**4. Time Series**

The time series analysis aims to: investigate the mechanism generating the time series; to forecast future values of the series, to describe the behavior of the series; seek relevant perio‐ dicities in the data. A model that describes a series does not necessarily lead to a procedure (or formula) prediction. You need to specify a function-loss, beyond the model, to get the procedure. A function-loss, which is often used, is the mean square error, although on some occasions, other criteria or loss functions are more appropriate (Morettin and Toloi, 2006; Camargo and Russo, 2011).

#### *Autocorrelation*

The autocorrelation is a measure of dependency between observations Same series separat‐ ed by a given range named retardation.

Be a time series*Yt*. The ratio between the covariance (*Yt*,*Yt*−*<sup>k</sup>* ) and variance (*Yt*) defines a autocorrelation coefficient simple (*rk* ), while the sequence of *rk*values is called autocorrela‐ tion function simple (AFS) (Camargo and Russo, 2006).

The graphical representation of this function is called correlogram. Formally, the autocorre‐ lation coefficients simple between *Yt* and their *Yt*−*<sup>k</sup>* lagged values, are defined by:

$$r\_k = \frac{\text{cov}(Y\_{t'}, Y\_{t-k})}{\text{var}(Y\_t)} = \frac{\sum\_{t=k+1}^n (Y\_t - \overline{Y})(Y\_{t-k} - \overline{Y})}{\sum\_{t=1}^n (Y\_t - \overline{Y})^2} \tag{14}$$

We can see the existence of unit root if the values of the autocorrelation function begin near to unit and decline slowly and gradually as increases the distance (number of lags, k) be‐ tween the two sets of observations to which they concern, calling himself, not stationary and follows a random walk. If these coefficients decline rapidly as this distance increases, there is a series of characteristics of stationary (Morettin and Toloi, 2006; Russo et al. 2006).

### *Stationary Processes*

Typical patterns of behavior are non-random (Lourenço Filho, 1964):

exchange of machines or operators.

brings motivation and improves performance.

and operator fatigue.

40 Practical Concepts of Quality Control

**4. Time Series**

Camargo and Russo, 2011).

ed by a given range named retardation.

tion function simple (AFS) (Camargo and Russo, 2006).

cov(*Yt*, *Yt*−*<sup>k</sup>* ) var(*Yt*) <sup>=</sup>

*rk* =

*Autocorrelation*

a) *Periodicity* - increases and decreases at regular intervals of time. The periodicity appears as one of the operating conditions of the process suffers periodic changes or when regular

b) *Trend* - when the points are directed substantially upwards, or downwards. The general trend indicates a gradual deterioration of a critical process. This "decay" can be a tool wear

c) *Shift* - changes in performance of the process. The cause of the change can be introduction of new machinery, new operators, new methods or even a quality program, which usually

The time series analysis aims to: investigate the mechanism generating the time series; to forecast future values of the series, to describe the behavior of the series; seek relevant perio‐ dicities in the data. A model that describes a series does not necessarily lead to a procedure (or formula) prediction. You need to specify a function-loss, beyond the model, to get the procedure. A function-loss, which is often used, is the mean square error, although on some occasions, other criteria or loss functions are more appropriate (Morettin and Toloi, 2006;

The autocorrelation is a measure of dependency between observations Same series separat‐

Be a time series*Yt*. The ratio between the covariance (*Yt*,*Yt*−*<sup>k</sup>* ) and variance (*Yt*) defines a autocorrelation coefficient simple (*rk* ), while the sequence of *rk*values is called autocorrela‐

The graphical representation of this function is called correlogram. Formally, the autocorre‐

∑ *t*=1 *n*

(*Yt* <sup>−</sup>*<sup>Y</sup>*¯)(*Yt*−*<sup>k</sup>* <sup>−</sup>*<sup>Y</sup>*¯)

(14)

(*Yt* <sup>−</sup>*<sup>Y</sup>*¯)2

lation coefficients simple between *Yt* and their *Yt*−*<sup>k</sup>* lagged values, are defined by:

∑ *t*=*k*+1 *n*

A common assumption in many time series techniques is that the data are stationary. A sta‐ tionary process has the property that the mean, variance and autocorrelation structure do not change over time. A process is considered stationary if its statistical characteristics do not change with time.

Stationarity is a assumption in time series analysis. It means that the main statistical proper‐ ties of the series remain unchanged over time. More precisely, a process {*Yt*} is said to be completely stationary or strict sense stationary (abbreviated as *SSS*) if the process *Yt* and *Yt*+*n* have the same statistics for any*n*. So, the characteristics*Y*(*t*+*n*) , for all n, will be the same as*Yt*.

#### *Non-Stationary Processes*

A big reason for using a stationary data sequence instead of a non-stationary sequence is that non-stationary sequences, usually, are more complex and take more calculations when forecasting is applied to a data series (Beusekom, 2003).

Where a series submit over time variation in your parameters, so, we have a series non-sta‐ tionary, which when submitted to differentiation process becomes stationary. If the time ser‐ ies is not stationary, we can often transform it to stationarity with one of the following way:

a) Difference the data, by create the new series
