Smooth one-dimensional data by error detection.

The type double function is imsl_d_smooth_1d_data.

A pointer to the vector of length ndata containing the smoothed data.

The function imsl_f_smooth_1d_data is designed to smooth a data set that is mildly contaminated with isolated errors. In general, the routine will not work well if more than 25% of the data points are in error. The routine imsl_f_smooth_1d_data is based on an algorithm of Guerra and Tapia (1974).

Setting ndata = n, ydata = f, sdata = s and xdata = x, the algorithm proceeds as follows. Although the user need not input an ordered xdata sequence, we will assume that x is increasing for simplicity. The algorithm first sorts the xdata values into an increasing sequence and then continues. A cubic spline interpolant is computed for each of the 6-point data sets (initially setting s = f)

where i = 4, …, n - 3. For each i the interpolant, which we will call Si, is compared with the current value of si, and a ‘point energy’ is computed as

Setting sc = sc, the algorithm terminates either if itmax iterations have taken place or if

If the above inequality is violated for any i, then we update the i-th element of s by setting si = si + d(pei), where d = dis. Note that neither the first three nor the last three data points are changed. Thus, if these points are inaccurate, care must be taken to interpret the results.

The choice of the parameters d, sc and itmax are crucial to the successful usage of this subroutine. If the user has specific information about the extent of the contamination, then he should choose the parameters as follows: d = 1, sc = 0 and itmax to be the number of data points in error. On the other hand, if no such specific information is available, then choose d =  5, itmax ≤ 2n, and

In any case, we would encourage the user to experiment with these values.

We take 91 uniform samples from the function 5 + (5 + t2 sin t)/t on the interval [1, 10]. Then, we contaminate 10 of the samples and try to recover the original function values.