Giới thiệu phân tích chuỗi thời gian

Toán ứng dụng
với chuỗi thời gian
TS. Nguy n Th Ng c Anhễ ị ọ
anh.nguyenthingoc@hust.edu.vn
SAMI, Hà N i, 15/11/2018ộ
15 Nov 2018 Toán ứng dung với chuỗi thời gian anh.nguyenthingoc@hust.edu.vn

ContentContent
1. Definition
2. Example
3. Objective of time series analysis
4. Steps in time series anaysis
5. Time series forecasting
6. Practical

•Set of evenly spaced numerical data
•Obtained by observing response variable at
regular time periods
•Forecast based only on past values
•Assumes that factors influencing past, present, &
future will continue
•Example
•Year: 1995 1996 1997 1998 1999
•Sales: 78.7 63.5 89.7 93.2 92.1
1. Definition1. Definition

Time series data is a sequence of observations
•collected from a process
•with equally spaced periods of time.

Contrary to restrictions placed on cross-sectional
data, the major purpose of forecasting with time
series is to extrapolate beyond the range of the
explanatory variables.

15 Nov 2018 Toán ứng dung với chuỗi thời gian 6
Time series is dynamic, it
does change over time.

When working with time series data, it is paramount
that the data is plotted so the researcher can view the
data.

•Discrete time series is one in which the set T0 at
which observations are made in a discrete set.
Continuous time series are obtained when
observations are recorded continuously over
some time interval.

•Data in business, economics, engineering,
environment, medicine, earth sciences, and other
areas of scientific investigations are often collected
in the form of time series.
•Hourly temperature readings
•Daily stock prices
•Weekly traffic volume
•Annual growth rate
•Seasonal ice cream consumption
•Electrical signals
915 Nov 2018 Toán ứng dung với chuỗi thời gian
2. Example2. Example

2.Example2.Example

2.Example2.Example (SUNSPOT NUMBERS)
20
The Wolf number (also known
as the International sunspot
number, relative sunspot
number, or Zürich number) is a
quantity that measures the
number of sunspots and
groups of sunspots present on
the surface of the sun. The
Sunspot Number is a crucial
tool used to study the solar
dynamo, space weather and
climate change.

3. Objective of time series3. Objective of time series
analysisanalysis
•Understanding the dynamic or time-dependent
structure of the observations of a single series
(univariate analysis)
•Forecasting of future observations
•Ascertaining the leading, lagging and feedback
relationships among several series (multivariate
analysis)

4. Steps in time series4. Steps in time series
analysisanalysis• Model Identification
• Time Series plot of the series
• Check for the existence of a trend or seasonality
• Check for the sharp changes in behavior
• Check for possible outliers
• Remove the trend and the seasonal component to get stationary
residuals.
• Estimation
• MME
• MLE
• Diagnostic Checking
• Normality of error terms
• Independency of error terms
• Constant error variance (Homoscedasticity)
• Forecasting
• Exponential smoothing methods
• Minimum MSE forecasting

5. Time series forecasting5. Time series forecasting
•Process of predicting a
future event
•Underlying basis of
all business decisions
•Production
•Inventory
•Personnel
•Facilities

•Used when situation is
vague & little data exist
• New products
• New technology
•Involve intuition,
experience
•e.g., forecasting sales on
Internet
Qualitative MethodsQualitative Methods Quantitative MethodsQuantitative Methods

vague & little data exist
• New products
• New technology
•Involve intuition,
experience
•e.g., forecasting sales on
Internet
‘stable’ & historical data
exist
• Existing products
• Current technology
•Involve mathematical
techniques
•e.g., forecasting sales of
color televisions
Quantitative MethodsQuantitative MethodsQualitative MethodsQualitative Methods

•Select several forecasting methods
•‘Forecast’ the past
•Evaluate forecasts
•Select best method
•Forecast the future
•Monitor continuously forecast accuracy

Quantitative
Forecasting

Quantitative
Forecasting
Time Series
Models

Causal
Models
Quantitative
Forecasting
Time Series
Models

Causal
Models
Quantitative
Forecasting
Time Series
Models
Exponential
Smoothing
Trend
Models
Moving
Average

Causal
Models
Quantitative
Forecasting
Time Series
Models
Regression
Exponential
Smoothing
Trend
Models
Moving
Average

5.1. Time series5.1. Time series
componentscomponents

TrendTrend

TrendTrend CyclicalCyclical

TrendTrend
SeasonalSeasonal
CyclicalCyclical

TrendTrend
SeasonalSeasonal
CyclicalCyclical
IrregularIrregular

Trend componentTrend component
•Persistent, overall upward or downward pattern
•Due to population, technology etc.
•Several years duration
Mo., Qtr., Yr.Mo., Qtr., Yr.
ResponseResponse
© 1984-1994 T/Maker Co.

