Iterative Method

Background

EMA requires calculation of the expected moments for censored flow values which includes the flow intervals and the perception thresholds. The expected moments depend on both the flow interval and the frequency curve (i.e. the parameter estimates for mean, standard deviation, and skew). In the equation below, the flow interval defines the limits of integration between the lower (X_L) and upper (X_U) flow values that define the flow interval. The parameter estimates (μ, σ, γ) define the value of the function f(x) which is just the LP3 frequency curve in the format of a probability density function. More details on f(x) will be covered later in this study guide.

$//<![CDATA[ \begin{array}{l}\mathrm{E}\left[\mathrm{X} \mid \mathrm{X}_{\mathrm{L}}< \mathrm{X} \leq \mathrm{X}_{\mathrm{H}} ; \mu, \sigma, \mathrm{\gamma}\right]=\frac{\int_{X_{l}}^{X_{u}}{x f}(x) d x}{\int_{X_{l}}^{X_{u}} f(x) d x}\end{array} //]]>$

Conversely, the frequency curve parameter estimates depend on the expected moments. In the equation below, the estimate of the LP3 mean (M) depends on the estimate of the first expected moment E[X].

$//<![CDATA[ \begin{array}{l}M_{k}=\frac{\sum_{i=1}^{N_{s}} X_{i}+\sum_{j=1}^{N_{c}} E\left[X_{j}\right]}{N_{s}+N_{c}}\end{array} //]]>$

The result is a system of equations for the expected moments and the parameter estimates that cannot be solved directly. A closed form expression cannot be derived from the system of equations.

https://en.wikipedia.org/wiki/Closed-form_expression

EMA uses the iterative method to solve the system of equations. The kth estimate of the mean, standard deviation, and skew are calculated using the expected moments from the previous (k-1) iteration.

https://en.wikipedia.org/wiki/Iterative_method

The general steps for EMA calculations are summarized below.

1. Calculate an initial estimate of the parameters using exact data values
2. Using the parameter estimates from Step 1 (or from the k-1 iteration), calculate the expected moments for the censored data values
3. Calculate a new estimate of the parameters using the exact data values and the expected moments from Step 2.
4. Iterate by repeating Steps 2 and 3 until the estimate of the parameters converges.

Exercise

Given the log-normal frequency curve shown below, estimate the mean and standard deviation using the iterative method.

A system of two equations in the form of Log(Q) = μ + zσ can be written based on estimating the AEP and flow value at two points on the frequency curve. The first point used for Equation 1 has an AEP of 0.001 and a flow estimate of 2670 cfs. The second point used for Equation 2 has an AEP of 0.1 and a flow estimate of 766 cfs. 

$//<![CDATA[ \begin{array}{l}3.427=\mu+3.090 \sigma\end{array} //]]>$   (Equation 1)

$//<![CDATA[ \begin{array}{l}2.884=\mu+1.282 \sigma\end{array} //]]>$   (Equation 2)

Values of μ and σ can be estimated using the iterative method. 

1. Make an initial estimate for μ. An initial value of 3.000 has been selected and entered in the table below.
2. Use Equation 1 to solve for σ using the estimate of μ from Step 1. The estimate of σ for iteration 0 has been calculated and entered in the table below.
3. Use Equation 2 to solve for μ using the estimate of σ from Step 2. The estimate of μ for iteration 1 has been calculated and entered in the table below.
4. Repeat Steps 2 and 3 until the parameter estimates converge

Question 5. Continue the calculations for 5 iterations and report the results to 3 decimal places.

Question 6. Compare these results to the results for the Exercise in Linearization.