2
$\begingroup$

Suppose we have time-series $ X_t $ and it has the following decomposition

$$X_t=\mu + \varepsilon_t,$$

where $\mu$ is a mean and $\varepsilon_t$ - the error term.

The model complexity will increase if we divide this time-series in to some segments,say $k$, and repeat above process. As the model complexity increases the approximation accuracy also increases. So I want to introduce a regularisation term here which will help in deciding the number of segments $k$ in which we need to divide the time-series. The error in approximation can be defined as

$$ \epsilon_t= \frac{1}{k} \sum_{i=1}^k (\mu_{1}-X_{i})+\frac{1}{n-k} \sum_{i=n-k}^n (\mu_{2}-X_{i}), $$

here I have divided the time-series in 2 segments and $\mu_{1}, \mu_{2}$ are their respective means. Now I want to find out the optimal number of segments in general. Please note, that here I want to introduce a "regularisation" term which will help in deciding optimal number of segments.

$\endgroup$
  • 1
    $\begingroup$ Although this question is stated quite differently, it appears to be identical to stats.stackexchange.com/q/2432 . $\endgroup$ – whuber May 24 '11 at 16:16
  • $\begingroup$ @whuber, good catch, judging from the comments to my answer, this looks like what OP actually wants. $\endgroup$ – mpiktas May 25 '11 at 7:00
2
$\begingroup$

Seems that you have a change point problem. Also look at change-point tag for related questions in this site. For fitting these type of models R for example has the packages segmented and strucchange. The relevant function to find the optimal number of segments in package strucchange is breakpoints. Here is the simple example:

e<-rnorm(100) ##errror term

##the mean, value 10 for first 20 observations, 
##then 30 for next 30 and 3 for last 50.
mu<-c(rep(10,20),rep(5,30),rep(3,50)) 

##generate time series X
x<-mu+e

> breakpoints(x~1,data=data.frame(x=x))

     Optimal 3-segment partition: 

Call:
breakpoints.formula(formula = y ~ 1, data = data.frame(y = y))

Breakpoints at observation number:
20 50 

Corresponding to breakdates:
0.2 0.5 
| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ thnx for the reply, i want to understand the mathematical formula for this and then may be implement. $\endgroup$ – Amit May 24 '11 at 11:56
  • $\begingroup$ @Amit, the reasoning behind this formula is not simple, you can get the references from the strucchange vignette. Why do you want to implement it yourself? $\endgroup$ – mpiktas May 24 '11 at 12:19
  • $\begingroup$ I want to understand how in general Regularisation can be applied to these kinds of problems $\endgroup$ – Amit May 24 '11 at 13:32
  • $\begingroup$ @Amit, why do you call this regularisation? Do you have any references? $\endgroup$ – mpiktas May 24 '11 at 13:37
  • $\begingroup$ Regularisation in general helps to overcome the problem of overfitting $\endgroup$ – Amit May 24 '11 at 14:40

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.