Here is a function that generates data from an AR$(1)$ model starting with the first value set to $0$ generate$\_$ar$1<-$ function$($n $=100,$ c $=0,$ phi, sigma $=1)$ # Generate errors error $<-$ rnorm$($n$,$ mean $=0,$ sd $=$ sigma$)$ # Set up vector for the response with initial values set to $0$ y $<-$ rep$(0,$ n$)$ # Generate remaining observations for$($i in seq$(2,$ length $=$ n$-1))$ y$[$i$]<-$ c $+$ phi $*$ y$[$i$-1]+$ error$[$i$]$ return$($y$)$ Here n is the number of observations to simulate, c is the constant, phi is the AR coefficient, and sigma is the standard deviation of the noise. The following example shows the function being used to generate $50$ observations library$($fpp$3)$ tsibble$($time $=1$:$50,$ y $=$ generate$\_$ar$1($n$=50,$ c$=1,$ phi$=0\mathrm{.}8),$ index $=$ time$)|>$ autoplot$($y$)$ Modify the generate$\_$ar$1$ function to generate data from any ARMA$($p$,$q$)$ model with parameters to be specified by the user. The first line of your function definition should be generate$\_$arma $<-$ function$($n $=100,$ c $=0,$ phi $=$ NULL, theta $=$ NULL, sigma $=1)$ Here phi and theta are vectors of AR and MA coefficients. Your function should return a numeric vector of length n$.$ For example generate$\_$arma$($n $=50,$ c $=2,$ phi $=$ c$(0\mathrm{.}4,-0\mathrm{.}6))$ should return $50$ observations generated from the model where $\backslash$epsi is iid N$(0,1).$ The noise should be generated using the rnorm$()$ function. Your function should check stationarity and invertibility conditions and return an error if either condition is not satisfied. You can use the stop$()$ function to generate an error. The model will be stationary if the following expression returns TRUE: $!$any$($abs$($polyroot$($c$(1,-$phi$)))<=1)$ The MA parameters will be invertible if the following expression returns TRUE: $!$any$($abs$($polyroot$($c$(1,$theta$)))<=1)$ Please submit your solution as a $.$R file. Here is a function that generates data from an A R$(1)$ model starting with the first value set to $0$ generate$\_$ar$1<-$ function $($n$=100,$ c$=0,$ phi, sigma $=1)\{$ # Generate errors error $<-$ rnorm$($n$,$ mean $=\backslash$theta $,$ sd $=$ sigma $)$ # Set up vector for the response with initial values set to $\backslash$theta y$<-$rep$(\backslash$theta $,$ n$)$ # Generate remaining observations for $($i in seq $(2,$ length $=$n$-1))\{$ y$[$i$]<-$c$+$ phi $*$ y$[$i$-1]+$error$[$i$]\}$ return$($y$)\}$ library $($fpp $3)$ tsibble$($time $=1$: $50,$ y$=$ generate$\_$ar$1($n$=50,$ c$=1,$ phi$=0\mathrm{.}8),$ index $=$ time$)|>$ autoplot $($y$)$ generate$\_$arma $<-$ function $($n$=100,$ c$=0,$ phi $=$ NULL, theta $=$ NULL, sigma $=1)$ Here phi and theta are vectors of AR and MA coefficients. Your function should return a numeric vector of length n$.$ For example generate$\_$arma $($n$=50,$ c$=2,$ phi $=$c$(0\mathrm{.}4,-0\mathrm{.}6))$ should return $50$ observations generated from the model

y$\_$t$=2+0\mathrm{.}4$ y$\_$t$-1-0\mathrm{.}6$ y$\_$t$-2+\backslash$epsi $\_$t

where $\backslash$epsi is iid N$(0,1)$ The noise should be generated using the rnorm$()$ function. $!$ any $($ abs $($ polyroot $($c$(1,-$ phi $)))<=1)$ The MA parameters will be invertible if the following expression returns TRUE: $!$ any $($ abs $($ polyroot $($c$(1,$ theta $)))<=1)$ Please submit your solution as a $.$R file.