Nayland College - Mathematics Home . Year 9 . Year 10 . Level 1 . Level 2 . L3 Statistics . L3 Maths . L3 Calculus . About . Links

# The Trend

REMEMBER: there are no indication of units and scales on the vertical axis. You MUST incorporate this, eg. Sea Ice is measured in millions of km2

Discussion: Of the Long Term Trend in very general terms but in context
eg. The Arctic Sea Ice area has decreased from about 9.5 million km2 in 1990 to about 8.5 million km2 in 2010

In more detail: The Arctic sea ice area changes from 9.5 million km2 in Jan 1990 to about 8.5 million km2 in March 2011, which is a change of 1 million km2 over 21 years and 3 months (which is 254 monthly increments) so the average sea ice change is about 4000 km2 per month (1 000 000 ÷ 254)

Trend Video (you tube video by Pricilla Allan)

How is the smoothing of the time series and future predictions done in iNZight?

Using the Holt - Winters LOWESS technique

Summary of what students need to know (by Rachel Passmore)

• Holt –Winters Additive model assumes seasonal pattern is reasonably constant

• Holt –Winters Multiplicative model is usually better when there is a change in the seasonal pattern - eg seasonal variation increasing (Find out more)

• Holt-Winters Model uses a technique of exponential smoothing, which is a weighted sum of previous values in a series. More weight is given to more recent values and less weight is given to values from the distant past. This method uses the seasonal LOWESS technique (which stands for LOcally WEighted Seasonal Smoother)

• Holt-Winters Additive model exponentially smooth's the series in order to produce predictions – the level, the trend and the seasonal sub-series.

• Students should be able to identify cyclical components and inconsistent seasonal patterns. They should note that such features are incompatible with assumptions underlying Holt-Winters Additive model and suggest a multiplicative model be considered instead. Such a comment would be expected at Excellence level only. Students are NOT expected to calculate a multiplicative model.

Examples:

Traveler numbers from different countries to NZ (Sept 1998 to October 2012 inclusive)

### Be Careful!!

The trend line can vary at the ends due to where the cycle starts and finishes. To avoid this error, use the trend line values half a cycle from the ends of the raw data as estimates, rather than the end of the trend line

Data set ending in April 2012: Notice the trend line INCREASES
As the data ends just after a peak so the trend line increases.

Data set ending in JULY 2012: Notice the trend line remains STEADY
As the data ends at a 'balanced point, so the trend line increases.

Data set ending in OCTOBER 2012: Notice the trend line DROPS
As the data ends just after a trough so the trend line drops.

How to cater for this?

To avoid this error, use the trend line values half a cycle from the ends of the raw data as estimates, rather than the end of the trend line. Often an insignificant difference, but avoid discussing the 'possible recent downturn' which isn't there (Excellence point)

This should be discussed if noticed in the trend line.