# Time series: probability to exceed a certain threshold

I'm a beginner and I have a generic time series and I want to know the probability that it reach a certain threshold in the future. For example, the time series is day average temperature and I want to know the probability that in the future there will be 40°.

I try to build a model (Holt, SES, ARIMA,...), fit past data and forecast future data, also calculate confindence intervals, but I didn't reach the point.

Is it right to say that if 40° is out the 95% intervals then his probability is less then 5%? But what if it is inside the interval? Imagine that in the previous days the temperature was 35,37,39,39,38... what is the probability that tomorrow there will be 40°?

P.S: I'm using software R.

The probability of exactly any one value occurring is 0.

The probability of a value falling in an interval can be computed by using the standard error/deviation of the forecast.

To compute the probability of getting at least 40 one could estimate the absolute value of

(40 $-$ predicted value)/(standard deviation of forecast)

... the $z$ score and look its value up in the normal tables.

Now the problem arises that the ARIMA formulation does not consider the uncertainty in the model estimated parameters but assumes that the estimated parameters are the global parameters; thus the standard error of the forecast is very presumptive and underestimated. This problem has been addressed here http://www.autobox.com/cms/index.php/blog but ignored elsewhere.

A second concern is that the residuals from the ARIMA model may not be normal; thus the assumed/implied probablity distribution around the forecast may be incorrect which is why Monte Carlo simulation (re-sampling the model's residuals) may be a better approach.

• Thank you for the suggestion. Can you please give me more info about how to use Monte Carlo to improve the forecast? Any sites or books? – Paolo Jan 5 '16 at 16:20
• What you might do is to take the residuals ( hopefully no outliers or step shifts or time trends or seasonal pulses ) that are free of auto-regressive structure and use them as the basis of the simulation. As an overview if you draw a random normal and convert it to 1 of the residuals and then use it to construct a possible forecast. Do this 1000 times , order the 1000 forecasts from low to high and find what % are less than 40 to determine the probability of < 40 – IrishStat Jan 5 '16 at 16:50