data science training in surat
                     




Forecasting 
Time Series 


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My Introduction 

PMP	
PMI-ACP	
PMI-RMP	
CSM	
LSSGB	

Project	Management	Professional	

Agile	Cer4fied	Prac44oner	

Risk	Management	Professional	

Cer4fied	Scrum	Master	

Lean	Six	Sigma	Green	Belt	

LSSBB	
SSMBB	
ITIL	

Lean	Six	Sigma	Black	Belt	

Six	Sigma	Master	Black	Belt	

Informa4on	Technology	Infrastructure	Library	

Agile	PM	 Dynamic	System	Development	Methodology	Atern	

Name:									Bharani	Kumar	
	
Educa+on:		IIT	Hyderabad	

				Indian	School	of	Business	
	
Professional	cer+fica+ons:	
	


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My Introduction 

HSBC	
Driven	using	UK	policies	

ITC	Infotech	
Driven	using	Indian	policies	SME	

Infosys	
Driven	using	Indian	policies	under	Large	enterprises	

DeloiHe	
Driven	using	US	policies	

1	
2	

3	
4	 RESEARCH	in	ANALYTICS,	DEEP	

LEARNING	&	IOT	
DATA	SCIENTIST	


Why	Forecas4ng	
Learn	about	the	various	examples	of	
forecasHng	

Forecas4ng	Strategy	
Learn	about	decomposing,	forecasHng	
&	combining	

EDA	&	Graphical	Representa4on	
Learn	about	exploratory	data	analysis,	
scaKer	plot,	Hme	plot,	lag	plot,	ACF	plot	

Forecas4ng	components	
Learn	about	Level,	Trend,	Seasonal,	
Cyclical,	Random	components	

Forecas4ng	Models	&	Errors	
Learn	about	various	forecasHng	
models	to	be	discussed	&	the	various	
error	measures	

AGENDA	

Why	
Forecas4ng

?	

Forecas4ng	
Strategy	

EDA	&	Graphical	
Representa4on	

Forecas4ng		
Decomposi4on	
components	

AGENDA 


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Why Forecasting 
•  Why	forecast,	when	you	would	know	the	outcome	eventually?	
•  Early	knowledge	is	the	key,	even	if	that	knowledge	is	imperfect	

–  For	seQng	producHon	schedules,	one	needs	to	forecast	sales		
–  For	staffing	of	call	centers,	a	company	needs	to	forecast	the	demand	for	service	
–  For	dealing	with	epidemic	emergencies,	naHons	should	forecast	the	various	flu	


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Types of forecast 

Micro	Scale	
or	Macro	
Scale	

Qualita4ve	or	
Quan4ta4ve	

Short	Term	
or	Long	Term	

Data	or	
Judgment	

Forecas4ng	
Classifica4on	

Point	
Forecast	

Density	
Forecast	

Interval	
Forecast	


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Who generates Forecast? 


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Who generates Forecast? 


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Time series vs Cross-sectional data 

01	 Cross-sec4onal	Data	

02	
Time	Series	Data	


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Dataset for further discussion 

Monthly	FooWalls	of	customers	from	
Jan	1991	to	March	2004		

t					=	1,	2,	3,…....=	Hme	period	index	
Yt					=	value	of	the	series	at	Hme	period	t	
Yt+k	=	forecast	for	Hme	period	t+k,	given	
data	unHl	Hme	t	
et					=	forecast	error	for	period	t	

Month Footfall in thousands 
Jan-91 1709 
Feb-91 1621 
Mar-91 1973 
Apr-91 1812 
May-91 1975 
Jun-91 1862 
Jul-91 1940 
Aug-91 2013 
Sep-91 1596 
Oct-91 1725 
Nov-91 1676 
Dec-91 1814 
Jan-92 1615 
Feb-92 1557 
Mar-92 1891 
Apr-92 1956 
May-92 1885 
Jun-92 1623 


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Forecasting Strategy 

01	
02	
03	
04	
05	

Define	Goal	

Data	Collec4on	

Explore	&	Visualize	Series	

Pre-Process	Data	

Par44on	Series	

06	
07	
08	

Apply	Forecas4ng	Method(s)	

Evaluate	&	Compare	Performance	

Implement	Forecasts	/	System	


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Forecasting Strategy – Step 1 

 

• DescripHve	=	Time	Series	
Analysis	

• PredicHve	=	Time	Series	
ForecasHng	

• How	far	into	the	future?	k	in	Yt+k	

• Rolling	forward	or	at	single	Hme	
point?	

 

#1 Is the goal descriptive or 
predictive? #2 What is the forecast horizon? 

• Who	are	the	stakeholders?	

• Numerical	or	event	forecast?	

• Cost	of	over-predicHon	&	
under-predicHon	

• In-house	forecasHng	or	
consultants?	

• How	many	series?	How	ofen?	

• Data	&	sofware	availability	

#3 How will the forecast be used? #4 Forecasting expertise & automation 

Define		
Goal	


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Forecasting Strategy – Step 2 

• Typically	small	sample,	so	need	
good	quality	

• Data	same	as	series	to	be	
forecasted	

• Should	we	use	real-Hme	Hcket	
collecHon	data?	

