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In statistical modeling (especially process modeling), polynomial functions and rational functions are sometimes used as an empirical technique for curve fitting.

Polynomial function models

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. A polynomial function is one that has the form

y = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{2} x^{2} + a_{1} x + a_{0}

where n is a non-negative integer that defines the degree of the polynomial. A polynomial with a degree of 0 is simply a constant function; 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.

Historically, polynomial models are among the most frequently used empirical models for curve fitting.

Advantages

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.

Disadvantages

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 asymptotic phenomena very well.
While no procedure is immune to the bias-variance tradeoff, polynomial models exhibit a particularly poor tradeoff between shape and degree. In order to model data with a complicated structure, the degree of the model must be high, indicating that the associated number of parameters to be estimated will also be high. This can result in highly unstable models.

When modeling via polynomial functions is inadequate due to any of the limitations above, the use of rational functions for modeling may give a better fit.

Rational function models

A rational function is simply the ratio of two polynomial functions.

y = \frac{a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{2} x^{2} + a_{1} x + a_{0}}{b_{m} x^{m} + b_{m - 1} x^{m - 1} + \dots + b_{2} x^{2} + b_{1} x + b_{0}}

with n denoting a non-negative integer that defines the degree of the numerator and m is a non-negative integer that defines the degree of the denominator. For fitting rational function models, the constant term in the denominator is usually set to 1. Rational functions are typically identified by the degrees of the numerator and denominator. For example, a quadratic for the numerator and a cubic for the denominator is identified as a quadratic/cubic rational function. A rational function model is a generalization of the polynomial model: rational function models contain polynomial models as a subset (i.e., the case when the denominator is a constant).

Advantages

Rational function models have the following advantages:

Rational function models have a moderately simple form.
Rational function models are a closed family. As with polynomial models, this means that rational function models are not dependent on the underlying metric.
Rational function models can take on an extremely wide range of shapes, accommodating a much wider range of shapes than does the polynomial family.
Rational function models have better interpolatory properties than polynomial models. Rational functions are typically smoother and less oscillatory than polynomial models.
Rational functions have excellent extrapolatory powers. Rational functions can typically be tailored to model the function not only within the domain of the data, but also so as to be in agreement with theoretical/asymptotic behavior outside the domain of interest.
Rational function models have excellent asymptotic properties. Rational functions can be either finite or infinite for finite values, or finite or infinite for infinite x values. Thus, rational functions can easily be incorporated into a rational function model.
Rational function models can often be used to model complicated structure with a fairly low degree in both the numerator and denominator. This in turn means that fewer coefficients will be required compared to the polynomial model.
Rational function models are moderately easy to handle computationally. Although they are nonlinear models, rational function models are particularly easy nonlinear models to fit.

Disadvantages

Rational function models have the following disadvantages:

The properties of the rational function family are not as well known to engineers and scientists as are those of the polynomial family. The literature on the rational function family is also more limited. Because the properties of the family are often not well understood, it can be difficult to answer the following modeling question: Given that data has a certain shape, what values should be chosen for the degree of the numerator and the degree on the denominator?
Unconstrained rational function fitting can, at times, result in undesired vertical asymptotes due to roots in the denominator polynomial. The range of x values affected by the function "blowing up" may be quite narrow, but such asymptotes, when they occur, are a nuisance for local interpolation in the neighborhood of the asymptote point. These asymptotes are easy to detect by a simple plot of the fitted function over the range of the data. These nuisance asymptotes occur occasionally and unpredictably, but practitioners argue that the gain in flexibility of shapes is well worth the chance that they may occur, and that such asymptotes should not discourage choosing rational function models for empirical modeling.

One common difficulty in fitting nonlinear models is finding adequate starting values. A major advantage of rational function models is the ability to compute starting values using a linear least squares fit. To do this, p points are chosen from the data set, with p denoting the number of parameters in the rational model. For example, given the linear/quadratic model

y = \frac{A_{0} + A_{1} x}{1 + B_{1} x + B_{2} x^{2}}

one would need to select four representative points, and perform a linear fit on the model

y = A_{0} + A_{1} x + \dots + A_{p_{n}} x^{p_{n}} - B_{1} x y - \dots - B_{p_{d}} x^{p_{d}} y

Here, p_n and p_d are the degrees of the numerator and denominator, respectively, and the x and y contain the subset of points, not the full data set. The estimated coefficients from this linear fit are used as the starting values for fitting the nonlinear model to the full data set.

