Deck for Sweetness

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Drawbacks of classical decomposition

Over-smooths data; loses data points; assumes seasonality repeats; not robust to unusual values

What is the purpose of simple forecasting methods?

An easy and straightforward way to assess the underlying data so that we can compare it to more complex models

List the 4 simple forecasting methods (i.e. the benchmarks)

1. The average method; 2. The naïve method; 3. The seasonal naïve method; 4. The drift method

What is the test data in terms of model forecasting?

The remainder of the split of the data set (i.e. data the model isn't exposed to)

Explain the average method

The forecasts of all future values are the mean of the historical data

When is the average method not ideal?

When there is seasonality and trend

Explain the naïve method

All future values are equal to the last observation in the time series

When is the naïve method preferred?

It performs well for many financial and economic time series

Explain the seasonal naïve method

All future values follow the last observed seasonal pattern

When is the seasonal naïve method best?

When there is strong seasonality

What is the best long-term simple forecasting method?

The drift method

How does the drift method work?

It is a continuation of the overall trend

How do you get the MSE in R?

RMSE²

In what order do you make transformations?

Check for constant variance first, then check for constant mean

What variance stabilising transformation do you use if lambda = 0?

log(Yt)

What variance stabilising transformation do you use if lambda ≠ 0?

(Yt^λ - 1) / λ

How do you get lambda for the variance stabilising transformation?

BoxCox.lambda(ts)

For what value of lambda is no variance stabilising transformation required?

Lambda = 1

Transformation when lambda ≈ 0.25

⁴√Yt (fourth root)

Transformation when lambda ≈ 0.33

³√Yt (cube root)

Transformation when lambda ≈ 0.5

√Yt (square root)

What transformation do you apply when constant mean is violated?

Detrend / deseasonalise via differencing

Detrend formula

Wt = Yt - Yt-1 (t = 2, 3, …, T)

What does it mean if ndiffs(ts_lam) = 1?

It requires first-order differencing at lag 1

R code for detrending

diff(ts_lam, 1), where ts_lam is the variance stabilised series

Formula to deseasonalise

Wt = Yt - Yt-g (t = g+1, g+2, …, T). g is the seasonal period (monthly data in a year: 12; daily data in a week: 7)

R code to deseasonalise

diff(ts_lam, 12)

What two R commands do you run to check if your transformations had an effect?

1. ur.kpss(ts_lam_dif); 2. checkresiduals(ts_lam_dif). We don't want to see autocorrelation

How do you know if deseasonalising is mandatory?

nsdiffs(ts_lam) = 1

What is SES used for?

Data with no trend or seasonality

What is Holt's method used for?

Trend, but no seasonality

What is Holt-Winters used for?

Trend + seasonality

Why do we use exponential smoothing?

It generates reliable forecasts quickly and for a wide range of time series, which is a great advantage and of major importance to applications in industry

Why is simple exponential smoothing similar to the naïve forecasting method?

All future forecasted values are set to the value of the last level component

What time series is SES used for?

Forecasting time series that do NOT display a trend or seasonal component

What do you use the AIC for in exponential smoothing?

To determine which model fits the data best

How do you get sigma values (to calculate CIs) for exponential smoothing methods in R?

modelName[["model"]]

What does a small beta mean in Holt-Winters?

The trend (slope) estimate is updated slowly, so it relies more on past trend values and changes little over time

What is cross-sectional data?

Data observed or measured at one point in time, e.g. final STA2020 marks

What is the definition of time series data?

A sequence of observations collected at regular, equally spaced intervals over a period of time (e.g. hourly, daily, weekly, monthly, quarterly or yearly)

Why do statistical techniques like regression do poorly on time series data?

They assume independence of observations, and since our data is collected at regular, equally spaced time intervals, the observations are likely to be dependent

What is autocorrelation?

The dependence that arises between observations g time periods apart

What is the main underlying assumption of time series forecasting?

Past patterns will continue into the future

What is the purpose of time series analysis?

To identify and isolate historic influencing factors in order to better understand the process underlying the time series, for predictive purposes

What is stationarity?

A time series is stationary if its statistical properties (mean, autocorrelation and variance) are constant over time, i.e. they are independent of time

Why should we make time series data stationary before conducting any statistical techniques?

So that we can better predict future values without the noise and complexity of handling changing statistical properties

What are the 4 components of time series data?

1. Trend; 2. Cyclical variation; 3. Seasonal variation; 4. Random variation

What are the assumptions of the additive time series model?

