Analysis of Sequential Data

As long as the to be predicted number is far away from 0 (like hundreds of thousends of better millions)

In a ts object:

• A list of numbers
• Information about times those numbers were recorded

A frequency argument (e.g. daily - 365, monthly - 12, quarterly - 4)

library(fpp2)

Loads:

• some data for use in examples and exercises
• forecast package (for forecasting functions)
• ggplot2 package (for graphics functions)
• fma package (for lots of  time series data)
• expsmooth package (for more time series data)

With seasonal plots:

• Data plotted against the individual "seasons" in which the data were observed. (in this case a "season" is a month.)
• Something like a time plot except that the data from each season are overlapped
• Enables the underlying sesonal pattern to be seen more clearly, and also allow any substantial departures form the seasonal pattern to be easily identified.
• In R: ggseasonplot()

• Trend
  • Pattern exists when there is a long-term increase or decrease in the data
• Seasonal
  • Pattern exists when a series in influenced by seasonal factors (e.g., the quarter of the year, the month, or day of the week)
• Cyclic
  • Pattern exists when data exhibit rises and falls that are not of fixed period (duration usually of at least 2 years)

• seasonal pattern constant length; cyclic pattern variable length
• Average length of cycle longer than length of seasonal pattern
• magnitude of cycle more variable than magnitude of seasonal pattern

The timing of peaks and troughs is predictable with seasonal data, but unpredictable in the long term with cyclic data.

Coraciance and correlation: measure extent of linear relationship between two variables (y and X).

Autocovariance and autocorrelation: measure linear relationship between lagged values of a time series y.

We measure the relationship between:

• yt and yt-1
• yt and yt-2
• yt and yt-3
• etc.

• r4 higher than for the other lags. This is due to the seasonal pattern in the data: the peaks tend to be 4 quarters apart and the troughs tend to be 2 quarters apart
• r2 is more negative than for the other lags because troughs tend to be 2 quarters behind peaks
• Together, the autocorrelations at lags 1, 2, ..., make up the autocorrelation or ACF
• The plot is known as a correlogram

• When data have a trend, the autocorrelations for small lags tend to be large and positive.
• When data sare seasonal , the autocorrelations willl be larger at the seasonal lags (e.g. at multiples of the seasonal frequency)
• When data are trended and seasonal, you see a combination of these effects

• Time plot shows clear trend and seasonality
• The same features are reflected in the ACF
• The slowly decaying ACF indicates trend
• The ACF peaks at lags 12, 24, 36, ... indicate seasonality of length 12

1. B; Trend and no seasonality
2. A; Seasonilty and no trend
3. D; Seasonality and a trend
4. C; Cyclic and no trend

95% interval

• We expext each autocorrelation to be close to zero
• All autocorrelation coefficients lie witihin the 95% interval, confirming that the data is white noise (more precisely, the data cannot be distinguished from white noise)

• The stock can be modelled by the radnom walk model yt+1 = yt + epsilont
• where epsilont = N(0, sigma2), epsilont is i.i.d.: hence epsilont is independen from epsilont-1, epsilont-2
• By differencing: yt+1 - yt = epsilont
  • which is indeed a white noise time series

• Average method
• Naive method
• Seasonal naive method

Residuals in Forecasting: Difference between observed value and its fittted value.

• Assumptions
  • Residuals are uncorrelated. If they aren't, then information left in residuals that should be used in computing forecasts.
  • Residuals have mean zero. If they don't, then forecasts are biased.
• Useful properties (for prediction intervals)
  • Residuals have constant variance
  • Residuals are normally distributed

Note: et are one-step-forecast residuals

• We assume that the residuals are white noise (uncorrelated, mean zero, constant variance). If they aren't, then there is information left in the residuals that should be used in computing forecasts.
• So a standard residual diagnostic is to check the ACF of the residuals of a forecasting method. Where the data should lay between the blue 95% boundary.
• We expect these to look like white noise

• Consider a whole set of rk values, and develop a test to see whether the set is significantly different from a zero set.
• If each rk is close to zero, Q will be small
• If some rk values are large (positive or negative), Q will be large
• Note mla: i assume rk are residuals

• h = 10 for non-seasonal data
• h = 2m for seasonal data, where m is the length of the season

Note: Gets automatically done with the checkresiduals() function

• The test checks the null hypothesis that the data is white noise
• Small p-values lead to rejecting the null hypothesis: They are evidence of significant auto-correlation
• Large p-values lead insted to axxepting the null hypothesis
• Typical threshold decision:
  • p-value > 0.05 -> accept he null hypothesis (white noise)
  • p-value < 0.05 -> reject the null hypothesis, concluding that there is a significant autocorrelation

• A model which firs the training data well will not necessarily forecast well
• A perfect fit can always be obtained by using a model with enough parameters
• Over-fitting a model to data is just as bad as failing to identify a systemtic pattern in the data
• The test set must not be used for any aspect of model decelopment or calculation of forecasts
• Forecast accuracy is based only on the test set

Additionally:

• MAE, MSE, RMSE are all scale dependent
• MAPE is scale independent but is only sensible if yt >> 0 for all t, and y has a natural zero

• Forecast accuracy averaged over test sets (time step for time step)
• Also known as "evaluation on a rolling forecasting origin"
• A good way to choose the best forecasting model is to find the model with the smallest RMSE computed using time series cross-validatoin

• Point forecasts are often useless without prediction intervals
• Prediction intervals require a stochastic model (with random errors, etc.)
• Multi-step forecasts for time series require a more sophisticated approach (with PI getting wider as the forecast horizon increases)

The expected value of yt+h is the same (yt) for any avalue of h, but its variance grows with h