[R] Random walk

[Sergio Andrés Estay Cabrera]

Dear Giovanni,

Thanks  so much for your answer, but your script returns gaussian white 
noise not a random walk, at least the time series generated don't have 
the expected periodogram for a random walk. That's the reason why I use 
cumsum, the sum of a white noise is a easy way produce a random walk 
which periodogram scales at 1/f.

finally I solve my problem with a brute-force algorithm. Searching in 
the web I found some pages about random walks and neural networks where 
people said that this task is very difficult and is in a exploratory 
step nowadays.

Thank you very much for your help

Sergio




Giovanni Petris wrote:
> Hi Sergio,
>
> Your function does not estimate what you want. In fact it does not
> estimate anything useful. A random walk is not stationary; in
> particular, the variance at time t is t. Therefore, estimating variances
> based on one run, averaging over time, does not make any sense. This is
> what you are doing with the command cor(cumsum(s[,1]),cumsum(s[,2]))
> (the correlation is obtained by standardizing the covariance, dividing
> by the square root of the variances).
>
> On the other hand, if you replicate your simulation (as you do) and
> compute the correlation of the simulated random walks at any _fixed_
> time, you can see that the result is what you expect. Here it is.
>
>   
>> require(MASS)
>> fz <- function(n, t, rho){
>>     
> +     f <- array(dim = c(t, 2, n))
> +     V <- matrix(c(1, rho, rho, 1), ncol = 2)
> +     for(i in 1 : n){
> +         f[,, i] <- mvrnorm(n = t, mu = c(0,0), Sigma = V)
> +     }
> +     return(f)
> + }
>   
>>     
>> set.seed(123)
>> out <- fz(1000, 20, rho = 0.7) 
>> apply(out, 1, function(x) cor(t(x))[1,2]) # cor at times 1,...,20
>>     
>  [1] 0.7306115 0.6810394 0.6879712 0.7095613 0.7325612 0.6922090
> 0.7007823
>  [8] 0.6828095 0.7144861 0.6932104 0.7061211 0.6781331 0.6898726
> 0.6926878
> [15] 0.6949656 0.6995899 0.7524582 0.6912938 0.6931928 0.6820243
>
> Best,
> Giovanni
>
> On Mon, 2010-05-10 at 14:55 -0400, Sergio Andrés Estay Cabrera wrote:
>   
>> Dear R users and specially Albyn and Giovanni,
>>
>> thanks for your answers, but in fact I supposed the same at the 
>> beginning of my problem. However, when I generate the data seldom I 
>> obtain the expected correlation. For example using this code:
>>
>> fz<-function(n,t,rho){
>> f<-NULL
>> for(i in 1:n){
>> s<-rmvnorm(n=t,mean=c(0,0),sigma=matrix(c(1,rho,rho,1),ncol=2),method='svd')
>> paso<-cor(cumsum(s[,1]),cumsum(s[,2]))
>> f<-c(f,paso)}
>> f<-f
>> }
>>
>> and then plot the histogram of the results, it is possible to observe 
>> that the distribution of the values is asymmetric with most of the 
>> simulations close to 1 when the value of rho is higher than 0.3 and 
>> looks like a uniform distribution with values below 0.3.
>>
>> I suspect than the only possibility is using a brute force algorithm.
>>
>>
>> Any advice would be helpful
>>
>> Sergio A. Estay
>> *CASEB *
>> Departamento de Ecología
>> Universidad Catolica de Chile
>>
>>
>>
>>
>> Albyn Jones wrote:
>>     
>>> Sums of correlated increments have the same correlation as the original
>>> variables...
>>>
>>>  library(mvtnorm)
>>>  X<- matrix(0,nrow=1000,ncol=2)
>>>  for(i in 1:1000){
>>>  Y <- rmvnorm(1000,mean=mu,sigma=S)
>>>  X[i,] <- apply(Y,2,sum)
>>>  }
>>>  cor(Y)
>>>           [,1]      [,2]
>>> [1,] 1.0000000 0.4909281
>>> [2,] 0.4909281 1.0000000
>>>
>>> So, unless you meant that you want the _sample_ correlation to be
>>> pre-specified, you are all set.
>>>
>>> albyn
>>>
>>> On Sun, May 09, 2010 at 09:20:25PM -0400, Sergio Andrés Estay Cabrera wrote:
>>>   
>>>       
>>>> Hi everybody,
>>>>
>>>>
>>>> I am trying to generate two random walks with an specific correlation, for 
>>>> example, two random walks of 200 time steps with a correlation 0.7.
>>>>
>>>> I built the random walks with:
>>>>
>>>> x<-cumsum(rnorm(200, mean=0,sd=1))
>>>> y<-cumsum(rnorm(200, mean=0,sd=1))
>>>>
>>>> but I don't know how to fix the correlation between them.
>>>>
>>>> With white noise is easy to fix the correlation using the function rmvnorm 
>>>> in the package mvtnorm
>>>>
>>>> I surfed in the web in the searchable mail archives in the R web site but 
>>>> no references appears.
>>>>
>>>> If you have some advices to solve this problems I would be very thankful.
>>>>
>>>> Thanks in advance.
>>>>
>>>> Sergio A. Estay
>>>> *CASEB *
>>>> Departamento de Ecología
>>>> Universidad Catolica de Chile
>>>>
>>>> -- 
>>>> “La disciplina no tiene ningún mérito en circunstancias ideales. ” – Habor Mallow
>>>>
>>>> ______________________________________________
>>>> R-help at r-project.org mailing list
>>>> https://stat.ethz.ch/mailman/listinfo/r-help
>>>> PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
>>>> and provide commented, minimal, self-contained, reproducible code.
>>>>
>>>>     
>>>>         
>>>   
>>>       
>>     
>
>
>
>
>
>   


-- 
“La disciplina no tiene ningún mérito en circunstancias ideales. ” – Habor Mallow