node4 Basic Models STAT 510 - Applied Time Series Analysis

node4 Basic Models STAT 510 - Applied Time Series Analysis...

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This is Google's cache of http://onlinecourses.science.psu.edu/stat510/node/3 . It is a snapshot of the page as it appeared on 9 Aug 2010 02:46:38 GMT. The current page could have changed in the meantime. Learn more Text-only version STAT 510 - Applied Time Series Analysis ANGEL Department of Statistics Eberly College of Science Home // Section 1: Introduction and Basics Basic Models Submitted by gfj100 on Mon, 03/22/2010 - 14:04 The analysis of data is always linked to a model. We cannot analyze data without a model. What we will talk about next are the models that are often used to analyze data in the context of time series. The basic time series model involves a list of random variables: ... x -2 , x -1 , x 0 , x , x 1 , x 2 , x 3 , . .. One thing different here is that they are not necessarily independent and not necessarily identically distributed. One defining characteristic of time series is that this is a list of observations where the ordering matters. For instance, if we have the list of pig weights that we have seen in our example earlier, would it matter if we switched the labels, i.e., the weight for one pig is now associated with a different pig. With independent identically distributed data labels can be switched. Pig #1 could be pig#2 or vice versa. The order is arbitrary. With time series data, however, you cannot do this, ordering is very important because there is dependency and this could change the meaning of the data. Another name for this type of models is a stochastic process . Stochastic processes are random variables that are indexed by time, (specifically the model above is called a discrete parameter or discrete time process). So, a typical time series is where x t is a continuous random variable but it is being indexed by discrete time. The models may start out at time negative infinity and go on out to infinity, so we have an infinite sequence in both directions. An example might be the surface temperature at a point on earth, (This is not infinite but it is a pretty long time!) we may have to consider this this type of model for theoretical purposes. Often, whatever process you are talking about it has been going on for a long time before you started looking at. This is not true for every process, but a lot things like weather, the economy, stocks, etc. have been active for a long time and at some point the researcher starts to look at this. This assumption simplifies many theoretical calculations.
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White Noise Our first example would be white noise. White noise a time series where the ordering matters, but the random variables are independent and identically distributed ( iid )with mean zero. Basic iid data could be a time series. In fact, this model will form a building block for our other time series models. We might have Gaussian white noise, which is white noise where the distribution is normal. (We will
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node4 Basic Models STAT 510 - Applied Time Series Analysis...

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