Polynomial regression
Daniel Borcard, Département de sciences biologiques, Université de Montréal
Reference: Legendre and Legendre (1998) p. 526
A variant form of multiple regression can be used to fit a nonlinear
model of an explanatory variable
x
(or several explanatory variables
x
j
) to a response variable
y
. The method consists in using the
explanatory variable
x
in different powers in the regression equation:
power 1 (which is the original variable), power 2, power 3, etc. The
equation becomes a polynomial function of order
k
of variable
x
:
ˆ
y
=
a
1
x
+
a
2
x
2
+
a
3
x
3
+
...
+
a
k
x
k
+
b
Adding an order to the equation adds a segment with a different slope
sign to the curve representing the fitted values. A firstorder equation
is a straight line; a secondorder equation is a parabola; a thirdorder
equation is represented by an Sshaped curve; and so on.
R
2
= 0.0139
0
0.5
1
1.5
2
2.5
3
0
10
20
30
R
2
= 0.2566
0
0.5
1
1.5
2
2.5
3
0
10
20
30
R
2
= 0.0836
0
0.5
1
1.5
2
2.5
3
0
10
20
30
R
2
= 0.7519
0
0.5
1
1.5
2
2.5
3
0
10
20
30
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6.3.2 Trend surface analysis
This technique is a
particular case of multiple regression
, where
the explanatory variables are geographical (xy) coordinates,
sometimes completed by higher order polynomials. When
applying this method, one generally supposes that the spatial
structure of the observed variable is a result of one or two
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 Spring '12
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 Regression Analysis, trend surface analysis, Daniel Borcard, Département de sciences

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