Stefano Sofia
2016-Jan-04 14:25 UTC
[R] Estimating MA parameters through arima or through package "dlm"
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 pu? contenere
informazioni confidenziali, pertanto ? 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 ?
il destinatario specificato, non leggere, copiare, inoltrare o archiviare questo
messaggio. Se si ? 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 necessit? ed urgenza, la
risposta al presente messaggio di posta elettronica pu? 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]]
Mark Leeds
2016-Jan-04 16:31 UTC
[R] Estimating MA parameters through arima or through package "dlm"
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 < stefano.sofia at regione.marche.it> 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 pu? contenere > informazioni confidenziali, pertanto ? 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 ? il destinatario specificato, non leggere, copiare, inoltrare o > archiviare questo messaggio. Se si ? 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 necessit? ed urgenza, la risposta al presente messaggio di posta > elettronica pu? 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]] > > ______________________________________________ > R-help at r-project.org 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]]