Statistical information encoding
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Statistical information encoding

Statistical measures of variation of certain parameters and functions can be embedded in a DAVE-ML model. This information is captured in a uncertainty element, which can be referenced by variableDef, griddedTableDef and ungriddedTableDef elements.

Uncertainty in the value of a parameter or function is given in one of two ways, depending on the appropriate probability distribution function (PDF): as a Gaussian or normal distribution (bell curve) or as a uniform (evenly spread) distribution. One of these distributions is selected by including either a normalPDF or a uniformPDF element within the uncertainty element.

Linear correlation between the randomness of two or more variables or functions can be specified. Although the correlation between parameters do not have a dependency direction (that is, the statistical uncertainty of either parameter is specified in terms of the other one so the calculation order doesn't matter) correlation is customarily specified as a dependency of one random variable on the value of another random variable. correlatesWith identifies variables or functions whose uncertainty 'depends' on the current value of this variable or parameter; the correlation sub-element specifies the correlation coefficient and identifies the (previously calculated) random variable or function on which the correlation depends.

These correlation sub-elements only apply to normal (Gaussian) probability distribution functions.

Each of these distribution description elements contain additional information, as described below. 

    uncertainty : effect=['additive'|'multiplicative'|'percentage'|'absolute']
      EITHER
        normalPDF : numSigmas=['1', '2', '3', ...]
            bounds :
           [correlatesWith : varID]
           [correlation : varID, corrCoef]
      OR
        uniformPDF : symmetric=['yes'|'no']
            bounds [, bounds]

uncertainty attributes:

effect

Indicates, by choice of four enumerated values, how the uncertainty is modeled: as an additive, multiplicative, or percentage variation about the nominal value, or an specific number (absolute).

uncertainty sub-elements:

normalPDF

If present, the uncertainty in the parameter value has a probability distribution that is Gaussian (bell-shaped). A single parameter representing the additive (± some value), percentage (± some %) of variation from the nominal value in terms of 1, 2, 3, or more standard deviations (sigmas or σs) is specified. Note here multiplicative and absolute bounds don't make much sense.

uniformPDF

If present, the uncertainty in the parameter or function value has a uniform likelihood of taking on any value between symmetric or asymmetric boundaries, which are specified in terms of additive (either ±x or +x/-y), multiplicative, percentage, or absolute variations. If absolute, the specified range of values must bracket the nominal value. For this element, the bounds sub-element may contain one or two values in which case the boundaries are symmetric or asymmetric.

Example 16. A variable with absolute uncertainty bounds

This example shows how to specify that a constant parameter that can take on a specified range of values with uniform probability distribution. The nominal value of the minimum drag coefficient is specified to be 0.005, but when performing parametric variations, it is allowed to take on values between 0.001 and 0.01. 

<DAVEfunc>
    <fileHeader>
      .
      .
      .
    </fileHeader>
    <variableDef name="CD zero" varID="CDo" units="ND" initialValue="0.005">     1
        <description>
            Minimum coefficient of drag with  
            symmetric uniform uncertainty band
        </description>
        <isOutput/>
        <uncertainty effect="absolute">    2
            <uniformPDF symmetric="yes">
                <bounds>0.001</bounds>
                <bounds>0.010</bounds>
            </uniformPDF>
        </uncertainty>
    </variableDef>
 </DAVEfunc>
1

We declare the parameter CDo as having a nominal value of 0.005.

2

When parametric variations are desired, the value of CDo can vary uniformly between 0.001 and 0.010.


Example 17. 10% uncertainty applied to output variable with uniform distribution

This example shows how to specify that a variable has a 10% uniformly distributed uncertainty band. In this example, the output variable comes from a non-linear one-dimensional function, but the uncertainty is applied downstream of the table. 

