# Multiple polynomial regression for EnergyPlus Curve Class

Generating EnergyPlus Curves from discrete points.

Dynamic simulations in **EnergyPlus** often require performance **data **of systems, not readily available, such as the COP of heat pumps in relation to water and air temperature.

Equipment **manufacturers** publish performance **data sheets** and work-field diagrams in tables with **discrete** **entries**, but **EnergyPlus** requires multidimensional continuous function in the form of the Curve family of classes. These classes build **continuous** functions given the coefficients of the equations.

What we need is a **conversion** between **discrete** and **continuous** data. Mathematically speaking we need to find the multiple polynomial **regression** of a set of points in space, in terms of a curve (2d case) or a surface (3d case).

## Manufacturers data

The following is an example of what a manufacturer may publish:

## EnergyPlus requirements

Here are the **Curve** classes specifications from EnergyPlus **Input Output Reference**:

### Quadratic curve

The equation is the following:

$$y = C_1 + C_2 x + C_3 x^2$$

The EnergyPlus string is the following:

`Curve:Quadratic,`

`WindACCBFFFF, ! name`

`-2.277, ! Coefficient1 Constant`

`5.2114, ! Coefficient2 x`

`-1.9344, ! Coefficient3 x\*\*2`

`0.0, ! Minimum Value of x`

`1.0; ! Maximum Value of x`

### Biquadratic surface

The equation is the following:

$$z = C_1 + C_2 x + C_3 x^2 + C_4 y + C_5 y^2 + C_6 x y$$

The EnergyPlus string is the following:

`Curve:Biquadratic,`

`WindACCoolCapFT, ! name`

`0.942587793, ! Coefficient1 Constant`

`0.009543347, ! Coefficient2 x`

`0.000683770, ! Coefficient3 x\*\*2`

`-0.011042676, ! Coefficient4 y`

`0.000005249, ! Coefficient5 y\*\*2`

`-0.000009720, ! Coefficient6 x\*y`

`15., 22., ! min and max of first independent variable`

`29., 47.; ! min and max of second independent variable`

## Multiple linear regression (Math alert!)

The problem we want to solve is to find a 2d **curve** that **approximates** a set of **points** in 2d space. In particular, we want a second-order equation (aka **quadratic**).

Moreover, we want to find a 3d **surface** that **approximates** a set of **points** in 3d space. In particular, we want a second-order equation (aka **biquadratic**).

To solve the problem we’ll exploit some **linear algebra**, using the polynomial **regression** model. This method works for both the cases we want to solve and more

$$y_i = \beta_0 + \beta_1 x_i + \beta_2 x_i^2 + \cdots + \beta_m x_i^m + \varepsilon_i \quad (i = 1,2,\dots,n)$$

where *n* is the number of input points and m is the order of the equation we want to use (2 in our case).

The previous equations can be expressed in **matrix form** in terms of a design matrix \(\bf X\), a response vector \(\vec y\), a parameter vector \(\beta\) (the beta vector is the same as the coefficients vector of the previous paragraph, it’s just a different notation), and a vector \(\vec\varepsilon\) of random errors. The *i*-th row of \(\bf X\) and \(\vec y\) will contain the *x* and *y* value for the *i*-th data sample.

We can write:

$$

\begin{bmatrix}

y_1 \\

y_2 \\

\vdots \\

y_n

\end{bmatrix}

=

\begin{bmatrix}

1 & x_1 & x_1^2 & \cdots & x_1^m \\

1 & x_2 & x_2^2 & \cdots & x_2^m \\

\vdots & \vdots & \vdots & \ddots & \vdots \\

1 & x_n & x_n^2 & \cdots & x_n^m

\end{bmatrix}

\begin{bmatrix}

\beta_0 \\

\beta_1 \\

\vdots \\

\beta_m

\end{bmatrix}

+

\begin{bmatrix}

\varepsilon_1 \\

\varepsilon_2 \\

\vdots \\

\varepsilon_n

\end{bmatrix}

$$

which when using pure** matrix notation** is written as

$$\vec y = \bf X \vec \beta + \vec \varepsilon$$

The **vector** of estimated polynomial regression **coefficients** (using ordinary** least squares** estimation) is

$$\hat{\vec \beta} = (\bf X^T \bf X)^{-1} \bf X^T \vec y $$

assuming *m* < *n* which is required for the matrix to be invertible; then since \(\bf X\) is a Vandermonde matrix, the invertibility condition is guaranteed to hold if all the x_{i} values are distinct. This is the unique least-squares solution.

