FLOOD08610

• County: Statewide
• Basin: Statewide
• Title: Techniques of Water-Resources Investigations of the US Geological Survey Some Statistical Tools in Hydrology
• Date: 1/1/1969
• Prepared By: USGS
• Floodplain - Doc Type: Educational/Technical/Reference Information

SOME STATISTICAL TOOLS IN HYDROLOGY

29

.

If the graphical line cannot be conveniently
extended to X=l or X=O, the coordinates of a
point on the curve can be substituted in the
equation and the intercept can be computed.
The standard equations given in analytic
geometry texts are of little use in empirical
analysis. More flexible mathematical expres-
sions are needed, and ones that may be put in
linear form are desirable because of ease in
computing the equation, If a transformation
caJinot be found that will make the relation
linear, then a model of the type

Y a+b,X+b,X"+b.x"+ . , . +b.X',

or some portion of it, will fit most plotted
smooth curves. If the curvature is only in one
direction, the X" term will introduce the needed
curvature. For a curve having a point of inflec-
tion, both the X" and the X" terms are needed.
Terms having higher exponents are rarely used
in empirical work.
The above model is equally applicable where
X is replaced by log X. A line curved in one
direction on log paper is expressed by

log Y=log z+b, log X+b, (log X)'.

Reducing this to power form gives

Y=aX"X" ,oa=aX"+" loa.

.

The general form of linear relations m
several variables is

Y=a+b,X,+b,x, . . . +b"x..

Sometimes the regression coefficient for one
variable changes with another variable. This is
known as an interaction, In the model

.

Y=a+b,X, +b,x,+b,x,X"

the last term is called a product interaction
term. Its use provides a systematic change in
slope.
Curvilinear relations in several variables may
be described by adding terms in powers of the
independent variables. The equation of a curved
line or of a multiple relation involving an inter-
action is not easily computed. The advantage
of recognizing the general form of equation
which would represent a particular graphical
relation lies in the need for a model if a least-
square regression is to be computed. The

definition of the equation of a graphical re-
gression is limited w linear regressions.

DeFinition of equations

The methods for defining the equation of a
graphical regression will be demonstrated by
two examples. The procedures used in these
examples can be adapted readily to other
problems. The first example, shown in figure
21, is a multiple linear regression by the method
of residuals. The equation of this relation is
obtained as follows. Consider first the relation
between Y, and X, where Y, is the curve value
from the left part of figure 21. Tbis relation is
of the form Y =a+bX, where a is the intercept
at X,=O and b is the slope of the line, For this
example,

Y,=5.4+0.86X,_

Tbe equation of the second line is obtained
similarly and is

Residual= -1O+2.78X,.

Tbe residual (call it R) is tbe individual point
value, Y, minus the value obtained from tbe
first equation; tbat is,

R=Y-Y,=-1O+2.78X,.

Substituting for Y, in the above equation gives

Y - (5.4+0.86X,) = -1O+2.78X,

from wbich the desired relation,

Y = -4,6+0.86X, +2.78X"

is obtained.
The second example (fig. 22) is a relatively
simple coaxial grapbical multiple regression
adapted from one made by the Hydraulic
Researcb Branch of the Bureau of Public
Roads. This regression is linear, and the lines
for Sand P are systematically spaced and
parallel. Under tbese conditions the equation
of the grapbical relation can be determined.
Tbe following facts are evident from a study
of fignre 22:

1. QlO is the dependent variable.
2. A is tbe principal independent variable,
3. Lines of equal P are linearly spaced on
logarithmic paper and are parallel.