STAT603: Time Series Decomposition, Transformations, and Adjustments 2A

What is the purpose of transformations and adjustments in forecasting?

To simplify patterns in historical data by removing known sources of variation or making the pattern more consistent.

What are the four types of adjustments used in forecasting?

Calendar adjustments, population adjustments, inflation adjustments, and mathematical transformations.

How can calendar adjustments improve forecasting accuracy?

By removing variations due to different numbers of trading days in each month, allowing for a clearer analysis of sales patterns.

What is an example of a calendar adjustment?

Calculating average sales per trading day instead of total monthly sales to eliminate calendar variation.

What is the benefit of population adjustments in data analysis?

They provide per-capita data, making it easier to interpret changes relative to population size.

How can population adjustments affect the interpretation of hospital bed data?

By showing the number of beds per thousand people, it clarifies whether increases in total beds are due to population growth.

What is the significance of inflation adjustments in financial time series?

They ensure that all values are stated in constant dollar values from a specific year, allowing for accurate comparisons over time.

How is the adjusted house price calculated using a price index?

Using the formula 𝑥𝑡 = 𝑦𝑡/𝑧𝑡 × 𝑧2000, where 𝑦𝑡 is the original price and 𝑧𝑡 is the price index.

What are some common mathematical transformations used in time series data?

Square root, cube root, and logarithm transformations.

Why are logarithmic transformations particularly useful?

They provide relative (percent) changes on the original scale, making interpretations easier.

What are Box-Cox transformations?

A family of transformations that includes logarithms and power transformations, defined by a parameter 𝜆.

What does a Box-Cox transformation look like when 𝜆 equals 0?

It becomes a logarithmic transformation: 𝑤𝑡 = log(𝑦𝑡).

What does a Box-Cox transformation look like when 𝜆 equals 1?

It results in no substantive transformation: 𝑤𝑡 = 𝑦𝑡.

What happens if 𝜆 is close to zero in a Box-Cox transformation?

The transformation behaves like a logarithm.

What is a potential issue when using Box-Cox transformations if some data values are zero?

A power transformation is not possible unless a constant is added to all values.

What should be checked when applying Box-Cox transformations?

The results should always be checked, as low values of 𝜆 can lead to large prediction intervals.

What is the relationship between population adjustments and GDP per capita?

GDP per capita provides a clearer picture of economic performance by adjusting for population size.

How can mathematical transformations stabilize variation in time series data?

By addressing different levels of variation across the series, making the data more consistent.

What is the effect of using power transformations like square roots and cube roots?

They can stabilize variance but may be less interpretable than logarithmic transformations.

What condition must be met for power transformation when some 𝑦𝑡 = 0?

𝜆 must be greater than 0.

What adjustment is necessary if some 𝑦𝑡 < 0?

All 𝑦𝑡 must be adjusted by adding a constant.

Why are simple values of 𝜆 preferred in transformations?

They are easier to explain.

How do results react to changes in 𝜆?

Results are relatively insensitive to 𝜆.

What is often the case regarding the need for transformation in time series?

Often no transformation (𝜆 = 1) is needed.

What effect can transformation have on prediction intervals?

Transformation can have a very large effect on prediction intervals.

What is a simple way to ensure forecasts are positive?

Choosing 𝜆 = 0.

What defines a trend pattern in time series data?

A long-term increase or decrease in the data.

What characterizes a cyclic pattern in time series data?

Rises and falls that are not of fixed period, usually lasting at least 2 years.

What is a seasonal pattern in time series data?

A series influenced by seasonal factors such as the quarter of the year or day of the week.

What does the equation 𝑦𝑡 = 𝑓(𝑆𝑡, 𝑇𝑡, 𝑅𝑡) represent?

It represents time series decomposition where 𝑦𝑡 is the data at period 𝑡.

What is the additive decomposition formula?

𝑦𝑡 = 𝑆𝑡 + 𝑇𝑡 + 𝑅𝑡.

What is the multiplicative decomposition formula?

𝑦𝑡 = 𝑆𝑡 × 𝑇𝑡 × 𝑅𝑡.

When is an additive model appropriate in time series decomposition?

When the magnitude of seasonal fluctuations does not vary with the level.

When is a multiplicative model appropriate in time series decomposition?

When seasonal fluctuations are proportional to the level of the series.

What is a Box-Cox transformation used for?

To transform data before applying additive decomposition.

What does taking the logarithm of a multiplicative relationship achieve?

It turns the multiplicative relationship into an additive relationship.

What is the purpose of seasonal adjustment in time series data?

To calculate seasonally adjusted data using past values.

What is the formula for seasonally adjusted data in additive decomposition?

𝑦𝑡 − 𝑆𝑡 = 𝑇𝑡 + 𝑅𝑡.

What is the formula for seasonally adjusted data in multiplicative decomposition?

𝑦𝑡 / 𝑆𝑡 = 𝑇𝑡 × 𝑅𝑡.

What should be used to look for turning points in time series data?

The trend-cycle component.