Model Type | Model Form | Description |
|---|---|---|
Exponential | y = a + becx | The pavement condition remains nearly constant for a number of years, and then rapidly deteriorates. |
Hyperbolic | y = a + 1 / (b + cx) | The pavement condition deteriorates steadily with the greatest rate of deterioration occurring in the early years. |
Inverse Exponential | y = a + becx | The pavement condition remains nearly constant for a number of years, and then rapidly deteriorates. |
Linear | y = a + bx | The rate of deterioration of the pavement condition is the same from year to year. |
Piecewise Linear | y = ai + bix | You specify up to six coordinates, and the system draws a straight line between each coordinate. This type allows you to specify points in time where the deterioration rate "jumps" to a new value, where it again remains constant until the next discontinuity. |
Power | y = a + bx^c | This type of model is a special form of the exponential type. Like an exponential-type graph, the pavement condition remains nearly constant for a large number of years, and then rapidly deteriorates. However, the rate of deterioration is based directly on time rather than a logarithm. |
Sigmoidal | y = a + (be(-c/x)^d) | The pavement condition remains nearly constant for a number of years, then rapidly deteriorates, and then remains nearly constant again. |
where:
y = condition attribute being predicted.
x = age (years).
a, b, c, and d = regression coefficients.
Linear regression is applied to the linear, exponential, inverse exponential, and hyperbolic types of models. To achieve a linear model form, two transformations are performed:
- Transform y to y' linearly to get it on a 100 to 0 scale where 100 is "good." This is achieved by using linear interpolation where the values corresponding to 100 and 0 are the model start and model finish values, respectively, in the Attributes pane of the Performance Models.
- Transform y' to y'' using the linear transformation shown in the table below:
Model Type | Model Form | Linear Transformation |
|---|---|---|
Linear | y = a + bx | None |
Exponential | y = a + becx | y'' = ln(y') |
Inverse Exponential | y = a + becx | y'' = ln(101 - y') |
Hyperbolic | y = a + 1 / (b + cx) | y'' = 1 / y' |
The general methodology for building performance models in the system is:
- Prepare a set of data of Y (condition index) and X (pavement age) from the performance master file.
- Transform Y (both the linear transformation, as applicable, and the 0 - 100 scale transformation).
- Run regression analysis (linear or non-linear, as applicable to the model type) without removing any data points.
- Identify outliers.
- Remove all outliers from the data set and run regression analysis again, if desired.
- Review the statistical outliers to determine if a user-defined set of "outliers" should replace the statistical outliers. Run the regression analysis again. Select the model or define user parameters until better data is available for the data set.
