Time series

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Time series
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Characteristics

• Non-independent observations (correlations
  structure)
• Systematic variation within a year (seasonal
  effects)
• Long-term increasing or decreasing level (trend)
• Irregular variation of small magnitude (noise)

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Where can time series be found?

• Economic indicators Sales figures, employment
  statistics, stock market indices,
• Meteorological data precipitation, temperature,
• Environmental monitoring concentrations of
  nutrients and pollutants in air masses, rivers,
  marine basins,

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Time series analysis

• Purpose Estimate different parts of a time
  series in order to
• understand the historical pattern
• judge upon the current status
• make forecasts of the future development

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• Methodologies

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Time series regression?
Let yt(Observed) value of times series at time
point t and assume a year is divided into L
seasons Regession model (with linear
trend) yt?0 ?1t?j ?sj xj,t?t where xj,t1
if yt belongs to season j and 0 otherwise,
j1,,L-1 and ?t are assumed to have zero mean
and constant variance (?2 )
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• The parameters ?0, ?1, ?s1,, ?s,L-1 are
  estimated by the Ordinary Least Squares method
• (b0, b1, bs1, ,bs,L-1)argmin ?(yt (?0
  ?1t?j ?sj xj,t)2
• Advantages
• Simple and robust method
• Easily interpreted components
• Normal inference (conf.intervals, hypothesis
  testing) directly applicable
• Drawbacks
• Fixed components in model (mathematical trend
  function and constant seasonal components)
• No consideration to correlation between
  observations

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Example Sales figures
jan-98 20.33 jan-99 23.58 jan-00 26.09 jan-01 28.4
3 feb-98 20.96 feb-99 24.61 feb-00 26.66 feb-01 29
.92 mar-98 23.06 mar-99 27.28 mar-00 29.61 mar-01
33.44 apr-98 24.48 apr-99 27.69 apr-00 32.12 apr-0
1 34.56 maj-98 25.47 maj-99 29.99 maj-00 34.01 maj
-01 34.22 jun-98 28.81 jun-99 30.87 jun-00 32.98 j
un-01 38.91 jul-98 30.32 jul-99 32.09 jul-00 36.38
jul-01 41.31 aug-98 29.56 aug-99 34.53 aug-00 35.
90 aug-01 38.89 sep-98 30.01 sep-99 30.85 sep-00 3
6.42 sep-01 40.90 okt-98 26.78 okt-99 30.24 okt-00
34.04 okt-01 38.27 nov-98 23.75 nov-99 27.86 nov-
00 31.29 nov-01 32.02 dec-98 24.06 dec-99 24.67 de
c-00 28.50 dec-01 29.78
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Construct seasonal indicators x1, x2, ,
x12 January (1998-2001) x1 1, x2 0, x3 0,
, x12 0 February (1998-2001) x1 0, x2 1,
x3 0, , x12 0 etc. December (1998-2001) x1
0, x2 0, x3 0, , x12 1 Use 11
indicators, e.g. x1 x11 in the regression model
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Regression Analysis sales versus time, x1,
... The regression equation is sales 18.9
0.263 time 0.750 x1 1.42 x2 3.96 x3 5.07
x4 6.01 x5 7.72 x6 9.59 x7
9.02 x8 8.58 x9 6.11 x10 2.24
x11 Predictor Coef SE Coef T
P Constant 18.8583 0.6467
29.16 0.000 time 0.26314 0.01169
22.51 0.000 x1 0.7495
0.7791 0.96 0.343 x2 1.4164
0.7772 1.82 0.077 x3
3.9632 0.7756 5.11 0.000 x4
5.0651 0.7741 6.54 0.000 x5
6.0120 0.7728 7.78 0.000 x6
7.7188 0.7716 10.00
0.000 x7 9.5882 0.7706
12.44 0.000 x8 9.0201 0.7698
11.72 0.000 x9 8.5819
0.7692 11.16 0.000 x10 6.1063
0.7688 7.94 0.000 x11
2.2406 0.7685 2.92 0.006 S 1.087
R-Sq 96.6 R-Sq(adj) 95.5
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Analysis of Variance Source DF
SS MS F P Regression
12 1179.818 98.318 83.26
0.000 Residual Error 35 41.331
1.181 Total 47 1221.150 Source
DF Seq SS time 1 683.542 x1
1 79.515 x2 1
72.040 x3 1 16.541 x4
1 4.873 x5 1 0.204 x6
1 10.320 x7 1
63.284 x8 1 72.664 x9
1 100.570 x10 1 66.226 x11
1 10.039
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Unusual Observations Obs time sales
Fit SE Fit Residual St Resid 12
12.0 24.060 22.016 0.583
2.044 2.23R 21 21.0 30.850
32.966 0.548 -2.116 -2.25R R
denotes an observation with a large standardized
residual Predicted Values for New
Observations New Obs Fit SE Fit
95.0 CI 95.0 PI 1 32.502
0.647 ( 31.189, 33.815) ( 29.934, 35.069)
Values of Predictors for New
Observations New Obs time x1
x2 x3 x4 x5 x6 1
49.0 1.00 0.000000 0.000000
0.000000 0.000000 0.000000 New Obs x7
x8 x9 x10 x11 1
0.000000 0.000000 0.000000 0.000000 0.000000
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What about serial correlation in data?
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Positive serial correlation Values follow a
smooth pattern Negative serial
correlation Values show a thorny
pattern How to obtain it? Use the residuals.
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Residual plot from the regression analysis
Smooth or thorny?
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Durbin Watson test on residuals Thumb rule
If d lt 1 or d gt 3, the conclusion is that
residuals (and orginal data( are correlated. Use
shape of figure (smooth or thorny) to decide if
positive or negative) (More thorough rules for
comparisons and decisions about positive or
negative correlations exist.)
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Durbin-Watson statistic 2.05 (Comes in the
output ) Value gt 1 and lt 3 ? No significant
serial correlation in residuals!
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Classical decomposition methods

