Dealing with Seasonality

1
Dealing with Seasonality

• Different seasons of the year may be a major
  source of variation in the Y variable.
• As with other exogenous variable, seasonal
  variation must be compensated for or removed in
  order to better discern the trend in Y over time.
• May also be interested in modelling seasonality
  to allow predictions of Y for different seasons.

2
Techniques for Dealing with Seasonality
3
Nonparametric method Seasonal Kendall Test
(Method 1)

• Accounts for seasonality by computing
  Mann-Kendall test on each of m seasons
  separately, then combining the results.
• For monthly seasons, January data are compared
  only with January, February only with February,
  etc.

4

• If product of number of years and number of
  seasons gt 25, normal distribution can be used.
• If Zsk gt Zcrit then reject null hypothesis of
  no trend.
• Zcrit 1.96 for ?0.05.

If Sk gt 0 If Sk 0 If Sk lt 0
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Estimate of trend slope

• Trend slope of Y over time T median of all
  slopes between data pairs within the same season.
• No cross season slopes contribute to the overall
  estimate of the Seasonal Kendall trend slope.
• Exogenous Variable
• Use LOWESS of Y on X to get R, then apply
  Seasonal Kendall on R, T.

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Mixture Methods Method 2a

• Apply seasonal Kendall test to R from a
  regression of Y on X. Must check for violation
  or regression assumptions.
• Method 2b
• Deseasonalize data by subtracting seasonal
  medians from all data within the season, and then
  regressing deseasonalized data against T. Less
  power to detect trend.

7
Parametric Method (Method 3)

• Multiple regression with periodic functions to
  describe seasonality.
• Other terms exogenous variables or dummy
  variables.
• If ?3 is significant, then there is trend.
• The term 2?T 6.2832.t When t is in years.
• 0.5236.m When m is in months
• 0.0172.d When d is in days.

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Comparison of methods

• Mann-Kendall and mixed approaches applicable to
  univariate data. Cannot be used for multiple Xs.
  Good for nonnormal data.
• Multiple regression does it all in one swoop.
  Fewer parameters but constrained by functional
  form (sine and cosine). Need close checking of
  regression assumptions. Can provide seasonal
  summary statistics.

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Presenting Seasonal Effects
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Introduction to Time Series Analysis

• When the Y or R values are dependent in time
  (auto or serial correlation).
• Two purposed a) Modelling and Simulation
• b) Forecasting
• Modelling and Simulation ARIMA, Fourier
  ARMA,
• Dynamic Regression
• Forecasting ARIMA, Exponential Smoothing,
  Dynamic Regression
• (Need a separate course to cover this topic)

E.g.
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Step TrendsStep Trends without Seasonality
12
Step Trends with Seasonality
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Summary

• First decide the type of trend to be analyzed
• step vs monotonic
• check assumptions
• nonparametric vs parametric
• Are there exogenous variables?
• Remove them first or model in one go
• Seasonality?
• Always plot the data - Boxplots, X-Y plots are
  most useful.