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# VINTERPOLATE procedure

Performs linear & inverse linear interpolation between variates (R.J. Reader).

### Options

`METHOD` = string token Type of interpolation required (`interval`, `value`): for `METHOD=value`, y-values are interpolated for each point in the `NEWINTERVALS` variates and stored in the `NEWVALUES` variates, while for `METHOD=interval`, x-values are estimated for the y-values in the `NEWVALUES` variates and stored in the `NEWINTERVALS` variates; default `inte` Whether the smallest value, largest value or the mean of the two is returned if more than one value is valid (`first`, `middle`, `last`); default `midd`

### Parameters

`OLDVALUES` = pointers Each one contains variates specifying the y-values (data values) with which an interpolation is to be carried out For `METHOD=value`, each pointer contains variates to store the results of an interpolation; for `METHOD=interval`, it contains either variates or scalars to specify y-values for which inverse interpolation is to be carried out Contains the x-values (intervals) corresponding to the variates in the `OLDVALUES` pointer For `METHOD=interval`, each pointer contains variates to store the results of an inverse interpolation; for `METHOD=value`, it contains either variates or scalars to specify x-values at which interpolation is to be performed

### Description

`VINTERPOLATE` performs linear interpolation and inverse linear interpolation between variates, as was given by the Genstat 4 functions `LINT` and `INLINT`. The y-values (or data values) are given in a set of variates, contained in the pointer specified by the `OLDVALUES` parameter. The corresponding x-values (intervals) are specified by the `OLDINTERVALS` parameter, in a variate with one value for each variate of the `OLDVALUES` pointer. For interpolation (`METHOD=value`), values are interpolated at the x-values specified by the variates or scalars contained in the `NEWINTERVALS` pointer, and are stored in variates contained in the `NEWVALUES` parameter. For inverse interpolation (`METHOD=interval`), x-values are estimated for the y-values specified by the variates or scalars contained in the `NEWVALUES` pointer, and are stored in variates contained in the `NEWINTERVALS` pointer. Where two or more successive `OLDINTERVALS` or `OLDVALUES` are the same, there is no unique solution to the interpolation; the `RANGEMETHOD` option allows the smallest (`RANGEMETHOD=first`), largest (`RANGEMETHOD=last`) or the mean of these two (`RANGEMETHOD=middle`) values or intervals to be selected.

Options: `METHOD`, `RANGEMETHOD`.

Parameters: `OLDVALUES`, `NEWVALUES`, `OLDINTERVALS`, `NEWINTERVALS`.

### Method

An estimate of the required value is calculated from each successive pair of points. If this estimate is between the two points from which it was calculated, it is a valid answer i.e. it was produced by interpolation not extrapolation. If it does not satisfy this condition it is set to missing. If the curve is horizontal or vertical at the point of interpolation, more than one point will satisfy the above condition. In this case the value returned will depend on the setting of the option `RANGEMETHOD`. This will determine whether the smallest value found, the largest value found, or the mean of these two values is returned. The default is to return the mean value.

### Action with `RESTRICT`

If any of the variates in the `OLDVALUES` pointer is restricted, the other variates in the pointer must either have the same restriction or be unrestricted. Missing values are then returned for the units excluded by the restriction. The `OLDINTERVALS` variate must not be restricted.

Directive: `INTERPOLATE`.

Procedure: `VREGRESS`.

Commands for: Calculations and manipulation.

### Example

```CAPTION 'VINTERPOLATE example'; STYLE=meta
VARIATE Y[1...10]; VALUES=!( 0,  0,  0, 22,  2,  0,  0,  2),\
!( 0,  3,  0, 19, 23,  0, 10, 60),\
!( 7, 53,  1, 10, 54,  7, 27, 84),\
!(40, 68, 11, 10, 63, 12, 31, 88),\
!(46, 69, 16,  6, 64, 12, 31, 88),\
!(51, 70, 17,  6, 65, 15, 33, 88),\
!(52, 71, 17,  3, 65, 15, 33, 88),\
!(52, 71, 18,  0, 65, 15, 33, 88),\
!(66, 71, 63,  0, 65, 15, 34, 88),\
!(66, 71, 65,  0, 65, 15, 34, 88)
&       [VALUES=1,4,7,10,13,16,19,22,25,28] X
FOR     [NTIMES=1]
CAPTION !t('The data are contained in the variates Y[1...10], whose',\
'values were observed at the times stored in the variate X.')
PRINT [RLWIDTH=1; ORIENTATION=across; SQUASH=yes] X; FIELD=7; DECIMALS=0
PRINT [INDENT=1]  Y[]; FIELD=7; DECIMALS=0
ENDFOR
CAPTION 'Interpolate values for times 2, 4 and 20.'
VINTERPOLATE [METHOD=value] OLDVALUES=Y; OLDINTERVALS=X;\
NEWVALUES=!P(Y2,Y4,Y20); NEWINTERVALS=!P(2,4,20)
PRINT   Y2[],Y4[],Y20[]
CAPTION !t('Use inverse interpolation to obtain the times',\
'at which the values 10, 50 and 75 were reached.',\
'RANGE option set to take the smallest value from ties.')
VINTERPOLATE [RANGE=first] OLDVALUES=Y; OLDINTERVALS=X;\
NEWVALUES=!P(10,50,75); NEWINTERVALS=!P(X10,X50,X75)
PRINT   X10[],X50[],X75[]
```
Updated on March 4, 2019