Functions

 In the Risk Assessment Methodology section of this technical reference manual, you'll read that risk is calculated by combining several summary relationships and integrating the final result. The summary relationship is a relationship between two random variables, and must meet the mathematical criteria for the definition of a function or our math breaks down. We'll refer to the two random variables as X and Y, where X is the independent variable and Y is the dependent variable. Our statistical library has two tools to track this type of data: uncertain paired data and paired data. 

Uncertain Paired Data

In the case of uncertain paired data, the Y variable is uncertain, and can be represented by one of the Continuous Distributions or an empirical distribution developed in Results Collection. Below is an example of an uncertain paired data summary relationship that represents a hypothetical stage-discharge function with uncertainty. The X variable is discharge, and the Y variable is the distribution of stages, and is provided as a Normal distribution for this example. 

The most important functionality of an uncertain paired data is its capability to be sampled. Given a random number p, the pth quantile of the uncertainty distributions is returned for each X value. Using the example above, we can sample a paired data summary relationship using p=0.75 to produce the 75th percentile of the uncertain paired data summary relationship. In some cases, the input functions are not monotonically increasing in the uncertainty, but the software requires evaluated paired data summary relationships to be monotonically increasing. In these situations, the software forces monotonicity of a paired data relationship after sampling. Monotonicity is generally forced from the bottom up, meaning that y values must be increasing as x increases, and if not, then a non-monotonic y value is forced to equal the preceding y value plus epsilon, a very very small number. In the case of graphical frequency functions, monotonicity is forced from the top down for sampled curves above the mean, navigate to Frequency Function Uncertainty to learn more. 

 Paired Data

In the case of paired data, the Y variable is deterministic. Below is an example paired data relationship that represents a hypothetical stage-discharge function, and was calculated by sampling the uncertain paired data relationship above at the 75th percentile. 

 The HEC-FDA statistical library involves substantial functionality to analyze a given paired data summary relationship. For example, the HEC-FDA software can calculate the value y of the dependent variable Y for a given value x of the independent variable X, calculating  $//<![CDATA[ \begin{array}{l}y = f(x)\end{array} //]]>$. For example, when evaluating a damage-frequency relationship, damage is the Y variable and exceedance probability is the X variable. Given an exceedance probability of interest, such as 0.01, we can obtain the corresponding damage by calculating $//<![CDATA[ \begin{array}{l}y = f(x)\end{array} //]]>$. This example calculation takes place when identifying the default threshold for evaluating system performance of Scenarios that do not involve levees. In some instances, the value x can be found exactly in the paired data summary relationship, in which case the software returns the value y at the corresponding index of the value x. In other instances, the value x is found between two coordinates of the summary relationship, in which case the value y must be linearly interpolated between the corresponding coordinates of interest. The software can also do this calculation in reverse, identifying the value x of the independent variable X for a given value y of the dependent variable Y by calculating $//<![CDATA[ \begin{array}{l}x = f^{-1}(y)\end{array} //]]>$. For example, the software evaluates a stage-frequency function (frequency is independent variable and stage is the dependent variable) for the annual exceedance probability of the threshold (or target) stage. Another important capability of the software to evaluate a given paired data relationship is integration. The software carries out numerical integration of a paired data relationship using the trapezoidal rule. The typical application of integration in the HEC-FDA software is integrating a damage-frequency function to calculate expected annual damage. See Expected Annual Damage for a terrific example of numerical integration using the trapezoidal rule. 

 The HEC-FDA statistical library also involves functionality for mathematical operations involving two paired data summary relationships: function composition and function multiplication. Function composition, also referred to as curve combination, is a workhorse of the HEC-FDA risk engine. Function composition or curve combination is the process of aligning the dependent variable of a given summary relationship with the independent variable of another summary relationship, which is defined mathematically for a function $//<![CDATA[ \begin{array}{l}y = f(x)\end{array} //]]>$ and a function $//<![CDATA[ \begin{array}{l}x = g(h)\end{array} //]]>$, defining Y as a function of H $//<![CDATA[ \begin{array}{l}y = f(g(h)).\end{array} //]]>$ The four plot diagram discussed in the page on Expected Annual Damage is an application of function composition, relating frequency of occurrence to flow, flow to stage, stage to damage, and damage to frequency of occurrence. Function multiplication is the process by which the dependent variables of two paired data sets are multiplied for the matching values of the independent variable. Multiplication is designed for scenarios involving levees, where a stage-damage function is multiplied by a system response function to produce a probability-weighted stage damage function which reflects the expected value of damage in the floodplain given the hazard loading and corresponding probability of levee failure. Navigate to this location of the risk measurement section to view an illustration of application of paired data multiplication in HEC-FDA.