Prediction of Annual Daylighting Performance Using Inverse Models

Abstract

This paper presents the results of a study that developed improved inverse models to accurately predict the annual daylighting performance (sDA and lighting energy use) of various window configurations. This inverse model is an improvement over previous inverse models because it can be applied to variable room geometries at different weather locations in the US. The room geometries can be varied from 3 m × 3 m × 2.5 m to 15 m × 15 m × 10 m (length × width × height). The other variables used in the model include orientation (N, E, S, W), window-to-floor ratio, window location in the exterior wall, glazing visible transmittance, ceiling visible reflectance, wall visible reflectance, shade type (overhangs, fins), shade visible reflectance, lighting power density (LPD) (W/m²), and lighting dimming setpoint (lux). Such models can quickly advise architects during the preliminary design phase about which daylighting design options provide useful daylighting, while minimizing the annual auxiliary lighting energy use. The inverse models tested and developed were multi-linear regression (MLR) models, which were trained and tested against Radiance-based annual daylighting simulation results. In the analysis, 482 cases with different model conditions were simulated, to develop and validate the inverse models. This study used 75% of the data to train the model and 25% of the data to validate the model. The results showed that the new inverse models had a high accuracy in the annual daylighting performance predictions, with an R² of 0.99 and an CV(RMSE) of 15.19% (RMSE of 58.91) for the lighting energy (LE) prediction, and an R² of 0.95 and an CV(RMSE) of 14.38% (RMSE of 8.02) for the sDA prediction. In addition, the validation results showed that the LE MLR model and sDA MLR model had an R² of 0.96 and 0.85, and RASE of 121.89 and 8.54, respectively, which indicate that the inverse models could accurately predict daylighting results for sDA and lighting energy use.

1. Introduction

Proper architectural daylighting design uses natural light to provide adequate indoor lighting levels and acceptable visual comfort during the daytime, which saves building energy use (i.e., auxiliary lighting energy and cooling energy use). During the past several decades, there have been many studies about daylighting simulation and prediction using both forward and inverse methods. The most widely-used daylighting simulations (i.e., forward methods) use either the split–flux (Bryan and Clear, 1980 [1]) (Hopkinson et al., 1954 [2]; Ref: Oh and Haberl, 2016 [3]), radiosity (Goral et al., 1984; O’brien, 1955 [4,5]), or ray-tracing forward methods (Ward, 1996 [6]) (Reinhart and Herkel, 2000 [7]). Both the split–flux and radiosity methods are based on a static daylight factor calculation, which generates an annual simulation result quickly (i.e., in less than 2 min). However, both methods have limitations.

The limitations of the split–flux method are as follows: (1) The method only works for square or rectangular shaped rooms; (2) The method cannot simulate specular materials/surfaces in the room or complex daylighting strategies, such as light shelves or reflective overhangs (Baker and Salem, 1990) [8]; (3) The method over-predicts the internally-reflected illuminance near the back of the room, because it assumes perfectly diffuse interior surfaces (Winkelmann and Selkowitz, 1985) [9]; (4) The split–flux method is also not sensitive to the window location in an exterior wall (Li and Haberl, 2020; Li, 2020) [10,11].

A limitation of the radiosity method is that it assumes all interior surfaces are perfectly diffuse. Therefore, it is difficult to simulate specular materials (Geebelen et al., 2005) [12]. In addition, the radiosity method does not correctly simulate varying window locations in the exterior wall (i.e., the midpoint location of the wall, top of the wall, etc.) (Li and Haberl, 2020; Li, 2020) [10,11].

One study showed that the ray-tracing method is the most accurate method for predicting the daylighting performance in a room under different sky conditions (Gibson and Krarti, 2015) [13]. The Radiance program uses the backward ray-tracing method (Ward, 1996; Ward and Rubinstein, 1988) [6,14]. However, Radiance has limitations when used for annual daylighting simulations. The first limitation is that a Radiance simulation can be very time-consuming when a high-quality analysis level is used. For example, in order to obtain high-accuracy annual daylighting results, a Radiance simulation takes several hours for one room on a typical desktop computer. Radiance’s parameters have four suggested options: min, fast, accur, and max (LBL, 2023) [15]. Of these options, the “min” value gives the fastest rendering with the least accurate annual daylighting results; the “fast” value gives a reasonably fast rendering; the “accur” value gives a more accurate results than the “fast” setting; and the “max” value gives the highest accuracy but requires the longest runtime.

Another limitation is that a Radiance simulation involves combinations of deterministic and stochastic processes (Ward, 1994) [16] (i.e., randomly determined), which means re-running the exact same case can produce a ±5% difference each time using the same “fast” simulation settings. However, using the more accurate “max” simulation setting reduces the difference between two exactly the same simulations. Therefore, a high-quality Radiance simulation is needed to obtain accurate annual daylighting results, which can require hours of computational runtime. Therefore, there is a need to find accurate daylighting predictions with reduced simulation runtimes.

2. Literature Review

Currently, selected forward models and inverse models have been developed to predict annual daylighting performance and reduce the simulation runtime. However, both methods have strengths and weakness.

2.1. Climate-Based Daylight Modelling (CBDM)

Climate-based daylight modelling (CBDM) is a forward method that was introduced by Mardaljevic (2000) [17] and Reinhart and Herkel (2000) [7]. The CBDM uses Radiance to predict various luminous quantities (e.g., irradiance, illuminance, radiance, and luminance) using sun and sky conditions derived from standard meteorological datasets (Mardaljevic, 2006) [18]. CBDM provides a framework for the annual evaluation of most daylighting environments in buildings (Mardaljevic, 2000) [17].

Later, based on the CBDM, more efficient methods were developed to reduce computation times, while accounting for direct and indirect components (Ayoub, 2020) [19]. These include the 4-component method (Mardaljevic, 1999) [20], the DAYSIM program (Reinhart and Walkenhorst, 2001) [21], the 2-phase method (Tregenza and Waters, 1983) [22], 3-Phase method (McNeil and Lee, 2013) [23], 5-Phase method (Geisler-Moroder et al., 2016) [24], 4-Phase method (Wang et al., 2018) [25], and 6-Phase method (Wang et al., 2018) [25]. These CBDM-based methods have encouraged the development of a number of additional software packages to evaluate daylight performance, such as DIVA-for-Rhino (Jakubiec and Reinhart, 2011) [26], ClimateStudio (Solemma, n.d.) [27], and OpenStudio (Guglielmetti et al., 2011) [28], which are packages based on Radiance.

