Autocorrelation

• Autocorrelation is one of the key features of a time series, lets start by looking at the lag plots.

library(fpp3)

── Attaching packages ─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────── fpp3 0.5 ──
✔ tibble      3.2.1     ✔ tsibble     1.1.3
✔ dplyr       1.1.3     ✔ tsibbledata 0.4.1
✔ tidyr       1.3.0     ✔ feasts      0.3.1
✔ lubridate   1.9.2     ✔ fable       0.3.3
✔ ggplot2     3.4.3     ✔ fabletools  0.3.3
── Conflicts ────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────── fpp3_conflicts ──
✖ lubridate::date()    masks base::date()
✖ dplyr::filter()      masks stats::filter()
✖ tsibble::intersect() masks base::intersect()
✖ tsibble::interval()  masks lubridate::interval()
✖ dplyr::lag()         masks stats::lag()
✖ tsibble::setdiff()   masks base::setdiff()
✖ tsibble::union()     masks base::union()

library(GGally)

Registered S3 method overwritten by 'GGally':
  method from   
  +.gg   ggplot2

Lag plots

aus_production |>
  filter(year(Quarter) >= 1992) -> new_production

new_production |> gg_lag(Beer, geom = "point")

• Each graph shows yt plotted aginst y(t-k) for different values of k

The autocorrelations are the correlations associated with these scatterplots * r1 = Correlation(yt,y(t-1)) * r2 = Corr(yt, y(t-2)) * r3 = Corr(yt, y(t-3)) * AND SO ON

Doing it in R

new_production |> 
  ACF(Beer, lag_max = 9) 

• Plotting it Correlogram : plot of the correlation of different lags

new_production |> 
  ACF(Beer) |>
  autoplot()

• When data has a trend autocorrelation for SMALL LAGS tend to be LARGE AND POSITIVE
• When data is SEASONAL, autocorrelations will be LARGER at SEASONAL LAGS (multiples of seasonal freq)
• IF BOTH, you see a combination of the effects

Example 2: US employment (BOTH)

retail <- us_employment |>
  filter(Title == "Retail Trade", year(Month) >= 1980)

retail |> autoplot(Employed)

Look at autocorrelation

retail |> ACF(Employed, lag_max = 48) |> autoplot()

• The effect of seasonality can be seen in lags 12, 24, 36 because sesonality is year
• The effect looks like WAVES
• All autocorrelation lags are quite positive because of the TREND

No seasonality no trend

gafa_stock |>
  filter(Symbol == "GOOG", year(Date) == 2015) |>
  select(Date, Close) -> google_2015

google_2015

The ! sing indicates that the frequency is irregular IRREGULARLY SPACED Timestamps show that weekends, holidays and other dates are missing

google_2015 |> autoplot()

Plot variable not specified, automatically selected `.vars = Close`

• Looking at the ACF

google_2015 |> ACF(Close, lag_max = 100) |> autoplot()

Warning: Provided data has an irregular interval, results should be treated with caution. Computing ACF by observation.

• This is typicall of STOCK Data and TREND DATA

THE END