# How does the number of free dimensions of a model affect its required size of sample?

Adding more variables to a model usually increases its accuracy. However, without adequate analysis it could also lead to curve fitting.

Another question (How much data is needed to validate a short-horizon trading strategy?) received answers related to the statistical significance of the standard error of the model. However, I wonder if anyone has results (or analysis) of what should be the ratio of sample data to dimensions used in a model. My intuition has led me to use at least 30 times more sample data points than variables implemented as dimensions but I am not happy with this approach.

I guess that this would depend on the characteristics of the model (it would be different for linear regressions, SVM, non-linear models, etc. and also dependent on the relationships among the variables used) but is there a general framework for estimating this?

## 2 Answers

The following is a good way to judge the quality of fits for a model.

http://en.wikipedia.org/wiki/Akaike_information_criterion

• Akaike's formula seems to be a good approach to compare several models in a relative way. However, I wonder if it's possible to discriminate a priori in an absolute manner if you are using too many parameters. Mar 26, 2013 at 19:30
• en.wikipedia.org/wiki/Overfitting I do the above. I have a training sample, a validation sample, and an out of sample Mar 26, 2013 at 22:16

In full generality this is a very difficult question. The closest you will get to a general framework is Vapnik-Chervonenkis theory. You can read about this in Chapter 7.9 of "The elements of statistical learning" by Hastie, Tibshirani and Friedman which can be downloaded from their website .

But be warned that this is a theoretical approach. Often more heuristic approaches will serve you better. Chapter 7 of the book covers those as well.

• The Elements of Statistical learning seems to be a very good resource, thanks! For the moment, I will wait to mark this answer to see if we can attract more people that might share some heuristics or other frameworks. Apr 3, 2013 at 3:54