• Decompose Analyse the observed time series in
  its different components
• Trend part (TR)
• Seasonal part (SN)
• Cyclical part (CL)
• Irregular part (IR)
• Cyclical part State-of-market in economic time
  series
• In environmental series, usually together with
  TR

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• Multiplicative model
• ytTRtSNt CLt IRt
• Suitable for economic indicators
• Level is present in TRt or in TCt(TRCL)t
• SNt , IRt (and CLt) works as indices
• ?Seasonal variation increases with level of yt

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• Additive model
• ytTRtSNt CLt IRt
• More suitable for environmental data
• Requires constant seasonal variation
• SNt , IRt (and CLt) vary around 0

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Example 1 Sales data
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Example 2
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Estimation of components, working scheme

• Seasonally adjustment/Deseasonalisation
• SNt usually has the largest amount of variation
  among the components.
• The time series is deseasonalised by calculating
  centred and weighted Moving Averages
• where LNumber of seasons within a year
  (L2 for ½-year data, 4 for quaerterly data och
  12 för monthly data)

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• Mt becomes a rough estimate of (TRCL)t .
• Rough seasonal components are obtained by
• yt/Mt in a multiplicative model
• yt Mt in an additive model
• Mean values of the rough seasonal components are
  calculated for eacj season separetly. ? L means.
• The L means are adjusted to
• have an exact average of 1 (i.e. their sum equals
  L ) in a multiplicative model.
• Have an exact average of 0 (i.e. their sum equals
  zero) in an additive model.
• ?Final estimates of the seasonal components are
  set to these adjusted means and are denoted

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• The time series is now deaseasonalised by
• in a multiplicative
  model
• in an additive model
• where is one of
• depending on which of the seasons t represents.

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• 2. Seasonally adjusted values are used to
  estimate the trend component and occasionally the
  cyclical component.
• If no cyclical component is present
• Apply simple linear regression on the seasonally
  adjusted values ?Estimates trt of linear or
  quadratic trend component.
• The residuals from the regression fit constitutes
  estimates, irt of the irregular component
• If cyclical component is present
• Estimate trend and cyclical component as a whole
  (do not split them) by
• i.e. A non-weighted centred Moving
  Average with length 2m1 caclulated over the
  seasonally adjusted values

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• Common values for 2m1 3, 5, 7, 9, 11, 13
• Choice of m is based on properties of the final
  estimate of IRt which is calculated as
• in a
  multiplicative model
• in an
  additive model
• m is chosen so to minimise the serial correlation
  and the variance of irt .
• 2m1 is called (number of) points of the Moving
  Average.

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Example, cont Home sales data Minitab can be
used for decomposition by Stat?Time
series?Decomposition
Val av modelltyp
Option to choose between two models
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Time Series Decomposition Data
Sold Length 47,0000 NMissing 0 Trend
Line Equation Yt 5,77613 4,30E-02t
Seasonal Indices Period Index 1
-4,09028 2 -4,13194 3 0,909722
4 -1,09028 5 3,70139 6
0,618056 7 4,70139 8 4,70139
9 -1,96528 10 0,118056 11
-1,29861 12 -2,17361
Accuracy of Model MAPE 16,4122 MAD
0,9025 MSD 1,6902
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Deseasonalised data have been stored in a column
with head DESE1. Moving Averages on these column
can be calculated by Stat?Time series?Moving
average
Choice of 2m1
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MSD should be kept as small as possible
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By saving residuals from the moving averages we
can calculate MSD and serial correlations for
each choice of 2m1.
A 7-points or 9-points moving average seems most
reasonable.
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Serial correlations are simply calculated
by Stat?Time series?Lag
and further Stat?Basic statistics?Correlation
Or manually in Session window MTB gt lag RESI4
c50 MTB gt corr RESI4 c50
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Analysis with multiplicative model
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Time Series Decomposition Data
Sold Length 47,0000 NMissing 0 Trend
Line Equation Yt 5,77613 4,30E-02t
Seasonal Indices Period Index 1
0,425997 2 0,425278 3 1,14238
4 0,856404 5 1,52471 6
1,10138 7 1,65646 8 1,65053
9 0,670985 10 1,02048 11
0,825072 12 0,700325
Accuracy of Model MAPE 16,8643 MAD
0,9057 MSD 1,6388
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additive
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additive
additive
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Classical decomposition, summary
Multiplicative model
Additive model
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Deseasonalisation

• Estimate trendcyclical component by a centred
  moving average

where L is the number of seasons (e.g. 12, 4, 2)
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• Filter out seasonal and error (irregular)
  components
• Multiplicative model

-- Additive model
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Calculate monthly averages
Multiplicative model
Additive model
for seasons m1,,L
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Normalise the monhtly means
Multiplicative model
Additive model
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Deseasonalise
Multiplicative model
Additive model
where snt snm for current month m
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Fit trend function, detrend (deaseasonalised) data
Multiplicative model
Additive model
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Estimate cyclical component and separate from
error component
Multiplicative model
Additive model