Many previous studies have investigated ways to speed-up a Radiance-based daylighting simulation. For example, (Mardaljevic, 1999) [20] introduced a 4-component method that decomposed daylight into four components: direct sunlight, indirect sunlight, direct skylight, and indirect skylight. The 4-component method was validated by comparing the simulated indoor illuminance with field measurements. The results showed that this 4-component method had more accurate daylight results and a shorter processing runtime than native Radiance simulation.

Reinhart and Herkel (2000) [7] introduced a new accelerated simulation method based on daylight coefficients (DAYSIM). Their new method reduced the number of raytracing runs for an annual daylight simulation. In their new method, a single ray-tracing run was used to calculate the daylight coefficient under an isotropic sky. In the raytracing calculation, a new program (rtrace.daylight) was written to calculate the diffuse daylight coefficients. To accomplish this, the study divided the celestial hemisphere into 145 sky segments and 3 ground segments to calculate the diffuse daylight coefficients, and 58 representative sun positions for Freiburg, Germany to calculate the direct daylight coefficients. The total indoor illuminance was then calculated based on 145 + 3 diffuse daylight coefficients and 58 direct daylight coefficients. The results showed that the new method had the lowest relative root mean square error (RMSE) and simulation runtimes comparable to other Radiance-based simulation method.

Jones and Reinhart (2015) [29] proposed a graphical processing unit (GPU)-based tool, Accelerad, which translated Radiance’s algorithms for use by the GPU. They compared daylighting simulation results from Radiance and Accelerad with measured data. The results showed that the Accelerad program generated solutions between 3- and 24-times faster than native radiance runs in selected daylight scenes.

ClimateStudio (Solemma, n.d.) [27] was introduced to replace DIVA and to deliver improved simulation runtimes. ClimateStudio is a Radiance-based simulation tool that implements Radiance in a progressive path tracing mode. In this mode, ClimateStudio only traces a few light paths, instead of all possible light paths. The progressive path tracing of ClimateStudio combined with GPU accelerator yielded accurate results at significantly reduced runtimes compared to DAYSIM, DIVA, and native Radiance. However, ClimateStudio is a proprietary software that requires a significant annual licensing fee. Therefore, there is still a need to develop publicly available analysis methods that can run quickly at low or no cost, yet deliver suitable accuracy.

2.2. Inverse Prediction Models

Recently, several studies have developed inverse models to predict daylighting performance. Previous studies have used different types of MLA, including artificial neural networks (ANN), linear regression (LR), multiple linear regression (MLR), auto regression (AR), random forests (RF), decision tree (DT), and support vector machine (SVM), etc.

Table 1. Previous Inverse and Data-Driven models for Predicting Daylighting Performance.

Author	Prediction Model	Input Variables	Output		Location	Shading Systems	Data Source	Model Accuracy
Krarti et al. (2005) [30]	Single variable Regression based on DOE-2 simulation data	4 building geometry, window area, window to perimeter floor area ratio, perimeter to total floor area ratio, glazing type, glass transmittance, building locations,	Annual lighting energy		4 cities (Atlanta, Chicago, Phoenix, and Denver)	Interior shade factor	DOE-2 simulation	Error <3%
Ihm et al. (2009) [31]					43 cities in US		Field measurements in Boulder, Colorado	Error <3.6%
Moret et al. (2013) [32]	Multiple Linear Regression based on EnergyPlus simulation data	luminous efficacy of light sources, average illuminance levels, occupant behaviour, annual hours of operation, window to wall ratio, solar gain ratio, location (CDD, HDD)	Lighting, cooling, and heating		Phoenix, AZ; Baltimore, MD; Minneapolis, MN	Switchable glazing, operable louvers	EnergyPlus simulation of DOE validation building models	R2 = 0.96
Inanici (2013) [34]	Least Squares Multiple Regression based on Radiance-based simulation data	time of the day, day of the year	Luminance		Seattle, Washington, US	NA	Radiance-based simulation	R2 = 0.89 (rtcontrib) R2 = 0.78 (rpict)
Verso et al. (2017) [35]	Multivariate regression model based on DAYSIM simulation data	orientation, room depth, window area, glazing transmittance, and obstruction (overhang) angles	DAcon DA, sDA, Energy Demand (ED)		Turin, Catania and Berlin in Germany	Indoor venetian blinds, outdoor overhangs	DAYSIM simulation	sDA RMSE [7.1,13.9]; DA RMSE [3.9, 6.6]; DAcon RMSE [4.4, 2.8]; ED RMSE [0.9, 2.4];
Ayoub (2019) [36]	Multiple Linear Regression based on DIVA simulation data	street width, building height, horizontal obstruction angle, external painting reflectivity, window width and height, window geometry, window position, window size	sDA, ASE, Lighting Energy		Alexandria, Egypt	NA	DIVA-for-Rhino simulation	sDA R2 = 0.90, RMSE = 8.13; ASE R2 = 0.87, RMSE = 5.28; LE R2 = 0.88, RMSE = 5.59
Kurian et al. (2008) [33]	Auto Regression model and ANN models based on the simulation data from ECOTECT	window dimension and location, room dimension and location, radiation, temperature, time, blinds, dimming control, sensor, sky and site definitions	Illuminance	Manipal, India,		blinds	Simulated form ECOTECT	Error < 0.16 RMSE < 1.12
Fonseca et al. (2013) [37]	Multiple Linear Regression and ANN models based on DAYSIM (daylight) + EnergyPlus (thermal) simulation data	orientation, SHGC, WWR, vertical/horizontal shading coefficient, depth, shading angle	Lighting + equipment + HVAC energy	Florianópolis, Brazil,		horizontal and vertical shading coefficients	DAYSIM + EnergyPlus simulation	ANN: MSE < 0.05, R2 = 0.99; MLR: R2 < 0.8
Nault et al. (2017) [38]	Gaussian Processes (GP) regression model and MLR based on DIVA simulation data	orientation, window geometry, room geometry, obstruction, global horizontal radiation, direct normal radiation, diffuse horizontal radiation	sDA	Geneva, Switzerland;		NO	Simulated form DIVA-for-Rhino	RMSE: MLR < 7.14; GP < 21.09
Kim et al. (2022) [39]	Tree-regression model based on survey	overall rating view content view access privacy	View satisfaction ratings				Survey	RMSE = 0.65

