Cubic Regression: Exponential Growth & Leveling Off

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Homework Help Overview

The discussion revolves around modeling population data over time using different mathematical functions, particularly focusing on cubic regression and logistic functions. Participants are examining the characteristics of the data and the appropriateness of various models to describe the observed growth and leveling off of the population figures.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the suitability of cubic regression for illustrating both initial exponential growth and subsequent leveling off of population data. Questions arise regarding the behavior of cubic functions versus logistic models, particularly in terms of long-term predictions.

Discussion Status

There is an ongoing exploration of different mathematical models, with some participants suggesting that a logistic function may be more appropriate than cubic regression. Concerns about the implications of using a cubic model, such as predicting population decline, are raised, indicating a productive dialogue about the limitations and assumptions of the models being considered.

Contextual Notes

Participants note discrepancies in the interpretation of the data intervals and the implications of the assignment requirements, which include developing a model function that fits the data points while considering domain restrictions.

ProPM
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Hi,

I have the following population figures for a five year interval:

554.8, 609, 657.5, 729.2, 830.7, 927.8, 998.9, 1070, 1155.3, 1220.5

The graph has an exponential growth from the first value to the fourth value and then the population starts to decay.

I found that a Cubic Regression best illustrates these figures but I have to describe it and, since I've never worked with them I am a bit wary.

Is it correct to say that the cubic correctly illustrates the initial exponential growth of the population but also manages to reflect the leveling off of the population in the latter segment of the plot?

Thanks
 
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ProPM said:
Hi,

I have the following population figures for a five year interval:

554.8, 609, 657.5, 729.2, 830.7, 927.8, 998.9, 1070, 1155.3, 1220.5

The graph has an exponential growth from the first value to the fourth value and then the population starts to decay.

I found that a Cubic Regression best illustrates these figures but I have to describe it and, since I've never worked with them I am a bit wary.

Is it correct to say that the cubic correctly illustrates the initial exponential growth of the population but also manages to reflect the leveling off of the population in the latter segment of the plot?

Thanks
A cubic would get steeper over time, not decay. A logistic function might be the better choice.
 
Yes, a Logistic is my next step, but this function here:

-0.0056755x^3+0.4186x^2+7.35529x+555.2542

Seems to me like it's leveling off towards the end, or is that impossible? It looks like it is possible from google images but you are probably a more trustworthy source :smile:
 
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No, that one isn't leveling off. Because the coefficient of x3 is negative, the graph of this function is heading to negative infinity as x gets large.
 
Um,

Look :redface:
 

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That's a pretty good fit, but is it likely that the population will die out in another 50 years? That's what modeling this data with a cubic spline is predicting. On the other hand, if the population is more likely to approach some stable value, then a logistic model is the way to go.
 
BTW, your first post says the data is for a five-year interval, but you graph uses about a 45-year interval. I suspect that you meant that the data represent populations at five year intervals.
 
Are you just wanting a curve of good fit for these points, or do you plan on extrapolating for the next couple of years?
 
This is what my assignment says:

What types of function could model the behavior of the graph

and a bit later:

Analytically develop one model function that fits the data points on your graph

Furthermore, I am restricting the domain of my graph too.
 
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