Estimating MA parameters through arima or through package "dlm"

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Estimating MA parameters through arima or through package "dlm"

Stefano Sofia
Dear list users,
I want to use apply a MA(2) process (x=beta1*epsilon_(t-1) + beta2*epsilon_(t-1) + epsilon_(t)) to a given time series (x), and I want to estimate the two parameters beta1, beta2 and the variance of the random variable epsilon_(t).

If I use
MA2_1 <- Arima(x, order=c(0,0,2))
I get the following result

[1] "MA2_1"
Series: x
ARIMA(0,0,2) with non-zero mean

Coefficients:
          ma1     ma2  intercept
      -0.0279  0.0783     5.3737
s.e.   0.0667  0.0622     0.0245

sigma^2 estimated as 0.1284:  log likelihood=-92.63
AIC=193.25   AICc=193.43   BIC=207.11
[1] 0 2 0 0 1 0 0

From this straightforward analysis V[epsilon]=0.1284, beta1=-0.0279 and beta2=0.0783.

I also tried to use a DLM representation of ARIMA models and estimate the unknown parameters by maximum likelihood through the dlm package (in particular applying the example at section 3.2.6, page 115, of "Dynamic Linear Models with R" by Petris, Petrone and Campagnoli:

arma_parameters <- function(x)
{
  buildGap <- function(u)
  {
    gap <- dlmModARMA(ma = u[2 : 3], sigma2 = u[1])
    return(gap)
   }
   init <- c(0.005, 0.004, 0.003)
   outMLE <- dlmMLE(x, init, buildGap)
   dlmGap <- buildGap(outMLE$par)
}

and this gives:
[1] "outMLE"
$par
[1] 1.00816794 0.02349296 0.02364788

$value
[1] 3089.196

$counts
function gradient
      10       10

$convergence
[1] 0

$message
[1] "CONVERGENCE: REL_REDUCTION_OF_F <= FACTR*EPSMCH"

[1] "dlmGap"
$FF
     [,1] [,2] [,3]
[1,]    1    0    0

$V
     [,1]
[1,]    0

$GG
     [,1] [,2] [,3]
[1,]    0    1    0
[2,]    0    0    1
[3,]    0    0    0

$W
           [,1]         [,2]         [,3]
[1,] 1.00816794 0.0236848488 0.0238410337
[2,] 0.02368485 0.0005564272 0.0005600964
[3,] 0.02384103 0.0005600964 0.0005637899

$m0
[1] 0 0 0

$C0
      [,1]  [,2]  [,3]
[1,] 1e+07 0e+00 0e+00
[2,] 0e+00 1e+07 0e+00
[3,] 0e+00 0e+00 1e+07

In this case
V[epsilon]=W[1,1]=1.00816794
beta1=W[2,1]/W[1,1]=0.02349296
beta2=W[3,1]/W[1,1]=0.02364788

I presume that these two approaches should give comparable results, but this does not happen.
Is the model that I used correct? And does it make sense to perform this kind of comparison?

This is the log of a rainfall time series (which has already been deseasonalised):
[1] 6.014937 4.978801 5.654592 5.616771 5.612398 5.837147 5.121580 5.832176
[9] 5.205654 5.355642 5.405376 6.257859 5.516247 5.500850 4.708629 5.482304
[17] 5.689684 5.727824 4.779123 5.289277 5.217107 5.976351 4.630838 5.683240
[25] 5.345678 5.906179 5.605434 5.497578 5.898801 5.660875 5.111988 5.571013
[33] 5.949340 5.374352 4.841033 5.995706 5.661223 5.458734 4.454347 5.795754
[41] 5.995706 5.596939 5.399971 5.908898 5.282696 5.438514 5.528635 6.022721
[49] 5.524257 5.519459 4.957235 5.547518 5.080783 5.411200 5.056883 5.798183
[57] 5.086361 5.536547 5.220356 5.141664 5.847017 5.052417 5.734635 5.340419
[65] 5.724238 5.634432 5.685958 5.307773 5.817706 5.134032 4.987708 5.110179
[73] 5.423628 5.347108 4.859037 5.556828 5.487283 5.661223 5.732370 5.469325
[81] 5.726848 5.419207 5.172187 5.608006 5.130490 5.586874 5.171052 5.683240
[89] 4.674696 5.286245 5.342813 5.370638 5.432411 5.748118 6.355239 5.557986
[97] 5.399067 5.222516 5.279644 5.425390 5.540871 5.917818 5.132853 5.689007
[105] 5.900993 5.007296 5.102911 5.778271 5.318120 5.927726 5.066385 5.716699
[113] 5.511815 4.714921 5.383577 5.319100 5.269403 5.354698 5.145749 5.204556
[121] 5.878296 5.070161 5.441552 5.213304 5.450180 5.695750 4.893352 5.425390
[129] 5.682559 5.487283 4.213608 5.751620 5.432411 5.379436 5.700444 5.580484
[137] 5.357529 5.319100 4.532599 5.603225 5.208393 5.254888 5.017280 5.349961
[145] 4.374498 5.187944 5.585374 5.716370 3.561046 5.119789 5.163070 5.422745
[153] 5.863915 5.651436 4.762174 5.655642 4.797442 5.735927 4.911183 5.240688
[161] 5.148076 5.477300 4.572647 5.493473 5.437644 4.854371 4.908233 4.755313
[169] 5.582744 5.527841 5.613128 5.211124 5.275049 5.462984 5.016617 5.981919
[177] 5.566817 5.094364 5.314191 5.712742 5.299317 5.452325 4.691348 5.851628
[185] 5.410753 5.488938 5.660179 5.900993 5.380819 5.256453 4.781641 5.531807
[193] 5.497578 5.274537 4.325456 5.271973 5.077047 5.258536 5.280662 5.247024
[201] 5.995208 4.700480 4.991113 5.457029 5.194622 5.487283 5.197391 5.747161
[209] 5.842094 5.372497 5.306781 5.641907 5.565286 5.259057 5.241218 4.759607
[217] 4.550714 5.230574 4.470495 5.664348 4.846547 5.771130 4.823502 5.598422
[225] 5.627621 5.547518 5.596939 5.468482 5.536940 5.606170 5.281680 5.656691
[233] 5.283204 5.752255 5.192401 4.550714