lists various inverse models that have been developed to predict daylighting performance. Krarti et al. (2005) [30] developed a simplified analysis method to predict auxiliary lighting energy usage reduced with daylighting. Ihm et al. (2009) [31] modified these regression models and extended the study to 43 cities in U.S. However, these two studies were based on the simulation results from DOE-2, which utilized the less accurate split–flux daylight analysis method for daylighting simulation. The production models from Moret et al. (2013) [32] and Kurian et al. (2008) [33] were also developed with high accuracy, but their models were based on a simulation using less accurate daylighting simulation tools.

Inanici (2013) [34], Verso et al. (2017) [35], Ayoub (2019) [36], Kurian et al. (2008) [33], Fonseca et al. (2013) [37], and Nault et al. (2017) [38] all used Radiance-based tools to predict daylighting performance. The results showed that the ANN model from Fonseca et al. (2013) [37] had the highest accuracy, with an R² of 0.99. The model from Inanici (2013) [34] had the lowest R².

Kim et al. (2022) [39] developed a tree-regression model that could predict indoor view satisfaction based on survey data. The results showed that their regression model accurately predicted the view satisfaction, with R² > 0.68, and could be used to improve view analysis in early design.

2.3. Issues Identified

During the last several decades, there have been many attempts to predict daylighting performance, which include the split–flux method, the radiosity method, and simulations using Radiance. However, the split–flux and radiosity method do not always correctly predict daylighting performance, especially for specular materials and certain complex facade systems. In contrast, Radiance can accurately simulate the daylighting results of most architectural configurations. However, it requires considerable runtimes, which can make the optimization process very time consuming. Therefore, there is a need to speed up accurate daylighting calculations based on Radiance, while reducing runtimes. With this goal in mind, the CBDM was introduced to improve the native/basic Radiance simulation with more accurate results and a shorter runtime. However, a CBDM simulation still takes significantly longer to run than a thermal simulation. Alternatively, many previous studies have used inverse models (i.e., machine learning algorithms, multiple variable regression) to predict daylighting performance, such as illuminance, daylight autonomy, sDA (sDA measures daylighting illuminance sufficiency for a given area, reporting a percentage of floor area that exceeds a specified illuminance level (e.g., 300 lux) for a specified amount of annual hours (e.g., 50% of hours from 8 a.m.–6 p.m.) (IESNA, 2023)) [40], ASE (ASE (annual sun exposure) measures the percentage of the work plane that exceeding the threshold of 1000 lux more than 250 occupied hours per year (IESNA, 2023)) [40], and light energy demand.

However, the prediction models in many of the previous studies were only applied to one fixed room geometry or a very limited number of room geometries (less than 4 building geometry dimensions). In addition, most of the MLA prediction models were only applied to one climate location. In one study, the model was applied to 43 cities in USA. However, the analysis was based on DOE-2 simulations, with only one variable in each equation (Ihm et al., 2009) [31].

More importantly, many of the previous MLA models were trained and tested using the split–flux or radiosity simulation models, which are not as accurate as Radiance-based simulations. Finally, one of previous studies (Fonseca et al., 2013) [37] showed that the ANN prediction models had an improved performance versus an MLR model. In addition, the metrices and goodness-of-fit criteria (i.e., difference, error, RMSE, R²) used to evaluate the model accuracy were inconsistent from one study to the next, which makes it difficult to compare the accuracy of the models between different studies.

Therefore, there is a need to develop improved inverse models to obtain a higher accuracy of annual daylighting performance for varying geometries at multiple locations with fast estimation. Fast and accurate results would yield suitable outcomes when compared against accurate Radiance-based simulations. Therefore, this study aimed to develop a fast and accurate method to predict sDA and lighting energy use. In addition, the prediction models can be applied to variable room geometries and all weather locations throughout the USA.

3. Methodology

3.1. Preliminary Analysis of the Choice of Radiance Parameters

As previously mentioned, there are four Radiance quality presets suggested in the Radiance manual (LBL, 2023) [15] to evaluate annual daylighting simulations: min, fast, accur, and max. However, the choice of Radiance quality preset significantly impacts the simulation runtime. Kharvari (2020) [41] analyzed these radiance parameters regarding the accuracy of simulation results by comparing the results against field measurements. Kharvari recommended “Maximum I” and “Maximum III” settings for accurate results with low bias. Therefore, the current study utilized the “Maximum I” parameter setting that was recommended from Kharvari (2020) [41], to obtain accurate annual daylighting results. In addition, this study also developed a new “proposed” preset to test the “rtrace” Radiance parameter to calculate the annual daylight performance. The Radiance parameter presets for this study are listed in

Table 2. Radiance Rendering Parameters (LBL, 2023) [15] (Kharvari, 2020) [41].

		Min Preset	Fast Preset	Accur Preset	Maximum I Preset	Proposed Preset
-aa	ambient accuracy	0.5	0.2	0.15	0.1	0.2
-ab	ambient bounces	0	0	2	8	2
-ad	ambient divisions	0	32	512	4096	512
-ar	ambient resolution	8	32	128	0	128
-as	ambient super-samples	0	32	512	1024	128
-dj	direct jittering	0	0	0.7	1	1
-ds	source substructuring	0.02	0.02	0.02	0.02	0.02
-dt	direct thresholding	0	0	0	0	0
-dc	direct certainty	1	1	1	1	1
-dr	direct relays	6	6	6	6	6
-dp	direct pretest density	0	0	0	0	0
-lr	limit reflection	0	4	8	16	4
-lw	limit weight of each ray	0.05	0.01	0.002	0	0.01
-ss	specular sampling	0	0.3	0.7	1	0.7
-st	specular threshold	1	0.85	0.15	0	0.85

.