Trend componentTrend component
• Overall Upward or Downward Movement
• Data Taken Over a Period of Years
Sales
Time
Upward trend

Cyclical componentCyclical component
•Repeating up & down movements
•Due to interactions of factors influencing economy
•Usually 2-10 years duration
Mo., Qtr., Yr.Mo., Qtr., Yr.
ResponseResponse
Cycle

Cyclical componentCyclical component
• Upward or Downward Swings
• May Vary in Length
• Usually Lasts 2 - 10 Years
Sales
Time
Cycle

Seasonal componentSeasonal component
•Regular pattern of up & down fluctuations
•Due to weather, customs etc.
•Occurs within one year
Mo., Qtr.Mo., Qtr.
ResponseResponse
SummerSummer
© 1984-1994 T/Maker Co.

Seasonal componentSeasonal component
• Upward or Downward Swings
• Regular Patterns
• Observed Within One Year
Sales
Time (Monthly or Quarterly)
Winter

Random or IrregularRandom or Irregular
componentcomponent
• Erratic, Nonsystematic, Random, ‘Residual’
Fluctuations
• Due to Random Variations of
•Nature
•Accidents
• Short Duration and Non-repeating

Time Series PlotTime Series Plot
01/93 01/94 01/95 01/96 01/97
Month/Year
16
17
18
19
20
Number of Surgeries
(X 1000)
Surgery Data
(Time Sequence Plot)
Source: General Hospital, Metropolis

Time Series Plot [Revised]Time Series Plot [Revised]
01/93 01/94 01/95 01/96 01/97
Month/Year
183
185
187
189
191
193
Number of Surgeries
(X 100)
Revised Surgery Data
(Time Sequence Plot)

Seasonality PlotSeasonality Plot
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Month
99.7
99.9
100.1
100.3
100.5
Monthly Index
(Seasonal Decomposition)

Trend AnalysisTrend Analysis
12/92 10/93 8/94 6/95 9/96 2/97 12/97
Month/Year
18
18.3
18.6
18.9
19.2
19.5
Number of Surgeries
(X 1000)
(Trend Analysis)

Time
Series

Time
Series
Trend?

Time
Series
Trend?Smoothing
Methods
NoNo

Time
Series
Trend?
Smoothing
Methods
Trend
Models
YesYesNoNo

Time
Series
Trend?
Smoothing
Methods
Trend
Models
YesYesNoNo
Exponential
Smoothing
Moving
Average

L in e a r
T i m e
S e r ie s
T r e n d ?
S m o o t h in g
M e t h o d s
T r e n d
M o d e ls
Y e sN o
E x p o n e n t ia l
S m o o t h in g
Q u a d r a t ic E x p o n e n t ia l
A u t o -
R e g r e s s iv e
M o v in g
A v e r a g e

Moving AverageMoving Average
MethodMethod
15 Nov 2018 Toán ứng dung với chuỗi thời gian Anh.nguyenthingoc@hust.edu.vn57

Time SeriesTime Series
ForecastingForecasting
L in e a r
T i m e
S e r ie s
T r e n d ?
S m o o t h in g
M e t h o d s
T r e n d
M o d e ls
Y e sN o
S m o o t h in g
A u t o -
R e g r e s s iv e
M o v in g
A v e r a g e

5.2. Moving Average5.2. Moving Average
MethodMethod
•Series of arithmetic means
•Used only for smoothing
•Provides overall impression of data over time

MethodMethod
•Series of arithmetic means
•Used only for smoothing
•Provides overall impression of data over time
Used for elementary forecastingUsed for elementary forecasting

Moving Average GraphMoving Average Graph
0
2
4
6
8
93 94 95 96 97 98
YearYear
SalesSales
ActualActual

MethodMethod
Example: You work for Firestone
Tire. You want to smooth
random fluctuations using a 3-
period moving average.
1995 20,000
1996 24,000
1997 22,000
1998 26,000
1999 25,000

Year Response Moving
Ave
1994 2 NA
1995 5 3
1996 2 3
1997 2 3.67
1998 7 5
1999 6 NA 94 95 96 97 98 99
8
6
4
2
0
Sales
MethodMethod

ExponentialExponential
Smoothing MethodSmoothing Method

5.3. Exponential5.3. Exponential
•Form of weighted moving average
•Weights decline exponentially
•Most recent data weighted most
•Requires smoothing constant (W)
•Ranges from 0 to 1
•Subjectively chosen
•Involves little record keeping of past data

Example: You’re organizing a Kwanza
meeting. You want to forecast attendance
for 1998 using exponential smoothing
(α = .20). Past attendance (00) is:
1995 4
1996 6
1997 5
1998 3
1999 7 © 1995 Corel Corp.

