IMSL_SMOOTHDATA1D

The IMSL_SMOOTHDATA1D function smooths one-dimensional data by error detection.

Note
This routine requires an IDL Advanced Math and Stats license. For more information, contact your ITT Visual Information Solutions sales or technical support representative.

Syntax

Result = IMSL_SMOOTHDATA1D(x, y [, DISTANCE=value] [, /DOUBLE] [, ITMAX=value] [, SC=value])

Return Value

One-dimensional array containing the smoothed data.

Arguments

x

One-dimensional array containing the abscissas of the data points.

y

One-dimensional array containing the ordinates of the data points.

Keywords

DISTANCE

Proportion of the distance the ordinate in error is moved to its interpolating curve. It must be in the range 0.0 to 1.0. Default: DISTANCE = 1.0

DOUBLE

If present and nonzero, double precision is used.

ITMAX

The maximum number of iterations allowed. Default: ITMAX = 500

SC

The stopping criterion. SC should be greater than or equal to zero. Default: SC = 0.0

Discussion

The IMSL_SMOOTHDATA1D function 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_SMOOTHDATA1D is based on an algorithm of Guerra and Tapia (1974).

Setting N_ELEMENTS(x) = n, Y = f, Result = s and X= x, the algorithm proceeds as follows. Although the user need not an ordered x sequence, we will assume that x is increasing for simplicity. The algorithm first sorts the x 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 ):

(x_j, s_j) j = i – 3, ... , i + 3 j ≠ i

where i = 4, ... , n – 3. For each i the interpolant, which we will call S_i, is compared with the current value of s_i, and a `point energy' is computed as:

pe_i= S_i(x_i) – s_i

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 s_i = s_i + d(pe_i), where d = DISTANCE. 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 DISTANCE, 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: DISTANCE = 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 DISTANCE = 0.5, ITMAX ≤ 2n, and:

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

Example

We take 91 uniform samples from the function 5 + (5 + t2 sin t )/t on the interval [1, 10]. First, define function F from which samples will be taken

FUNCTION F, xdata 
   RETURN, (xdata*xdata*SIN(xdata) + 5)/xdata + 5 
END

Next, we contaminate 10 of the samples and try to recover the original function values.

isub  =  [5, 16, 25, 33, 41, 48, 55, 61, 74, 82] 
rnoise  =  [2.5, -3.0, -2.0, 2.5, 3.0, -2.0, -2.5, 2.0, -2.0, 3.0] 
 
; Example 1: No specific information available. 
dis  =  0.5 
sc  =  0.56 
itmax  =  182 
; Set values for xdata and fdata.      
xdata  =  1 + 0.1*FINDGEN(91) 
fdata  = f(xdata) 
 
; Contaminate the data. 
fdata(isub)  =  fdata(isub) + rnoise 
 
; Smooth the data. 
sdata  = IMSL_SMOOTHDATA1D(xdata, fdata, Itmax = itmax, $ 
   Distance = dis, Sc = sc) 
 
; Output the results. 
PM, [[f(xdata(isub))], [fdata(isub)], [sdata(isub)]], $ 
   Title  =  '        F(X)    F(X) + noise     sdata'       
           F(X)    F(X) + noise     sdata 
         9.82958      12.3296      9.87030 
         8.26338      5.26338      8.21537 
         5.20083      3.20083      5.16823 
         2.22328      4.72328      2.26399 
         1.25874      4.25874      1.30825 
         3.16738      1.16738      3.13830 
         7.16751      4.66751      7.13076 
         10.8799      12.8799      10.9092 
         12.7739      10.7739      12.7075 
         7.59407      10.5941      7.63885 
 
 
; Example 2: Specific information available. 
dis  =  1.0 
sc  =  0.0 
itmax  =  10.0 
; A warning message is produce because the maximum number  
; of iterations is reached. 
sdata = IMSL_SMOOTHDATA1D(xdata, fdata, Itmax = itmax, $ 
   Distance = dis, Sc = sc) 
% IMSL_SMOOTHDATA1D: Warning: MATH_ITMAX_EXCEEDED 
; Maximum number of iterations limit 'ITMAX' = 10 exceeded. The 
; best answer found is returned. Output the results. 
PM, [[f(xdata(isub))], [fdata(isub)], [sdata(isub)]], $ 
         Title  =  '        F(X)    F(X) + noise     sdata'       
           F(X)    F(X) + noise     sdata 
         9.82958      12.3296      9.83127 
         8.26338      5.26338      8.26223 
         5.20083      3.20083      5.19946 
         2.22328      4.72328      2.22495 
         1.25874      4.25874      1.26142 
         3.16738      1.16738      3.16958 
         7.16751      4.66751      7.16986 
         10.8799      12.8799      10.8779 
         12.7739      10.7739      12.7699 
         7.59407      10.5941      7.59194

Version History


6.4	Introduced