Solved – Forecasting Interventions (pulse) with ARIMA Model

arimaforecastingintervention-analysistime series

Q1: Is there any Arima (p,d,q) model that can forecast interventions (pulse) itself? I know that I can use xreg or even xtransf arguments as the covariates to include the intervention over the observed time series.

The problem is that I don’t know the new values of the covariates to include in the predict function to do the forecasts.

Q2: Can we simulate (estimate) these new covariates? If yes, how? I think that depends on the probability of having pulse. But how we can estimate these probabilities?

To be more precise, I want to develop a time series model that can capture the interventions and also be able to forecast them! Is that even possible? Maybe I am asking too much 🙂 I appreciate any comment, suggestion or reference.

Best Answer

In response to your second question. I had found this paper a few months ago which is trying to exactly what you are proposing.

http://www.upo.es/eps/troncoso/papers/PRL2011.pdf

Related Solutions

Time-Series Intervention Analysis – Intervention Analysis in Time-Series Regression with Seasonal ARIMA Errors

The differencing implied by the denominator of your error term must be applied to $Y_t$, $S_t$ and $P_t$. That is, your model is equivalent to $$ \nabla\nabla_{12}Y_t=\frac{\omega \nabla\nabla_{12}S_t}{1-\delta B}+\frac{\omega \nabla\nabla_{12}P_t}{1-\delta B}+\frac{\Theta(B)}{\Phi(B)} \eta_t, $$ where $\nabla\nabla_{12} = (1-B)(1-B^{12})$. This is a transfer function model with ARMA errors which is how it would actually be estimated.

If you intended that the pulse and step apply to the differenced $Y$ series, then you need to doubly integrate $S$ and $P$ in the model (as suggested by @IrishStat). That is $$ Y_t=\frac{\omega S_t}{\nabla\nabla_{12}(1-\delta B)}+\frac{\omega P_t}{\nabla\nabla_{12}(1-\delta B)}+\frac{\Theta(B)}{\nabla\nabla_{12}\Phi(B)} \eta_t. $$

Solved – Intervention Analysis – Pulse over several periods

Lets say you have a time series data 42 observations. Observation 7 to 13 have step(level shift) intervention and observations 29 to 35 have step intervention of 5 units. The remainder observations are 0. As an example, see below chart for simulated data with actual and actual + error. enter image description here

If you would like to model this data as a intervention analysis using arima transfer function, you simply need to create a dummy coded variable with 1's for 7 to 13 and 1's for 29 to 35 and pass this is a step intervention in ARIMAX model. I don't know how to use R for transfer function modeling, so I used SAS. See below the actual and predicted data. The ARIMAX approximated the data very well. See below the SAS code that I used to create the data and as well as the ARIMAX.

If your second intervention has a first intervention has a decaying effect, all you need to do is create a second intervention variable at the end of first step function and code it appropriately.

Hopefully this helped.

enter image description here

data temp;
    input actual @@;
    error = rand("NORMAL");
    datalines;
0 0 0 0 0 0 5
5 5 5 5 5 5 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
5 5 5 5 5 5 5
0 0 0 0 0 0 0
;
run;

data temp;
    set temp;
      actual_error = actual+error; *add random error;

      if 7 <= _n_ <= 14 or 29 <= _n_ <= 35 then step = 1; * First Intervention step;
            else step = 0;

run;

ods graphics on;
proc arima data = temp plots = all;
    identify var=actual_error crosscorr = (step); 
     estimate input = (step) method = ml outest = mm1; /*Estimate Model using Transfer Function*/
     forecast out = out1 lead = 0;
run;
ods graphics off;

EDIT: as b_miner would like to know how to model this with temporary level shift with a linear decay, I have modified the example. I have shown only the first level shift, the second level shift could be easily added. If you are looking for a decay other than a linear decay then it is fairly easy to do, just use a step+gradual permanent decay function using transfer function. Intervention modeling is more of an art than science. You assume a shape, test the hypothesis, if acceptable move on else change the shape. Intervention modeling is deterministic in nature. If you are interested in learning more about transfer function modeling/intervention detection, I would highly recommend Forecasting-Dynamic-Regression-Models-Pankratz. This text has all the different types of intervention models possible, in addition to transfer function modeling.

Hope this helps.

data temp;
    input actual @@;
    error = rand("NORMAL");
    datalines;
0 0 0 0 0 0 5
5 5 5 5 5 5 4.1
4 3.5 3 2.5 2 1.75 1.25
0.75 0.25 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
;
run;

data temp;
    set temp;
    retain ramp 0;
      actual_error = actual+error; *add random error;

      if 7 <= _n_ <= 13 then step = 1;
            else step = 0;


      if 14 <= _n_ <= 23  then do;
            step = (_n_ - 24)/(13-24); 

      end;

run;



ods graphics on;
proc arima data = temp plots = all;
    identify var=actual_error crosscorr = (step); 
     estimate input = (step) method = ml outest = mm1; /*Estimate Model using Transfer Function*/
     forecast out = out1 lead = 0;
run;
ods graphics off;

enter image description here

Best Answer

Related Solutions

Time-Series Intervention Analysis – Intervention Analysis in Time-Series Regression with Seasonal ARIMA Errors

Solved – Intervention Analysis – Pulse over several periods

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