There are a few good books on it actually. You should look it up.
You need the data to be homoscedastic, the errors to be independent. The independent variable(s) should lie within (or very close to) the range of observed values.
Dan Henderson vs. Rashad Evans Prediction
They are interval.
interval
The valid prediction range is the range of the "predictor."
There are many ways one might use Exponential Smoothing. Basically, Exponential Smoothing is a simple calculation one uses to collect data that allows one to predict future events.
When implemented digitally, exponential smoothing is easier to implement and more efficient to compute, as it does not require maintaining a history of previous input data values. Furthermore, there are no sudden effects in the output as occurs with a moving average when an outlying data point passes out of the interval over which you are averaging. With exponential smoothing, the effect of the unusual data fades uniformly. (It still has a big impact when it first appears.)
That, my friend, is not a question.
1) forecasting for stationary series A- Moving average B- Exponential Smoothing 2) For Trends A- Regression B- Double Exponential Smoothing 3) for Seasonal Series A- Seasonal factor B- Seasonal Decomposition C- Winters's methode
Confidence interval considers the entire data series to fix the band width with mean and standard deviation considers the present data where as prediction interval is for independent value and for future values.
Exponential Smoothing Model
what exponential function is the average rate of change for the interval from x = 7 to x = 8.
G. Kallianpur has written: 'White noise theory of prediction, filtering, and smoothing' -- subject(s): Gaussian processes, Kalman filtering, Prediction theory
The linear function changes by an amount which is directly proportional to the size of the interval. The exponential changes by an amount which is proportional to the area underneath the curve. In the latter case, the change is approximately equal to the size of the interval multiplied by the average value of the function over the interval.
Joseph V. Reilly has written: 'A dynamic inventory model using exponential smoothing'
Exponential moving average is a running average of a set of observations, where the weight of each observation is inversely exponentially weighted as a function of how old it is. It is a relatively simple thing to do. Given a set of observations O1, O2, O3, ... ON the running exponential moving average A1, A2, A3, ... AN can be calculated in real time, at each time N, with the expression ... AN = AN-1 (1 - X) + ON X ... where X is a weighting factor that determines that amount of smoothing. For instance, if X were zero, then the smoothing is infinite, and O does not contribute at all to A, and if X were one, then smoothing is zero, and A follows O with no smoothing at all. In a more useful example, if X were 0.2, then the smoothing would be five, and A would follow O with a time constant of five iterations, i.e. after five iterations we would be at about 63% of one step change and after 25 iterations we would be at about 95% of one step change.
Lets define exponential smoothing first... Exponential smoothing, or exponential moving average, is a running average of a set of observations, where the weight of each observation is inversely exponentially weighted as a function of how old it is. It is a relatively simple thing to do. Given a set of observations O1, O2, O3, ... ON the running exponential moving average A1, A2, A3, ... AN can be calculated in real time, at each time N, with the expression ... AN = AN-1 (1 - X) + ON X ... where X is a weighting factor that determines that amount of smoothing. For instance, if X were zero, then the smoothing is infinite, and O does not contribute at all to A, and if X were one, then smoothing is zero, and A follows O with no smoothing at all. In a more useful example, if X were 0.2, then the smoothing would be five, and A would follow O with a time constant of five iterations, i.e. after five iterations we would be at about 63% of one step change and after 25 iterations we would be at about 95% of one step change. Some people swap the position of X and (1 - X) in the above equation. Its their choice, but the discussion that follows will have to change accordingly. X is the smoothing factor. It is simply the number of iterations that you want for your time constant. If you were to model this as an electronic circuit, for instance, with a capacitor and a resistor, the exponential curve would be in the form ... e-T/RC ... where RC was your time constant. The same thing applies here. If you evaluated the first equation once per second, with an X value of 0.2, you would have a time constant of 5 seconds. If you, on the other hand, evaluated it 100 times per second, with X being 0.002, you would still have a time constant of 5 seconds, but it would much more closely approximate the second equation, which is a continuous equation, rather than a discrete equation. In summary, then, the smoothing factor, or X, is one over the number of iterations that you want to be your time constant.