This study used a typical office space with daylighting provided by one or more vertical windows to test the Radiance parameter presets listed in Table 2 in predicting annual daylighting performance (i.e., sDA, ASE, lighting energy). The office model parameters used in the analysis of the daylighting simulation are listed in

Table A1. Model Parameters in the Daylighting Simulation.

Parameters		Value
Length (m)		3.0
Width (m)		4.6
Height (m)		2.6
Window area (m2)		2.76
Reference point height above floor (m)		0.76
Lighting dimming setpoint (lux)		538
Lighting Power Density (W/m2)		11.95
Visual Transmittance		0.41
Floor Visible Reflectance	Dark	0.2
	Bright	0.8
Wall Visible Reflectance		0.7
Roof Visible Reflectance		0.7
Shading Visible Reflectance	No Shades	NA
	Bright Shades	0.9

in Appendix A of this paper. In the current analysis, the total window-to-wall ratio was 20%, and the window visual transmittance was 0.41. The different conditions tested were a dark floor, a bright floor, and a model with and without shading devices (i.e., shades with bright surface). This study used the Radiance-based daylighting simulation tools Ladybug and Honeybee (Roudsari, 2023) [42] to obtain annual lighting energy use and sDA. All simulations were conducted on a laptop computer equipped with an Intel i7-8650U processor with 4 cores at 2.11 GHz and 16 GB RAM. The test sensors were set above the floor at 0.76 m with a grid of 0.6 m facing upward (Figure 1). The lighting control system in the simulation used photosensor controlled dimming. The photosensor controlled dimming assumed the dimming control had perfect knowledge of the illuminance from the daylight into the space, and dimmed the auxiliary lighting to meet the lighting target from a continuous dimming sensor with a user-defined setpoint. For example, when the interior illuminance from daylighting was set to meet the setpoint 500 lux, the lighting control system (photosensor controlled dimming) was turned off to save lighting energy. The lighting dimming control sensors were 0.3 m below the ceiling, facing downward. The grid for the lighting control sensors was 2 m (Figure 1). The illuminance setpoint in the simulation for the dimming control was set at 538 lux (50 fc), and the lighting power density was set at 1.11 W/ft² (12 W/m²) based on ASHRAE Standard 90.1-2016 [43]. The lighting schedule and occupancy schedule used the same schedule, which was fully on from 8:00 a.m. to 6:00 p.m. daily for all days of the week.

In Radiance, the daylighting matrix for the ASE is affected by the -dj, -dt, and -dc, which relate to how the solar discs are sampled in the direct calculation. As the parameters -dj, -ds, -dt, -dc, -dr, and -dp do not affect the simulation runtime, all these parameters for the min, fast, accur, and proposed runs were set to match the recommended “Maximum I” presets (Table 2). Therefore, in this study, the ASE results were the same for the min, fast, accur, Maximum I, and proposed parameter presets. The results of the simulated sDA value and the annual auxiliary lighting energy use are shown in

Table 3. Daylighting Results of the Different Presets for Radiance Rendering.

Condition	Preset Setting	Runtime	sDA	Lighting Energy (kWh)
No shades + bright floor	Maximum	60 min	100	78.8
	Accur	9 min	100	84.1
	Proposed	0.5 min	100	89.5
	Fast	0.18 min	80	130.2
	Min	0.08 min	46.7	245.8
No shades + dark floor	Maximum	60 min	100	86.6
	Accur	9 min	100	94.3
	Proposed	0.5 min	100	106.4
	Fast	0.18 min	66.7	140.1
	Min	0.08 min	46.7	251.4
Shades + bright floor	Maximum	60 min	100	102.4
	Accur	10 min	80	136.9
	Proposed	0.5 min	73.3	143.8
	Fast	0.18 min	6.7	290.4
	Min	0.08 min	0	458.5
Shades + dark floor	Maximum	60 min	100	106.1
	Accur	10 min	66.7	139.8
	Proposed	0.5 min	60	164.2
	Fast	0.18 min	6.7	292.4
	Min	0.08 min	0	453

and Figure 2 and Figure 3 for the five Radiance settings (maximum, accur, proposed, fast, min). In the analysis, the maximum I preset took 60 min to obtain the most accurate simulation results. The min and fast preset settings took less than one minute to obtain less accurate results. The simulation runtime for the accur preset setting was 10 min, while for the proposed preset setting this was 30 s (Table 3). The analysis showed that the annual lighting energy use increased as the Radiance rendering time decreased (Figure 2), while the sDA reduced as the Radiance rendering time decreased (Figure 3). In addition, the results showed the maximum I preset had the highest sDA and lowest annual lighting energy use, while the min preset had the lowest sDA and highest lighting energy use. The analysis also showed that the room configuration without shades had a reduced annual auxiliary lighting energy use versus the simulation with shades. The results also showed the annual daylighting performance results for the accur preset and the proposed preset were similar. However, there were gaps between the annual daylighting results for the Maximum I preset and the results for the accur and proposed presets. Therefore, this study used a MLR model to determine the correlations between maximum I preset and proposed preset, to further reduce simulation runtime, while maintaining the most accurate results.

3.2. Method to Develop the MLR Models

This study aimed to predict the annual daylighting performance using an inverse model that consisted of a multi-linear regression (MLR) to reduce the daylighting simulation runtime. Previous studies that used inverse models did not consider room size and weather location as variables. Therefore, this study covered locations in all 50 states in the US with varying room sizes. In this study, the variables used in the MLR model included orientation, window-to-floor ratio, window location in the exterior wall, glazing visible transmittance, ceiling visible reflectance, wall visible reflectance, shade types (overhangs, fins), shade visible reflectance, lighting power density (LPD) (W/m²), and lighting dimming setpoint (lux). Figure 4 shows the process that was used to develop the MLR model. In this study, Radiance parameters were carefully reviewed and compared, which revealed there was always a gap between the results from the maximum I preset and proposed preset (Figure 2 and Figure 3). Therefore, in this study, the annual daylighting results of the proposed preset and the maximum I preset were compared and analyzed to develop an MLR model to determine the statistical correlations between the proposed preset and the maximum I preset. In this way, the sDA and lighting energy (LE) usage were predicted using MLR models.

