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nortsTest:An R Package for Assessing Normality of Stationary Process

nortsTest is anR package for assessingnormality of stationary processes, it tests if a given data follows astationary Gaussian process. The package works as an extension of thenortest package that performs normality tests in randomsamples (independent data). The package’s principal functionsare:

Additionally, inspired in the functioncheckresiduals()of theforecastpackage, we provide thecheck_residuals methods forchecking model’s assumptions using the estimated residuals. The functionchecks stationarity, homoscedasticity and normality, presenting a reportof the used tests and conclusions.

Checking normalityassumptions

library(nortsTest)

Classic hypothesis tests for normality such as Shapiro & Wilk,Anderson & Darling, or Jarque & Bera, do not perform well ondependent data. Therefore, these tests should not be used to checkwhether a given time series has been drawn from a Gaussian process. As asimple example, we generate a stationary ARMA(1,1) process simulatedusing an t student distribution with 7 degrees of freedom, and performthe Anderson-Darling test from thenortest package.

x=arima.sim(100,model =list(ar =0.32,ma =0.25),rand.gen = rt,df =7)nortest::ad.test(x)#>#>  Anderson-Darling normality test#>#> data:  x#> A = 0.50769, p-value = 0.1954

The null hypothesis is that the data has a normal distribution andtherefore, follows a Gaussian Process. At\(\alpha = 0.05\) significance level thealternative hypothesis is rejected and wrongly concludes the datafollows a Gaussian process. Applying the Lobato and Velasco’s test ofour package, the null hypothesis is correctly rejected.

lobato.test(x)#>#>  Lobatos and Velascos test#>#> data:  x#> lobato = 16.864, df = 2, p-value = 0.0002177#> alternative hypothesis: x does not follow a Gaussian Process

Example: stationary AR(2)process

In the next example we generate a stationary AR(2) process, using anexponential distribution with rate of 5, and perform theEppsandRP withk = 5 random projections tests. With asignificance level at\(\alpha=0.05\),the null hypothesis of non-normality is rejected.

set.seed(298)# Simulating the AR(2) processx=arima.sim(250,model =list(ar =c(0.2,0.3)),rand.gen = rexp,rate =5)# testsepps.test(x)#>#>  epps test#>#> data:  x#> epps = 38.158, df = 2, p-value = 5.178e-09#> alternative hypothesis: x does not follow a Gaussian Processrp.test(x,k =5)#>#>  k random projections test#>#> data:  x#> k = 5, lobato = 188.771, epps = 28.385, p-value = 0.0007823#> alternative hypothesis: x does not follow a Gaussian Process

Example: stationary VAR(1)process

In the next example we generate a stationary VAR(1) process ofdimensionp = 2, using two independent Gaussian AR(1)processes, and perform theEl Bouch’s test. With a significancelevel of\(\alpha = 0.05\), thealternative hypothesis of non-normality is rejected.

set.seed(298)# Simulating the VAR(2) processx1=arima.sim(250,model =list(ar =c (0.2)))x2=arima.sim(250,model =list(ar =c (0.3)))## testelbouch.test(y = x1,x = x2)#>#>  El Bouch, Michel & Comon's test#>#> data:  w = (y, x)#> Z = 0.1438, p-value = 0.4428#> alternative hypothesis: w = (y, x) does not follow a Gaussian Process

Checking model’sassumptions:cardox data

As an example, we analyze the monthly mean carbon dioxide (inppm) from theastsa package, measured at Mauna LoaObservatory, Hawaii. from March, 1958 to November 2018. The carbondioxide data measured as the mole fraction in dry air, on Mauna Loaconstitute the longest record of direct measurements of CO2 in theatmosphere. They were started by C. David Keeling of the ScrippsInstitution of Oceanography in March of 1958 at a facility of theNational Oceanic and Atmospheric Administration.

library(astsa)data("cardox")autoplot(cardox,xlab ="years",ylab =" CO2 (ppm)",color ="darkred",size =1,main ="Carbon Dioxide Levels at Mauna Loa")

The time series clearly has trend and seasonal components, foranalyzing thecardox data we proposed a Gaussian linear statespace model. We use the model’s implementation from theforecast package asfollows:

library(forecast)#>#> Attaching package: 'forecast'#> The following object is masked from 'package:astsa':#>#>     gasmodel=ets(cardox)summary(model)#> ETS(M,A,A)#>#> Call:#>  ets(y = cardox)#>#>   Smoothing parameters:#>     alpha = 0.5591#>     beta  = 0.0072#>     gamma = 0.1061#>#>   Initial states:#>     l = 314.6899#>     b = 0.0696#>     s = 0.6611 0.0168 -0.8536 -1.9095 -3.0088 -2.7503#>            -1.2155 0.6944 2.1365 2.7225 2.3051 1.2012#>#>   sigma:  9e-04#>#>      AIC     AICc      BIC#> 3136.280 3137.140 3214.338#>#> Training set error measures:#>                     ME     RMSE       MAE         MPE       MAPE      MASE#> Training set 0.0232403 0.312003 0.2430829 0.006308831 0.06883992 0.1559102#>                    ACF1#> Training set 0.07275949

The best fitted model is amultiplicative level,additive trend andseasonality state space model. Ifthe model’s assumptions are satisfied, then the model’s errors behavelike a Gaussian stationary process. These assumptions can be checkedusing ourcheck_residuals functions.

In this case, we use an Augmented Dickey-Fuller test for stationaryassumption, and a random projections test for normality.

check_residuals(model,unit_root ="adf",normality ="rp",plot =TRUE)#>#>  ***************************************************#>#>  Unit root test for stationarity:#>#>  Augmented Dickey-Fuller Test#>#> data:  y#> Dickey-Fuller = -9.7249, Lag order = 8, p-value = 0.01#> alternative hypothesis: stationary#>#>#>  Conclusion: y is stationary#>  ***************************************************#>#>  Goodness of fit test for Gaussian Distribution:#>#>  k random projections test#>#> data:  y#> k = 2, lobato = 3.8260, epps = 1.3156, p-value = 0.3328#> alternative hypothesis: y does not follow a Gaussian Process#>#>#>  Conclusion: y follows a Gaussian Process#>#>  ***************************************************

After all the model’s assumptions are checked, the model can be usedto forecast.

autoplot(forecast(model,h =12),include =100,xlab ="years",ylab =" CO2 (ppm)",main ="Forecast: Carbon Dioxide Levels at Mauna Loa")

How to installnortsTest?

The current development version can be downloaded from GitHub via

if (!requireNamespace("remotes"))install.packages("remotes")remotes::install_github("asael697/nortsTest",dependencies =TRUE)

Additional test functions

ThenortsTest package offers additional functions fordescriptive analysis in univariate time series.

For visual diagnostic, we offer ggplot2 methods for numeric andtime-series data. Most of the functions were adapted from Rob Hyndman’sforecastpackage.

Accepted models for residualcheck

Currently ourcheck_residuals() andcheck_plot() methods are valid for the current models andclasses:

For overloading more functions, methods or packages, please make apull request or send a mail to:asael_am@hotmail.com.

References

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