Simulating White Noise

Hi. This page focuses on simulating white noise in R. White Noise is a very simple stationary process in the field of time series analysis. Weak white noise has a fixed constant mean over time and a fixed constant variance and there is no correlation over time.

I use Datacamp Slides as my reference but the plots are in base R graphics. The plots here are generated with ggplot2.

To simulate observations from a White Noise process, the arima.sim() function is used. Make sure to use model = list(order = c(0, 0, 0).

# Simulating White Noise (Time Series)

# Reference: Datacamp Slides

# In the slides, a base R graphics plot is used. I want to use ggplot2 as ggplot2 looks
# nicer.

library(ggplot2)

# White Noise is a very simple stationary process.

# Weak white noise has a fixed constant mean, a fixed constant variance and
# no correlation over time.

#### Simulate 100 observations in White Noise model.

wn_sim_100 <- arima.sim(model = list(order = c(0, 0, 0)), n = 100)

head(wn_sim_100)

## [1] -0.8333583  1.0140553  0.2805254  1.4033654  1.0580948  0.3120627

ts_wn100 <- data.frame(Time = seq(0, 99, 1), WN = wn_sim_100)

head(ts_wn100); tail(ts_wn100)

##   Time         WN
## 1    0 -0.8333583
## 2    1  1.0140553
## 3    2  0.2805254
## 4    3  1.4033654
## 5    4  1.0580948
## 6    5  0.3120627

##     Time         WN
## 95    94  0.1729552
## 96    95  1.1674964
## 97    96 -0.5647851
## 98    97 -0.5251206
## 99    98  0.1787511
## 100   99  1.1456842

From the ts_wn100 data frame, a plot can be generated with the help of ggplot2 graphics.

# Ggplot2 line plot of simulated white noise observations:

ggplot(data = ts_wn100, aes(x = Time, y = WN)) + 
  geom_line() +
  labs(x = "\n Time ", y = "WN \n", title = "Simulated Observations From A White Noise Process \n") +
  theme(plot.title = element_text(hjust = 0.5), 
        axis.ticks.x = element_blank(),
        axis.title.x = element_text(face="bold", colour="darkblue", size = 12),
        axis.title.y = element_text(face="bold", colour="darkblue", size = 12))

## Don't know how to automatically pick scale for object of type ts. Defaulting to continuous.

Adding More Observations In White Noise Simulation

### Simulate 10000 observations in White Noise model.

wn_sim10000 <- arima.sim(model = list(order = c(0, 0, 0)), n = 10000)

head(wn_sim10000)

## [1] -1.3416160 -2.0917630 -0.9199639 -1.2969639 -0.6874203 -0.7232486

ts_wn10000 <- data.frame(Time = seq(0, 9999, 1), WN = wn_sim10000)

head(ts_wn10000); tail(ts_wn10000)

##   Time         WN
## 1    0 -1.3416160
## 2    1 -2.0917630
## 3    2 -0.9199639
## 4    3 -1.2969639
## 5    4 -0.6874203
## 6    5 -0.7232486

##       Time         WN
## 9995  9994 -0.2686607
## 9996  9995  0.6225617
## 9997  9996 -0.3849075
## 9998  9997  0.7504948
## 9999  9998 -1.2778687
## 10000 9999  0.1074248

# Ggplot2 line plot of simulated white noise observations:

ggplot(data = ts_wn10000, aes(x = Time, y = WN)) + 
  geom_line() +
  labs(x = "\n Time ", y = "WN \n", title = "Simulated Observations From A White Noise Process \n") +
  theme(plot.title = element_text(hjust = 0.5), 
        axis.ticks.x = element_blank(),
        axis.title.x = element_text(face="bold", colour="darkblue", size = 12),
        axis.title.y = element_text(face="bold", colour="darkblue", size = 12))

## Don't know how to automatically pick scale for object of type ts. Defaulting to continuous.

# Simulating observations from white noise model with different means and variance.

# Simulate 50 observations from the WN model with mean = 2, sd = 2 (var = 4)

wn_2 <- arima.sim(model = list(order = c(0, 0, 0)), n = 50, mean = 4, sd = 2)

head(wn_2)

## [1] 5.774795 4.005328 0.233911 3.116922 3.786954 4.811082

ts_wn_2 <- data.frame(Time = seq(0, 49, 1), WN = wn_2)

head(ts_wn_2); tail(ts_wn_2)

##   Time       WN
## 1    0 5.774795
## 2    1 4.005328
## 3    2 0.233911
## 4    3 3.116922
## 5    4 3.786954
## 6    5 4.811082

##    Time       WN
## 45   44 4.501007
## 46   45 3.668701
## 47   46 6.973405
## 48   47 5.156340
## 49   48 5.085812
## 50   49 4.844296

# Ggplot2 line plot of simulated white noise observations:

ggplot(data = ts_wn_2, aes(x = Time, y = WN)) + 
  geom_line() +
  labs(x = "\n Time ", y = "WN \n", title = "Simulated Observations From A \n White Noise Process (Mean = 4, SD = 2) \n") +
  theme(plot.title = element_text(hjust = 0.5), 
        axis.ticks.x = element_blank(),
        axis.title.x = element_text(face="bold", colour="darkblue", size = 12),
        axis.title.y = element_text(face="bold", colour="darkblue", size = 12))

## Don't know how to automatically pick scale for object of type ts. Defaulting to continuous.

# Fit second white noise model with arima():

arima(wn_2, order = c(0, 0, 0))

## 
## Call:
## arima(x = wn_2, order = c(0, 0, 0))
## 
## Coefficients:
##       intercept
##          4.4938
## s.e.     0.2810
## 
## sigma^2 estimated as 3.948:  log likelihood = -105.28,  aic = 214.55

# Mean and variance of white noise:

mean(wn_2)

## [1] 4.493833

var(wn_2)

## [1] 4.028461