The detailed research method is listed below:

• Custom presets (proposed preset) were tested and developed for different Radiance parameters to reduce the Radiance runtime, while maintaining a suitable accuracy;
• A series of sensitivity tests with different variables were then conducted, to collect data;
• The annual daylighting simulation results from both the proposed preset and maximum I preset were then compared to determine the correct prediction;
• Finally, 75% of the dataset was used to train MLR models to predict sDA and annual auxiliary lighting energy (LE), and 25% of the dataset was used to validate the MLR models.

3.3. Daylighting Simulation Settings

Grid subdivisions in Radiance also affect the accuracy of the annual daylighting simulation results, since smaller grid subdivisions improve the accuracy of the model but increase the runtime. In this study, the sDA and ASE calculation grids were set to 0.6 m on each side, based the requirement of LEED 4.1 (USGBC, 2023) [44]. The location of the illuminance sensors and lighting dimming control sensors are shown in Figure 1.

In order to construct the MLR model, a series of simulations were conducted with the different model variables listed in

Table 4. Daylighting Simulation Parameters and Variables.

Parameters				Range	Interval	Number of Cases	Number of Changes
Room Geometry	Room Length (L)			3–15 m	1	482	16 room sizes
	Room Width (W)			3–15 m	1
	Room Height (H)			2–10 m	0.5
Window to Floor Ratio				1–100%	3%	482	33
Reference Point Height above Floor (m)				0.762		482	1
Lighting Dimming Setpoint (lux)				100–1000	100	482	10
Lighting Power Density (W/m2)				2–20	1	482	19
Ground Visible Reflectance				0.2		482	1
Glazing Visible Transmittance				0.1 to 0.9	0.1	482	9
Wall Visible Reflectance (WVR)				0.2–0.8	0.1	482	7
Roof Visible Reflectance (RVR)				0.2–0.8	0.1	482	7
Floor Visible Reflectance (FVR)				0.2–0.8	0.1	482	7
Orientation				N		113	4
				S		221
				E		70
				W		77
Window Position				Centered		240	4
				Top		114
				Down		68
				Mixed		60
Shading Visible Reflectance		No shades	0			191
		Overhangs	0.2–0.9		0.1	191	8
		Fins				50	8
		Overhang + Fin				50	8
Weather Locations				50 cities in USA		482	50
Radiance Parameter Preset				Maximum I		482	1
				Proposed		482	1

. In this study, a total of 482 cases were simulated with the Radiance-based tools Ladybug and Honeybee (Roudsari, 2023) [42], to obtain the annual lighting energy use and sDA. The simulated data were then used to train and test the MLR models. All daylighting simulation parameters, including the range, interval, and number of cases used in the simulations, are listed in Table 4.

In this study, different weather locations were analyzed from 50 cities within the continental US (listed in Appendix A 

Table A2. Weather Locations Used in this Study.

Location	Latitude	Longitude	Location	Latitude	Longitude
Atlanta, GA	33.63	−84.43	Kansas, MO	39.12	−94.60
Anchorage, AK	61.18	−150	Las Vegas, NV	36.08	−115.15
Ann Arbor, MI	42.22	−83.75	Los Angeles, CA	33.93	−118.4
Baltimore, MD	39.17	−76.68	Louisville, KY	38.18	−85.73
Bangor, ME	44.80	−68.82	Madison, WI	43.13	−89.33
Boston, MA	42.37	−71.02	Manchester, NH	42.93	−71.43
Burlington, VT	44.47	−73.15	Medford-Rogue, OR	42.19	−122.70
Charleston, SC	32.90	−80.03	Memphis, TN	35.07	−89.98
Charleston, WV	38.38	−81.58	Miami, FL	25.82	−80.3
Charlotte-Douglas, NC	35.22	−80.95	Minneapolis, MN	44.88	−93.23
Cheyenne, WY	41.15	−104.80	New Orleans, LA	30.00	−90.25
Chicago, IL	41.98	−87.92	New York, NY	40.78	−73.88
Cleveland, OH	41.40	−81.85	Newark, NJ	40.72	−74.18
Denver, CO	39.83	−104.65	Oklahoma City, OK	35.38	−97.60
Des Moines, IA	41.53	−93.67	Omaha-Eppley, NE	41.32	−95.90
Fargo-Hector, ND	46.93	−96.82	Philadelphia, PA	39.87	−75.23
Fort Smith, AR	35.33	−94.37	Phoenix, AZ	33.45	−111.98
Fort Wayne, IN	41.00	−85.20	Providence, RI	41.72	−71.43
Glasgow, MT	48.22	−106.62	Salt Lake City, UT	40.77	−111.97
Hartford Bradley, CT	41.93	−72.68	Seattle, WA	47.47	−122.32
Honolulu, HI	21.32	−157.93	Sioux Falls, SD	43.58	−96.75
Houston, TX	30.00	−95.37	Sterling-Washington, VA	38.98	−77.47
Huntsville, AL	34.65	−86.78	Topeka, KS	39.07	−95.63
Idaho, ID	43.52	−112.07	Wilmington, DE	39.67	−75.60
Jackson, MS	32.32	−90.08

), using sixteen different office room sizes L × W × H (Figure 5, Appendix A 

Table A3. Room Geometries.