<DAVEfunc>
    <fileHeader>
       .
       .
       .
    </fileHeader>
    <variableDef name="Alpha_deg" varID="Alpha_deg" units="d"/>
    <variableDef name="Cm_u" varID="Cm_u" units="ND">
        <description>
            Coefficient of pitching moment with 10 percent 
            symmetric uniform uncertainty band
        </description>
        <isOutput/>
        <uncertainty effect="percentage">    1
            <uniformPDF symmetric="yes">
                <bounds>10.0</bounds>
            </uniformPDF>
        </uncertainty>
    </variableDef>
    <breakpointDef bpID="ALP">
        <bpVals>0, 5, 10, 15, 20, 25, 30, 35</bpVals>
    </breakpointDef>
    <function name="Nominal Cm">
        <description>
            Nominal pitching moment values prior to application 
            of uncertainty
        </description>
        <independentVarRef varID="Alpha_deg"/>
        <dependentVarRef varID="Cm_u"/>
        <functionDefn>   2
            <griddedTableDef>
                <breakpointRefs>
                    <bpRef bpID="ALP"/>
                </breakpointRefs>
                <dataTable>
5.2, 4.3, 3.1, 1.8, 0.3, 0.1, 0.0, -0.1
                </dataTable>
            </griddedTableDef>
        </functionDefn>
    </function>
</DAVEfunc>
1

We declare the output variable Cm_u as having the uncertainty of ±10% uniform distribution

2

This function gives the nominal values of Cm_u as a one-dimensional function of angle of attack α.


Example 18. Asymmetric additive uncertainty applied to output variable with uniform distribution

This example shows how to specify that a variable has an asymmetric, uniformly distributed, additive uncertainty band. In this example, the output variable comes from a non-linear one-dimensional function, but the uncertainty is applied downstream of the table. 

<DAVEfunc>
    <fileHeader>
      .
      .
      .
    </fileHeader>
    <variableDef name="Alpha_deg" varID="Alpha_deg" units="d"/>
    <variableDef name="Cm_u" varID="Cm_u" units="ND">
        <description>
            Coefficient of pitching moment with an 
            asymmetric uniform uncertainty band
        </description>
        <isOutput/>
        <uncertainty effect="additive">  1
            <uniformPDF symmetric="yes">
                <bounds>0.50</bounds>
                <bounds>0.00</bounds>
            </uniformPDF>
        </uncertainty>
    </variableDef>
    <breakpointDef bpID="ALP">
        <bpVals>0, 5, 10, 15, 20, 25, 30, 35</bpVals>
    </breakpointDef>
    <function name="Nominal Cm">
        <description>
            Nominal pitching moment values prior to application 
            of uncertainty
        </description>
        <independentVarRef varID="Alpha_deg"/>
        <dependentVarRef varID="Cm_u"/>   2
        <functionDefn>
            <griddedTableDef>
                <breakpointRefs>
                    <bpRef bpID="ALP"/>
                </breakpointRefs>
                <dataTable>
5.2, 4.3, 3.1, 1.8, 0.3, 0.1, 0.0, -0.1
                </dataTable>
            </griddedTableDef>
        </functionDefn>
    </function>
</DAVEfunc>
1

We declare the output variable Cm_u varies by as much as -0.5 to +0.0 about the nominal value. This delta value is in the same units as the nominal value, i.e. it is not a multiplier or percentage, but an additive delta to the nominal value which comes from the one-dimensional Cm_u function table description.

2

This function gives the nominal values of Cm_u as a one-dimensional function of angle of attack α.


Example 19. A one dimensional table, point-by-point, Gaussian

In this example, a Gaussian (normal) distribution function is applied to each point in a one-dimensional function table, with the 3σ value expressed as a multiplier of the nominal value. 

<DAVEfunc>
    <fileHeader>
       .
       .
       .
    </fileHeader>
    <variableDef name="Alpha_deg" varID="Alpha_deg" units="d"/>
    <variableDef name="Cm_u" varID="Cm_u" units="ND">
        <description>
            Coefficient of pitching moment with 10 percent 
            symmetric uniform uncertainty band
        </description>
        <isOutput/>
    </variableDef>
    <breakpointDef bpID="ALP">
        <bpVals>0, 5, 10, 15, 20, 25, 30, 35</bpVals>
    </breakpointDef>
  <function name="Uncertain Cm">
    <independentVarRef varID="Alpha_deg"/>
    <dependentVarRef varID="Cm_u"/>
    <functionDefn>
      <griddedTableDef>
        <breakpointRefs>
          <bpRef bpID="ALP"/>
        </breakpointRefs>
        <uncertainty effect="multiplicative">  1
          <normalPDF numSigmas="3">  2
            <bounds>
              <dataTable>  3
                0.10, 0.08, 0.06, 0.05, 0.05, 0.06, 0.07, 0.12
              </dataTable>
            </bounds>
          </normalPDF>
        </uncertainty>
        <dataTable>   4
          5.2, 4.3, 3.1, 1.8, 0.3, 0.1, 0.0, -0.1
        </dataTable>
      </griddedTableDef>
    </functionDefn>
  </function>
</DAVEfunc>
1

This declares the statistical uncertainty bounds of the Cm_u dependent variable will be expressed as a multiplication of the nominal value.

