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Time Series Analysis and Forecasting I

Introduction

- A time series is a set of observations generated

sequentially in time - Continuous vs. discrete time series
- The observations from a discrete time series,

made at some fixed interval h, at times ?1, ?2,,

?N may be denoted by z(?1), z(?2),, z(?N)

Introduction (cont.)

- Discrete time series may arise in two ways
- 1- By sampling a continuous time series
- 2- By accumulating a variable over a period of

time - Characteristics of time series
- Time periods are of equal length
- No missing values

Components of a time series

Zt Ft at

Areas of application

- Forecasting
- Determination of a transfer function of a system
- Design of simple feed-forward and feedback

control schemes

Forecasting

- Applications
- Economic and business planning
- Inventory and production control
- Control and optimization of industrial processes
- Lead time of the forecasts
- is the period over which forecasts are needed
- Degree of sophistication
- Simple ideas
- Moving averages
- Simple regression techniques
- Complex statistical concepts
- Box-Jenkins methodology

Approaches to forecasting

- Self-projecting approach

- Cause-and-effect approach

Approaches to forecasting (cont.)

- Self-projecting approach
- Advantages
- Quickly and easily applied
- A minimum of data is required
- Reasonably short-to medium-term forecasts
- They provide a basis by which forecasts developed

through other models can be measured against - Disadvantages
- Not useful for forecasting into the far future
- Do not take into account external factors

- Cause-and-effect approach
- Advantages
- Bring more information
- More accurate medium-to long-term forecasts
- Disadvantages
- Forecasts of the explanatory time series are

required

Some traditional self-projecting models

- Overall trend models
- The trend could be linear, exponential,

parabolic, etc. - A linear Trend has the form
- Trendt A Bt
- Short-term changes are difficult to track
- Smoothing models
- Respond to the most recent behavior of the series
- Employ the idea of weighted averages
- They range in the degree of sophistication
- The simple exponential smoothing method

Some traditional self-projecting models (cont.)

- Seasonal models
- Very common
- Most seasonal time series also contain long- and

short-term trend patterns - Decomposition models
- The series is decomposed into its separate

patterns - Each pattern is modeled separately

Drawbacks of the use of traditional models

- There is no systematic approach for the

identification and selection of an appropriate

model, and therefore, the identification process

is mainly trial-and-error - There is difficulty in verifying the validity of

the model - Most traditional methods were developed from

intuitive and practical considerations rather

than from a statistical foundation - Too narrow to deal efficiently with all time

series

ARIMA models

- Autoregressive Integrated Moving-average
- Can represent a wide range of time series
- A stochastic modeling approach that can be used

to calculate the probability of a future value

lying between two specified limits

ARIMA models (Cont.)

- In the 1960s Box and Jenkins recognized the

importance of these models in the area of

economic forecasting - Time series analysis - forecasting and control
- George E. P. Box Gwilym M. Jenkins
- 1st edition was in 1976
- Often called The Box-Jenkins approach

Transfer function modeling

- Yt ?(B)Xt where
- ?(B) ?0 ?1B ?2B2 ..
- B is the backshift operator
- BmXt Xt - m

Transfer function modeling (cont.)

- The study of process dynamics can achieve
- Better control
- Improved design
- Methods for estimating transfer function models
- Classical methods
- Based on deterministic perturbations
- Uncontrollable disturbances (noise) are not

accounted for, and hence, these methods have not

always been successful - Statistical methods
- Make allowance for noise
- The Box-Jenkins methodology

Process control

- Feed-forward control

- Feedback control

Process control (cont.)

Process control (cont.)

- The Box-Jenkins approach to control is to typify

the disturbance by a suitable time series or

stochastic model and the inertial characteristics

of the system by a suitable transfer function

model - The Control equation, allows the action which

should be taken at any given time to be

calculated given the present and previous states

of the system - Various ways corresponding to various levels of

technological sophistication can be used to

execute a control action called for by the

control equation

The Box-Jenkins model building process

Model identification

Model estimation

Is model adequate ?

No

Modify model

Yes

Forecasts

The Box-Jenkins model building process (cont.)

- Model identification
- Autocorrelations
- Partial-autocorrelations
- Model estimation
- The objective is to minimize the sum of squares

of errors - Model validation
- Certain diagnostics are used to check the

validity of the model - Model forecasting
- The estimated model is used to generate forecasts

and confidence limits of the forecasts

Important Fundamentals

- A Normal process
- Stationarity
- Regular differencing
- Autocorrelations (ACs)
- The white noise process
- The linear filter model
- Invertibility

A Normal process (A Gaussian process)

- The Box-Jenkins methodology analyze a time series

as a realization of a stochastic process. - The observation zt at a given time t can be

regarded as a realization of a random variable zt

with probability density function p(zt) - The observations at any two times t1 and t2 may

be regarded as realizations of two random

variables zt1, zt2 and with joint probability

density function p(zt1, zt2) - If the probability distribution associated with

any set of times is multivariate Normal

distribution, the process is called a normal or

Gaussian process

Stationary stochastic processes

- In order to model a time series with the

Box-Jenkins approach, the series has to be

stationary - In practical terms, the series is stationary if

tends to wonder more or less uniformly about some

fixed level - In statistical terms, a stationary process is

assumed to be in a particular state of

statistical equilibrium, i.e., p(zt) is the same

for all t

Stationary stochastic processes (cont.)

- the process is called strictly stationary
- if the joint probability distribution of any m

observations made at times t1, t2, , tm is the

same as that associated with m observations made

at times t1 k, t2 k, , tm k - When m 1, the stationarity assumption implies

that the probability distribution p(zt) is the

same for all times t

Stationary stochastic processes (cont.)

