Guest contribution by Willis Eschenbach
Commenting on my previous post got me thinking about cumulative sums of a series of numbers. In a “cumulative total” we start with the first value in the series and create a new series by adding each number in the old series to the sum of the new series.
So if the series is say (1, 3, 7, 10), the new series, which is the cumulative sum of the old series, is (1, 4, 11, 21). It is calculated as
- 1
- 1 + 3 = 4
- 4 + 7 = 11
- 11 + 10 = 21
Why is there a cumulative sum of interest? It can reveal underlying trends and changes in data sets. For example, consider the Pacific Decadal Oscillation, also known as the PDO. This is a slow change in the Pacific, with the northernmost part alternately getting cooler and warmer.
Figure 1. The two extremes of sea surface temperatures in the Pacific Decadal Oscillation.
And here is a chart of the PDO index that measures the status of the PDO.
Figure 2. PDO index and date of the “Great Pacific Climate Shift” from 1976-1977.
And what is the Great Pacific Climate Shift when he’s at home? It was the date of the first shift in the Pacific Decadal Oscillation identified by scientists. (As a longtime commercial salmon fisherman, I really appreciate the fact that the PDO was first noticed in records of salmon catches in the Pacific Northwest … but I digress). From the Journal of Climate article “The Importance of Pacific Climate Change 1976 in Alaskan Climatology”:
In 1976, the North Pacific region, including Alaska, saw a dramatic shift towards a climatic regime in which winter and spring temperatures rose sharply in summer and fall compared to the previous 25 years.
And what does 1976 Pacific climate change have to do with accumulated sums? It becomes obvious when we graph the cumulative sum of the PDO index as shown below.
Figure 3. Cumulative sum of the PDO index and the 1977 Pacific Climate Shift date
As you can see, the cumulative sum of the PDO index clearly shows the date of the shift of the Pacific climate into the warm phase of the PDO.
What else can cumulative sums do? You can show us whether two records are linked. Here are the cumulative sums of
together with the PDO.
Figure 4. Cumulative sum of the PDO index and the 1977 Pacific Climate Shift date
These four climate indices are based on very different things. The PDO index is the first major component of sea surface temperatures north of 20 ° N. The Southern Ocean Index is based on the difference in air pressure between Tahiti and Australia. The NINO34 index is based on the sea surface temperatures in the region of 5 ° N-5 ° S and 170 ° W-120 ° W. The North Pacific index is based on the area-weighted sea level pressure in the region of 30 ° N-65 ° N, 160 ° E -140 ° W.
And yet they all clearly show the climate change in the Pacific from 1976 …
What else can we do with accumulated amounts? Well, we can use them to show what records are as well Not related … here are a few examples.
Figure 5 Cumulative Sums, Pacific Decadal Oscillation Index, North Atlantic Oscillation Index, and Monthly Sunspots
Not much in common in these three. This implicitly shows that sunspots and the North Atlantic Oscillation are also not closely related to the El Nino indices shown in Figure 4.
w.
Technical note: The accumulated amounts are very sensitive to initial conditions and anomalies. When the data is trending up, or when the zero point is lower than the values, a cumulative sum is directed towards the sky and vice versa. However, the anomaly of the same data behaves very differently. All of the above cumulative sums were initially expressed as an anomaly over the mean of the data set in question. For trendless indices, this makes little difference. It ensures that they return to the value they started at. If you think of them as anomalies around another zero point, it will result in an overall trend depending on the point you choose. As a result, the trend of a cumulative sum is generally meaningless, but as shown above, the variations in the cumulative sum can be quite significant.
My usual tip: Please, I beg you, quote the exact words you are discussing. Without that, it is often impossible to tell who or what you are referring to, and misunderstandings pile up.
Like this:
Loading…
Comments are closed.