Forecasting Transformations

Motivating Transformations and Adjustments

• Adjusting any historical data can often help make forcasting easier and more accurate

• In general, there are four main adjustments:

  • Calendar adjustments
  • Population adjustments
  • Inflation adjustments
  • Mathematical transformations
• Making patterns simpler (by performing transformations) usually lead to more accurate forecasts

Defining Calendar Adjustments

• Calendar adjustments include transforming data to account for seasonality trends
• Forecasts become more accurate by removing any seasonality variation

• For example, monthly flower sales will vary between months for two different reasons:

  • Each month has a different number of days
  • Seasonal variation across the year
• In this example, we could adjust monthly flower sales using the following formula:

$$\text{monthly sales}^{*} = \frac{\text{monthly sales}}{\text{number of days in month}}$$

Defining Population Adjustments

• Some data is impacted by yearly population changes
• Thus, population data should be normalized using per-capita data
• For example, yearly housing sales will always vary, since the population increases every year
• As a result, forecasts would be more accurate by removing the yearly effects of population changes
• Then, we can determine whether there have been real changes in housing sales relative to the population
• Thus, we can normalize sales using the following formula:

$$\text{yearly sales}^{*} = \frac{\text{yearly sales}}{\text{population for that year}}$$

Defining Inflation Adjustments

• Some data is impacted by the fluctuations in the value of money
• Thus, financial data should be inflation-adjusted

• For example, yearly housing prices will fluctuate slightly each decade due to inflation

  • A $200,000 house in 2000 will be worth$300,000 in 2020 after adjusting for inflation

Defining Mathematical Transformations

• Some data can be transformed to account for non-linear trends

• For example, some data contains exponential growth

  • Normalizing this data by taking the log (of this data) will lead to more accurate and interpretable forecasts

• Also, a Box-Cox is a common transformation

  • A Box-Cox transformation attempts to adjust a non-normal variable into a normal shape
• Essentially, we cycle through different sets of power exponents $\lambda$ in the Box-Cox formula, until we eventually find a $\lambda$ that best transforms our non-normal data into a normal shape
• Note, looping through different power exponenets $\lambda$ implies we're looping through different transformations, such as:

$\lambda$	Box-Cox Transformation
$-3$	$Y^{-3} = \frac{1}{Y^{3}}$
$-2$	$Y^{-3} = \frac{1}{Y^{2}}$
$-1$	$Y^{-1} = \frac{1}{Y}$
$-0.5$	$Y^{-0.5} = \frac{1}{\sqrt{Y}}$
$0$	$\log (Y)$
$0.5$	$Y^{0.5} = \sqrt{Y}$
$1$	$Y$
$2$	$Y^{2}$
$3$	$Y^{3}$

References

• Textbook for Forecasting: Principles and Practices