Room Number	Length	Width	Height
1	3	4.6	2.6
2	3	7	4
3	3	9	5
4	4	12	6
5	4	4	3
6	6	4	5
7	7	3	6
8	8	4	3
9	9	4	5
10	10	10	6
11	11	4	3
12	13	5	5
13	14	7	7
14	15	15	10
15	15	3	8
16	15	6	3

). In the analysis, the floor visible reflectance (FVR), wall visible reflectance (WVR), and roof/ceiling visible reflectance (RVR) variables were changed from 0.2 (dark) to 0.8 (bright). The simulations were run for north (N), south (S), east (E), and west (W) orientations. In addition, the window size was varied from a window-to-floor ratio of 1% to 200% (window-to-wall ratio 1% to 70%). The glazing visible transmittance was changed from 0.1 to 0.9, and the window position was divided into four categories: centered, top, down, and mixed (Figure 6) as defined in Li (2020) [11]. Finally, the analysis had a rule for an exterior wall with more than one window; specifically, if all the window locations were the same, then set the window position as “the same”, otherwise set the window position as “mixed” (i.e., the windows having different top or down positions).

For the shading systems, this study applied three types of shade with different sizes, including: overhangs, fins, and overhang + fins (Figure 7). For all the fins and overhangs, the ratio (A/B) of the projection of the shading surface “A” and the gap between shades “B” was between 1.0 and 1.1 (Figure 8).

4. Multi-Linear Regression Models

Statistical software (SAS, 2019) [45] was used to predict the annual lighting energy (LE) and sDA using the multi-linear regression model (Equation (1)). Therefore, the dependent variable (Y) of the MLR model was LE_maximum and sDA_maximum, respectively. The independent variables (X) for both the LE and sDA MLR prediction models are listed in Table 5.

Y = β₀ + β₁ × X₁ + β₂ × X₂ + β₃ × X₃ + β₄ × X₄ + … + β₁₆ × X₁₆ + β₁₇ × X₁₇ + Ɛ

(1)

• where Y = LE_maximum or sDA_maximum
• ${\mathrm{X}}_{\mathrm{j}}$ = Factor, Predictor; j = 1, 2…, p (p = number of factors)
• ${\mathsf{\beta }}_0$ = Intercept,
• ${\mathsf{\beta }}_{\mathrm{j}}$ = Coefficients

Table 5. Independent Variables.

Variables		Variable Value or Range	Interval	Number of Cases
X1	LE_proposed or	Continuous number	1	482
	sDA_proposed	0–100%	1	482
X2	Orientation	South		221
		North		113
		East		70
		West		77
X3	Window position	Top		114
		Centered		240
		Down		68
		Mixed		60
X4	Window to Floor Ratio (WFR)	1–100%	3%	482
X5	Glazing visible transmittance	0.1–0.9	0.1	482
X6	Floor Visible Reflectance (FVR)	0.2–0.8	0.1	482
X7	Roof Visible Reflectance (RVR)	0.2–0.8	0.1	482
X8	Wall Visible Reflectance (WVR)	0.2–0.8	0.1	482
X9	Room size—Width	3–15 m	1	482
X10	Room size—Length	3–15 m	1
X11	Room size—Height	2–10 m	1
X12	Shade Types	No shades		191
		Overhangs		191
		Fins		68
		Overhangs + Fins		60
X13	Shade Reflectance	0.2–0.9	0.1	291
X14	Weather Location—Latitude	Continues number		482
X15	Weather Location—Longitude	Continues number		482
X16	Lighting Power Density (W/m2)	2–20	1	482
X17	Lighting dimming setpoint (lux)	100–1000	100	482

Table 5. Independent Variables.

Variables		Variable Value or Range	Interval	Number of Cases
X1	LE_proposed or	Continuous number	1	482
	sDA_proposed	0–100%	1	482
X2	Orientation	South		221
		North		113
		East		70
		West		77
X3	Window position	Top		114
		Centered		240
		Down		68
		Mixed		60
X4	Window to Floor Ratio (WFR)	1–100%	3%	482
X5	Glazing visible transmittance	0.1–0.9	0.1	482
X6	Floor Visible Reflectance (FVR)	0.2–0.8	0.1	482
X7	Roof Visible Reflectance (RVR)	0.2–0.8	0.1	482
X8	Wall Visible Reflectance (WVR)	0.2–0.8	0.1	482
X9	Room size—Width	3–15 m	1	482
X10	Room size—Length	3–15 m	1
X11	Room size—Height	2–10 m	1
X12	Shade Types	No shades		191
		Overhangs		191
		Fins		68
		Overhangs + Fins		60
X13	Shade Reflectance	0.2–0.9	0.1	291
X14	Weather Location—Latitude	Continues number		482
X15	Weather Location—Longitude	Continues number		482
X16	Lighting Power Density (W/m2)	2–20	1	482
X17	Lighting dimming setpoint (lux)	100–1000	100	482

4.1. MLR Model to Predict the Annual Auxiliary Lighting Energy (LE)

This study used the minimum corrected Akaike information criterion (Minimum AICc) to choose the best model (SAS, 2019) [45]. The Akaike information criterion (AIC) is an estimator of the out-of-sample prediction error and thereby yields a relative quality of statistical models for a given set of data (Aho et al., 2014; Akaike, 1998) [46,47].

Based on minimum AICc, the best MLR model for predicting the LE is shown in Equation (2). This MLR model showed that the room length, width, height, WFR, glazing visible transmittance, WVR, orientations, LE_proposed, lighting power density, and lighting dimming setpoint variables had statistically significant linear trends. However, the weather location, FVR, RVR, shade types, and shade reflectance variables did not have statistically significant linear trends. In addition, since the weather locations were already built into the proposed preset simulation (variable LE_proposed), the weather location variable did not play a significant role in the final regression model.

Table 6. Associations between the Y Response and Independent Variables from the LE MLR Model.

Term	Estimate	Std. Error	t Ratio	Prob. > |t|
Intercept	796.67183	77.29099	10.31	<0.0001
Length	−57.02912	4.707136	−12.12	<0.0001
Width	−33.37372	6.296278	−5.30	<0.0001
Height	48.823022	16.02964	3.05	0.0027
WFR	25.551495	159.4338	0.16	0.8729
Glazing Visible Transmittance	−137.3583	53.53022	−2.57	0.0111
Wall Visible Reflectance	−475.4641	52.03353	−9.14	<0.0001
Lighting Power Density (W/m2)	−14.4767	3.066264	−4.72	<0.0001
Lighting Dimming Setpoint (lux)	−0.022589	0.064885	−0.35	0.7282
Floor Visible Reflectance	−150.4627	44.77568	−3.36	0.0010
LE_proposed	0.6106068	0.017132	35.64	<0.0001
WFR × Height	−144.4815	52.80161	−2.74	0.0069
WFR × WFR	928.47189	318.9507	2.91	0.0041

shows the variables that were selected for the final model, and the Prob > ∣t∣(P-value) shows the influence of the independent variables on the dependent variables; when the P-value was smaller than 0.05, the independent variable was considered highly significant.

