I am trying to develop a simple measure of "smoothness" of a trajectory. For instance, given an AR(1) process and another AR(3) process, how does one compare the two processes and say that one is "smoother" than the other one?

You might try calculating the fractal dimension. Smaller fractal dimension -> "smoother" curve

Related to 2fish's reply, calculate the length of the curve, simply summing the lines that would join successive points.The rougher curve will be longer, an ideal Brownian curve will be inifinitely long.

If the fractal dimension is the same for two curves then an obvious way of defining roughness is by the usual measure of volatility!

Although volatility works well for aggregate variation it gives no idea of the path.Consider4,5,6,7,8,9,10,11rearranged to4,11,5,10,6,9,7Identical vol, but very different "roughness".

Convert your implicit price numbers to returns, and the first series is zero vol. If these are already returns then the vols are identical.If these are returns and you mean that the first one is quadratic in price while the second one isn't then I agree with your comment. It gets down to what you want to call roughness.

QuoteOriginally posted by: twofishYou might try calculating the fractal dimension. Smaller fractal dimension -> "smoother" curveBut how do you define/measure fractal dimension for a discrete curve? I am assuming it _is_ discrete, as AR processes are mentioned, and also all data is discrete.

QuoteOriginally posted by: ZmeiGorynychBut how do you define/measure fractal dimension for a discrete curve? I am assuming it _is_ discrete, as AR processes are mentioned, and also all data is discrete. Thank you! Can we get back to the AR(1) and AR(3) time series? I'm just talking about simple discrete financial time series here, such as the earnings of a company for 10 years. Fractal dimensions are not relevant. The basic question is: How do I compare the earnings time series of two companies and say that Wal-Mart (for example) has "smoother" or "more robust" earnings than Coke (for example)?

You can try to calculate autocorrelation of your curve (according to your lag ). It should give you a good idea if you also consider volatility...To get this value just shift your serie and calculate your correlation.

QuoteOriginally posted by: gjossYou can try to calculate autocorrelation of your curve (according to your lag ). It should give you a good idea if you also consider volatility...To get this value just shift your serie and calculate your correlation. I already stipulated that we have AR(N) (or autocorrelated time series), so we already know the autocorrelation. I need a measure of "long run persistence or robustness" that is not redundant to saying they are AR(N).

If you are interested in the robustness of your prediction of earnings, not the robustness of your prediction of the change in earnings, then volatility is the best measurement of robustness. Otherwise, volatility in the rate of change of earnings is the best measure of robustness.So if they rise 1% every single period, making volatility zero, that is very robust.Alternately, you can add the predicted 1-period followthrough from any change in earnings to the current data point. Then the volatility of the rate of change from this adjusted or predicted data point, to the next data point, is inverse to your robustness.In other words, the volatility around your predicted rate of change. But as you looked more periods ahead, your predicted rate of change would converge to zero, as the volatility around your immediate prediction would rise. So the robustness is dependent on how many periods ahead you need to look right now.

"serie" is not a word. "series" is not always a plural, despite the "s" at the end.

It sounds like you want to predict here. In which case you should be testing how well the prediction works, say based on AR(1) or an AR(3) series. Look at how close you came to prediciting the actual number divided by the predicted uncertainty of your estimate. For smoothness/robustness you can use the uncertainty of your estimate.

QuoteOriginally posted by: Frashe"serie" is not a word. "series" is not always a plural, despite the "s" at the end. Even in australia?

Why not? Us quants are one big happy anglophone community, only divided by our common language.