# What is the purpose of finding a model to best fit data?

So given the following question:

What is the purpose of finding a model to best fit data?

Someone answers: The purpose of fitting data to a curve is so that you can give an exact statement about what will happen in a future situation like it for which you do not have the data

Is this answer valid? If not, explain and what would you add?

I personally believe it is a valid answer, does anyone have an opinion or objection?

In addition to what was already revealed in other answers, it is important to know that the fitting results, specially in real life, are not unique. That is, you don't get one and only one relationship between your x and y. The relationship you obtain depends on the method you choose to fit the data. For example, you may use the method of Linear Least Squares or a Non-Linear Least Square method. If the phenomena is not linear, you will get different relations even thought the input pairs are the same. For this reason and others, you can't always depend on the relationship obtained a 100% nor for the future or even for the current set of data. The relationship obtained in many cases represent a good formula with some compromises.

Another value of representing a data set as a concrete mathematical expression is to be able to describe a phenomena concisely so that further study can be applied, such as finding probability, finding average rate of change at any given point, etc. effectively. Again, the accuracy of the result depends on several factors.

Wikipedia Least Squares - Some information about Least Squares mentioned here.

You can use it in at least two ways. One is for interpolaton/extrapolaton. You have data at certain values of the independent variable (say certain times) and want an approximation (I wouldn't call it an exact statement) for the value at other values/times. A fitted model can give you that. Another use is to guide making a theory. If you collect some data and fit it, the functional form might guide your theory. For example, if you measure drag on an object as a function of air speed, you find it is quadratic. You can then think about what physics would cause it to be so.

Someone answers: The purpose of fitting data to a curve is so that you can give an exact statement about what will happen in a future situation like it for which you do not have the data

Is this answer valid? If not, explain and what would you add?

As an instructor, I would probably give half credit for this answer. Here's why: Yes, a primary purpose of the regressed model is to make future predictions. I wouldn't add anything to that -- if anything, the quoted response is already unnecessarily wordy to the point that it seems suspect. (Nonetheless, I'd be charitable and give half credit for that part.)

The real problem is the phrase "give an exact statement about what will happen in a future". That's incorrect and simply impossible. Any predictions we make are necessarily estimates (possibly expressed with a likely range of error). It should be obvious that no one is able to make "exact" predictions about the future, whether one thinks of examples like weather forecasts, financial markets, sporting events, etc.

From an engineering standpoint, it's useful to have a model fit to your data so that you can interpolate between points. If I'm measuring some variable $x$ over the course of, let's say, an airplane runway, I don't have the time, money, or patience to measure it finely. I might measure the variable every yard.

If I know what a good model for the variable is, like $y = ax^2$, I can use it to calculate what the variable would have been at some location I didn't measure. And now I can do fancy things like integrate the variable analytically, when before all I had was a set of discrete points.