• Balance	between	signal	&	noise	
• AggregaHon	/	DisaggregaHon	

#1	Data	Quality	 #2	Temporal	Frequency	

• Coverage	of	the	data	–	
Geographical,	populaHon,	Hme,…	

• Should	be	aligned	with	goal	
	

• Necessary	informaHon	source	
• Affects	modeling	process	from	start	
to	end	

• Level	of	communicaHon/
coordinaHon	between	forecasters	&	
domain	experts	

#3	Series	Granularity?	 #4.	Domain	exper4se	

Data		
Collec-on	


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Forecasting Strategy Step3 (Explore Series) 

Seasonal	
PaHerns	

SYSTEMATIC	
PART	

Level	
Trend	

Season
al	
PaHern
s	

NON-SYSTEMATIC	PART	
	
Noise	

Addi4ve:		
	
Yt	=	Level	+	Trend	+	Seasonality	+	Noise		

Mul4plica4ve:	
	
Yt	=	Level	x	Trend	x	Seasonality	x	Noise		


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Trend Component 

•  Persistent,	overall	upward	or	downward	paKern	
•  Due	to	populaHon,	technology	etc.	
•  Overall	Upward	or	Downward	Movement	
•  Several	years	duraHon		

Mo., Qtr., Yr. 

Response 


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Seasonal Component 
•  Regular	paKern	of	up	&	down	fluctuaHons	
•  Due	to	weather,	customs	etc.	
•  Occurs	within	one	year		
•  Example:	Passenger	traffic	during	24	hours	

Mo., Qtr. 

Response 
Summer 


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Irregular/Random/Noise Component 
•  ErraHc,	unsystemaHc,	‘residual’	fluctuaHons	
•  Due	to	random	variaHon	or	unforeseen	events	

–  Union	strike	
–  War	

•  Short	duraHon	&	nonrepeaHng		


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Time Series Components 


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Time Plot 

•  Plots	a	variable	against	Hme	index	

•  Appropriate	for	visualizing	serially	collected	data	(Hme	series)	

•  Brings	out	many	useful	aspects	of	the	structure	of	the	data	

•  Example:		Electrical	usage	for	Washington	Water	Power	
(Quarterly	data	from	1980	to	1991)	


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Time plot 

400	

500	

600	

700	

800	

900	

1000	

1100	

1980	 1982	 1984	 1986	 1988	 1990	

Po
w
er
	u
sa
ge
	(K

ilo
w
aH

s)
	

Year	

Electrical	power	usage	for	Washington	Water	Power:	1980-1991	


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Observations 
•  There	is	a	cyclic	trend	

•  Maximum	demand	in	first	quarter;	minimum	in	third	quarter	

•  There	may	also	be	a	slowly	increasing	trend	(to	be	examined)	

•  Any	reasonable	forecast	should	have	cyclic	fluctuaHons		

•  Trend	(if	any)	need	to	be	uHlized	for	forecasHng	

•  Forecast	would	not	be	exact	–	there	would	be	some	error	


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Time plot 


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Quarterly Sales of Ice-cream 


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Scatter Diagram 

•  Plots	one	variable	against	another	
•  One	of	the	simplest	tools	for	visualizaHon	

Cost	 Age	
859	 8	
682	 5	
471	 3	
708	 9	
1094	 11	
224	 2	
320	 1	
651	 8	
1049	 12	

ž  Example:	Maintenance	cost	and	Age	
for	nine	buses	(Spokane	Transit)	

	

ž  This	is	an	example	of	cross-secHonal	
data	(observaHons	collected	in	a	single	
point	of	Hme)	

	


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Scatter Plot 

0	

200	

400	

600	

800	

1000	

1200	

0	 2	 4	 6	 8	 10	 12	 14	

Ye
ar
ly
	c
os
t	o

f	m
ai
nt
en

an
ce
	(U

S	
$)
	

Age	of	bus	


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Observations 
•  Older	buses	have	higher	cost	of	maintenance	

•  There	is	some	variaHon	(case	to	case)	

•  The	rise	in	cost	is	about	$	80	per	year	of	age	

•  It	may	be	possible	to	use	‘age’	to	forecast	maintenance	
cost	

•  Forecast	would	not	be	a	‘certain’	predicHon	–	there	would	
be	some	error	


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Lag plot 

•  Plots	a	variable	against	its	own	lagged	sample	

•  Brings	out	possible	associaHon	between	successive	samples	

•  Example:	Monthly	sale	of	VCRs	by	a	music	store	in	a	year	

=	Number	of	VCRs	sold	in	Hme	period	t	
	
=	Number	of	VCRs	sold	in	Hme	period	t	–	k	
	


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Example of lagged variables 
Number of VCRs sold in a month 

Time	
	

Original		
	

Lagged	one	step		
	

Lagged	two	steps		
	

1	 123	 	 	
2	 130	 123	 	
3	 125	 130	 123	
4	 138	 125	 130	
5	 145	 138	 125	
6	 142	 145	 138	
7	 141	 142	 145	
8	 146	 141	 142	
9	 147	 146	 141	
10	 157	 147	 146	
11	 150	 157	 147	
12	 160	 150	 157	


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Lag plot (k = 1) 