Table 7. Summary of Fit for LE MLR Model.

R2	0.99
R2 Adj	0.98
Root Mean Square Error (RMSE)	58.91
CV (RMSE)	15.19%
Mean of Response	387.8

shows the statistical results of the MLR lighting energy (LE) model. R² is the statistical measure that represents the proportion of the variance for a dependent variable that is explained by the independent variables in a regression model. The R² of this model was 0.99, the root mean square error (RMSE) was 58.91, and the CV(RMSE) was 15.19%. The MLR predicted lighting energy versus the Radiance calculated LE_maximum plot is shown in Figure 9 and Figure 10, which indicated that the MLR model could successfully predict the lighting energy (LE) usage. Figure 9 shows the categories of different shading types, and Figure 10 shows the categories of different window positions. Therefore, the results also demonstrated that the lighting energy prediction was evenly distributed across different window location and shading types.

$$\begin{array}{ll}{LE}_{-maximum}& =796.67-57.03\ast Length-33.37\ast Width-48.82\ast Height\\ & +25.55\ast WFR-137.36\ast Glazing Visible transmittance-475.46\ast WVR\\ & +0.61\ast {LE}_{proposed}-144.48\ast WFR\ast Height+928.47\ast WFR\ast WFR\\ & -14.48\ast Lighting Power Density-0.02\ast Lighting dimming setpoint\\ & +match \left(Orientation\right)\left(\begin{array}{c}\genfrac{}{}{0pt}{}{N=67.48}{S=-54.08}\\ E=-64.77\\ W=51.37\end{array}\right)\end{array}$$

(2)

4.2. MLR Model to Predict the sDA

Based on the minimum AICc method, the best MLR model for predicting the sDA is listed in Equation (3). This analysis showed that the weather position, RVR, and shade reflectance did not have statistical significance. The variables sDA_proposed, room length, width, WFR, glazing visible transmittance, WVR, FVR, window position, orientations, room height, shade types, and cross variables (i.e., Length × WFR, Width × WFR, WFR × Glazing visible transmittance, WVR × FVR, sDA_proposed × WFR, sDA_proposed × Glazing visible transmittance, and sDA_proposed × WFR × Glazing visible transmittance) were selected for the final MLR model, to predict the sDA_maximum (

Table 8. Associations between Y Response and Variables from the sDA MLR Model.

Term	Estimate	Std Error	t Ratio	Prob > |t|
Intercept	1.120608	29.20131	0.04	0.9694
sDA_proposed	−0.281909	0.453798	−0.62	0.5355
(Room) Length	2.0051692	0.61101	3.28	0.0013
(Room) Width	−4.209144	1.247138	−3.38	0.0010
WFR	−214.932	111.3362	−1.93	0.0556
Glazing visible transmittance	−82.75078	38.24644	−2.16	0.0322
FVR	−94.90968	51.55686	−1.84	0.0678
WVR	42.190196	23.89045	1.77	0.0796
Shading type (Fins)	−1.091208	6.711314	−0.16	0.8711
Shading type (NO)	3.4272667	2.975727	1.15	0.2514
Shading type (Overhang + fin)	−8.154601	2.965519	−2.75	0.0068
Orientation (E)	0.0722852	1.479427	0.05	0.9611
Orientation (N)	−13.80994	1.959216	−7.05	<0.0001
Orientation (S)	10.279483	1.679657	6.12	<0.0001
Position (Centered)	−2.860371	1.302068	−2.20	0.0297
Position (Down)	−9.722868	1.968187	−4.94	<0.0001
Position (Mix)	4.6962141	1.524803	3.08	0.0025
Length × WFR	−4.585675	3.235156	−1.42	0.1586
Width × WFR	17.709521	5.326667	3.32	0.0011
WFR × Glazing visible transmittance	946.35935	199.1677	4.75	<0.0001
FVR × WVR	182.51127	78.0294	2.34	0.0208
WFR × sDA_proposed	6.3691657	2.08662	3.05	0.0027
Glazing visible transmittance × sDA_proposed	3.9982993	0.948308	4.22	<0.0001
WFR × Glazing visible transmittance × sDA_proposed	−23.52474	4.417013	−5.33	<0.0001

).

Table 9 shows the results of the sDA MLR model. The R² of this model was 0.86, the root mean square error (RMSE) was 0.84, and the CV(RMSE) was 14.37%. MLR predicted sDA_maximum versus the Radiance simulated sDA_maximum plots are shown in Figure 11 and Figure 12, which show that the accuracy of predictions was similar across the different window locations and shade types.

$$\begin{gathered} \begin{array}{ll}sD{A}_{maximum}& =1.12-0.28\ast {sDA}_{proposed}+2.01\ast Length-4.21\ast Width-214.93\ast WFR\\ & -4.59\ast Length\ast WFR+17.71\ast Width\ast WFR\\ & -82.75\ast Glazing visible transmittance\\ & +946.36\ast WFR\ast Glazing visible transmittance\\ & -94.91\ast FVR+42.19\ast WVR+182.51\ast FVR\ast WVR\\ & +6.37\ast {sDA}_{proposed}\ast WFR\\ & +4.00\ast {sDA}_{proposed}\ast Glazing visible transmittance\\ & -23.52\ast {sDA}_{proposed}\ast WFR\ast Glazing visible transmittance\end{array} \\ \begin{array}{l}+Match (Shades Type)\left(\genfrac{}{}{0pt}{}{NO=3.43}{\begin{array}{c}Fins=-1.09\\ Overhangs=5.82\\ Overhang+Fin=-8.15\end{array}}\right)\\ +Match \left(Orientation\right)\left(\genfrac{}{}{0pt}{}{N=-13.81}{\begin{array}{c}S=10.28\\ E=0.07\\ W=3.46\end{array}}\right)\\ +Match \left(Position\right)\left(\genfrac{}{}{0pt}{}{Centered=-2.86}{\begin{array}{c}Top=7.89\\ Down=-9.72\\ Mix=4.70\end{array}}\right)\end{array} \end{gathered}$$

(3)

Note: The maximum predicted sDA_maximum should not be higher than 100%, if the calculated sDA_maximum was higher than 100%, it was set equal to 100%.