Note: This type of fit, with the response variable appearing on both sides of the function, should only be used to obtain starting values for the nonlinear fit. The statistical properties of fits like this are not well understood.

The subset of points should be selected over the range of the data. It is not critical which points are selected, although obvious outliers should be avoided.

Bibliography

20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
Box, G. E. P. and Draper, Norman. 2007. Response Surfaces, Mixtures, and Ridge Analyses, Second Edition [of Empirical Model-Building and Response Surfaces, 1987], Wiley.
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
R. H. Hardin and N. J. A. Sloane, "A New Approach to the Construction of Optimal Designs", Journal of Statistical Planning and Inference, vol. 37, 1993, pp. 339-369
R. H. Hardin and N. J. A. Sloane, "Computer-Generated Minimal (and Larger) Response Surface Designs: (I) The Sphere"
R. H. Hardin and N. J. A. Sloane, "Computer-Generated Minimal (and Larger) Response Surface Designs: (II) The Cube"
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 (Modeling with rational functions)

Historical

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The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

External links

Rational Function Models

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+In [[statistical modeling]] (especially [[process modeling]]), polynomial functions and rational functions are sometimes used as an empirical technique for [[curve fitting]].
+==Polynomial function models==
+{{Main|polynomial interpolation}}
+A [[polynomial function]] is one that has the form
+:<math>
+y = a_{n}x^{n} + a_{n-1}x^{n-1} + \cdots + a_{2}x^{2} + a_{1}x + a_{0}
+</math>
+where ''n'' is a non-negative [[integer]] that defines the degree of the polynomial. A polynomial with a degree of 0 is simply a [[constant function]]; with a degree of 1 is a [[linear function|line]]; with a degree of 2 is a [[quadratic function|quadratic]]; with a degree of 3 is a [[cubic function|cubic]], and so on.
+Historically, polynomial models are among the most frequently used empirical models for [[curve fitting]].
+===Advantages===
+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. [[translation (geometry)|Changes of location]] and [[scaling (geometry)|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 (mathematics)|metric]].
+#Polynomial models are computationally easy to use.
+===Disadvantages===
+However, polynomial models also have the following limitations.
+#Polynomial models have poor [[interpolation|interpolatory]] properties. High-degree polynomials are notorious for [[Runge's phenomenon|oscillations between exact-fit values]].
+#Polynomial models have poor [[extrapolation|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 [[asymptote|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 asymptotic phenomena very well.
+#While no procedure is immune to the [[bias of an estimator|bias]]-[[variance]] tradeoff, polynomial models exhibit a particularly poor tradeoff between shape and degree. In order to model data with a complicated structure, the degree of the model must be high, indicating that the associated number of [[statistical parameter|parameter]]s to be [[estimation theory|estimated]] will also be high. This can result in highly unstable models.
+When modeling via polynomial functions is inadequate due to any of the limitations above, the use of rational functions for modeling may give a better fit.
+==Rational function models==
+A [[rational function]] is simply the ratio of two polynomial functions.
+:<math>
+y = \frac{a_{n}x^{n} + a_{n-1}x^{n-1} + \ldots + a_{2}x^{2} + a_{1}x + a_{0}} {b_{m}x^{m} + b_{m-1}x^{m-1} + \ldots + b_{2}x^{2} + b_{1}x + b_{0}}
+</math>
+with ''n'' denoting a non-negative integer that defines the degree of the numerator and ''m'' is a non-negative integer that defines the degree of the denominator. For fitting rational function models, the constant term in the denominator is usually set to 1. Rational functions are typically identified by the degrees of the numerator and denominator. For example, a quadratic for the numerator and a cubic for the denominator is identified as a quadratic/cubic rational function. A rational function model is a generalization of the polynomial model: rational function models contain polynomial models as a subset (i.e., the case when the denominator is a constant).
+===Advantages===
+Rational function models have the following advantages:
+#Rational function models have a moderately simple form.