Time Yi
Smoothed Value, Ei
(W = .2)
Forecast
Yi + 1
1995 4 4.0 NA
1996 6 (.2)(6) + (1-.2)(4.0) = 4.4 4.0
1997 5 (.2)(5) + (1-.2)(4.4) = 4.5 4.4
1998 3 (.2)(3) + (1-.2)(4.5) = 4.2 4.5
1999 7 (.2)(7) + (1-.2)(4.2) = 4.8 4.2
2000 NA NA 4.8
Ei = W·Yi + (1 - W)·Ei-1
^

0
2
4
6
8
93 96 97 98 99
YearYear
AttendanceAttendance
ActualActual

Forecast Effect ofForecast Effect of
Smoothing CoefficientSmoothing Coefficient
((WW))
Weight
W is... Prior Period
2 Periods
Ago
3 Periods
Ago
W W(1-W) W(1-W)2
0.10 10% 9% 8.1%
0.90 90% 9% 0.9%
YYii+1+1 == W·YW·Yii ++ W·W·(1-(1-WW))·Y·Yii-1-1 ++ W·W·(1-(1-WW))22
·Y·Yii-2-2 +...+...
^^

Linear Time-SeriesLinear Time-Series
Forecasting ModelForecasting Model

Time Series ForecastingTime Series Forecasting
L in e a r
T i m e
S e r ie s
T r e n d ?
S m o o t h in g
M e t h o d s
T r e n d
M o d e ls
Y e sN o
S m o o t h in g
A u t o -
R e g r e s s iv e
M o v in g
A v e r a g e

5.4. Linear Time-Series5.4. Linear Time-Series
•Used for forecasting trend
•Relationship between response variable Y & time X is
a linear function
•Coded X values used often
•Year X: 1995 1996 1997 1998 1999
•Coded year: 0 1 2 3 4
•Sales Y: 78.7 63.5 89.7 93.2 92.1

Y
T im e , X 1
Y b b Xi i= +0 1 1
bb11 > 0> 0
bb11 < 0< 0

Example: You’re a marketing analyst for Hasbro Toys.
Using coded years, you find Yi = .6 + .7Xi.
1995 1
1996 1
1997 2
1998 2
1999 4
Forecast 2000 sales.

Year Coded Year Sales (Units)
1995 0 1
1996 1 1
1997 2 2
1998 3 2
1999 4 4
2000 5 ?
2000 forecast sales: Yi = .6 + .7·(5) = 4.1
The equation would be different if ‘Year’ used.

The Linear Trend ModelThe Linear Trend Model
iii X..XbbYˆ 743143210 +=+=Year Coded Sales
94 0 2
95 1 5
96 2 2
97 3 2
98 4 7
99 5 6
0
1
2
3
4
5
6
7
8
1993 1994 1995 1996 1997 1998 1999 2000
Projected to
year 2000
Coefficients
Intercept 2.14285714
X Variable 1 0.74285714
Excel Output

Quadratic Time-SeriesQuadratic Time-Series

5.5. Quadratic Time-5.5. Quadratic Time-
Series Forecasting ModelSeries Forecasting Model
•Relationship between response variable Y & time X
is a quadratic function
•Coded years used

5.5. Quadratic Time-5.5. Quadratic Time-
•Relationship between response variable Y & time X
is a quadratic function
•Coded years used
•Quadratic model
Y b b X b Xi i i
= + +0 1 1 11
2
1

Y
Y e a r , X 1
Y
Y e a r , X 1
Y
Y e a r , X 1
Quadratic Time-SeriesQuadratic Time-Series
Model Relationships
Y
Y e a r , X 1
bb1111 > 0> 0bb1111 > 0> 0
bb1111 < 0< 0bb1111 < 0< 0

Quadratic Trend ModelQuadratic Trend Model
2
210 iii XbXbbYˆ ++=
2
2143308572 iii X.X..Yˆ +−=
Excel Output
Year Coded Sales
94 0 2
95 1 5
96 2 2
97 3 2
98 4 7
99 5 6
Coefficients
X Variable 1 -0.3285714
X Variable 2 0.21428571