2

This declares that the probability distribution is a normal distribution and the bounds represent 3-σ (99.7%) confidence bounds.

3

This table defines ±3-σ bounds of the Cm_u function as a table. The table must have the same dimensions and independent variable arguments as the nominal function; it is in effect an overlay to the nominal function table, but the values represent the bounds as multiples of the nominal function value.

4

This table defines the nominal values of the function; these values will be used if the random variable associated with the uncertainty of this function is zero or undefined by the application.


Example 20. Two nonlinear functions with correlated uncertainty

In this example, uncertainty in pitching-moment coefficient varies in direct correlation with lift coefficient uncertainty. 

<DAVEfunc>
    <fileHeader>
      .
      .
      .
    </fileHeader>
    <variableDef name="Alpha_deg" varID="Alpha_deg" units="d"/>
    <variableDef name="CL_u" varID="CL_u" units="ND">
        <description> Coefficient of lift with a symmetric Gaussian uncertainty of 20%; correlates
            with Cm uncertainty. </description>
        <uncertainty effect="multiplicative">    1
            <normalPDF numSigmas="3">
                <bounds>0.20</bounds>
                <correlatesWith varID="Cm_u"/>   2
            </normalPDF>
        </uncertainty>
    </variableDef>
    <variableDef name="Cm_u" varID="Cm_u" units="ND">
        <description> Coefficient of pitching moment with a symmetric Gaussian uncertainty
            distribution of 30%; correlates directly with lift uncertainty. </description>
        <isOutput/>
        <uncertainty effect="percentage">        3
            <normalPDF numSigmas="3">
                <bounds>30</bounds>
                <correlation varID="CL_u" corrCoef="1.0"/>  4
            </normalPDF>
        </uncertainty>
    </variableDef>
    <breakpointDef bpID="ALP">
        <bpVals>0, 5, 10, 15, 20, 25, 30, 35</bpVals>
    </breakpointDef>
    <function name="Nominal CL">
        <description> Nominal lift coefficient values prior to application of uncertainty </description>
        <independentVarRef varID="Alpha_deg"/>
        <dependentVarRef varID="CL_u"/>
        <functionDefn>
            <griddedTableDef>
                <breakpointRefs>
                    <bpRef bpID="ALP"/>
                </breakpointRefs>
                <dataTable> 0.0, 0.1, 0.2, 0.3, 0.4, 0.45, 0.5, 0.45, 0.30 </dataTable>
            </griddedTableDef>
        </functionDefn>
    </function>
    <function name="Nominal Cm">
        <description> Nominal pitching moment values prior to application of uncertainty </description>
        <independentVarRef varID="Alpha_deg"/>
        <dependentVarRef varID="Cm_u"/>
        <functionDefn>
            <griddedTableDef>
                <breakpointRefs>
                    <bpRef bpID="ALP"/>
                </breakpointRefs>
                <dataTable> 5.2, 4.3, 3.1, 1.8, 0.3, 0.1, 0.0, -0.1 </dataTable>
            </griddedTableDef>
        </functionDefn>
    </function>
</DAVEfunc>
1

Lift coefficient has a nominal value which varies with angle of attack according to a non-linear one-dimensional table defined later. When performing parametric variations, CL_u can take on a multiplicative variation of up to 20% (3σ) with a Gaussian distribution.

2

This element indicates that the variation of pitching moment Cm_u correlates with the variation of lift coefficient.

3

Pitching-moment coefficient has a nominal value which varies as a function of angle of attack, according to a non-linear one-dimensional table defined later. When performing parametric variations, Cm_u can take on a 30% variation (3σ) with a Gaussian distribution.

4

This element indicates that the variation of pitching moment correlates directly with the variation in lift coefficient.


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