+#Rational function models are a closed family. As with polynomial models, this means that rational function models are not dependent on the underlying metric.
+#Rational function models can take on an extremely wide range of shapes, accommodating a much wider range of shapes than does the polynomial family.
+#Rational function models have better interpolatory properties than polynomial models. Rational functions are typically smoother and less oscillatory than polynomial models.
+#Rational functions have excellent extrapolatory powers. Rational functions can typically be tailored to model the function not only within the domain of the data, but also so as to be in agreement with theoretical/asymptotic behavior outside the domain of interest.
+#Rational function models have excellent asymptotic properties. Rational functions can be either finite or infinite for finite values, or finite or infinite for infinite x values. Thus, rational functions can easily be incorporated into a rational function model.
+#Rational function models can often be used to model complicated structure with a fairly low degree in both the numerator and denominator. This in turn means that fewer coefficients will be required compared to the polynomial model.
+#Rational function models are moderately easy to handle computationally. Although they are [[nonlinear regression|nonlinear models]], rational function models are particularly easy nonlinear models to fit.
+===Disadvantages===
+Rational function models have the following disadvantages:
+#The properties of the rational function family are not as well known to engineers and scientists as are those of the polynomial family. The literature on the rational function family is also more limited. Because the properties of the family are often not well understood, it can be difficult to answer the following modeling question: ''Given that data has a certain shape, what values should be chosen for the degree of the numerator and the degree on the denominator?''
+#Unconstrained rational function fitting can, at times, result in undesired vertical [[asymptote]]s due to roots in the denominator polynomial. The range of ''x'' values affected by the function "blowing up" may be quite narrow, but such asymptotes, when they occur, are a nuisance for local interpolation in the neighborhood of the asymptote point. These asymptotes are easy to detect by a simple plot of the fitted function over the range of the data. These nuisance asymptotes occur occasionally and unpredictably, but practitioners argue that the gain in flexibility of shapes is well worth the chance that they may occur, and that such asymptotes should not discourage choosing rational function models for empirical modeling.
+One common difficulty in fitting nonlinear models is finding adequate starting values. A major advantage of rational function models is the ability to compute starting values using a [[Ordinary least squares|linear least squares]] fit. To do this, ''p'' points are chosen from the data set, with ''p'' denoting the number of parameters in the rational model. For example, given the linear/quadratic model
+:<math>
+y=\frac{A_0 + A_1x} {1 + B_1x + B_2x^{2}}
+</math>
+one would need to select four representative points, and perform a linear fit on the model
+:<math>
+y = A_0 + A_1x + \ldots + A_{p_n}x^{p_n} - B_1xy - \ldots - B_{p_d}x^{p_d}y
+</math>
+Here, ''p<sub>n</sub>'' and ''p<sub>d</sub>'' are the degrees of the numerator and denominator, respectively, and the ''x'' and ''y'' contain the subset of points, not the full data set. The estimated coefficients from this linear fit are used as the starting values for fitting the nonlinear model to the full data set.
+Note: This type of fit, with the response variable appearing on both sides of the function, should only be used to obtain starting values for the nonlinear fit. The statistical properties of fits like this are not well understood.
+The subset of points should be selected over the range of the data. It is not critical which points are selected, although obvious outliers should be avoided.
+==See also==
+* [[Response surface methodology]]
+==Bibliography==
+* {{cite book |author=[http://stats.lse.ac.uk/atkinson/ Atkinson, A. C.] and [http://www.maths.manchester.ac.uk/~adonev/  Donev, A. N.] and [http://support.sas.com/publishing/bbu/companion_site/index_author.html#tobias Tobias, R. D.]|title=Optimum Experimental Designs, with SAS |url=http://books.google.se/books?id=oIHsrw6NBmoC|publisher=[http://www.us.oup.com/us/catalog/general/subject/Mathematics/ProbabilityStatistics/~~/dmlldz11c2EmY2k9OTc4MDE5OTI5NjYwNg==  Oxford University Press]|year=2007 |pages=511+xvi |isbn=978-0-19-929660-6 |oclc= |doi=}}