- In particular, if zt is a stationary process,

then the first difference ?zt zt - zt-1and

higher differences ?dzt are stationary - Most time series are nonstationary

Achieving stationarity

- Regular differencing (RD)
- (1st order) ?zt (1 B)zt zt zt-1
- (2nd order) ?2zt (1 B)2zt zt 2zt-1 zt-2
- B is the backward shift operator
- It is unlikely that more than two regular

differencing would ever be needed - Sometimes regular differencing by itself is not

sufficient and prior transformation is also needed

Some nonstationary series

Some nonstationary series (cont.)

Some nonstationary series (cont.)

How can we determine the number of regular

differencing ?

Autocorrelations (ACs)

- Autocorrelations are statistical measures that

indicate how a time series is related to itself

over time - The autocorrelation at lag 1 is the correlation

between the original series zt and the same

series moved forward one period (represented as

zt-1)

Autocorrelations (cont.)

- The theoretical autocorrelation function

- The sample autocorrelation

Autocorrelations (cont.)

- A graph of the correlation values is called a

correlogram - In practice, to obtain a useful estimate of the

autocorrelation function, at least 50

observations are needed - The estimated autocorrelations rk would be

calculated up to lag no larger than N/4

A correlogram of a nonstationary time seies

After one RD

After two RD

The white noise process

- The Box-Jenkins models are based on the idea that

a time series can be usefully regarded as

generated from (driven by) a series of

uncorrelated independent shocks at

- Such a sequence at, at-1, at-2, is called a

white noise process

The linear filter model

- A linear filter is a model that transform the

white noise process at to the process that

generated the time series zt

The linear filter model (cont.)

- ?(B) is the transfer function of the filter

The linear filter model (cont.)

- The linear filter can be put in another form

- This form can be written

Stationarity and invertibility conditions for a

linear filter

- For a linear process to be stationary,

- If the current observation zt depends on past

observations with weights which decrease as we go

back in time, the series is called invertible - For a linear process to be invertible,

Model building blocks

- Autoregressive (AR) models
- Moving-average (MA) models
- Mixed ARMA models
- Non stationary models (ARIMA models)
- The mean parameter
- The trend parameter

Autoregressive (AR) models

- An autoregressive model of order p

- The autoregressive process can be thought of as

the output from a linear filter with a transfer

function ?-1(B), when the input is white noise at - The equation ?(B) 0 is called the

characteristic equation

Moving-average (MA) models

- A moving-average model of order q

- The moving-average process can be thought of as

the output from a linear filter with a transfer

function ?(B), when the input is white noise at - The equation ?(B) 0 is called the

characteristic equation

Mixed AR and MA (ARMA) models

- A moving-average process of 1st order can be

written as

- Hence, if the process were really MA(1), we would

obtain a non parsimonious representation in terms

of an autoregressive model

Mixed AR and MA (ARMA) models (cont.)

- In order to obtain a parsimonious model,

sometimes it will be necessary to include both AR

and MA terms in the model - An ARMA(p, q) model

- The ARMA process can be thought of as the output

from a linear filter with a transfer function

?(B)/?(B), when the input is white noise at

The Box-Jenkins model building process

- Model identification
- Autocorrelations
- Partial-autocorrelations
- Model estimation
- Model validation
- Certain diagnostics are used to check the

validity of the model - Model forecasting

Partial-autocorrelations (PACs)

- Partial-autocorrelations are another set of

statistical measures are used to identify time

series models - PAC is Similar to AC, except that when

calculating it, the ACs with all the elements

within the lag are partialled out (Box Jenkins,

1976)

Partial-autocorrelations (cont.)

- PACs can be calculated from the values of the ACs

where each PAC is obtained from a different set

of linear equations that describe a pure

autoregressive model of an order that is equal to

the value of the lag of the partial-autocorrelatio

n computed - PAC at lag k is denoted by ?kk
- The double notation kk is to emphasize that ?kk

is the autoregressive parameter ?k of the

autoregressive model of order k

Model identification

- The sample ACs and PACs are computed for the

series and compared to theoretical

autocorrelation and partial-autocorrelation

functions for candidate models investigated

Stationarity and invertibility conditions

- For a linear process to be stationary,

- For a linear process to be invertible,

Stationarity requirements for AR(1) model

- For an AR(1) to be stationary
- -1 lt ?1 lt 1
- i.e., the roots of the characteristic equation 1

- ?1B 0 lie outside the unit circle - For an AR(1) it can be shown that
- ?k ?1 ?k 1 which with ?0 1 has the solution
- ?k ?1k k gt 0
- i.e., for a stationary AR(1) model, the

theoretical autocorrelation function decays

exponentially to zero, however, the theoretical

partial-autocorrelation function has a cut off

after the 1st lag

Invertibility requirements for a MA(1) model

- For a MA(1) to be invertible
- -1 lt ?1 lt 1
- i.e., the roots of the characteristic equation 1

- ? 1B 0 lie outside the unit circle - For a MA(1) it can be shown that

- i.e., for an invertible MA(1) model, the

theoretical autocorrelation function has a cut

off after the 1st lag, however, the theoretical

partial-autocorrelation function decays

exponentially to zero

Higher order models

- For an AR model of order p gt 1
- The autocorrelation function consists of a

mixture of damped exponentials and damped sine

waves - The partial-autocorrelation function has a cut

off after the p lag - For a MA models of order q gt 1
- The autocorrelation function has a cut off after

the q lag - The partial-autocorrelation function consists of

a mixture of damped exponentials and damped sine

waves

Permissible regions for the AR and MA parameters

Theoretical ACs and PACs (cont.)

Theoretical ACs and PACs (cont.)

Model identification

Model estimation

Model verification

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