Basis
Functions

The Basics of Basis Functions

The Characteristics of Spline Functions

The Characteristics of Spline Functions II

Goods

Dissecting the U.S. Nondurable Goods Index

Smoothing the Nondurable Goods Index

-Smoothing with a fourth order roughness penalty
-A first look at the seasonal variation

Phase-Plane Plotting of the Goods Index

Growth

The Features of Human Growth Data

From Raw Data to Functional Data

Smoothing the Growth Data

Monotone Smoothing of the Growth Data

Weather

Weather Introduction

Summarizing the Functional Data

Weather Data Analysis

Smoothing the Nondurable Goods Index
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Expertise: Intermediate

A first look at the seasonal variation
Figure 1 zooms in on the data and the fitted curve for the period 1964 through 1967. We see a tendency to three peaks per year, with the largest being in the fall. The low point of the index in each year is in mid December. We can also see the linear trend over this period.

Figure 1: The log nondurable goods index for 1964 through 1966. The circles are the observed
values, and the smooth curve is the smoothing spline fit to the data.

In order to look more closely at seasonal variation, it is desirable to first estimate the long-term nonseasonal trend. The straight line in the plot of the log index is too crude; we need to estimate the finer scale wiggles in the curve without trying to track variation within each year. To do this, we return to spline smoothing, but this time place a knot at each year, and smooth with order 4 splines. This fit does not have the resolving power to show much curvature within any twelve-month period. We then subtract this long-term trend from the fit to the data computed above to get a better view of the strictly seasonal variation. This is shown in Figure 2.

Figure 2: The seasonal variation in the log nondurable goods index estimated by estimating the
long-term trend and then subtracting it from the high-resolution smooth of the data.

Now we can see that the major events of the century are also reflected in the seasonal trend. Seasonal variation balloons to 6% during the depression, and again around the Treasury Board closure of the money supply, but drops to its lowest level during World War II. The sixties are a remarkably orderly period, but the end of the Vietnam War first shows up as increased variability, and then in the following years as decreased stability. The size of seasonal variation seems to be on the decrease in recent years.

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