Tuesday, November 22, 2011

Averages of Averages

This is how I thought of it at first:

The Double-3: Take a zero-phase 3-year moving average of a zero-phase 3-year moving average of some economic data... I think this would be like assigning weights to the numbers, heaviest at the current value, lightest two years ahead or behind.

When I looked at it in Excel, a Double-5 gave a result more like what I had in mind:

Looking at any point on a trend line, the actual value at that point should have more significance than neighboring points; and more distant neighbors should have even less. I think this is a decent way to generate a trend line for an economic dataset.

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Jazzbumpa said...

This gives you double smoothing, but no current smoothed data. How do you intend to use it?

Cheers
JzB

Jazzbumpa said...

Try this.

JzB

The Arthurian said...

Good post, Jazz.

I intend to use the data to obtain a smoother picture of past trends. I do not really need "current" data. I am studying the past. I am not making investment decisions in the current moment.

A 3-year moving average contains one-third of each of the values (let us say) for 1961, 1962, and 1963. These values are unweighted. These observations are true regardless of whether the point is plotted at 1962 (for a zero-phase picture) or at 1963 (for a causal picture).

A Double-Three moving average contains one-ninth of each of the values for 1961, 1962, 1963, and 1962, 1963, 1964, and 1963, 1964, 1965. In sum: the Double-Three contains one ninth of the 1961 and 1965 values; two ninths of the 1962 and 1964 values, and three ninths of the 1963 value. In other words, the Double-Three uses weighted data.

jim said...

Hi Art,

You have progressed from a rectangular filter to a triangle filter.
triangle filter

-jim

The Arthurian said...

Jim,
I have heard of a somebody-prescott filter, which is I think supposed to strip out minor variations (small business cycles maybe) and provide a picture of longer-term trends (longwave cycles maybe). But that isn't intuitive for me, and what I found on the internet required more work than I have yet put into it.

What I am looking for is to determine longer-term trends objectively, I guess to see the forest despite the trees.

Thanks... and have a good long weekend.

jim said...

Hi Art,

The late response is because I was too busy eating turkey yesterday.

My familiarity with filters is mostly in the area of audio and image processing not statistics.

I think what you are doing is about as good as it will get if your goal is to look at the smoothed data and find long-term peaks and troughs in the data. You don't want the peaks shifted in time by filtering so you don't want a causal filter.

If you want to get a little more fancy you might view this as a downsampling and reconstruction problem. The basic point behind a filter designed for downsampling is you want to remove the high frequency content that can't be supported at the lower sample rate. You aren't actually interested in a lower sample rate, but the low-pass filters designed for that purpose will be the best filter for removing high frequency content above a certain threshold.
However, for your purposes that is likely to be overkill.