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Polynomial Functions
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A polynomial function is one that has the form
where n is a non-negative integer that defines
the order of the polynomial. An order of 0 is simply
a constant, an order of 1 is a line, an order of 2 is a
quadratic, an order of 3 is a cubic, and so on.
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Polynomial Models: Advantages
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Historically, polynomial models are among the most frequently
used empirical models for fitting functions. These models
are popular for the following reasons.
- Polynomial models have a simple form.
- Polynomial models have well known and understood
properties.
- Polynomial models are theoretically exact for high
enough degree. Specifically, n distinct response
values can be fit exactly with an order (n-1)
polynomial.
- Polynomial models have moderate flexibility of shapes.
- Polynomial models are a closed family. Changes of
location and scale in the raw data result in a polynomial
model being mapped to a polynomial model. That is,
polynomial models are not dependent on the underlying
metric.
- Polynomial models are computationally easy to apply.
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Polynomial Model: Limitations
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However, polynomial models also have the following limitations.
- Polynomial models have poor interpolatory properties.
High degree polynomials are notorius for oscillations
between exact-fit values.
- Polynomial models have poor extrapolatory properties.
Polynomials may provide good fits within the range of
data, but they will frquently deteriorate rapidly outside
the range of the data.
- Polynomial models have poor asymptotic properties. By
their nature, polynomials have finite response for finite
X values and have infinite response if and only if
the X value is infinite. Thus polynomials cannot
model phenomenon which have infinite response for finite
X values, or have finite response for infinite
X values.
- Polynomial models have a shape/degree tradeoff. In order
to model data with complicated structure, the degree of
the model becomes unduly inflated, and so the associated
number of coefficients becomes unduly large. This
can result in highly unstable models.
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Example
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The load cell calibration
case study contains an example of fitting a quadratic
polynomial model.
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Specific Polynomial Functions
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- Straight Line
- Quadratic Polynomial
- Cubic Polynomial
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