Basic Statistics in Time Series - examples

Let's use some of the statistics I mentioned before to describe some Time Series. We can start with Dow Jones dataset which are in fpp library. Dataset containing the Dow Jones Index is a stock market index that measures the stock performance of 30 large companies listed on stock exchanges in the United States.

library(fpp)
dowjones # It is our dataset, which has class ts so we don't have to convert it. 

Time Series:
Start = 1 
End = 78 
Frequency = 1 
 [1] 110.94 110.69 110.43 110.56 110.75 110.84 110.46 110.56 110.46 110.05 109.60 109.31 109.31 109.25
[15] 109.02 108.54 108.77 109.02 109.44 109.38 109.53 109.89 110.56 110.56 110.72 111.23 111.48 111.58
[29] 111.90 112.19 112.06 111.96 111.68 111.36 111.42 112.00 112.22 112.70 113.15 114.36 114.65 115.06
[43] 115.86 116.40 116.44 116.88 118.07 118.51 119.28 119.79 119.70 119.28 119.66 120.14 120.97 121.13
[57] 121.55 121.96 122.26 123.79 124.11 124.14 123.37 123.02 122.86 123.02 123.11 123.05 123.05 122.83
[71] 123.18 122.67 122.73 122.86 122.67 122.09 122.00 121.23

Let's check now some basic statistic on this data.

mean(dowjones)
[1] 115.6833
median(dowjones) 
[1] 113.755

Mean and median are close to each other.

sort(dowjones) # We can sort our time series
 [1] 108.54 108.77 109.02 109.02 109.25 109.31 109.31 109.38 109.44 109.53 109.60 109.89 110.05 110.43
[15] 110.46 110.46 110.56 110.56 110.56 110.56 110.69 110.72 110.75 110.84 110.94 111.23 111.36 111.42
[29] 111.48 111.58 111.68 111.90 111.96 112.00 112.06 112.19 112.22 112.70 113.15 114.36 114.65 115.06
[43] 115.86 116.40 116.44 116.88 118.07 118.51 119.28 119.28 119.66 119.70 119.79 120.14 120.97 121.13
[57] 121.23 121.55 121.96 122.00 122.09 122.26 122.67 122.67 122.73 122.83 122.86 122.86 123.02 123.02
[71] 123.05 123.05 123.11 123.18 123.37 123.79 124.11 124.14

quantile(dowjones)
      0%      25%      50%      75%     100% 
108.5400 110.5925 113.7550 121.8575 124.1400 

Extracting the deciles we can do as follow:

quantile(dowjones,prob=seq(0,1,length=11),type=5) 
   0%     10%     20%      30%         40%      50%      60%     70%      80%     90%     100% 
108.540   109.398   110.470   110.831   111.834   113.755  118.202  120.986  122.629 123.041 124.140 

var(dowjones)
[1] 30.31672

Visualization of Time Series 

plot(dowjones) 

It seems that this dataset is moving towards a direction. It has a trend.

We are checking now stationarity with Augmented Dickey-Fuller Test

adf.test(dowjones)

Augmented Dickey-Fuller Test
data:  dowjones
Dickey-Fuller = -1.8053, Lag order = 4, p-value = 0.6552
alternative hypothesis: stationary

As we can see the p-value is above 0.05 therefore data is not stationary.

Let's check the autocorrelation.

acf(dowjones)

Slowly deceasing ACF indicates trend, no seasonality.

pacf(dowjones)

It looks like no seasonal data but lets check it with one of our function

ggseasonplot(dowjones)
Error in ggseasonplot(dowjones) : Data are not seasonal

Let's take now the seasonal Time Series like usdeaths data. This time series present the monthly total of accidental deaths in the United States( Jan 1973-Dec 1978).

usdeaths

     Jan   Feb   Mar   Apr   May   Jun   Jul   Aug   Sep   Oct   Nov   Dec

1973  9007  8106  8928  9137 10017 10826 11317 10744  9713  9938  9161  8927

1974  7750  6981  8038  8422  8714  9512 10120  9823  8743  9129  8710  8680

1975  8162  7306  8124  7870  9387  9556 10093  9620  8285  8433  8160  8034

1976  7717  7461  7776  7925  8634  8945 10078  9179  8037  8488  7874  8647

1977  7792  6957  7726  8106  8890  9299 10625  9302  8314  8850  8265  8796

1978  7836  6892  7791  8129  9115  9434 10484  9827  9110  9070  8633  9240

mean(usdeaths)

[1] 8787.736

median(usdeaths) 

[1] 8728.5

Again mean and median close to each other

quantile(usdeaths)

      0%      25%      50%      75%     100% 

 6892.00  8089.00  8728.50  9323.25 11317.00 

var(usdeaths)

[1] 918411.7

plot(usdeaths) 

We can see seasonal data set, no trend.

adf.test(usdeaths)

Augmented Dickey-Fuller Test

data:  usdeaths

Dickey-Fuller = -3.8111, Lag order = 4, p-value = 0.02318

alternative hypothesis: stationary

Checking the stationary. The p-value is below 0.05, the data is stationary.

acf(usdeaths) # checking the autocorrelation

pacf(usdeaths)

ggseasonplot(usdeaths) #checking the seasonality

monthplot(usdeaths)

plot(decompose(usdeaths))

What conclusions can we have based on above plots? It seems seasonality is evident in all plots however no cyclicity or trend.