Exponential Time-SeriesExponential Time-Series
ModelModel

5.6. Exponential Time-5.6. Exponential Time-
•Relationship is an exponential function
•Series increases (decreases) at increasing (decreasing)
rate

Exponential Time-SeriesExponential Time-Series
ModelModel RelationshipsRelationships
Y
Y e a r , X 1
bb11 > 1> 1
0 <0 < bb11 < 1< 1

Exponential WeightExponential Weight
[Example Graph]
94 95 96 97 98 99
8
6
4
2
0
Sales
Year
Data
Smoothed

Coefficients
X Variable 10.08068544
Exponential Trend ModelExponential Trend Model
iX
i bbYˆ 10= or 110 blogXblogYˆlog i +=
Excel Output of Values in logs
iX
i ).)(.(Yˆ 21172=
Year Coded Sales
94 0 2
95 1 5
96 2 2
97 3 2
98 4 7
99 5 6
antilog(.33583795) = 2.17
antilog(.08068544) = 1.2

Autoregressive ModelingAutoregressive Modeling

Time Series
Forecasting
L in e a r
T i m e
S e r ie s
T r e n d ?
S m o o t h in g
M e t h o d s
T r e n d
M o d e ls
Y e sN o
S m o o t h in g
A u t o -
R e g r e s s iv e
M o v in g
A v e r a g e

5.7. Autoregressive5.7. Autoregressive
ModelingModeling
•Like regression model
•Independent variables are lagged response
variables Yi-1, Yi-2,Yi-3etc.
•Assumes data are correlated with past data values
•1st
Order: Correlated with prior period
•Estimate with ordinary least squares

Time Series Data PlotTime Series Data Plot
09/83 07/86 05/89 03/92 01/95
Month/Year
0
2
4
6
8
10
12
Number of Passengers
(X 1000)
Intra-Campus Bus Passengers
Data collected by Coop Student (10/6/95)

Auto-correlation PlotAuto-correlation Plot
0 5 10 15 20 25
Lag
-1
-0.5
0
0.5
1
± 2 σ
Intra-Campus Bus
Passengers(Auto Correlation Function
Plot

The Office Concept Corp. has acquired a number of office
units (in thousands of square feet) over the last 8 years.
Develop the 2nd order Autoregressive models.
Year Units
92 4
93 3
94 2
95 3
96 2
97 2
98 4
99 6
5.7. Autoregressive Model5.7. Autoregressive Model
[An Example]

Year Yi Yi-1 Yi-2
92 4 --- ---
93 3 4 ---
94 2 3 4
95 3 2 3
96 2 3 2
97 2 2 3
98 4 2 2
99 6 4 2
Coefficients
Intercept 3.5
X Variable 1 0.8125
X Variable 2 -0.9375
Excel Output
21 9375812553 −− −+= iii Y.Y..Y
•Develop the 2nd order
table
•Use Excel to run a
regression model
5.7. Autoregressive Model5.7. Autoregressive Model
[Example Solution]

Evaluating ForecastsEvaluating Forecasts

Quantitative ForecastingQuantitative Forecasting
StepsSteps
•Select several forecasting methods
•‘Forecast’ the past
•Evaluate forecasts
•Select best method
•Forecast the future
•Monitor continuously forecast accuracy


Forecasting GuidelinesForecasting Guidelines
•No pattern or direction in forecast error
•ei = (Actual Yi - Forecast Yi)
•Seen in plots of errors over time
•Smallest forecast error
•Measured by mean absolute deviation
•Simplest model
•Called principle of parsimony

Pattern of Forecast ErrorPattern of Forecast Error
T im e ( Y e a r s )
E r r o r
0
Trend Not FullyTrend Not Fully
Accounted forAccounted for Desired PatternDesired Pattern
T im e ( Y e a r s )
E r r o r
0

Residual AnalysisResidual Analysis
Random errors
Trend not accounted for
Cyclical effects not accounted for
Seasonal effects not accounted for
T T
T T
e e
e e
0 0
0 0

SummarySummary
•Described what forecasting is
•Explained time series & its components
•Smoothed a data series
•Moving average
•Exponential smoothing
•Forecasted using trend models

Questions?

SAMI
Email:
anh.nguyenthingoc@hust.edu.vn

Giới thiệu phân tích chuỗi thời gian

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Similar to Giới thiệu phân tích chuỗi thời gian

Similar to Giới thiệu phân tích chuỗi thời gian (20)

More from NguyenThiNgocAnh9

More from NguyenThiNgocAnh9 (14)

Recently uploaded

Recently uploaded (20)

Giới thiệu phân tích chuỗi thời gian

Editor's Notes