Components are additive and independent of each other

What are the assumptions of the multiplicative time series model?

The four components of a time series are not necessarily independent and they can affect one another

What is the purpose of moving average smoothing?

It gives values for the components of the time series by dampening the error terms to provide a trend component (i.e. removing random variation)

When plotting MA(K) on a chart, which period is the computed value associated with?

The middle period (for odd K); middle period - 1 (for even K), then compute the CMA

When plotting CMA(K) on a chart, which period is the computed value associated with?

Middle period + 1

What is an advantage of moving average smoothing?

It is quick and easy to apply

What are the disadvantages of moving average smoothing?

1. You lose observations; 2. Once an observation falls out of the window its value is never considered again, so it cannot be used to predict future values like exponential smoothing

What is the point of classical decomposition?

To get numeric values for the trend component (via regression analysis), the seasonal component (via SIs) and the random component

Steps of classical decomposition

1. Estimate the trend by calculating the MA/CMA. 2. Detrend each observation with its CMA (Yt / CMA(K)t). 3.1 Calculate seasonal indices of the detrended data (group each quarter, month or day and average each one). 3.2 Adjust the SIs by the CF if the sum of SIs ≠ m. 4. Compute the random component by dividing the ORIGINAL Yt by the product of the trend and seasonal estimates. 5. Deseasonalise the ORIGINAL Yt by dividing it by its adjusted SI. 6. If the deseasonalised data looks linear, regress it to get the REAL trend (not an estimate)

What should the m seasonal indices sum to?

The number of seasons in the time series (m), e.g. quarterly m = 4, monthly m = 12

What is the formula for the correction factor of seasonal indices?

CF = m / (sum of the m seasonal indices)

What would St be if there was no seasonal component in a time series?

1 for every season

How would you interpret an adjusted seasonal index of 0.839 for Q1 (each seasonal cycle is one year)?

On average, earnings in the first quarter are 100 × |0.839 - 1| = 16.1% below the annual average earnings

Why do we perform transformations on time series?

To make the data a stationary series, isolating the true internal signal (autocorrelation) from external noise like trend and changing variance, so ARIMA can model stable relationships for more accurate predictions

In what order do we perform transformations?

1. First check for constant variance; 2. Then check for constant mean

How do you decide which variance stabilising transformation to use?

Use lambda

What is the R code for obtaining a lambda value?

BoxCox.lambda(data)

Which variance stabilising transformation do you use if lambda ≈ 0.25?

⁴√Yt

Which variance stabilising transformation do you use if lambda ≈ 0.33?

³√Yt

Which variance stabilising transformation do you use if lambda ≈ 0.5?

√Yt

How do you apply lambda to a time series in R?

BoxCox(data, lambda)

How is first-order differencing at lag 1 performed?

Subtract the previous value from the current value

How do you know what lag value to use for differencing?

Run both ndiffs() and nsdiffs(). If nsdiffs() ≠ 0, use the seasonal lag; if nsdiffs() = 0, use lag = 1

TRUE OR FALSE: Differencing can remove both trend and seasonality

True

What is the null hypothesis for the autocorrelation function?

There is no autocorrelation between Yt and Yt-g

What is the telltale sign that a time series has been over-differenced?

The first lag autocorrelation is more negative than -0.5

What is the null hypothesis of the KPSS test?

The time series is stationary

What are the ideal properties of the model residuals?

1. No autocorrelation (otherwise patterns remain in the data); 2. Mean of zero; 3. Constant variance; 4. Normally distributed

What are the null and alternative hypotheses for the Ljung-Box test?

H0: There is no autocorrelation in any of the first m lags. H1: There is autocorrelation in at least one of the first m lags

What does it mean if the ACF plot of residuals has no values outside the confidence interval?

The model has captured all the relevant information/patterns in the data

If it is important to avoid large errors, which forecast accuracy measure would you use and why?

MSE, because squaring penalises large deviations more heavily than MAE

What does the confidence interval for predictions represent?

An interval within which we expect forecasted values to lie with a specified probability

Why are confidence intervals so powerful?

They express the uncertainty in the forecasts. With only point forecasts, there is no way of telling how accurate the forecasts are

Define cyclicality

Long-term, irregular wavelike movements (peaks and troughs), usually lasting longer than a year

Define seasonality

Regular, short-term, repetitive wavelike movements, usually repeating during the same calendar periods