120	

125	

130	

135	

140	

145	

150	

155	

160	

120	 125	 130	 135	 140	 145	 150	 155	 160	

ScaHer	plot	of	VCR	sales	with	1-step	lagged	VCR	sales	


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Observations 

•  There	is	a	reasonable	degree	of	associaHon	
between	the	original	variable	and	the	lagged	one	

•  Value	of	lagged	variable	is	known	beforehand,	so	
it	is	useful	for	predicHon	

•  AssociaHon	between	original	and	lagged	variable	
may	be	quan+fied	through	a	correlaHon	


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Autocorrelation 

•  CorrelaHon	between	a	variable	and	its	lagged	
version	(one	Hme-step	or	more)	

	
	
=	ObservaHon	in	Hme	period	t	
=	ObservaHon	in	Hme	period	t	–	k	
=	Mean	of	the	values	of	the	series	
=	AutocorrelaHon	coefficient	for	k-step	lag	
	


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Standard error of rk 
•  The	standard	error	is	

•  Increases	progressively	with	k,	but	eventually	reaches	a	
maximum	value	

•  If	the	‘true’	autocorrelaHon	is	0,	then	the	esHmate	rk	should	
be	in	the	interval	(–	2SE(rk),	2SE(rk))	95%	of	the	Hme	

•  SomeHmes	SE(rk)	is	approximated	by	

The	 standard	 error	 of	 the	 mean	 esHmates	 the	
variability	 between	 samples	 whereas	 the	
standard	deviaHon	measures	the	variability	within	
a	single	sample.	


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Correlogram or ACF plot 

•  Plots	the	ACF	or	AutocorrelaHon	funcHon	(rk)	
against	the	lag	(k)	

•  Plus-and-minus	two-standard	errors	are	
displayed	as	limits	to	be	exceeded	for	staHsHcal	
significance	

•  Reveals	lagged	variables	that	can	be	potenHally	
useful	for	forecasHng	


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Correlogram for VCR data 


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ACF plot for electricity usage data 


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Observations 
•  Every	alternate	sample	is	large,	many	of	them	

staHsHcally	significant	also	

•  ACFs	at	lags	4,	8,	12,	etc	are	posiHve	

•  ACF	at	lags	2,6,10	etc	are	negaHve	

•  All	these	pick	up	the	seasonal	aspect	of	the	data	

•  The	data	may	be	re-examined	afer	‘removing’	
seasonality	


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ACF of de-seasoned KW data 


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Observations 

•  De-seasoned	series	has	small	ACFs	

•  This	part	of	the	data	has	liKle	forecasHng	value	


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Typical questions in exploratory analysis 

All	the	plots	contain	
informaHon	regarding	

these	quesHons	

Is there a TREND? 

Is there a 
SEASONALITY? Are the data RANDOM? 


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Time series plots 


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Effect of omission of data on the  
Time series plot 


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Effect of omission of data on the  
Time series plot 


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Confusing kind of trend due to other type  
of scaling  

0 
20

 
40

 
60

 
80

 
y 

0 5 10 15 20 
t 

0 
20

 
40

 
60

 
80

 
y 

0 1 2 3 
Log   t 

2.
5 

3 
3.

5 
4 

4.
5 

Lo
g 

  y
 

0 5 10 15 20 
t 

2.
5 

3 
3.

5 
4 

4.
5 

Lo
g 

  y
 

0 1 2 3 
Log   t 


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Few points on Plots 

Plot	helps	us	to	summarize	&	reveal	paKerns	
in	data	

Graphics	help	us	to	idenHfy	anomalies	in	data	

Plot	helps	us	to	present	a	huge	amount	of	data	in	
small	space	&	makes	huge	data	set	coherent	

To	get	all	the	advantages	of	plot,	the	“Aspect	RaHo”	
of	plot	is	very	crucial	

The	raHo	of	Height	to	Width	of	a	plot	is	called	
the	ASPECT	RATIO	


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Aspect Ratio 

•  Generally	aspect	raHo	should	be		around	0.618	

•  However,	for	long	Hme	series	data	aspect	raHo	should	be	
around	0.25.	To	understand	the	impact	of	aspect	raHo	see	the	
two	plots	in	the	next	two	slides	


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Aspect ratio 


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Aspect ratio 


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Should	we	use	all	
historical	data	for	
forecas4ng	

?	
Preliminaries for Step 3 of 8-Step forecasting strategy 

Training	Data	

Valida4on	Data	

Fit	the	model	only	to	
TRAINING	period	

Assess	performance	on	
VALIDATION	period	

Solu4on	=	DATA	PARTIONING	


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Partitioning 

Deploy	model	by	joining	Training	+	ValidaHon	to	forecast	the	Future	


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How to choose a Validation Period? 

