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I've implemented an incremental F-test program that evaluates the fit of an unrestricted model $M_{UR}$ against the restricted model $M_R$ using the F statistic $\frac{SSE_{R} - SSE_{UR}}{SSE_{UR}}\frac{n-p-1}{j}$. In this instance, I'm interested in comparing a polynomial with order $p$, against another (the restricted model) with order $p-1$. It is worth noting that this necessarily makes $j = 1 $.

In order to validate this program, I create data using randomly generated polynomials, add gaussian noise to it, and see if the incremental F-test identifies the correct polynomial order (i.e. if the data is created from a $3^{rd}$ order polynomial, I would expect to get order $3$). In detail, the framework is as follows:

For i = 1 : $n_trials$:
    1. Randomly choose a polynomial order between 2 and 10
    2. Populate the coefficients of this polynomial with values between -5 and 5
    3. Evaluate this polynomial at abscissa values X = [0,0.01,0.02,...3.00]
    4. Add gaussian noise ~N(0,0.01) to each output of P(X)
    5. For p = 3:10 :
           a. Fit the tuples (X,P(X)) using polynomials of order p-1 and p
           b. Compare the results using the FTest, if it fails, exit. If it passes
              try increasing p = p+1
    6. Return the last polynomial order p-1 that passed the F-Test (at p-value 0.05)

Having done this for $n_{trials} = 3000$, I'm finding that the algorithm incorrectly identifies the order on average $200$ to $300$ times. However, if I've chosen a p-value of $0.05$, shouldn't I only expect to see errors $5\%$ of the time, that is $0.05\cdot3000 = 150$?

I also noticed that, if I change the range of X from $[0, 0.01, ... ,3.00]$ to $[0, 0.1, ... , 30.0]$, the F-test fails much more frequently, even though the number of data points is the same between the two experiments! Is this an artifact of the multicollinearity problem with polynomials?

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There are a lot of issues here. The question specifically is about a difference in performance based on the range of values of $x$. This is easily explained. These tests compare amounts of variation of residuals compared to the fits. A polynomial of degree $d$ and coefficients bounded in absolute value by $k$ (equal to $5$ here) can have a range over the coordinates from $0$ to $u$ at least equal to $k\left(u + u^2 + \cdots + u^d\right)$ = $k u\left(u^{d}-1\right)/\left(u-1\right)$. When you change $u$ from $3$ to $30$ the change in potential ranges is huge. E.g., for $d=10$ the maximum in one case is on the order of $3^{11}$ and in the other case it is $10^{10}$ times as great. At this point, the noise (whose standard deviation is a tiny $0.01$) is inconsequential. Thus, even when the coefficient of $p^{10}$ is incredibly tiny, it will have an important (and therefore detectable) effect on the data.

Here is a plot of ten of your random polynomials (all of order $10$). Note the astronomical scale on the y-axis and observe how the highest term dominates the values.

Figure 1

You ought to consider a different universe of models. For instance, use polynomials of the form

$$p(x) = \sum_{i=0}^d \alpha_i \left(\frac{x}{u}\right)^i$$

defined on the range $[0,u]$. Here is a collection of them, once more with the coefficients varying randomly in $[-5,5]$ and all still of tenth order:

Figure 2

A rigorous test would add noise with standard deviation about the same as the variation in the polynomial values: around $10$ or so.

There are other concerns here: please read the replies by @gung and @jbowman. Consider, too, that you are using a restricted version of forward stepwise regression and do some research on the pros and cons of that approach for model building. Finally, note that in general, unless theory specifically indicates a polynomial model and suggests its order, fitting polynomials to data can be a deceptively poor approach: a tiny bit of overfitting can result in models that are grossly bad because higher degree polynomials can (and often do) vary so wildly in between the data values and will be horrible extrapolators.

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    $\begingroup$ +1, this is really nice. It addresses some of the points I was trying to make much more clearly and thoroughly than I did, and also brings in additional ideas that are quite germane. $\endgroup$ – gung - Reinstate Monica Jan 20 '12 at 20:32
  • $\begingroup$ @Gung, your recommendation to "plot the data generating processes" was excellent and led immediately to my writing this reply. $\endgroup$ – whuber Jan 20 '12 at 20:48
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First of all, kudos for thinking of simulating what you're doing first to see if it works before applying it.

As whuber says, the p-value indicates the fraction of the time you'd expect to make a false positive determination, the "false" part of which implies that your "baseline" model is true. For all those situations where the true polynomial is of order $q$ and you are testing for order $p$ vs a null hypothesis of order $p-1 \ne q$, your baseline model is incorrect, so the p-value can't be interpreted that way any more. This is because its distribution is no longer that of a p-value under a true null hypothesis.

What might be better for your purposes is using a criterion such as AIC, which measures model fit and penalizes it for the number of parameters (in your case, the degree of the polynomial.) Your simulation could then be used to help inform you of how far off in degree your model might be from the true model, assuming there is such a thing in the space of polynomials.

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Let me say that I can't quite read your code, so take all this with a grain of salt perhaps. However, I don't think your testing strategy is very solid (although, I do really like that you are checking how this works by simulating). You are leaving too many aspects to chance to be able to interpret what happens very well.

  • I would suggest you start by setting the random seed to a specific value to that you can reproduce the results.
  • Next, I would pick a couple of fixed models that you want to be able to differentiate amongst. For example, create models with polynomial orders 2, 3, and 4.
  • Furthermore, decide in advance what you want the coefficients to be.
  • In particular, I would plot these data generating processes without having added the error and look at them. Remember that it can be easy for two different functions to be almost identical within some specific range of x values, depending on what the coefficients are. For instance, within a certain range, a parabola is well approximated by a steep, but straight, line.
  • Once you have models that you might realistically be interested in differentiating between, and which are possible to do so, then you can start your simulation. What is important to hold in mind at this point, is that you need to be thinking about two different kinds of errors: cases where the test should be 'significant', and cases where it should not. In other words, when you say in the question that you find more errors than you expect, I can't tell what kind of errors they are, and whether it is really appropriate to say that they are out of line.
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