6.
Process or Product Monitoring and Control
6.4. Introduction to Time Series Analysis


Smoothing data removes random variation and shows trends and cyclic components 
Inherent in the collection of data taken over time is some form
of random variation. There exist methods for reducing of canceling
the effect due to random variation. An oftenused technique in
industry is "smoothing". This technique, when properly applied,
reveals more clearly the underlying trend, seasonal and cyclic
components.
There are two distinct groups of smoothing methods


Taking averages is the simplest way to smooth data 
We will first investigate some averaging methods, such as the
"simple" average of all past data.
A manager of a warehouse wants to know how much a typical supplier delivers in 1000 dollar units. He/she takes a sample of 12 suppliers, at random, obtaining the following results:
Is this a good or bad estimate? 

Mean squared error is a way to judge how good a model is 
We shall compute the "mean squared error":


MSE results for example 
The results are:
The estimate = 10


Table of MSE results for example using different estimates 
So how good was the estimator for the amount spent for each supplier?
Let us compare the estimate (10) with the following estimates: 7,
9, and 12. That is, we estimate that each supplier will spend $7,
or $9 or $12.
Performing the same calculations we arrive at:
The estimator with the smallest MSE is the best. It can be shown mathematically that the estimator that minimizes the MSE for a set of random data is the mean. 

Table showing squared error for the mean for sample data 
Next we will examine the mean to see how well it predicts net income
over time.
The next table gives the income before taxes of a PC manufacturer between 1985 and 1994.
The MSE = 1.8129. 

The mean is not a good estimator when there are trends 
The question arises: can we use the mean to forecast income
if we suspect a trend? A look at the graph below shows clearly
that we should not do this.


Average weighs all past observations equally 
In summary, we state that