+* Box, G. E. P. and Draper, Norman. 2007. ''Response Surfaces, Mixtures, and Ridge Analyses'', Second Edition [of ''Empirical Model-Building and Response Surfaces'', 1987], Wiley.
+* {{cite book |author=[[Jack Kiefer (mathematician)|Kiefer, Jack Carl]]. |title=[[Jack Kiefer (mathematician)|Jack Carl Kiefer]] Collected Papers III Design of Experiments | editor=[[Lawrence D. Brown|L. D. Brown]] et al.|publisher=Springer-Verlag|year=1985|isbn=0-387-96004-X}}
+* R. H. Hardin and [[Neil Sloane|N. J. A. Sloane]], [http://neilsloane.com/doc/design.pdf "A New Approach to the Construction of Optimal Designs", ''Journal of Statistical Planning and Inference'', vol. 37, 1993, pp. 339-369]
+* R. H. Hardin and [[Neil Sloane|N. J. A. Sloane]], [http://neilsloane.com/doc/doeh.pdf "Computer-Generated Minimal (and Larger) Response Surface Designs: (I) The Sphere"]
+* R. H. Hardin and [[Neil Sloane|N. J. A. Sloane]], [http://neilsloane.com/doc/meatball.pdf "Computer-Generated Minimal (and Larger) Response Surface Designs: (II) The Cube"]
+*  {{Cite book| title=Design and Analysis of Experiments | series=Handbook of Statistics| volume=13|editor=Ghosh, S. and [[Calyampudi Radhakrishna Rao|Rao, C. R.]] | publisher=North-Holland| year=1996| isbn=0-444-82061-2}}
+**  {{Cite book| author=Draper, Norman and Lin, Dennis K. J.| chapter=Response Surface Designs |pages=343–375}}
+**  {{Cite book| author=Gaffke, N. and Heiligers, B| chapter=Approximate Designs for [[Linear regression|Polynomial Regression]]: [[Invariant estimator|Invariance]], [[Admissible decision rule|Admissibility]], and [[Optimal design|Optimality]] |pages=1149–1199}}
+* {{cite book |author=Melas, Viatcheslav B.|title=Functional Approach to Optimal Experimental Design |series=Lecture Notes in Statistics| volume=184 | publisher=Springer-Verlag | year=2006 |isbn=0-387-98741-X}} (Modeling with rational functions)
+===Historical===
+*{{cite journal
+|title=Application de la méthode des moindre quarrés a l'interpolation des suites<!--  [The application of the method of least squares to the interpolation of sequences] -->
+|author=[[Joseph Diaz Gergonne|Gergonne, J. D.]]
+|journal=Annales de mathématiques pures et appliquées
+|volume=6
+|year=1815
+|pages=242–252
+}}
+*{{cite journal
+|title=The application of the method of least squares to the interpolation of sequences
+|author=[[Joseph Diaz Gergonne|Gergonne, J. D.]]
+|journal=Historia Mathematica
+|volume=1
+|issue=4 <!-- |month=November -->
+|year=1974 |origyear=1815
+|pages=439–447
+|edition=Translated by Ralph St. John and [[Stephen M. Stigler|S. M. Stigler]] from the 1815 French
+|doi=10.1016/0315-0860(74)90034-2
+|url=http://www.sciencedirect.com/science/article/B6WG9-4D7JMHH-20/2/df451ec5fbb7c044d0f4d900af80ec86
+}}
+*{{cite journal
+|title=Gergonne's 1815 paper on the design and analysis of polynomial regression experiments
+|author=[[Stephen M. Stigler|Stigler, Stephen M.]]
+|journal=Historia Mathematica
+|volume=1
+|issue=4 <!-- |month=November -->
+|year=1974
+|pages=431–439
+|doi=10.1016/0315-0860(74)90033-0
+|url=http://www.sciencedirect.com/science/article/B6WG9-4D7JMHH-1Y/2/680c7ada0198761e9866197d53512ab4}}
+* {{cite journal
+|author=[http://www.webdoe.cc/publications/kirstine.php Smith, Kirstine]
+|title=On the Standard Deviations of Adjusted and Interpolated Values of an Observed Polynomial Function and its Constants and the Guidance They Give Towards a Proper Choice of the Distribution of the Observations
+|year=1918
+|journal=[http://biomet.oxfordjournals.org/cgi/content/citation/12/1-2/1 ''Biometrika'']
+|volume=12
+|issue=1/2
+|pages=1–85
+|jstor=2331929}}
+==External links==
+*[http://www.itl.nist.gov/div898/handbook/pmd/section6/pmd642.htm Rational Function Models]
+{{Least squares and regression analysis}}
+{{Statistics}}
+{{NIST-PD}}
+[[Category:Regression analysis]]
+[[Category:Interpolation]]
+[[Category:Statistical ratios]]

Predicate functor logic: Difference between revisions

Revision as of 01:28, 19 February 2013

Contents

Polynomial function models

Advantages

Disadvantages

Rational function models

Advantages

Disadvantages

See also

Bibliography

Historical

External links

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Predicate functor logic: Difference between revisions

Revision as of 01:28, 19 February 2013

Polynomial function models

Advantages

Disadvantages

Rational function models

Advantages

Disadvantages

See also

Bibliography

Historical

External links

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