Appropriateness of Constrained Segmented Univariate Polynomial Regression Model

In summary: Jp7ACvyNtAe26ZS0k1HmVQIn summary, unconstrained polynomial regression can be determined using two F tests, one for the overall regression and the other for the significance of higher coefficients. However, for a segmented polynomial fit with C1 continuity and boundary conditions, a test statistic needs to be derived. This can be found in Gallant and Fuller's work, but other papers, such as the one by Huang and Chen, also discuss this topic and provide references for further reading.
  • #1
midnite131
1
0
Hi all,

I've learned that in unconstrained polynomial regression, the optimal order can be determined using two F tests : one to test for the significance of the overall regression, the other to test for the significance of the higher coefficients (assuming the first test passed of course).

However in my application, I'm interested in testing for the appropriateness of a segmented polynomial fit subjected to be first order continuous, with boundary conditions placed at the end of the entire domain as well. The polynomials do not have to have the same order between segments. I should also mention that the join points are known, so I don't have to estimate them.

The closest paper I could find that broaches this topic is Gallant and Fuller's work [1]. Here, they also have a segmented polynomial fit with C1 continuity, but with no constraints. Frustratingly, they make up a test statistic for the appropriateness of their fit "by analogy to linear models theory", yet they provide no references.

I've tried to search for other papers on this topic but to no avail. This leads me to question - is this test statistic trivial to derive for the constrained case, and if so, could you please point me to resources that could help me understand how to do it?

Thanks in advance for your help!

References

[1] Gallant, A.R. and Fuller, W.A. (1973). Fitting segmented polynomial regression models whose join points have to be estimated. J. Amer. Statist. Assoc., 68, 144-147

p.s. If it helps, I'm an engineer who regrettably didn't pay too much attention in his stats course, so any material no matter how trivial it may seem would be appreciated!
 
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  • #2
midnite131 said:
Hi all,

I've learned that in unconstrained polynomial regression, the optimal order can be determined using two F tests : one to test for the significance of the overall regression, the other to test for the significance of the higher coefficients (assuming the first test passed of course).

That's probably true under a certain set of assumptions that are commonly made. Without stating the assumptions, it sounds like too grand a claim.

Speaking only as an operator of Google, there is this paper online:


ANALYSIS OF VARIANCE, COEFFICIENT OF DETERMINATION AND F -TEST FOR LOCAL POLYNOMIAL REGRESSION By Li-Shan Huang and Jianwei Chen

It has other references.

http://www.google.com/url?sa=t&rct=...zLHsDg&usg=AFQjCNEZsaUitjQ7MjxvKmLElrpxNPOOyw
 

What is a constrained segmented univariate polynomial regression model?

A constrained segmented univariate polynomial regression model is a statistical technique used to analyze the relationship between a single dependent variable and one or more independent variables. It involves fitting a series of polynomial curves to different segments of the data, with the constraint that the curves must join smoothly at specific points.

How is this model different from traditional regression models?

This model differs from traditional regression models in that it allows for more flexibility in the relationship between the dependent and independent variables. Instead of assuming a linear or curvilinear relationship throughout the entire dataset, this model allows for distinct relationships within different segments of the data.

What are the advantages of using a constrained segmented univariate polynomial regression model?

One advantage of this model is that it can capture non-linear relationships between variables that traditional regression models may miss. It also allows for more precise interpretation of the data by identifying specific points where the relationship changes. Additionally, this model can handle complex datasets with multiple segments and non-linear patterns.

How do I determine the appropriate degree of polynomial to use in this model?

The appropriate degree of polynomial to use in this model can be determined through various methods, such as visual inspection of the data or using statistical tests to compare the fit of different polynomial degrees. It is important to balance the complexity of the model with its ability to accurately represent the data.

Are there any limitations to using a constrained segmented univariate polynomial regression model?

Like all statistical models, this model has some limitations. It may not be suitable for datasets with a small number of data points or when there is high variability in the data. It also relies on the assumption that the relationship between variables within each segment is smooth and continuous, which may not always be the case. Additionally, the interpretation of the model results may be more complex compared to traditional regression models.

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