Forecast	
Horizon	

Seasonality	

Length	of	
series	

Underlying	
condi4ons	
affec4ng	series	

Strategy	to	choose	
Valida4on	Data	
Period	


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Rolling-forward forecasts 


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NAÏVE Forecasts 

Forecast	method:	Last	sample	
	
	
k-step	ahead	
	
	
Seasonal	series	(	M	series	)	

Ft+k		=		Yt	

Ft+k		=		Yt-M+k	


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Forecast error 
•  Forecast	error	is	
	
•  If	model	is	adequate,	forecast	error	should	contain	no	informaHon	
•  Plots	of	et	should	resemble	that	of	‘white	noise’	or	uncorrelated	random	

numbers	with	0	mean	and	constant	variance	
	(There	should	be	NO	PATTERN)	


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Forecast error 
•  Forecast	error	can	follow	different	distribuHons	based	on	business	context		


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Forecasting Errors 


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Evaluating Predictive Accuracy 

•  Mean error      

•  Mean absolute deviation 

•  Mean squared error 

•  Root mean squared error 

•  Mean percentage error 

•  Mean absolute percentage error   


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Typical plots of ‘White noise’ 
Time	plot	

Lag	plot	

ACF	plot		

Histogram		


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Mean error (ME) 
•  If the ME is around zero, forecasts are called unbiased. Model 

is unbiased to overestimation or the underestimation. Certainly 
this is a desirable property of a model 

Actual	data	 Forecast	based	
on	Model	1	

Error	from	
model	1	

Forecast	based	
on	Model	2	

Error	from	
model	2	

100	 101	 1	 110	 10	

200	 199	 -1	 190	 -10	

300	 301	 1	 310	 10	

400	 399	 -1	 390	 -10	

ME	 0	 0	


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Mean error 

•  Mean	error	has	the	disadvantage	that	small	amount	and	
large	amount	of	error	may	have	same	effect	

•  To	overcome	this	problem	we	may	define	two	different	
forecast	performance	measure	

•  1.	Mean	Absolute	DeviaHon:		

•  2.	Mean	Square	Error:	


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MAD & MSE 
Actual	data	 Forecast	based	

on	Model	1	
Error	from	
model	1	

Forecast	based	
on	Model	2	

Error	from	
model	2	

100	 101	 1	 110	 10	

200	 199	 -1	 190	 -10	

300	 301	 1	 310	 10	

400	 399	 -1	 390	 -10	

MAD	 	 1	 	 10	

MSE	 	 1	 	 100	

ME	 	 0	 	 0	


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Problem with ME, MAD, MSE 
•  All	these	three	measures	are	not	unit	free	and	also	not	scale	free	

•  Just	 think	of	a	case	 that	one	 is	 forecasHng	sales	figures.	Someone		
in	India	using	rupee	figure,	and	somebody	else	in	USA	is	expressing	
the	 same	 sales	 figure	 in	 dollar.	 Both	 are	 using	 the	 same	 model.	
However	 forecast	 measure	 will	 differ.	 This	 	 is	 a	 very	 awkward	
situaHon	

•  MSE	 has	 the	 added	 disadvantage	 that	 its	 unit	 is	 in	 square.	 RMSE	
does	not	have	this	added	disadvantage	

•  So	we	need	unit	free	measure	


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MPE and MAPE---Unit free measure 

•  Both	expressed	in	percentage	form	

•  Both	are	unit	free	


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Last Sample: Number of customers requiring repair work 

Customers 
Y_t 

Fitted value 
 

Residual 
 e_t |e_t| e_t^2 e_t/Y_t |e_t/Y_t| 

58       
54 58 -4 4 16 -0.07407 0.074074 
60 54 6 6 36 0.1 0.1 
55 60 -5 5 25 -0.09091 0.090909 
62 55 7 7 49 0.112903 0.112903 
62 62 0 0 0 0 0 
65 62 3 3 9 0.046154 0.046154 
63 65 -2 2 4 -0.03175 0.031746 
70 63 7 7 49 0.1 0.1 
  MAD MSE RMSE MPE MAPE 
  4.25 23.5 4.85 0.0203 0.0695 

Forecast	method:	Last	sample	


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MA: Number of customers requiring repair work 
Customers 

Y_t 
Fitted 
value 

Residual 
 e_t |e_t| e_t^2 e_t/Y_t |e_t/Y_t| 

58       
54       
60       
55 57.3333 -2.3333 2.3333 5.4444 -0.0424 0.0424 
62 56.3333 5.6667 5.6667 32.1111 0.0914 0.0914 
62 59.0000 3.0000 3.0000 9.0000 0.0484 0.0484 
65 59.6667 5.3333 5.3333 28.4444 0.0821 0.0821 
63 63.0000 0.0000 0.0000 0.0000 0.0000 0.0000 
70 63.3333 6.6667 6.6667 44.4444 0.0952 0.0952 
  MAD MSE RMSE MPE MAPE 
  3.83 19.91 4.46 0.0458 0.0599 

Forecast	method:	3-point	moving	average	


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Challenges 

Zero	Counts	
MAE/RMSE:	no	problem	
Cannot	compute	MAPE	

Exclude	Zero	count	
Use	alternate	measure	-	MASE	

	

Missing	values	
Compute	average	metrics	
Exclude	missing	values	


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Forecast / Prediction Interval 

1	
2	

Probability	of	95%	that	the	
value	will	be	in	the	range	[a,b]	

If	the	forecast	errors	are	normal,	predic4on	
interval	is	
σ	=	es4mated	standard	devia4on	of	forecast	errors	
k	=	some	mul4ple		
(k=2	corresponds	to	95%	probability)			

