I have got a model for forecasting using holt-winters. However the parameters confuse me... The parameters show that there is no trend or seasonality even though there is definite trend and seasonality in the data, the forecasts also match the pattern of the data so the parameters really do not make sense.

M<-matrix(c("08Q1", "08Q2", "08Q3", "08Q4", "09Q1", "09Q2", "09Q3", "09Q4", "10Q1", "10Q2", "10Q3", "10Q4", "11Q1", "11Q2", "11Q3", "11Q4", "12Q1", "12Q2", "12Q3", "12Q4", "13Q1", "13Q2", "13Q3", "13Q4", "14Q1", "14Q2", "14Q3",  5403.676,  6773.505,  7231.117,  7835.552,  5236.710, 5526.619,  6555.782, 11464.727,  7210.069,  7501.610,  8670.903, 10872.935,  8209.023,  8153.393, 10196.448, 13244.502,  8356.733, 10188.442, 10601.322, 12617.821, 11786.526, 10044.987, 11006.005, 15101.946, 10992.273, 11421.189, 10731.312),ncol=2,byrow=FALSE)
Nu <- M[, length(M[1,])] 
Nu <- ts(Nu, deltat=1/4, start = c(8,1))
HWMb <- ets(N, model = "MAM", damped = FALSE, opt.crit = "lik", ic="aic", lower = c(0.001, 0.001, 0.001, 0.001), 
            upper = c(0.999, 0.999, 0.999, 0.999), bounds = "admissible", restrict = FALSE)

Smoothing parameters:
alpha = 0.0183 
beta  = 0.0056 
gamma = 0.0027

This shows the time series with the forecasts where you can see definite seasonality and trend

full data

Are these parameters normal for holt-winters?

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    $\begingroup$ You should probably mention you're using the forecast package. Also your third line is incomplete so your example doesn't run. Please give code that actually works. $\endgroup$ – Glen_b -Reinstate Monica Oct 28 '14 at 9:46

The small values for $\beta$ and $\gamma$ show that the trend and seasonality do not change much over time. They do not tell you that there is no trend or seasonality.

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  • $\begingroup$ Ahhh so there isn't actually anything wrong with them? $\endgroup$ – Summer-Jade Gleek'away Oct 27 '14 at 10:33
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    $\begingroup$ Youre one of the authors of the forecasting books i was given when i started my job. It has been such a big help to me in the past three months! $\endgroup$ – Summer-Jade Gleek'away Oct 27 '14 at 10:43

All parameters, $\alpha$, $\beta$ and $\gamma$, have values between 0 and 1. In broad terms, a simple exponential smoothing model looks like this (though the idea also works for double and triple exponential smoothing):

$$ \mathit{smoothed_t} = \color{blue}{\mathit{parameter}} \cdot \mathit{observation_t} + (\color{blue}{1 - \mathit{parameter}}) \cdot \mathit{smoothed_{t-1}} $$

So the closer a parameter is to 0, the lower the weight of the present observation and the higher the weight of previous estimatives in determining the updated statistic. Your model still recognized the presence of level, trend and seasonality, it just weighs older observations higher than newer ones.

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