This page provides more detailed information on the statistical models used in InfoAsset Planner's Deterioration Models.

# Cohort Model: Weibull

Data Inputs: User-defined life expectancies (per Cohort), or failure Data.

Processes: Estimates input parameters (typically referred to as the shaping factors) for the Weibull distribution and then calculates the probability (for a given Cohort) as a function of time.

#### Distribution Function: Where:

f(t) is pipe failure probability density function (pdf);

t is the pipe age;

k is the shape parameter;

λ is the scale parameter;

c is the resistance time or the elapsed time from the construction to the first observed failure (t > c).

Outputs: Failure Probability Density, Cumulative Failure Probability, Survival Probability, Residual Life Expectancy

Comments: Network and Cohort level analysis, more general failure equation designed for deterioration analysis for many different applications, easier to converge and generally requires less data

# Cohort Model: Herz

Data Inputs: User-defined life expectancies (per Cohort), or failure Data

Processes: Estimates input parameters (typically referred to as the shaping factors) for the Herz distribution and then calculates the probability (for a given Cohort) as a function of time.

#### Distribution Function: Where:

f(t) is pipe failure probability density function (pdf);

t is the pipe age;

a is the aging factor;

b is the final failure rate at very old age;

c is the resistance time or the elapsed time from the construction to the first observed failure (t > c).

Outputs: Failure Probability Density, Cumulative Failure Probability, Survival Probability, Residual Life Expectancy

Comments: Network and Cohort level analysis, water distribution specific failure model, slightly more difficult in converging than Weibull model

# Cohort Model: NHPP and HPP

Data Inputs: User-defined life expectancies (per Cohort), or failure data.

Processes: Estimates input parameters for the Homogeneous Poisson Process and Non-Homogeneous Poisson Process. These processes are developed for repairable system and therefore can be applied to model the probability distribution of pipe failure interval time.

#### Distribution Function:

The time to the first failure has the following distribution: The time to the next failure given a failure at time T has the following distribution: Where:

f(t) is pipe failure probability density function (pdf);

t is the time to next pipe failure;

a and b are model parameters (a > 0 and b > 0) of which b = 1 for HPP;

It should be noted that the failure rate or repair rate of a pipe at age t is: For HPP process, a pipe has constant failure rate as a.

Outputs: Failure Probability Density, Cumulative Failure Probability, Survival Probability, Residual Life Expectancy

Comments: Network and Cohort level analysis, system specific

# Regression Model: Cox Proportional Hazard Model

Data Inputs: Survey data (from WW systems), Failure data (such as breaks), Inspection data, and user-calculated Likelihood of Failure scores are all inputs. The basic premise here is that: as the user you are defining for the tool what criteria is indicative of pipes that are no longer reliable, for example: “pipes with any CCTV score equal to or above a certain score are no longer reliable.” The input failure data is critical to this analysis, specifically in terms of the date of the observed ‘failure.’ The tool needs to know when a pipe began its life, and when failure occurred (Survey date, Inspection data, failure date, etc). Furthermore, pipe failure anomalies should be omitted from the analysis so as not to skew the results (for example: pipes that failed soon after install, due perhaps to bad installation, should be omitted).

Processes: Regression analysis on user-defined covariates, uses three parameters to define the distribution shape.  This deterioration is modeled by the following equation:

#### Function: Where:

t is the pipe age;

p is the number of covariables;

f(t) is the probabilty density function (pdf);

F(t) is the cumulative density function (cdf);

h0(t) is the baseline hazard function;

xi is the ith covariables;

βis the regression coefficient for ith covariables

Outputs: Covariate significance, Survival Probability, Cumulative Failure Probability, Failure Probability Density

# Regression Model: Non-Homogeneous Poisson Process (NHPP)

Data Inputs:  Survey data (from WW systems), Failure data (such as breaks), Inspection data, and user-calculated Likelihood of Failure scores are all inputs. The basic premise here is that: as the user you are defining for the tool what criteria is indicative of pipes that are no longer reliable, for example: “pipes with any CCTV score equal to or above a certain score are no longer reliable.” The input failure data is critical to this analysis, specifically in terms of the date of the observed ‘failure.’ The tool needs to know when a pipe began its life, and when failure occurred (Survey date, Inspection data, failure date, etc). Furthermore, pipe failure anomalies should be omitted from the analysis so as not to skew the results (for example: pipes that failed soon after install, due perhaps to bad installation, should be omitted).

Processes: The NHPP regression model determines failure risk using factors such as pipe age, past breaks, and failure factors (e.g., soil type, buried depth, diameter, etc.). This model explicitly assumes that the conditional pipe failure rate, given pipe failure history, is a function of pipe age, previous pipe failure, and pipe covariates.

#### Function: Where:

t is the pipe age;

N(t) is the number of pipe failures up to time t;

p is the number of covariables;

xi is the ith covariables;

βis the regression coefficient for ith covariables;

P is the probabilty.

Outputs: Covariate significance, Survival Probability, Cumulative Failure Probability, Failure Probability Density

# Regression Model: Linear Extended Yule Process (LEYP)

Data Inputs:  Survey data (from WW systems), Failure data (such as breaks), Inspection data, and user-calculated Likelihood of Failure scores are all inputs. The basic premise here is that: as the user you are defining for the tool what criteria is indicative of pipes that are no longer reliable, for example: “pipes with any CCTV score equal to or above a certain score are no longer reliable.” The input failure data is critical to this analysis, specifically in terms of the date of the observed ‘failure.’ The tool needs to know when a pipe began its life, and when failure occurred (Survey date, Inspection data, failure date, etc). Furthermore, pipe failure anomalies should be omitted from the analysis so as not to skew the results (e.g, pipes that failed soon after install, due perhaps to bad installation, should be omitted).

Processes: The LEYP regression model determines failure risk using factors such as pipe age, past breaks, and failure factors (e.g., soil type, buried depth, diameter, etc.). This model explicitly assumes that the conditional pipe failure probability, given pipe failure history, is a function of pipe age, previous pipe failure, and pipe covariates.

#### Function: Where:

t is the pipe age;

N(t) is the number of pipe failures up to time t;

p is the number of covariables;

xi is the ith covariables;

βis the regression coefficient for ith covariables;

P is the probabilty.

Outputs: Covariate significance, Survival Probability, Cumulative Failure Probability, Failure Probability Density

# Regression Model: Non Homogeneous Markov Chain (NHMC)

Data Inputs:  Survey data (from WW systems), Failure data (such as breaks), Inspection data, and user-calculated Likelihood of Failure scores are all inputs. The basic premise here is that: as the user you are defining for the tool what criteria is indicative of pipes that are no longer reliable, for example: “pipes with any CCTV score equal to or above a certain score are no longer reliable.” The input failure data is critical to this analysis, specifically in terms of the date of the observed ‘failure.’ The tool needs to know when a pipe began its life, and when failure occurred (Survey date, Inspection data, failure date, etc). Furthermore, pipe failure anomalies should be omitted from the analysis so as not to skew the results (e.g, pipes that failed soon after install, due perhaps to bad installation, should be omitted).

Processes: Using pipes that have been CCTV’d (CCTV has a ranking for pipes from 1-5, with 5 being the worst) to predict the state of pipes that have not been CCTV’d.via the DTMC distribution

#### Function: Where:

t is the pipe age;

k is the condition index, k is from 1 to c-1 and c is the number of pipe condition states;

p is the number of covariables;

xi is the ith covariables;

βis the regression coefficient for ith covariables;

P is the probabilty.

At any t, the probability of pipe state can be calculated as:

Outputs: Probability of a given pipe having a given condition score at a given time