3	
Challenges	to	formula	
•		Errors	ofen	non-normal		
•		If	model	is	biased	(over/under-forecasts),	symmetric	interval	around	Ft+k?		
•		EsHmaHng	the	error	standard	deviaHon	is	tricky		
	

	One	soluHon	is	transforming	errors	to	normal		
	


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Forecast / Prediction Interval – Non-Normal 
To	construct	predicHon	interval	for	1-step-ahead	forecasts		

1.		Create	roll-forward	forecasts	(Ft+1)	on	validaHon	period		
2.		Compute	forecast	errors		
3.		Compute	percenHles	of	error	distribuHon		

	(e(5)=5th	percenHle;	e(95)=95th	percenHle)		
4.		PredicHon	interval:		

	[	Ft+1	+	e(5)	,	Ft+1	+	e(95)	]		

In	Excel	=percen+le		
	
5th	percenHle	=	-307.0		
95th	percenHle	=	292.8		
	
95%	predicHon	interval	for	1-step	ahead	forecast	Ft+1:		
[Ft+1	–	307	,	Ft+1	+	292.8]		


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Forecasting Different Methods 

Model	
based	

Data	
driven	

•  Linear	regression		
•  Autoregressive	models		
•  ARIMA		
•  LogisHc	regression		
•  Econometric	models		

	
•  Naïve	forecasts		
•  Smoothing		
•  Neural	nets		


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Forecasting Different Methods 
Linear	Model:	Yt	=	βo	+	β1t	+	ε	

ExponenHal	Model:	Log	(Yt)	=	βo	+	β1t	+	ε	

QuadraHc	Model:	Yt	=	βo	+	β1t	+	β2t2	+	ε	
		

AddiHve	Seasonality:	Yt	=	βo	+	β1DJan	+	β2DFeb	+	β3DMar	+	…...+	β11DNov	+	ε		

AddiHve	Seasonality	with	QuadraHc	Trend:	
Yt	=	βo	+	β1t	+	β2t2	+	β3DJan	+	β4DFeb	+	β5DMar	+	…...+	β13DNov	+	ε		
	MulHplicaHve	Seasonality:	
Log	(Yt)	=	βo	+	β1DJan	+	β2DFeb	+	β3DMar	+	…...+	β11DNov	+	ε		
	
	


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Irregular Component 

• Outliers 

• Special Events 

• Interventions 

• Remove unusual periods 
from the model 

• Model separately 

• Keep in the model, using 
dummy variable 

Irregular Components Solutions 


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External Information 

ForecasHng	
Internet	Sales	

ForecasHng	Airline	
Ticket	Price

		

Fuel	price	impacts	
the	airline	Hcket	

Amount	spend	in	
adverHsements	

Sales(t)	=		
g{	f(sales(t-1,	t-2,	...	,	t-6),	
a1*SQRT[AdSpend(t-1)]	+	...	+	
a6*SQRT[AdSpend(t-6)]	}		

Airfaret	=	b0	+	b1	(Petrol	Price)t	+	e		
	
																										Must	be	forecasted		


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Linear Regression for forecasting 

Global	Trend	
	
•  Linear	Trend	

(constant	growth)	
•  ExponenHal	Trend	

(%	growth)	

1	
Seasonality	
	
•  AddiHve	(Y)	

•  MulHplicaHve	
log(Y)	

Irregular	PaHerns	

2	 3	


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Autoregressive (AR) Models 

•  AR	model	is	used	to	forecast	errors	

•  AR	model	captures	autocorrelaHon	directly	

•  AutocorrelaHon	measures	how	strong	the	values	of	a	Hme	series	are	related	to	
their	own	past	values	

•  Lag(1)	autocorrelaHon	=	correlaHon	between		
																																																	(y1,	y2,	…,	yt-1	)	and	(y2,y3,…,	yt)	

•  Lag(k)	autocorrelaHon	=	correlaHon	between		
																																																	(y1,	y2,	…,	yt-k)	and	(yk+1,yk+2,…,yt)			


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Autocorrelation & its uses 
Check forecast errors for 

independence 

Model remaining 
information 

Evaluate 
predictability 


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Autoregressive Model 
•  MulH-layer	model	

•  Model	the	forecast	errors,	by	treaHng	them	as	a	Hme	series		

•  Then	examine	autocorrelaHon	of	“errors	of	forecast	errors”	?	
	

ü  If	autocorrelaHon	exists,	fit	an	AR	model	to	the	forecast	errors	series	
ü  If	autocorrelated,	conHnue	modeling	the	level-2	errors	(not	pracHcal)	

•  AR	model	can	also	be	used	to	model	original	data		
Yt	=	α	+	β1Yt-1	+	β2Yt-2	+	εt							->	AR(2),	order	=	2	

	
1-step	ahead	forecast:	 	Ft+1	=	α	+	β1Yt			+	β2Yt-1	
2-steps	ahead:	 	 	Ft+2	=	α	+	β1Ft+1	+	β2Yt	
3-steps	ahead:	 	 	Ft+3	=	α	+	β1Ft+2	+	β2Ft+1		


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Autoregressive Model 
	
•  Use	level	1	to	forecast	next	value	of	series	Ft+1		
•  Use	AR	to	forecast	next	forecast	error	(residual)	Et+1	
•  Combine	the	two	to	get	an	improved	forecast	F*t+1	
	

	F*t+1	=	Ft+1	+	Et+1	

^	

^	


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Random Walk 
•  Specific	case	of	AR(1)	model	

•  If	β1	=	1	in	AR(1)	model	then	it	is	called	as	Random	Walk	
•  EquaHon	will	be	Yt	=	a	+	Yt-1	+	εt	

	a	=	drif	parameter	
	σ(std	of	ε)	=	volaHlity	

	
•  Changes	from	one	period	to	the	next	are	random	

•  How	to	find	out	whether	there	in	random	walk	to	not	in	the	data?	
•  Run	AR(1)	model	&	check	for	the	value	of	β1	
•  Do	a	differenced	series	and	run	ACF	plot	

•  How	to	esHmate	drif	&	volaHlity?	


