Initial Parameter Estimates Using Exact (Systematic) Data

Background

The first step in an EMA analysis is to calculate an estimate of the LP3 parameters (mean, standard deviation, and skew) using the exact flow values. This typically includes flows with a systematic data type. However, HEC-SSP allows the user to enter a flow interval for a systematic data type and allow the user to enter an exact flow value for a historic data type. This means that systematic flow values are not always exact values and historic flow values are not always flow intervals. All exact flow values should be used for the initial parameter estimates regardless of the data type assigned in HEC-SSP.

The equations for mean, standard deviation, and skew are provided below. In general, the variable X is used throughout this study guide and in the literature to indicate the logarithm of flow for an LP3 frequency curve where X=Log(Q). The number of exact flow values is N_s.

$//<![CDATA[ \begin{array}{l}M=\frac{\sum_{i=1}^{N_{S}} X_{i}}{N_{s}}\end{array} //]]>$

$//<![CDATA[ \begin{array}{l}S=\sqrt{\frac{\frac{N_{S}}{N_{S}-1} \sum_{i=1}^{N_{S}}\left[X_{i}-M\right]^{2}}{N_{S}}}\end{array} //]]>$

$//<![CDATA[ \begin{array}{l}G=\frac{\frac{\frac{N_{S}{ }^{2}}{\left(N_{S}-1\right)\left(N_{S}-2\right)} \sum_{i=1}^{N_{S}}\left[X_{i}-M\right]^{3}}{S^{3}}}{N_{S}}\end{array} //]]>$

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

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

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

Exercise

Perform the first step of an EMA analysis for the flow dataset provided in Part 1 using the workbook EMA_Exercise_ManualCalculations v2.0.0.xlsb. Cells that require the user to enter a value or a formula are highlighted. Use the "EMA Iteration 0 – Step 1" worksheet and the equations shown above to calculate the initial parameter estimates for the mean (μ), standard deviation (σ), and skew (γ) using the exact flow values. This flow dataset defines all exact flow values as a systematic data type and all flow intervals as a historical data type. Perception thresholds have a censored data type. In general, this is good practice for assigning the data type in HEC-SSP.

Question 4. Calculate the mean, standard deviation, and skew using the built in Excel functions AVERAGE, STDEV.S, and SKEW.  Report the results rounded to three decimal places.

The solution to all Excel exercises can be found within the EMA_Exercise_ManualCalculations_Solution v2.0.0.xlsb workbook.