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Polynomial Functions
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A polynomial function is one that has the form,
$$ y = a_{n}x^{n} + a_{n-1}x^{n-1} + ... + a_{2}x^{2} + a_{1}x + a_{0} \, , $$
with \(n\)
denoting a non-negative integer that defines
the degree of the polynomial. A polynomial with a degree
of 0 is simply a constant, with a degree of 1 is a line, with a
degree of 2 is a quadratic, with a degree 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 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 use.
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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 notorious 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 frequently deteriorate rapidly outside
the range of the data.
- Polynomial models have poor asymptotic properties. By
their nature, polynomials have a finite response for finite \(x\)
values and have an infinite response if and only if the \(x\)
value is infinite. Thus polynomials may not model
asympototic phenomena very well.
- Polynomial models have a shape/degree tradeoff. In order
to model data with a complicated structure, the degree of
the model must be high, indicating and the associated
number of parameters to be estimated will also be high.
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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