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Random Walk 

•  One-step-ahead	forecast:	Ft+1	=	a	+	Yt		

•  Two-step-ahead	forecast:	Ft+2	=	a	+	Yt+1=	2a	+	Yt		

•  k-step-ahead	forecast	:	Ft+k	=	ka	+	Yt		

•  If	the	drif	parameter	is	0,	then	the	k-step-ahead	forecast	is	Ft+k	=	Yt	for	all	k	


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Model based approaches & drawbacks 


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Model vs Data based approaches 

01	
Model	Based	Approach	

02	
Data	Based	Approach	

Past	is	SIMILAR	to	
Future	

Past	is	NOT	SIMILAR	
to	Future	


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Forecast methods based on smoothing 
	There	are	two	major	forecasHng	techniques	based	on	smoothing	

–  Moving	averages	

–  ExponenHal	smoothing	

•  Success	depends	on	
choosing	window	width	

•  Balance	between	over	&	
under	smoothing	


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Smoothing – Moving Average 
Smoothing	
Noise	

Data		
VisualizaHon	

Removing	Seasonality		
&	CompuHng	seasonal	
indexes	

ForecasHng	

•  Forecast	future	points	by	using	an	average	of	several	
past	points	

	
•  More	suitable	for	series	with	no	Trend	&	no	seasonality	

4	uses	

•  A	Hme-plot	of	the	MA	reveals	the	
Level	&	Trend	of	a	series	

	
•  It	 filters	 out	 the	 seasonal	 &	

random	components	


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Moving Average - Calculations 

1500

2250

3000

3750

4500

5250

6000

Q1
-86
Q3
-86
Q1
-87
Q3
-87
Q1
-88
Q3
-88
Q1
-89
Q3
-89
Q1
-90
Q3
-90
Q1
-91
Q3
-91
Q1
-92
Q3
-92
Q1
-93
Q3
-93
Q1
-94
Q3
-94
Q1
-95
Q3
-95
Q1
-96

Quarter

Sa
le
s

Centered	Moving	Average	
It	is	calculated	based	on	a	
window	centered	around	
Hme	‘t’	

Trailing	Moving	Average	
It	is	calculated	based	on	
a	window	from	Hme	‘t’	&	
backwards	


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Calculation – Trailing MA 
1.  Choose	window	width	(W)	

2.  For	MA	at	Hme	t,	place	window	on	Hme	points	t-W+1,…,t	

3.  Compute	average	of	values	in	the	window:	

W
yyyyMA ttWtWtt

++++
= −+−+− 121

!

t-4      t-3      t-2      t-1      t 
W=5 


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Calculation – Centered MA 

5
2112 ++−− ++++= tttttt

yyyyyMA
2/

4

4
211

112

⎟
⎟
⎟
⎟

⎠

⎞

⎜
⎜
⎜
⎜

⎝

⎛

+++

+
+++

=
++−

+−−

tttt

tttt

t yyyy

yyyy

MA

	
Compute	average	of	values	in	window	(of	width	W),	which	is	centered	at	t	
	

Odd	width:	center	window	on	Hme	t	and	average	the	values	in	the	window	
	
Even	width:	take	the	two	“almost	centered”	windows	and	average	the	values	in	

them	

t-2      t-1      t      t+1      t+2 
W=5 

W=4 W=4 


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Moving Average Hands On 


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Exponential Smoothing 

•  Assigns	more	
weight	to	most	
recent	observaHons		

•  Assigns	less	weight	
to	farthest	
observaHons	

Simple	Exponen4al	Smoothing	

•  No	Trend	
•  No	Seasonality	
•  Level	
•  Noise	(cannot	be	modeled)	

Holt’s	method	

•  Also	called	double	exponenHal	
•  Trend		
•  No	Seasonality	

Winter’s	method	

•  Trend	
•  Seasonality	
•  Variants	are	possible	
	


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Simple Exponential Smoothing 
Forecasts	=	es+mated	level	at	most	recent	Hme	point:	Ft+k	=	Lt	
	
AdapHve	algorithm:	adjusts	most	recent	forecast	(or	level)	based	
	
on	the	actual	data:	

	
	
	
	
		

α	=	the	smoothing	constant	(0<α≤ 1)	
	
IniHalizaHon:	F1	=		L1	=		Y1		

	

Lt =  αYt + (1-α) Lt-1 


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Simple Exponential Smoothing 
The formula: Lt =  αYt + (1-α) Lt-1  
 
Substitute Lt with its own formula: 