Thank you for your attention and your help
Stefano


________________________________

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Re: Estimating MA parameters through arima or through package "dlm"

mark leeds
Hi: I don't have time to look at the details of what you're doing but the
"equivalence"
between state space and arima ( as paul gilbert pointed out a few weeks ago
) is not a true equivalence.

 if you are in an area of the parameter space that the state space
formulation
 can't reach, then you won't get the same parameter estimates. so, what
you're doing
might be okay or might not be, depending on whether the state space
formulation
can reach that area of the parameter space. there's another state space
formulation that is truly equivalent which is called the SSOE formulation
or innovations representation but
I don't know if you want to get into that. google "SSOE state space" if
you're interested.


Mark






On Mon, Jan 4, 2016 at 9:25 AM, Stefano Sofia <
[hidden email]> wrote:

> Dear list users,
> I want to use apply a MA(2) process (x=beta1*epsilon_(t-1) +
> beta2*epsilon_(t-1) + epsilon_(t)) to a given time series (x), and I want
> to estimate the two parameters beta1, beta2 and the variance of the random
> variable epsilon_(t).
>
> If I use
> MA2_1 <- Arima(x, order=c(0,0,2))
> I get the following result
>
> [1] "MA2_1"
> Series: x
> ARIMA(0,0,2) with non-zero mean
>
> Coefficients:
>           ma1     ma2  intercept
>       -0.0279  0.0783     5.3737
> s.e.   0.0667  0.0622     0.0245
>
> sigma^2 estimated as 0.1284:  log likelihood=-92.63
> AIC=193.25   AICc=193.43   BIC=207.11
> [1] 0 2 0 0 1 0 0
>
> From this straightforward analysis V[epsilon]=0.1284, beta1=-0.0279 and
> beta2=0.0783.
>
> I also tried to use a DLM representation of ARIMA models and estimate the
> unknown parameters by maximum likelihood through the dlm package (in
> particular applying the example at section 3.2.6, page 115, of "Dynamic
> Linear Models with R" by Petris, Petrone and Campagnoli:
>
> arma_parameters <- function(x)
> {
>   buildGap <- function(u)
>   {
>     gap <- dlmModARMA(ma = u[2 : 3], sigma2 = u[1])
>     return(gap)
>    }
>    init <- c(0.005, 0.004, 0.003)
>    outMLE <- dlmMLE(x, init, buildGap)
>    dlmGap <- buildGap(outMLE$par)
> }
>
> and this gives:
> [1] "outMLE"
> $par
> [1] 1.00816794 0.02349296 0.02364788
>
> $value
> [1] 3089.196
>
> $counts
> function gradient
>       10       10
>
> $convergence
> [1] 0
>
> $message
> [1] "CONVERGENCE: REL_REDUCTION_OF_F <= FACTR*EPSMCH"
>
> [1] "dlmGap"
> $FF
>      [,1] [,2] [,3]
> [1,]    1    0    0
>
> $V
>      [,1]
> [1,]    0
>
> $GG
>      [,1] [,2] [,3]
> [1,]    0    1    0
> [2,]    0    0    1
> [3,]    0    0    0
>
> $W
>            [,1]         [,2]         [,3]
> [1,] 1.00816794 0.0236848488 0.0238410337
> [2,] 0.02368485 0.0005564272 0.0005600964
> [3,] 0.02384103 0.0005600964 0.0005637899
>
> $m0
> [1] 0 0 0
>
> $C0
>       [,1]  [,2]  [,3]
> [1,] 1e+07 0e+00 0e+00
> [2,] 0e+00 1e+07 0e+00
> [3,] 0e+00 0e+00 1e+07
>
> In this case
> V[epsilon]=W[1,1]=1.00816794
> beta1=W[2,1]/W[1,1]=0.02349296
> beta2=W[3,1]/W[1,1]=0.02364788
>
> I presume that these two approaches should give comparable results, but
> this does not happen.
> Is the model that I used correct? And does it make sense to perform this
> kind of comparison?
>
> This is the log of a rainfall time series (which has already been
> deseasonalised):
> [1] 6.014937 4.978801 5.654592 5.616771 5.612398 5.837147 5.121580 5.832176
> [9] 5.205654 5.355642 5.405376 6.257859 5.516247 5.500850 4.708629 5.482304
> [17] 5.689684 5.727824 4.779123 5.289277 5.217107 5.976351 4.630838
> 5.683240
> [25] 5.345678 5.906179 5.605434 5.497578 5.898801 5.660875 5.111988
> 5.571013
> [33] 5.949340 5.374352 4.841033 5.995706 5.661223 5.458734 4.454347
> 5.795754
