i having trouble getting the concept of smoothing. Is it possoble do it on calculator?
if you cannot please explain how do it by hand with table and graph
There are two very different concepts in time series analysis, one is smoothing, and the other is filtering. They are closely related but fundamentally different. I'll try illustrate what each of the two are.
Assume some model where
, for
, denotes the observation at time
and then
denotes the state at time
. Let's look at the simplest model of all, the dynamic linear model (DLM). It is characterized by:
Observation (measurement) equation:
System (State) equation:
As you can see, the observation equation is the observed data value and noisy components. The state equation only consists of the underlying noisy components that drive the observed data. Now
, and
can all be vector valued. We assume that the DLM error terms
and
are distributed as:
where
and
are both symmetric variance-covariance matrices for
and
, respectively.
Also it is important to note that the error term at time t is independent of all information at earlier times. Formally,
')
is independent of
for
.
Under the above assumptions, note the probability distribution of
is completely characterized by
, that is,
 = p(y_t \mid \alpha_t) )
. Similarly,
 = p(\alpha_{t+1} \mid \alpha_t))
. This result will be very useful in deriving the filtering and smoothing recursions later.
Note that we need to start off the process somehow, so assume
)
.
FilteringFiltering means we update the state at each
, conditional on the data observed at time
. More formally, we use
)
to produce the updated (or filtered) state,
)
. To do this, start off with
)
and use the state equation
to predict the one step ahead state distribution
)
, with
and
.
Next, note the joint distribution of
')
can be viewed as a function of
so that:
This implies that:
Now conditioning on
, we have the filtered state of
as:
SmoothingSmoothing means to find the probability distribution of
given ALL information, that is, finding
)
. Can see you the subtle differences? Filtering means to find
)
while smoothing is conditioned on
(CAPITAL t). To prove the smoothing procedure is much much harder than filtering, so I won't bother, but if you're interested you can take a look at:
http://en.wikipedia.org/wiki/Kalman_filter#Fixed-lag_smoother
For your second query, I don't think there are any calculator packages that can do high-tech smoothing, it requires quite a bit of computing power depending on how much data you have. Also you would need to program a bit depending on what model you use. By hand, smoothing would probably take your forever, so unless you only have 2-3 datapoints to smooth (why would you smooth if you had that little data points anyway
), then I'd use some computing software. I use R to implement smoothing. Here are some plots to illustrate the differences between filtering and smoothing. So I have data on the Nile river flow as follows:
The following shows the constant level DLM
Next we apply filtering!
The following shows the one step ahead prediction distribution
Finally, some summary statistics to compare
I forgot to mention, the filtered states are generally "wider" than smoothed states at the beginning of the procedure. Think about this intuitively. You have lots of data points, by conditioning on the first few, you're not gaining as much information as you would by conditioning on ALL data points (smoothing). Thus, your variance would be larger and hence, your one step ahead distribution will be less accurate. But as the series progresses, your filtered states will become better and better (conditioned on more information), in fact the last filtered state IS a smoothed state (since both are conditioned on all information). So the filtered state should "converge" to a smoothed state as you have more and more data.
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