 

Lt    = αYt + (1-α)[ αYt-1 + (1-α) Lt-2] = 
 
       = αYt + α (1-α)Yt-1  + (1-α)2 Lt-2 = … 
 
       = αYt + α (1-α)Yt-1 + α (1-α)2 Yt-2 +… 


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Simple Exponential Smoothing 
The formula: Lt =  αYt + (1-α) Lt-1  

 

Yt+1 = Lt = Lt-1 + α (Yt - Lt-1 ) 
 

      = Yt   + α (Yt - Yt ) 
 

      = Yt + α Et 

update	previous	forecast		
By	an	amount	that	depends	on	the	

error	in	the	previous	forecast	

α	controls	the	degree	of	“learning”	

^ ^

^

^


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Smoothing Constant ‘α’ 
α determines how much weight is given to the past 

α =1: past observations have no influence over forecasts (under-
smoothing) 

α→0: past observations have large influence on forecasts (over-
smoothing) 

 
Selecting α

“Typical” values: 0.1, 0.2 
Trial & error: effect on visualization 
Minimize RMSE or MAPE of training data 

=	αYt	+	α	(1-α)Yt-1	+	α	(1-α)2	Yt-2	+…	


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Exponential Smoothing Hands On 


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MA vs ES 

MA	 ES	
•  Assigns	equal	weights	to	

all	past	observaHons	

•  BeKer	to	forecast	when	
data	&	environment	is	
not	volaHle	

•  Window	width	is	key	to	
success	

•  Assigns	more	weight	to	
recent	observaHons	than	
past	observaHons	

•  BeKer	to	forecast	when	
data	&	environment	is	
volaHle	

•  Smoothing	constant	(α)	
value	is	key	to	success	


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De-trending & De-seasoning 

	

•  Simple	&	popular	for	removing	trend	and	/	or	seasonality	from	a	Hme	series	
•  Lag-1	difference:	Yt	–	Yt-1	(For	removing	trend)	;	Lag-M	difference:	Yt	–	Yt-M	(For	

removing	seasonality)	

•  Double	–	differencing:	difference	the	differenced	series	
	

•  Uses	moving	average	to	remove	seasonality	
•  Generates	seasonal	indexes	as	a	byproduct	

•  To	remove	trend	and/or	seasonality,	fit	a	regression	model	with	
trend	and/or	seasonality	

•  Series	of	forecast	errors	should	be	de-trended	&	de-seasonalized		

1	
Regression	

Ra4o	to		
Moving		
average	

3	

2	
Differencing	


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Seasonal Indexes 

For	a	series	with	M	seasons:	
	
Sj	=		seasonal	index	for	the	jth	season	

	indicates	the	exceedance	of	Y	on	season	j	above/below	the	average	of	Y	in	a	
complete	cycle	of	seasons	

	
Make	sense	out	of	this	statement:	

	“Daily	sales	at	retail	store	shows	that	Friday	has	a	seasonal	index	of	1.30	
and	Monday	has	an	index	of	0.65”	
	
Let	us	put	in	easy	terms:	

	“Friday	sales	is	30%	higher	than	the	weekly	average,	and	Monday	sales	is	35%	lower	
than	the	weekly	average	sales”	
	
Average	of	the	M	seasonal	indexes	is	1	(they	must	sum	to	M)	


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Seasonal Indexes 
1.  Construct the series of centered moving averages of span M 

2.  For each t, compute the raw seasonals = Yt / Mat 
3.   Sj = average of raw seasonals belonging to season j (normalize to ensure 

that seasonal indexes have average=1) 

De-seasonalized (=seasonally-adjusted) series: 
 
•  If done appropriately, de-seasonalized series will not exhibit seasonality 
•  If so, examine for trend and fit a model 
•  This model will yield de-seasonalized forecasts 
•  Convert forecasts by re-seasonalizing, i.e. multiply them by the 

appropriate seasonal index 


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The seasonally-adjusted sales for Q1-86 are in 
the range 

1734.83 / 0.8785 = $1974.8 mil 
Quarter Sales Centered MA with W=4

Q1_86 1734.83
Q2_86 2244.96 raw seasonal s(j) seasonal index
Q3_86 2533.80 2143.76 1.18194269 1.063872992 1.062660535
Q4_86 2154.96 2102.82 1.02479749 0.96423378 0.963134878
Q1_87 1547.82 2020.32 0.76612585 0.879530967 0.878528598
Q2_87 2104.41 1934.99 1.08755859 1.096926116 1.09567599
Q3_87 2014.36 1954.74 1.03050222
Q4_87 1991.75 2021.05 0.9855033
Q1_88 1869.05 2061.44 0.90667088
Q2_88 2313.63 2080.07 1.11228431
Q3_88 2128.32 2089.65 1.01850452
Q4_88 2026.83 2097.04 0.96651998
Q1_89 1910.60 2109.01 0.90592533
Q2_89 2331.16 2137.18 1.09076712

1.  $1500-$1700 (million) 
2.  $1700-$1800 (million) 
3.  $1800-$1900 (million)  
4.  $1900-$2000 (million)   


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Advanced Exponential Smoothing 


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Holt’s / Double Exponential Method 
																			Forecasts	=	most	recent	es+mated	level	+	trend	
	

Yt+k	=	Lt	+	k	Tt	
^	

Lt	=		αYt	+	(1-α)(	Lt-1	+	Tt-1)	 	Tt	=	β (Lt -Lt-1)	+	(1-	β)	Tt-1	
•  Global	Trend	=	Linear	Regression	Model	
•  Local	Trend	=	ExponenHal	Model	

•  It	is	always	beKer	to	choose	default	‘α’ & ‘β’	values	(0.2,	0.15)	

•  What	happens	when	α = 0 ?

