The SPLINE Method

The SPLINE method fits a cubic spline curve to the input values. A cubic spline is a segmented function consisting of third-degree (cubic) polynomial functions joined together so that the whole curve and its first and second derivatives are continuous.

For point-in-time input data, the spline curve is constrained to pass through the given data points. For interval total or average data, the definite integrals of the spline over the input intervals are constrained to equal the given interval totals.

For boundary constraints, the not-a-knot condition is used by default. This means that the first two spline pieces are constrained to be part of the same cubic curve, as are the last two pieces. Thus the spline used by PROC EXPAND by default is not the same as the commonly used natural spline, which uses zero second-derivative endpoint constraints. While DeBoor (1981) recommends the not-a-knot constraint for cubic spline interpolation, using this constraint can sometimes produce anomalous results at the ends of the interpolated series. PROC EXPAND provides options to specify other endpoint constraints for spline curves.

To specify endpoint constraints, use the following form of the METHOD= option.

METHOD=SPLINE( constraint < , constraint > )

The first constraint specification applies to the lower endpoint, and the second constraint specification applies to the upper endpoint. If only one constraint is specified, it applies to both the lower and upper endpoints.

The constraint specifications can have the following values:

NOTAKNOT
specifies the not-a-knot constraint. This is the default.

NATURAL
specifies the natural spline constraint. The second derivative of the spline curve is constrained to be zero at the endpoint.

SLOPE= value
specifies the first derivative of the spline curve at the endpoint. The value specified can be any positive or negative number, but extreme values may produce unreasonable results.

CURVATURE= value
specifies the second derivative of the spline curve at the endpoint. The value specified can be any positive or negative number, but extreme values may produce unreasonable results. Specifying CURVATURE=0 is equivalent to specifying the NATURAL option.

For example, to specify natural spline interpolation, use the following option in the CONVERT or PROC EXPAND statement:

   method=spline(natural)

For OBSERVED=BEGINNING, MIDDLE, and END series, the spline knots are placed at the beginning, middle, and end of each input interval, respectively. For total or averaged series, the spline knots are set at the start of the first interval, at the end of the last interval, and at the interval midpoints, except that there are no knots for the first two and last two midpoints.

Once the cubic spline curve is fit to the data, the spline is extended by adding linear segments at the beginning and end. These linear segments are used for extrapolating values beyond the range of the input data.

For point-in-time output series, the spline function is evaluated at the appropriate points. For interval total or average output series, the spline function is integrated over the output intervals.

The JOIN Method

The JOIN method fits a continuous curve to the data by connecting successive straight line segments. For point-in-time data, the JOIN method connects successive nonmissing input values with straight lines. For interval total or average data, interval midpoints are used as the break points, and ordinates are chosen so that the integrals of the piecewise linear curve agree with the input totals.

For point-in-time output series, the JOIN function is evaluated at the appropriate points. For interval total or average output series, the JOIN function is integrated over the output intervals.

The STEP Method

The STEP method fits a discontinuous piecewise constant curve. For point-in-time input data, the resulting step function is equal to the most recent input value. For interval total or average data, the step function is equal to the average value for the interval.

For point-in-time output series, the step function is evaluated at the appropriate points. For interval total or average output series, the step function is integrated over the output intervals.

The AGGREGATE Method

The AGGREGATE method performs simple aggregation of time series without interpolation of missing values.

If the input data are totals or averages, the results are the sums or averages, respectively, of the input values for observations corresponding to the output observations. That is, if either TOTAL or AVERAGE is specified for the OBSERVED= option, the METHOD=AGGREGATE result is the sum or mean of the input values corresponding to the output observation. For example, suppose METHOD=AGGREGATE, FROM=MONTH, and TO=YEAR are specified. For OBSERVED=TOTAL series, the result for each output year is the sum of the input values over the months of that year. If any input value is missing, the corresponding sum or mean is also a missing value.

If the input data are point-in-time values, the result value of each output observation equals the input value for a selected input observation determined by the OBSERVED= attribute. For example, suppose METHOD=AGGREGATE, FROM=MONTH, and TO=YEAR are specified. For OBSERVED=BEGINNING series, January observations are selected as the annual values. For OBSERVED=MIDDLE series, July observations are selected as the annual values. For OBSERVED=END series, December observations are selected as the annual values. If the selected value is missing, the output annual value is missing.

The AGGREGATE method can be used only when the FROM= intervals are nested within the TO= intervals. For example, you can use METHOD=AGGREGATE when FROM=MONTH and TO=QTR because months are nested within quarters. You cannot use METHOD=AGGREGATE when FROM=WEEK and TO=QTR because weeks are not nested within quarters.

In addition, the AGGREGATE method cannot convert between point-in-time data and interval total or average data. Conversions between TOTAL and AVERAGE data are allowed, but conversions between BEGINNING, MIDDLE, and END are not.

Missing input values produce missing result values for METHOD=AGGREGATE. However, gaps in the sequence of input observations are not allowed. For example, if FROM=MONTH, you may have a missing value for a variable in an observation for a given February. But if an observation for January is followed by an observation for March, there is a gap in the data, and METHOD=AGGREGATE cannot be used.

When the AGGREGATE method is used, there is no interpolating curve, and therefore the EXTRAPOLATE option is not allowed.

Alternate methods for aggregating or accumulating time series data are supported by the TIMESERIES procedure. See Chapter 27, The TIMESERIES Procedure, for more information.

METHOD=NONE

The option METHOD=NONE specifies that no interpolation be performed. This option is normally used in conjunction with the TRANSFORMIN= or TRANSFORMOUT= option.

When METHOD=NONE is specified, there is no difference between the TRANSFORMIN= and TRANSFORMOUT= options; if both are specified, the TRANSFORMIN= operations are performed first, followed by the TRANSFORMOUT= operations. TRANSFORM= can be used as an abbreviation for TRANSFORMIN=. METHOD=NONE cannot be used when frequency conversion is specified.