Analysis of Sequential Data

• Automatically chooses a model by default using the AIC, AICc or BIC
• Can handle any combination of trend, seasonality and damping
• Ensures the parameters are admissible ( equivalent to invertible)
• Produces an object of class "ets"

• A stationary series is
  • roughly horizontal
  • constant variance
  • no patterns predictable in th elong-term (for instance no seasonality)
• Necessaty for arima models

No

No

No

No

It is kind of regular. But the problem is that the peaks between the peaks are not constant and therefore we can not apply seasonal differencies.

So no

• Time plot
• The ACF of stationary data drops to zero relatively quickly
• The ACF of non-stationary data decreases slowly
• For non-stationary data, the value of r1 is often large and positive

• Used when arima models are applied to forecast the time series
• Differencing helps to stailize the mean
• the differenced series is the change between each observation in the original series
  • y't = yt - yt-1
• The differenced series will have only T-1 values since it is not possible to calculate a difference y'1 for the first observation

If we simulate many random walks, the average value of the simulatoin at time t is 0, while the spread of the simulations increase with t -> variance of random walk is non-stationary

Occasionally the first-order differenced data will not appear stationary and it may be necessary to difference the data a second time.

• Some thing is easier to forecast if:
  • we have a good understanding of the factors that contribute to it
  • There is lots of data available
  • the forecasts cannot affect the thing we are trying to forecast
  • There is relatively low natural/unexplainable random variation
  • the future is somewhat similar to the past

• Daily stock prices
• Monthly rainfall
• Annual business profits
• Production, e.g. quartely australien beer production

Forecasting is estimating how the sequence of observations will continue into the future

An uncerainty range

If you have just the probabilty (e.g. 50%) you do not know the deviation. So you could for example produce the 80% of the forecast. The worst case scenario is that customer could order or demand more. If you can not provide the asked product of piece, you have the bigger loss (in image) than if you have produced a bit too much, which you can hold and put in the sale in the end.

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