•  What	happens	when	β = 0 ?


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Winter’s Method 
																			Forecasts	=	most	recent	es+mated	level	+	trend	+	Seasonal	
	

Yt+k	=	(Lt	+	k	Tt)	*	St-k+M	
^	

•  St	=	seasonal	index	of	period	‘t’	
•  M	=	number	of	seasons	

	
	
	
						 	 		

	 	 	Level:	
	
					Trend	(same	as	Holt’s):	
	
					Seasonality	(mulHplicaHve):	

)TL )((1-  Y   L 1-t1-ttt ++=
−

αα
MtS

1-t1-ttt T )(1-  )LL (   T ββ +−=

-Mt
t

t S )(1-  
Y   S γγ +=
tL


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All 3 models – Generic  

IniHalizaHon	(technical):	
	

•  L1	=	Y1		or	L1=a	from	esHmated	model	Yt	=	a	+	bt	

•  T1	=	Y2-Y1	or	T1	=	(YT-Y1)	/	T		(avg	overall	trend)		

•  IniHal	seasonal	indexes	=	MA	indexes	(that	we	saw	earlier)	

All	three	smoothing	constants	(α,	β,	γ)	will	be	in	the	range:	0	to	1	
	
It	is	always	beKer	to	choose	default	‘α’, ‘β’, ‘γ’ values	(0.2,	0.15,	
0.05)	
	
	
	


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AR(1) model 

•  Yt = φ0 + φ1Yt – 1 + εt ,   εt white noise 

ACF	plot	 PACF	(parHal	ACF)	plot	


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AR(p) model 

•  Yt = φ0 + φ1Yt – 1 + φ2Yt – 2 + … + φpYt – p + εt ,   εt white noise 
•  Such a model has non-zero ACF at all lags 
•  However, only the first p PACFs are non-zero; the rest are zero 
•  If PACF plot shows large PACFs only at a few lags, then AR model 

is appropriate  
 
•  If an AR model is to be fitted, the parameters  φ0, φ1, φ2,…, φp have 

to be estimated from the data, under the restriction that the estimated 
values should guarantee a stationary process 


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MA(1) model 

•  Yt = θ0 + εt + θ1 εt – 1 ,   εt white noise 

ACF	plot	 PACF	(parHal	ACF)	plot	

θ1	=	0.8	

θ1	=	–	0.8	


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MA(q) model 

•  Yt = θ0 + εt + θ1 εt – 1 + θ2 εt – 2 + … + θq εt – q ,   εt 
white noise 

•  Such a model has non-zero PACF at all lags 
•  However, only the first q ACFs are non-zero; the rest are 

zero 
•  If ACF plot shows large ACFs only at a few lags, then MA 

model is appropriate  

•  If an MA model is to be fitted, the parameters  θ0, θ1, θ2,…, 
θq have to be estimated from the data 


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ARMA(p,q) model 

•  Yt =  φ0 + φ1Yt – 1 + φ2Yt – 2 + … + φpYt – p  
   + εt + θ1 εt – 1 + θ2 εt – 2 + … + θq εt – q ,  εt white noise 

 
•  Such a model has non-zero ACF and non-zero PACF at all lags 

•  If an ARMA(p,q) model is to be fitted, the parameters φ0, φ1, φ2,…, 
φp, θ1, θ2,…, θq have to be estimated from the data, under the 
restriction that the estimated values produce a stationary process 

•  AR(p) is ARMA(p,0) 
•  MA(q) is ARMA(0,q) 


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ARIMA(p,d,q) model 

•  If d-times differenced series is ARMA(p,q), then original series is 
said to be ARIMA(p,d,q). 

•  ARIMA stands for ‘Autoregressive Integrated Moving average’. 
•  If Wt  is the differenced version of Yt, i.e., Wt = Yt – Yt – 1,  

 then Yt can be written as   
  Yt = Wt + Wt – 1 + Wt – 2 + Wt – 3 + … . 
 Thus, the series Yt  is an ‘integrated’ (opposite of ‘differenced’) 
version of the series Wt. 

•  If Yt is ARIMA(p,d,q), it is non-stationary.  
•  However, its d-times differenced version, an ARMA(p,q) process, 

can be stationary. 


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Box-Jenkins ARIMA model-building 

•  Model identification 
–  If the time plot ‘looks’ non-stationary, difference it until the plot 

looks stationary 
–  Look at ACF and PACF plots for possible clue on model order 

(p, q) 
–  When in doubt (regarding choice of p and q), use the principle of 

parsimony: A simple model is better than a complex model 
•  Estimate model parameters 
•  Check residuals for health of model 
•  Iterate if necessary 
•  Forecast using the fitted model 


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THANK YOU