# What is the interpretation for quadratic functions?

I am working through the book Applied Predictive Modeling and came across something that was a bit confusing.

It discussed adding non linearity to a model to improve its fit - I get this part.

For example: $$x^2 + 2x - 4$$

What is the interpretation of these values though?

When we are using just a normal linear regression or a multivariate regression, we would say that the coefficients like 2 would suggest its relative importance relative to the other features being included in the model. However, what does this mean in the context of quadratic functions?

ie. Fuel efficiency of a car based on 2 Displacement + Displacement$$^2$$ -4

What exactly does displacement squared mean?

Any help would be greatly appreciated.

Thanks.

## 3 Answers

$$Displacement$$ component gives us a "line" to fit over the data points. To get more freedom add $$Displacement^{2}$$ a "curve" element. This adds to the flexibility - with same feature/ variable - to map the data points. Please refer page 90, 91 on Introduction to Statistical Learning in R - Hastie, Tibshirani

• I understand that we are using a non linear fit vs a linear fit. My question is what is the conceptual understanding for displacement$^2$ in words? – user67797 Jun 20 at 23:56
• Seems that even Hastie, Tibshirani did not provide answer your query. Also, I'm sure that know the meaning/ difference of x-y relationship in: y = x, y = x$\^2$ curves on a graph. Further ahead, you definitely know what is meaning of x in both graphs/ curves. If you're clear about all of the above, and you've referred the free pdf of book, then I'll curious to see an answer, that resolves your query. Regards. – Continue2Learn Jun 21 at 1:56

Generally, I would say that in statistics (and even more so in math in general) it only makes so much sense to look for an intuitive understanding of everything. Sometimes this will just give you a lot of headache while it might be much easier to just take something like, say a function, for what it is: a function. You plug in numbers and get something out (ok, that is not really the formal definition but I hope it gets my point across).

Having said that, for the equation

$$efficiency = 2 displacements + displacements^2$$

(neglecting the $$-4$$ here for simplicity) you can actually find an intuitive perspective: The quadratic equation assumes that efficiency increases "more than just linearly" with the number of displacements.

If you have a car $$a$$ with $$displacements_a = 2$$ and car $$b$$ with $$displacements_b=4$$, then in the linear case of

$$efficiency = 2 displacements$$

the $$efficiency$$ of car $$a$$ would be two times the $$efficiency$$ of car $$b$$.

But assuming the above mentioned quadratic relationship means that the number of $$displacements$$ increases $$efficiency$$ additionally by the quadratic term. $$efficiency$$ of car $$b$$ would now be three times the $$efficiency$$ of car $$a$$!

So as you can see from that simple example in the quadratic case any change of the value of $$displacements$$ has a bigger impact on $$efficiency$$. And the quadratic term just defines this "more" mathematically.

Your ways of putting coefficient value denotes the relative importance of such variable is wrong. You need to also say that predictor variable is standardized(zero-centred and s.d. of 1) to make it a better argument.

Personally I don't think . If you are discussing this on data science stack exchange we only care about how the relationship between efficiency and fuel. So we start of with prior assumption of $$efficiency = a \cdot displacement^2 + b\cdot displacement + c$$ and how we can find such $$a,b,c$$ and whether quadratic relationship could be a fit that could also generalize well with our data. Whether or not the relationship makes any sense is subject to people that has better domain expertise to interpret.

Predictor importance reference