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# PE Model Equations and Regression Methods

The table below shows the model forms used for models. Seven forms are available. You can calculate and select from each of the regression equations to choose the best model.

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
(with intersection at the end points)

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:

1. 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.
2. 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:

1. Prepare a set of data of Y (condition index) and X (pavement age) from the performance master file.
2. Transform Y (both the linear transformation, as applicable, and the 0 - 100 scale transformation).
3. Run regression analysis (linear or non-linear, as applicable to the model type) without removing any data points.
4. Identify outliers.
5. Remove all outliers from the data set and run regression analysis again, if desired.
6. 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.

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