R help - Feb 2009 - Problem about SARMA model forcasting

Hello, Guys:
I'm from China, my English is poor and I'm new to R. The first message I
sent to R help meets some problems, so I send again.
Hope that I can get useful suggestions from you warm-hearted guys.
Thanks.

I builded a multiplicative seasonal ARMA model to a series named
"cDownRange".
And the order is (1,1)*(0,1)45
The regular AR=1; regular MA=1; seasonal AR=0; seasonal MA=1; seasonal
period=45.
I fitted the model in R and get the result as below:
Call:arima(x = cDownRange, order = c(1, 0, 1), seasonal = list(order = c(0, 1,
1), period = 45))
Coefficients:         
    ar1      ma1     sma1      
    0.7364  -0.5046  -0.9511
s.e.    0.0458   0.0594   0.0130
When I use the predict command of this model in R, it gives the right
forcasting.

So I think the forcast formula of this SARMA model should be written as below:
X(t)=ar1*X(t-1)-ma1*a(t-1)-sma1*a(t-45)+ma1*sma1*a(t-46)

But when I use this forcast formula in Excel, it gives a totally different
predict from R. And I don't know why?
I guess the expression of the forcast formula of this SARMA(1,1)*(0,1)45 is
wrong, but I don't know the right form.
Can anybody help me with this?Thank, again!

saji from Shanghai


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Saji,

This may help.

Your model is

(1,0,1)X(0,1,1)S

giving difference polynomials

nonseasonal (1,0,1) = (1-ar1*B)   =  (1-ma1*B)
seasonal (0,1,1)S   = (1-B**S)    =  (1-sma1*B**S)

giving:  (1-ar1*B)X(1-B**S) x_t = (1-ma1*B)X(1-sma1*B**S) a_t

multiplying out:

         x_t - x_(t-S) - ar1*x_(t-1) + ar1*x_(t-S-1) = a_t - ma1*a_(t-1) -
sma1*a_(t-S) + ma1*sma1*a_(t-S-1)

or

         x_t =  x_(t-S) + ar1*x_(t-1) - ar1*x_(t-S-1) + a_t - ma1*a_(t-1) -
sma1*a_(t-S) + ma1*sma1*a_(t-S-1)


Hopefully I've multiplied all the bits out correctly and I think R uses
this (Box-Jenkins) form for ARIMA models.

Gerard




                                                                           
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             .com>                                                      To 
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                                                                   Subject 
                                       [R] Problem about SARMA model       
             03/02/2009 10:22          forcasting                          
                                                                           
                                                                           
                                                                           
                                                                           
                                                                           
                                                                           





Hello, Guys:
I'm from China, my English is poor and I'm new to R. The first message I
sent to R help meets some problems, so I send again.
Hope that I can get useful suggestions from you warm-hearted guys.
Thanks.

I builded a multiplicative seasonal ARMA model to a series named
"cDownRange".
And the order is (1,1)*(0,1)45
The regular AR=1; regular MA=1; seasonal AR=0; seasonal MA=1; seasonal
period=45.
I fitted the model in R and get the result as below:
Call:arima(x = cDownRange, order = c(1, 0, 1), seasonal = list(order = c(0,
1, 1), period = 45))
Coefficients:
    ar1      ma1     sma1
    0.7364  -0.5046  -0.9511
s.e.    0.0458   0.0594   0.0130
When I use the predict command of this model in R, it gives the right
forcasting.

So I think the forcast formula of this SARMA model should be written as
below:
X(t)=ar1*X(t-1)-ma1*a(t-1)-sma1*a(t-45)+ma1*sma1*a(t-46)

But when I use this forcast formula in Excel, it gives a totally different
predict from R. And I don't know why?
I guess the expression of the forcast formula of this SARMA(1,1)*(0,1)45 is
wrong, but I don't know the right form.
Can anybody help me with this?Thank, again!

saji from Shanghai


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First of all, sorry to *Gerard.
*I have changed my email account, and I don't know how to reply to my posted
thread before. So I just create a new message here.
Thanks again for your help! Now I realized where my mistake is.
I forgot to include the seasonal differencing order.

After I corrected the formula as below:
(*S-ARIMA(p,d,q)*(P,D,Q)* models, where *p=1,d=0,q=1; P=0,D=1,Q=1;* and *the
seasonal period S=45*.)

X(t) = X(t-45) + ar1*X(t-1) - ar1*X(t-46) - ma1*a(t-1) - sma1*a(t-45) +
ma1*sma1*a(t-46) + a(t)

Thus, we get:

a(t) = X(t) - ar1*X(t-1) - X(t-45) + ar1*X(t-46) + ma1*a(t-1) + sma1*a(t-45)
- ma1*sma1*a(t-46),
*when t>=46*;

Now my question is that: *What is the initial value of a(t) when t<46?
*And what is the initial setting in R?

Because R gives a very good forcasting of the analyzed data series, and I
just can not reproduced the results in other software  (like EXCEL).

Hope some one to help! Thanks!

saji from Shanghai

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Somebody said that ARIMA models like discussed above are easy to implement on a
spreadsheet.
The prediction formula is simply a linear equation that refers to past values of
original time series and past values of the errors.
Thus, setting up an spreadsheet by stroing the data in one column, the
forcasting values computed bu the formula in another column, and the errors(data
minus forcasts) in the third column can give the results just as the same as any
statistical computing software(like R).
But I just can't, and I think there must be some mistakes in my procedure
but just can't find it by myself.
Really need helps!
Thanks again!

saji from Shanghai