> [41] 5.995706 5.596939 5.399971 5.908898 5.282696 5.438514 5.528635
> 6.022721
> [49] 5.524257 5.519459 4.957235 5.547518 5.080783 5.411200 5.056883
> 5.798183
> [57] 5.086361 5.536547 5.220356 5.141664 5.847017 5.052417 5.734635
> 5.340419
> [65] 5.724238 5.634432 5.685958 5.307773 5.817706 5.134032 4.987708
> 5.110179
> [73] 5.423628 5.347108 4.859037 5.556828 5.487283 5.661223 5.732370
> 5.469325
> [81] 5.726848 5.419207 5.172187 5.608006 5.130490 5.586874 5.171052
> 5.683240
> [89] 4.674696 5.286245 5.342813 5.370638 5.432411 5.748118 6.355239
> 5.557986
> [97] 5.399067 5.222516 5.279644 5.425390 5.540871 5.917818 5.132853
> 5.689007
> [105] 5.900993 5.007296 5.102911 5.778271 5.318120 5.927726 5.066385
> 5.716699
> [113] 5.511815 4.714921 5.383577 5.319100 5.269403 5.354698 5.145749
> 5.204556
> [121] 5.878296 5.070161 5.441552 5.213304 5.450180 5.695750 4.893352
> 5.425390
> [129] 5.682559 5.487283 4.213608 5.751620 5.432411 5.379436 5.700444
> 5.580484
> [137] 5.357529 5.319100 4.532599 5.603225 5.208393 5.254888 5.017280
> 5.349961
> [145] 4.374498 5.187944 5.585374 5.716370 3.561046 5.119789 5.163070
> 5.422745
> [153] 5.863915 5.651436 4.762174 5.655642 4.797442 5.735927 4.911183
> 5.240688
> [161] 5.148076 5.477300 4.572647 5.493473 5.437644 4.854371 4.908233
> 4.755313
> [169] 5.582744 5.527841 5.613128 5.211124 5.275049 5.462984 5.016617
> 5.981919
> [177] 5.566817 5.094364 5.314191 5.712742 5.299317 5.452325 4.691348
> 5.851628
> [185] 5.410753 5.488938 5.660179 5.900993 5.380819 5.256453 4.781641
> 5.531807
> [193] 5.497578 5.274537 4.325456 5.271973 5.077047 5.258536 5.280662
> 5.247024
> [201] 5.995208 4.700480 4.991113 5.457029 5.194622 5.487283 5.197391
> 5.747161
> [209] 5.842094 5.372497 5.306781 5.641907 5.565286 5.259057 5.241218
> 4.759607
> [217] 4.550714 5.230574 4.470495 5.664348 4.846547 5.771130 4.823502
> 5.598422
> [225] 5.627621 5.547518 5.596939 5.468482 5.536940 5.606170 5.281680
> 5.656691
> [233] 5.283204 5.752255 5.192401 4.550714
>
>
> Thank you for your attention and your help
> Stefano
>
>
> ________________________________
>
> AVVISO IMPORTANTE: Questo messaggio di posta elettronica può contenere
> informazioni confidenziali, pertanto è destinato solo a persone autorizzate
> alla ricezione. I messaggi di posta elettronica per i client di Regione
> Marche possono contenere informazioni confidenziali e con privilegi legali.
> Se non si è il destinatario specificato, non leggere, copiare, inoltrare o
> archiviare questo messaggio. Se si è ricevuto questo messaggio per errore,
> inoltrarlo al mittente ed eliminarlo completamente dal sistema del proprio
> computer. Ai sensi dell’art. 6 della DGR n. 1394/2008 si segnala che, in
> caso di necessità ed urgenza, la risposta al presente messaggio di posta
> elettronica può essere visionata da persone estranee al destinatario.
> IMPORTANT NOTICE: This e-mail message is intended to be received only by
> persons entitled to receive the confidential information it may contain.
> E-mail messages to clients of Regione Marche may contain information that
> is confidential and legally privileged. Please do not read, copy, forward,
> or store this message unless you are an intended recipient of it. If you
> have received this message in error, please forward it to the sender and
> delete it completely from your computer system.
>
>         [[alternative HTML version deleted]]
>
> ______________________________________________
> [hidden email] mailing list -- To UNSUBSCRIBE and more, see
> 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.

        [[alternative HTML version deleted]]

______________________________________________
[hidden email] mailing list -- To UNSUBSCRIBE and more, see
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PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
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