data science training

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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:

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

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

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

Who generates Forecast?

Time series vs Cross-sectional data
01 Cross-sec4onal Data
02
Time Series Data

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

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

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

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

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

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

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

Irregular/Random/Noise Component
•  ErraHc, unsystemaHc, ‘residual’ ﬂuctuaHons
•  Due to random variaHon or unforeseen events
–  Union strike
–  War
•  Short duraHon & nonrepeaHng

Time Series Components

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)

Time plot
400
500
600
700
800
900
1000
1100
1980 1982 1984 1986 1988 1990
Power usage (KilowaHs)
Year
Electrical power usage for Washington Water Power: 1980-1991

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

Time plot

Quarterly Sales of Ice-cream

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)

Scatter Plot
0
200
400
600
800
1000
1200
0 2 4 6 8 10 12 14
Yearly cost of maintenance (US $)
Age of bus

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

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

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

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

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

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

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.

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

Correlogram for VCR data

ACF plot for electricity usage data

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

ACF of de-seasoned KW data

Observations
•  De-seasoned series has small ACFs
•  This part of the data has liKle forecasHng value

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?

Time series plots

Effect of omission of data on the
Time series plot

Confusing kind of trend due to other type
of scaling
020406080
y
0 5 10 15 20
t
020406080
y
0 1 2 3
Log t
2.533.544.5
Logy
0 5 10 15 20
t
2.533.544.5
Logy
0 1 2 3
Log t

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

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

Aspect ratio

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

Partitioning
Deploy model by joining Training + ValidaHon to forecast the Future

How to choose a Validation Period?
Forecast
Horizon
Seasonality
Length of
series
Underlying
condi4ons
aﬀec4ng series
Strategy to choose
Valida4on Data
Period

Rolling-forward forecasts

NAÏVE Forecasts
Forecast method: Last sample

k-step ahead

Seasonal series ( M series )
Ft+k = Yt
Ft+k = Yt-M+k

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)

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

Forecasting Errors

Evaluating Predictive Accuracy
•  Mean error
•  Mean absolute deviation
•  Mean squared error
•  Root mean squared error
•  Mean percentage error
•  Mean absolute percentage error

Typical plots of ‘White noise’
Time plot
Lag plot
ACF plot
Histogram

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

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:

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

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

MPE and MAPE---Unit free measure
•  Both expressed in percentage form
•  Both are unit free

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

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

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

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

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]

Forecasting Different Methods
Model
based
Data
driven
•  Linear regression
•  Autoregressive models
•  ARIMA
•  LogisHc regression
•  Econometric models

•  Naïve forecasts
•  Smoothing
•  Neural nets

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

Irregular Component
• Outliers
• Special Events
• Interventions
• Remove unusual periods
from the model
• Model separately
• Keep in the model, using
dummy variable
Irregular Components Solutions

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

Linear Regression for forecasting
Global Trend

•  Linear Trend
(constant growth)
•  ExponenHal Trend
(% growth)
1
Seasonality

•  AddiHve (Y)
•  MulHplicaHve
log(Y)
Irregular PaHerns
2 3

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)

Autocorrelation & its uses
Check forecast errors for
independence
Model remaining
information
Evaluate
predictability

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

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
^
^

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?

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

Model based approaches & drawbacks

Model vs Data based approaches
01
Model Based Approach
02
Data Based Approach
Past is SIMILAR to
Future
Past is NOT SIMILAR
to Future

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

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

Moving Average - Calculations
1500
2250
3000
3750
4500
5250
6000
Q
1-86Q
3-86
Q
1-87Q
3-87
Q
1-88Q
3-88Q
1-89
Q
3-89Q
1-90
Q
3-90Q
1-91Q
3-91
Q
1-92Q
3-92Q
1-93
Q
3-93Q
1-94
Q
3-94Q
1-95Q
3-95
Q
1-96
Quarter
Sales
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

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
yyyy
MA ttWtWt
t
++++
= −+−+− 121 !
t-4 t-3 t-2 t-1 t
W=5

Calculation – Centered MA
5
2112 ++−− ++++
= ttttt
t
yyyyy
MA
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

Moving Average Hands On

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

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

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 +…

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”
^ ^
^
^

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 +…

Exponential Smoothing Hands On

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

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

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)

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

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

Advanced Exponential Smoothing

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 ?

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-t
t
t ++=
−
αα
MtS
1-t1-ttt T)(1-)LL(T ββ +−=
-Mt
t
t S)(1-
Y
S γγ +=
tL

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)

AR(1) model
•  Yt = φ0 + φ1Yt – 1 + εt , εt white noise
ACF plot PACF (parHal ACF) plot

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

MA(1) model
•  Yt = θ0 + εt + θ1 εt – 1 , εt white noise
ACF plot PACF (parHal ACF) plot
θ1 = 0.8
θ1 = – 0.8

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

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)

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.

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

THANK YOU

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