November 17th 2021

Multiple Regression Equation as an Analysis Tool

Multiple Regression Equation as an Analysis Tool

Buildings now are becoming ever more innovative and dynamic simulation energy models provide the ideal platform to test these solutions. An energy model (otherwise known as dynamic thermal model) is the perfect route to predicting building performance. The value of a design is entirely at risk unless key design decisions are assessed in advance. In this article we will be assessing energy model inputs and outputs and converting its data into a multi regression equation, testing the equations strength and verifying its predictive power.

The International Performance Measurement and Verification Protocol (IPMVP) proposes four options to determine and quantify building energy savings. These four options (A, B, C, D) depend on the economic, legal, or technical context of the project.

If performance reporting needs to be at the Building level, option C or D is favorable.  If only the performance of the specific (Energy Conservation Measure) retrofit needs to be reported, options A, B are more suitable.

Option A - Retrofit Isolation (Key Parameter Measurement) 

Savings are quantified by field measurement of only the key performance parameter(s) which define the energy use of the systems affected by the ECM retrofit.

Option B - Retrofit Isolation (All Parameter Measurement)

Savings are quantified by field measurement of the actual energy use of the systems affected by the ECM retrofit.

Option C - Whole Facility

Savings are based on actual energy consumption as measured by the utility meters, this is usually combined with simple regression modeling to accommodate variables such as weather, occupancy, etc. which allows factors to be applied to the model to adjust the measured data, being that environmental factors change throughout the year.

Option D - Calibrated Simulation:

Savings are determined by calibrating computer simulation models of a component or whole-building energy consumption to determine energy savings. The simulation aims to demonstrate and model actual projected energy performance.

Considering the above four options,

Can a VE model be used to generate a multiple regression equation? - YES

Can this equation be used to predict and forecast annual energy consumption? - YES

Can it be used to deduce relationships between the independent and dependent variables? - YES

Regression analysis is often used in energy engineering analysis but results can be less than ideal for many cases. Multiple regression is an extension of simple linear regression and is used when seeking to predict the value of a variable based on the value of two or more other variables.

Organizations might want to know how much of the variation in annual building energy consumption can be explained by the Heating Degree Days (HDD), Cooling Degree Days (CDD), Global Horizontal Irradiation (GHI), Cooling and Heating setpoint temperatures … "as a whole", but also the "relative contribution" of each independent variable in explaining the variance.

A starting point for this exercise type would be to determine the independent and dependent variables. Independent variables are variables that are influencing the dependent variable (cause and effect). In an energy model exercise, the independent variables are the model inputs and the dependent variable is the annual energy consumption.

For example, a 29,000m2 building located in Paris, France (ASHRAE Climatic zone 4A) has a simulated annual energy consumption of around 2,200 MWh.

From this a total of 7 independent variables (listed in the table below) that are influencing the dependent variable (Annual Energy Consumption)

Some of these variables cannot be directly obtained from the VE simulation model such as the HDD and CDD and do require some post-processing.  A simpler way would be to use custom variables within VistaPro to create our own results variables.

The multiple regression equation will have the following form:

Using multiple regression functions, we can determine the regression coefficients.

The next step in a multi-regression analysis would be to perform statistical tests to verify the strength and validity of the equation. This confirms the equation provides a solid description of the underlying model.

Key linear regression model tests:

Example of Key tests:

Coefficient of determination checks: R2 is a statistical measure of the variation in the dependent variable as explained by the linear model. By definition, it is only explanatory and not predictive. (R2 = 0.87)

Coefficient of Variation Root Mean Squared Error Check: CV (RMSE) is a statistical measure that allows us to quantify the predictive capability of the model. This indicates the absolute fit of the model and shows how close the predicted values are to the actual data points. It gives an objective representation of the predictive accuracy of the model. (CV (RMSE) = 0.04)

Interpreting P-values:

The smaller the p-value the stronger the relationship between independent and dependent variable. Standard practice is to use the coefficient p-values to decide whether to include the independent variables in the final model. The p-value for each independent variable test whether or not there is a correlation between the independent variable and the dependent variable.

Predictive power check:

After passing the key linear regression model tests, the multi regression equation is validated and could be used to predict the building's annual energy consumption and to calibrate the model.

In this test, we want to verify the predictive power of the multi regression equation. A simple test is changing the weather file from Paris to Glasgow and then rerun the dynamic simulation.

The graph below shows the monthly energy profile when the Glasgow weather file is used. Having checked the validity of the regression model we anticipate the simulated energy consumption and predicted energy consumption to line up and the resulting percentage difference was found to be approximately 3.5%.

We can conclude the multi regression formula presented here captures most of the facility’s energy consumption behavior and viable for energy use prediction and forecasting.

Although not common practice, generating a regression analysis of the VE model at the early stages of the design would allow designers and stakeholders to forecast the building's energy consumption following changes in weather and design conditions. It would also show the independent variables' statistical significance and their impact on the dependent variable.

How can we help?

IES Consulting are the experts in energy modelling across the full spectrum of building types and climate zones. We have produced thousands of dynamic simulation energy models focused on enhancing building performance through all the design stages to identify opportunities and minimise the risks. IES can help you explore the opportunities for energy optimization at the design stage and model calibration. We can work with you right from concept using the latest in analysis technology to communicate visually and statistically the impact of a potential design strategy.

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