Adding It Up

Guest Post by Willis Eschenbach

A comment on my previous post got me to thinking about cumulative sums of a series of numbers. In a “cumulative sum”, we start with the first value in the series, and make a new series by adding each number in the old series to the total of the new series.

So if the series is say (1, 3, 7, 10), the new series that is the cumulative sum of the old series is (1, 4, 11, 21). It’s calculated as

• 1
• 1 + 3 = 4
• 4 + 7 = 11
• 11 + 10 = 21

Why is a cumulative sum of interest? It can reveal underlying trends and changes in datasets. For example, consider the “Pacific Decadal Oscillation”, also known as the “PDO”. This is a slow change in the Pacific, in which the northernmost part alternately gets cooler and warmer.

Figure 1. The two extremes of sea surface temperatures in the Pacific Decadal Oscillation.

And here is a graph of the PDO Index, which measures the state of the PDO.

Figure 2. PDO Index, and the date of the “Great Pacific Climate Shift” of 1976-1977.

And what is the “Great Pacific Climate Shift” when it is at home? It was the date of the first shift in the Pacific Decadal Oscillation that was identified by scientists. (As a long-time commercial salmon fisherman, I greatly 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 Significance of the 1976 Pacific Climate Shift in the Climatology of Alaska“:

In 1976, the North Pacific region, including Alaska, underwent a dramatic shift to a climate regime that saw great increases in winter and spring temperatures, and lesser increases in summer and autumn, when compared to the previous 25 yr.

And what does the 1976 Pacific Climate Shift have to do with cumulative 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 date of the 1977 Pacific Climate Shift

As you can see, the cumulative sum of the PDO index clearly shows the date of the shift in the Pacific climate to the warm phase of the PDO.

What else can cumulative sums do? They can show us if two datasets are related to each other. Here are the cumulative sums of

• The NINO34 Index
• The Southern Oscillation Index
• The North Pacific Index

along with the PDO.

Figure 4. Cumulative sum of the PDO Index, and the date of the 1977 Pacific Climate Shift

These four indices of the climate are based on very different things. The PDO Index is the first Principal Component of sea surface temperatures north of 20°N. The Southern Oscillation Index is based on the difference in barometric pressure between Tahiti and Australia. The NINO34 Index is based on sea surface temperatures in the region 5°N-5°S and 170°W-120°W. And the North Pacific Index is based on the area-weighted sea level pressure over the region 30°N-65°N, 160°E-140°W.

And despite that, they all clearly show the 1976 Pacific climate shift …

What else can we do with cumulative sums? Well, we can also use them to show which datasets are not related … here’s a couple of examples.

Figure 5 Cumulative Sums, Pacific Decadal Oscillation Index, North Atlantic Oscillation Index, and Monthly Sunspots

Not a whole lot of commonality in those three. And by implication, this shows that sunspots and the North Atlantic Oscillation are also not closely related to the El Nino indices shown in Figure 4 … and that’s it for cumulative sums for now.

[UPDATE] A commenter pointed out that I hadn’t included the Atlantic Multidecadal Oscillation (AMO) in the analysis. Here’s that graph:

w.

Technical Note: Cumulative sums are very sensitive to initial conditions and anomalies. If there is an upward trend in the data, or if the zero point is lower than the values, a cumulative sum will head for the sky, and vice versa. However, the anomaly of that same data will behave very differently. All of the cumulative sums above were first expressed as an anomaly about the mean of the dataset in question. For trendless indices, this makes little difference. It does ensure that they return to the value at which they started. Taking them as anomalies around any other zero point will lead to an overall trend depending on the point chosen. 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 meaningful.

My Usual Note: Please, I implore you, quote the exact words you are discussing. Without that it is often impossible to tell either who or what you are referring to, and misunderstandings multiply.