Feasibility of finding "Governing Equation" with unknown BCs and known values in a domain

[AlanWu]

Sorry for bringing up an open question.

Let's say I have the price(p) and location(x,y) of all the properties with respect to time(t) in an area .
Thus, the price, p, is express as follow: p = f(x, y, t)



I know in normal cases we have a governing equation and boundary conditions related to a question, then we calculate the values in the domain.
In my case, however, the governing equation and BCs are unknown while the values are given. Would it be possible going reverse to find the governing equation f?
Or at least I need to make an reasonable assumption for the BCs to make it work?

Any advice is appreciated!

 

[rb1957]

Is this an FEA question, or a mathematical one ?

It sounds like you need to fit a surface (or a bunch of curves) to your data.

another day in paradise, or is paradise one day closer ?

 

[AlanWu]

Hi rb1957,

Yes, I am trying to fit a surface and I feel it is essentially a mathematical problem.
In the other hand, due to the variation of property density, I assume it is reasonable to have elements with different sizes and shapes.
That why I am wondering if FEA can address my question.

Thanks for asking!

 

[IRstuff]

I think you misunderstand FEA. FEA implements governing equations, it can't find an an arbitrary unknown equation. Finite Element Analysis (FEA) is a process of discretizing continuous equations that are typically too difficult to solve in their continuous time forms. By discretizing the equations, they can be solved using quasi-linear approaches.

It seems to me that companies like Zillow and Redfin are more likely to have what you want, although I would dispute the accuracy of their algorithms.

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[IRstuff]

Rather than finding a "governing equation" which I think is more or less a wild goose chase, you might consider a brute-force machine learning approach:

https://yalantis.com/blog/predictive-algorithm-for-house-price/

https://towardsdatascience.com/regression-predict-house-price-lesson-2-5e23bec1c09d

https://nycdatascience.com/blog/stu...use-prices-using-machine-learning-algorithms/

https://www.kaggle.com/c/house-prices-advanced-regression-techniques#description

https://www.kaggle.com/dgawlik/house-prices-eda

https://www.kaggle.com/erick5/predicting-house-prices-with-machine-learning

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[AlanWu]

Hi IRstuff,

Thanks for pointing out my misunderstanding.
I feel I kind of get your point to some extent.
So the possible solutions in my case are infinite and even if I manage to find one that fits, it has nothing to do with FEA?

If I do not get it right, would it be possible for you to further explain your tip or recommend some reading material for me?

Again, thanks for the help!

 

[IRstuff]

Is this for school? Student posting is generally not allowed.

Again, FEA is a general class of implementations of discretized continuous equations that are difficult, or impossible, to solve in a closed-form. You can easily find this sort of stuff in Wikipedia:

https://en.wikipedia.org/wiki/Finite_element_method

In general, we can readily identify factors that affect the price of a house
Intrinsic

# of rooms​

size​

location/lot size​

amenities​

upgrades​

age​

construction/builder quality​

lot/view premiums​

Extrinsic

market forces​

neighborhoods​

comparables​

interest rates​

schools​

crime​

geologicals​

The likelihood of finding a plausible equation that encompasses all these factors is quite low.

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[AlanWu]

No, I am not a student anymore(although I hope so).

The price p here is actually already the result of machine learning models.
Since the models already took the factors you listed above, now I just reduce the inputs to x,y,t.
I am just naively wondering if I can come up the "governing equation" that fits p with the inputs.

Thanks for the help!

 

[rb1957]

No, FEA won't answer your question. There are many curve fitting approaches that should get close, including the min error solution of overdefined equations (more equations than unknowns).

yes, there are an infinite number of solutions to an unbounded problem. If you had just two points than a single line will fit, or an infinite number of planes (flat or curved/warped).

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[jhardy1]

For the problem you are describing (value of property as a function of location and time), there is no "governing equation" as such. (Or at least, if there is one, it would involve VASTLY more parameters than you have defined - interest rates, inflation, wage growth, stock exchange indices, natural disasters, transport infrastructure, fuel costs, political climate and government policy, etc would all be factors.)

The best you can hope to do is to do a "curve fit" or "look-up" approach to an acceptable fit, and using a format that is useful for the intended application - e.g. does your data set have any unusual discontinuities (such as high-priced suburbs in very close proximity to low-priced suburbs); will you be using your generated function to only interpolate between existing data points, or will you also use it to extrapolate beyond the data set; and how will it work across discontinuous or rapidly varying parts of the data-set?

As a trivial example - suppose you have a set of 20 (x,y) data points, which visually approximate to a straight line when plotted in Excel. You apply a linear trend-line, and look at the resulting Equation (which is your first estimate of the "governing equation") and r² value (which tells you how well the trend-line fits the data set). Now change the trend-line to a 2^nd order polynomial (i.e. a parabola) - chances are, the r² term will be slightly higher than you got for the linear fit.

Hmmm ... that's interesting - what happens if I go to higher and higher order polynomials? As you increase the order of the polynomial, r² gets bigger and bigger (Excel will let you go up to 6^th order), but the resulting trend-line gets more and more wiggly - is that actually a better fit to your data?

More importantly - look at the function expressions that are generated for your linear fit and the 6^th order polynomial fit, and calculate what would happen if you try to extrapolate a y-value for an x-value which lies just outside the bounds of your data set. The linear fit will generate predictable values when you extrapolate in either direction, but your 6^th order polynomial is likely to veer off dramatically at one or both ends when you go beyond the bounds of your input data.

Other statistical analysis packages will allow other function fitting approaches - e.g. using cubic splines will allow your interpolated data to fit smoothly and exactly to EVERY point in your data set - but is that actually a "better" fit than a linear trend-line?

Only you can decide which of these approximations to the "governing equation" are actually useful.

http://julianh72.blogspot.com

 

[rb1957]

of course then you have to worry about linking correlation with causation.

and whilst good for interpolation, how good are the extrapolations.

another day in paradise, or is paradise one day closer ?