
8
9
10
11
10
7
11
4
6
3
1
5
2
3
1
-4
-3
-2
-1
0
1
2
3
4
5
6
4
1
5
-2
0
-3
-5
-1
-4
-3
-5
-1.000
- .333
-2.500
2.000
---
-3.000
-2.500
- .333
-1.000
- .600
- .833
6
10
3
4
11
1
2
9
5
8
7
-— ; xl x.
Ix;
I‘2
I‘3
‘4
‘5
‘6
‘7
= -15
=+1
-14
=+1
7
= +’4
-5
= +4
(subscript denotes ranking)
- 1 ‘-T
-1
Note sign change at rank 7
=+6
%
Then for the first iteration: (b= y /X7 = -.833.)
This line passes through X = 11, Y = ~. Use this point
as reference for next iteration
(continued next page)Table 2 (continued)
(Second iteration)
x Y
(x-h) [Y-;)
y/x Ranking
1
2
3
4
5
6
7
8
9
10
11
10
7
11
4
6
3
1
5
2
3
1
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
9
6
10
3
5
2
0
4
1
2
0
-e900
-.667
-1.250
-.429
-.833
-0400
0
-1.333
-.500
-2.000
---
4
6
3
8
5
9
10
2
7
1
11
b;l = +1*O (Subscriptdenotes ranking)
-26.5
lx I = + 8.0
3 -15.5
lx I = +10.0
4 - 5.57
_-
Note sign change at rank 5
x= + 6.0
5’ + 0.5 ~,~-.+.%.
Then for the second iteration: (b=y5/x5= -.833)
This is the same as the result of the first iteration. The fitted
line reflects back through X = 5, Y = 6 and is the final unconstrained
solution.
b= -.833
a = 6+ (.833) (5) = 10.165
f = 1 - (17.167 /29.000) = .4089
Figure 1: Fitted MSAD lines and sample data for illustrative example
v
11
10
9
8
7
6
5
4
3
2
1
0
l Unconstrained
constrained~
@
a
e
m
o 1 2 3 4 5 6 ‘7 8 9 10 11References
10
2.
3.
4.
5.
6.
7.
8.
Ashar, V. G. and T. D. Wallace., “A Sampling
Deviations Estimators,” Operations Research,
October 1963, pp. 747-758.
Study of Minimum Absolute
No. 5, Vol. 1, September -
Edgeworth, F. Y., “On a New Method of Reducing Observations Relating to
Secular Quantities,” philosophical Maqazine, Series 5, Vol. 25? No. 154,
March 1888, pp. 184-191.
Fisher, W. D., “A Note on Curve Fitting with Minimum Deviations by Linear
Programming,” Journal of American Statistical Association? Vol. 56, 1961,
pp. 359-362.
Jones, Harold B., and Jack C. Thompson., “Squared versus Unsquared Deviations
for Lines of Best Fit,” Agricultural Economics Research, Vol. 20, No. 29
April 1968, pp. 64-69.
Kaarst, Otto J., “Linear Curve Fitting Using Least Deviations,” Journal of the
American Statistical Association, No. 281, Vol. 53, March 1958, pp. 118-132,
Rhodes$ E. C., “Reducing Observations by The Method of Minimum DeviationsJ”
Philosophical Maqazine, Series 7, Vol. 9, No. 60, May 1930.
Singleton, R. R., “A Method for Minimizing The Sum of Absolute Values of De-
viations,” Annals of Mathematical Statisticsz
Waugh, A. E., Elements of Statistical Methodl
P“ 1140
vol. 11, 1940, pp. 301-305.
Third edition, (McGraw-Hill: 1952)7

USING ABSOLUTE DEVIATIONS TO COMPUTE LINES OF BEST FIT

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