Tuesday, 24 June 2008

Can't see the tress... Because there's no forest

Climate has varied on every time scale to which we have any observational access. Ice ages come and go on time scales of tens of thousands of years, for example. . . . Climate changes. It changes on all time scales. What's different between our time and our grandparents' time is that now humankind, which has been a passive spectator at this great natural pageant, has become an actor and is up on the stage. And what we—all 6 billion of us—do can affect the climate”. (Quoted from PBS, What's Up with the Weather?)

Speculation into CO2 changing the whether is the least of the world’s problems. We are doing our best to both deforest the planet and to reduce the algal plankton. Both factors do impact climate. More importantly, they have a positive effect to existing situations that can be modelled now without speculation.

Paleoclimatic Glacial Varves

Melting glaciers deposit yearly layers of sand and silt during the spring melting seasons, which can be reconstructed yearly a period ranging from the time deglaciation began in New England (about 12,600 years ago) to the time it ended (about 6,000 years ago). Such sedimentary deposits, called varves, can be used as proxies for paleoclimatic parameters, such as temperature, because, in a warm year, more sand and silt are deposited from the receding glacier. Dataset varve.sav contains glacial varve thicknesses from one location in Massachusetts for 634 years, beginning 11,834 years ago. The timeplot of the data is given in the figure 1 below. Because the variation in thicknesses increases in proportion to the amount deposited, a logarithmic transformation could improve the nonstationarity observable in the variance as a function of time.

Figure 1 - A time plot of the varve thickness

The variance of log(varve) is more stable. Outliers are less obvious. Some minor spikes (see observation numbers 568 and 572 remain unusually large and are unusually high). The plot suggests non-stationarity.

The slowly decaying autocorrelations of log(varve) indicate nonstationarity which will be examined to see if it can be removed by differencing.

A time plot of first differences of log(varve) is given below.
The first difference looks like a stationary series. The autocorrelogram is given below. In the differenced series, autocorrelations decay rapidly. In fact, only the lag 1 autocorrelation is significant, suggesting the possibility of an ARIMA(0,1,1) model. First, however, We examine the partial autocorrelations of the first difference of log(varve).




The partial autocorrelations decline slowly, supporting a moving average, rather than an autoregressive process.

From this we see that a good starting model would be an ARIMA(0,1,1) model of logarithmically-transformed data. Other models could be ARIMA(p,1,1), for p=1,2,3,4,5. Calculating the AIC and SBC, we obtain the most minimum AIC and SBC with the ARIMA(1,1,1) model without constant.

To be continued... Post Part 2

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