#### Examples

So, for example, the quadratic problem for n points would be:

$$

\begin{bmatrix}

y_1 \\

y_2 \\

\vdots \\

y_n

\end{bmatrix}

=

\begin{bmatrix}

1 & x_1 & x_1^2 \\

1 & x_2 & x_2^2 \\

\vdots & \vdots & \vdots \\

1 & x_n & x_n^2

\end{bmatrix}

\begin{bmatrix}

\beta_0 \\

\beta_1 \\

\beta_2

\end{bmatrix}

$$

and the biquadratic:

$$

\begin{bmatrix}

z_1 \\

z_2 \\

\vdots \\

z_n

\end{bmatrix}

=

\begin{bmatrix}

1 & x_1 & x_1^2 & y_1 & y_1^2 & x_1 y_1 \\

1 & x_2 & x_2^2 & y_2 & y_2^2 & x_2 y_2 \\

\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\

1 & x_n & x_n^2 & y_n & y_n^2 & x_n y_n \\

\end{bmatrix}

\begin{bmatrix}

\beta_0 \\

\beta_1 \\

\beta_2 \\

\beta_3 \\

\beta_4 \\

\beta_5

\end{bmatrix}

$$

### Least square error

The **error** can be estimated with the least square method, by computing the coefficient of determination \(R^2\), which is defined as

$$R^2 \equiv 1 – {SS_{res} \over SS_{tot}}$$

where

$$SS_{tot} = \sum_i (y_i – \bar y)^2$$

$$SS_{res} = \sum_i (y_i – f_i)^2 = \sum_i \varepsilon_i^2$$

\(SS_{tot}\) (**total sum**) indicates how far each \(y_i\) `(the dependent variable of the input data) is from the average of the dependent variables;`

\(SS_{res}\) (**residual sum**) indicates how far each \(y_i\)` is from the approximation calculated from the regression function.`

The **closer** the value is to **1**, the **better** the approximation.

## Geogebra

To get a **graphical** feel of what these coefficients do to edit the **curve/surface**, here are two **Geogebra** canvases you can play with (click on the images to open)

### Quadratic

### Biquadratic

*NOTE: keep in mind that what we use as a fitting function is a small part of the full dominium, usually fairly close to the origin and that our coefficients will be close to zero, so the curve/surface will be quite flat.*

## Grasshopper (with IronPython)

For the practical implementation of the regression, we use IronPython, the version of the language embedded in Rhino Grasshopper. An alternative and valid approach which only uses Python is possible using libraries such as Numpy and Scipy, or any codebase that supports matrix-vector manipulation.

### Quadratic

#### Download

The following link contains the grasshopper EP_Curve_Quadratic cluster for free download:

click me to get the quadratic cluster

### Biquadratic

Here is the node tree which makes use of a custom Cluster:

#### Download

The following link contains the grasshopper EP_Curve_Biquadratic cluster for free download:

click me to get the biquadratic cluster

## Useful References

- EnergyPlus Input Output Reference | https://energyplus.net/sites/all/modules/custom/nrel_custom/pdfs/pdfs_v9.3.0/InputOutputReference.pdf
- Polynomial regression on Wikipedia | https://en.wikipedia.org/wiki/Polynomial_regressionhttps://en.wikipedia.org/wiki/Polynomial_regression
- Coefficient of determination (least square error) | https://en.wikipedia.org/wiki/Coefficient_of_determination

Mattia Bressanelli

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