Quantifying nonlinearity from data

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In summary, the conversation discusses a function with the form y = ax + b + f(x), where f(x) is a non-linearity in x that cannot be absorbed into the ax+b part. The speaker is wondering if it is possible to extract f(x) by only measuring x and y and if it can be connected to the deviation from linearity. It is noted that without knowing the form of f(x), it may be difficult to distinguish it from adding values to a and b. The speaker is fine with redefining a and b and is only interested in any deviation from linearity. The other speaker suggests using linear regression to estimate y and then modeling the non-linear piece with a parameterized function. Noise in the
  • #1
BillKet
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Hello! I have a function of the form:

$$y = ax + b + f(x)$$
and I can measure experimentally only x and y. I also know that ##f(x)<<ax,b##, where ##f(x)## is some non-linearity in x i.e. it can't be absorbed into the ##ax+b## part (for example ##f(x) = cx^2##), but I don't know its form. Is there a way to extract ##f(x)##, by measuring only ##x## and ##y##? I am basically wondering if I can quantify the deviation of the expression above from linearity and connect that to the value of x. Thank you!
 
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  • #2
As a first pass, you could be in a lot of trouble if ##f(x)=x^2+2x+1=(x+1)^2##, which is going to be literally indistinguishable from adding 2 to a, 1 to b, and ##f(x)=x^2##.

That said, it sounds like maybe you don't care, and you are happy to just think of ##f(x)## as ##x^2## in this case. Is that right?
 
  • #3
Office_Shredder said:
As a first pass, you could be in a lot of trouble if ##f(x)=x^2+2x+1=(x+1)^2##, which is going to be literally indistinguishable from adding 2 to a, 1 to b, and ##f(x)=x^2##.

That said, it sounds like maybe you don't care, and you are happy to just think of ##f(x)## as ##x^2## in this case. Is that right?
Yes! I am fine with redefining a and b if needed (i.e. absorbing those terms you mentioned above). I am purely interested in any deviation from linearity, regardless of the actual value of a and b.
 
  • #4
And the thing you care about specifically is trying to estimate y given a value of x? Are we assuming your measurements are perfect with no noise?

I think you would start with doing linear regression to get ##y \approx ax+b## for some ##a## and ##b##. Then compute ##y-ax-b##, and attempt to model it with your favorite parameterized function. If you have a specific example, just drawing a plot of that would probably be a good start for guessing the shape of the non linear piece
 

FAQ: Quantifying nonlinearity from data

1. What is nonlinearity and why is it important to quantify it from data?

Nonlinearity refers to the relationship between variables that is not directly proportional. In other words, the output does not change at a constant rate as the input changes. It is important to quantify nonlinearity from data because it can affect the accuracy and reliability of statistical models and predictions.

2. How is nonlinearity quantified from data?

Nonlinearity can be quantified using various statistical methods such as correlation coefficients, regression analysis, and nonlinear regression. These methods help to determine the strength and direction of the relationship between variables and identify any nonlinear patterns in the data.

3. What are the common challenges in quantifying nonlinearity from data?

One of the main challenges in quantifying nonlinearity from data is identifying the appropriate statistical methods to use. It is also important to consider the assumptions and limitations of these methods. Another challenge is dealing with missing or incomplete data, which can affect the accuracy of the results.

4. Can nonlinearity be present in all types of data?

Yes, nonlinearity can be present in all types of data, including numerical, categorical, and time series data. It can also exist in both continuous and discrete variables. Therefore, it is important to assess nonlinearity in any type of data before conducting statistical analysis.

5. How can quantifying nonlinearity from data be useful in practical applications?

Quantifying nonlinearity from data can be useful in various practical applications, such as predicting future trends, identifying patterns and relationships in data, and improving the accuracy of statistical models. It can also help to identify outliers and influential data points, which can affect the overall analysis and conclusions drawn from the data.

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