Variable Gradient Smoothing

 

This form of smoothing groups data from neighboring bins together with the bin factor being determined. The level of smoothing is determined by a smoothing parameter called the radius. This method of smoothing is appropriate where there is believed to be a positive relationship between neighboring bins, but where there is not an expectation of a linear relationship between the target and either the bin number, the exposure or the log of the exposure.

 

All bins contribute to the calculation of the bin factor, with decreasing weight as the distance from the target bin increases (except for the NULL bin, which is calculated separately from the other bins). The weight is equal to the radius2 / (distance2 + radius2).

 

The following is an example of factor determination for bin 3 with a radius = 3 and the Exposure Method with 50% Credibility Level = 30:

 

 

Exposure (E)

Modified Exposure (M)

Target (T)

Distance from target bin

Weight (w)

w*E

w*M

w*T

Bin 1

10

5

8

2

0.692

6.92

3.46

5.54

Bin 2

10

6

7

1

0.900

9.00

5.40

6.30

Bin 3

10

3

3

0

1.000

10.00

3.00

3.00

Bin 4

10

7

6

1

0.900

9.00

6.30

5.40

Bin 5

10

5

6

2

0.692

6.92

3.46

4.15

Bin 6

10

7

6

3

0.500

5.00

3.50

3.00

Bin 7

10

3

2

4

0.360

3.60

1.08

0.72

Bin 8

10

5

4

5

0.265

2.65

1.32

1.06

Bin 9

10

4

2

6

0.200

2.00

0.80

0.40

Bin 10

10

6

5

7

0.155

1.55

0.93

0.78

Total

100

51

51

8

 

56.64

29.26

30.35

 

Credibility = 56.64 / (30 + 56.64) = 0.654

Credibility Weighted Factor = 0.654 * (30.35/29.26) + (1-0.654) * 1 = 1.024

 

This calculation is performed for each bin, 7 times with various radii. The default radii are (0, 0.414, 1, 1.828, 3, 4.657, 7). These radii can be modified by the user in the Characteristic Selection window. 

 

Continuing with the example above:

Radius

Factor before applying credibility

Credibility

Credibility Weighted Factor

0.000

1.000

0.250

1.000

0.414

1.021

0.321

1.007

1.000

1.041

0.468

1.019

1.828

1.048

0.581

1.028

3.000

1.037

0.654

1.024

4.657

1.016

0.701

1.011

7.000

0.995

0.731

0.996

 

The radius that generates the most significant credibility weighted result (i.e. most different from 1.0) is selected, which in this case is the radius of 1.828 generating a bin 3 factor of 1.028.

 

Each bin is allowed to have a different radius.