Modeling help - estimating relative likelihood of different time series models

[andy...@cubist-asia.com]

Hi all,

I have a set of time series values (positive) of y:

vector<lower=0>[N] y

I want to estimate the relative likelihood of the following 3 scenarios:

a. diverging AR(1):     y_n ~ normal (alpha + y_(n-1), sigma), alpha > 0
b. converging AR(1):  y_n ~ normal (alpha + y_(n-1), sigma), alpha < 0
c. stationary:             y_n ~ normal (mu, sigma)

What would be the best way to do this in STAN?

Many thanks,
Andy

[Daniel Lee]

Write 3 Stan programs? That's what I would do and use the loo package. 

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[Michael Betancourt]

To discriminate between (a) and (b) you can just

run a model with alpha unconstrained and consider

p(alpha > 0) = n(alpha > 0) / n.

The alpha=0 case is tricker because it has zero

measure compared to the others and you need

to do proper model comparison.  Cross validation,

as approximated by the loo package, would be

one such approach.

[Daniel Lakeland]

You might think about defining what it means for alpha to have no practical significance for your timeseries. Like for example if abs(alpha) < SomeValue

For example if you are observing for a year and the cumulative effect of alpha over a year is less than the measurement error in one observation, or something like that.

Also, I think there's something amiss with your model. y is positive, but if y_0 is small enough and alpha is negative, then the average value of the next step is negative!

[Seth Flaxman]

hmm, maybe I'm missing something but don't you need an autoregressive parameter beta, e.g.:

y_n ~ normal (alpha + beta * y_(n-1), sigma)

In which case, couldn't you just fit one model, and see whether alpha > 0 or < 0 and whether beta is "significant"?

Seth

[andy...@cubist-asia.com]

Michael, many thanks. Will take a look at loo.

On Thursday, August 13, 2015 at 10:55:51 PM UTC+8, Michael Betancourt wrote:

To discriminate between (a) and (b) you can just

run a model with alpha unconstrained and consider

p(alpha > 0) = n(alpha > 0) / n.

The alpha=0 case is tricker because it has zero

measure compared to the others and you need

to do proper model comparison.  Cross validation,

as approximated by the loo package, would be

one such approach.

On Aug 13, 2015, at 4:21 AM, Daniel Lee <bea...@alum.mit.edu> wrote:

Write 3 Stan programs? That's what I would do and use the loo package. 

On Thursday, August 13, 2015, <andy...@cubist-asia.com> wrote:

Hi all,

I have a set of time series values (positive) of y:

vector<lower=0>[N] y

I want to estimate the relative likelihood of the following 3 scenarios:

a. diverging AR(1):     y_n ~ normal (alpha + y_(n-1), sigma), alpha > 0
b. converging AR(1):  y_n ~ normal (alpha + y_(n-1), sigma), alpha < 0
c. stationary:             y_n ~ normal (mu, sigma)

What would be the best way to do this in STAN?

Many thanks,
Andy

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[andy...@cubist-asia.com]

Hi Daniel,

My goal is to avoid some hard limit/cut-off set-up because, a priori, I don’t know what these limits are.

re -ve next step: thanks for the heads-up, let me take a look at the data again and see if this could happen or not

Andy