Figure 11. Radiance Calculated sDA versus MLR Predicted sDA Plot (RMSE = 8.02) in the Categories of Different Shading Types.

Figure 11. Radiance Calculated sDA versus MLR Predicted sDA Plot (RMSE = 8.02) in the Categories of Different Shading Types.

Figure 12. Radiance Calculated sDA versus MLR Predicted sDA Plot (RMSE = 8.02) in the Categories of Different Window Locations.

Figure 12. Radiance Calculated sDA versus MLR Predicted sDA Plot (RMSE = 8.02) in the Categories of Different Window Locations.

Table 9. Summary of Fit for the sDA MLR Model.

R2	0.86
R2 Adj	0.84
Root Mean Square Error (RMSE)	8.02
CV (RMSE)	14.38%
Mean of Response	55.78

Table 9. Summary of Fit for the sDA MLR Model.

R2	0.86
R2 Adj	0.84
Root Mean Square Error (RMSE)	8.02
CV (RMSE)	14.38%
Mean of Response	55.78

4.3. Validation

This study used 75% of the data for developing the MLR models, and used 25% of the simulated data to validate the two models.

Table 10. Validation Test Results of the MLR models.

MLR model	Source	Data size	R2	RASE
LE prediction MLR model	Training Set	374	0.99	56.82
	Validation Set	108	0.96	121.89
sDA prediction MLR model	Training Set	374	0.86	7.10
	Validation Set	108	0.85	8.54

provides the R² and root average square error (RASE) values of the validation tests. The validation of the LE MLR model and sDA MLR model were successful, with an R² of 0.96 and 0.85, respectively. The validation results showed that the MLR models performed well in predicting the annual daylighting performance.

5. Discussion

The MLR models that were developed in this study showed that a fast runtime and accurate prediction could be accomplished, which is useful in daylighting optimization of the annual lighting energy (LE) usage and spatial daylight autonomy (sDA). The daylighting model configurations changes included orientation, room size, weather location, window-to-floor ratio, window location in the exterior wall, glazing visible transmittance, ceiling visible reflectance, wall visible reflectance, shade types (overhangs, fins), and shade visible reflectance. In order to reduce the daylighting simulation runtime, new fast customized Radiance rendering parameters (called “proposed preset”) were suggested to replace the time-consuming Radiance rendering parameters (Maximum I) using the MLR models. This “proposed preset” took only 30 s to obtain annual daylighting results, while the most accurate preset “Maximum I” took over one hour to complete one simulation. The MLR models that were used to predict the LE and sDA were trained and validated using the daylighting simulated data. The MLR models were trained based on 75% of the data, and validated using the remaining 25% of the data.

The training results showed that the MLR models had high accuracy in the annual daylighting result predictions, with an R² of 0.99 and CV(RMSE) of 15.19% (RMSE of 58.91) for the LE MLR model, and a R² of 0.86 and CV (RMSE) of 14.38% (RMSE of 8.02) for the sDA MLR model. The validation results showed that the LE MLR model and sDA MLR model had an R² of 0.96 and 0.85, and RASE of 121.89 and 8.54, respectively, which indicate that the MLR models performed well in the prediction of the annual daylighting performance for lighting energy and sDA.

This study demonstrated that a carefully-constructed inverse MLR model is a useful method to quickly predict accurate annual daylighting performance for varying daylighting configurations at different weather locations. However, this study has limitations. The first limitation is that the MLR models in this paper would need new input data from the newly design model that uses a fast-daylighting simulation with the proposed preset in Ladybug and Honeybee. The second limitation of this study is that it did not cover all the possible shading systems. For example, movable shading systems, complex facade systems, and shading control systems would need to be added to the dataset.

This first limitation originates from the small size of the dataset used to construct the MLR models. For the MLR models to be more inclusive, a larger dataset would be required. In addition, any modification to the MLR models would need to avoid non-linear relationships, which would require more advanced methods to improve the model performance, such as kernel ridge regression (KRR), ElasticNet, support vector regression (SVR), gradient boosting (GB), and deep learning models.

6. Conclusions

This paper discussed the development and application of models to accurately predict the annual daylighting performance of different daylighting configurations at different weather locations. This study conducted 482 daylighting simulations using the Radiance-based tools Ladybug and Honeybee with varying daylighting configurations to construct the MLR models. The testing and validation results showed that these MLR models performed well in the prediction of the annual daylighting performance (i.e., lighting energy, sDA).

The new MLR models can be applied in different weather locations in the USA and for various room geometries, with the condition of fast simulations (“proposed preset”). The new MLR models are an improvement over the previous models, because they can be applied to variable room geometries and weather locations. These MLR models can be applied with the following steps:

(1)

Conduct a fast-daylighting simulation (30 s) to obtain the annual daylighting performance results for LE_proposed and sDA_proposed;

(2)

Using the results of LE_proposed and the input variables, the MLR model in Equation (2) can be applied to predict LE_maximum.

(3)

Using the results of sDA_proposed and the input variables, the MLR model in Equation (3) can be applied to predict sDA_maximum.

These MLR models can help architects to understand which daylighting features (variables) perform better with a shorter simulation runtime, especially when this applies to the optimization of a design with thousands of parameter changes. As mentioned in the previous section, an annual daylighting simulation can take hours to obtain LE_maximum and sDA_maximum results on a typical PC. Therefore, these MLR models can help users to reduce a simulation run to 30 s, to obtain correct sDA and LE results.

The limitations mentioned in the previous section of this paper are mainly from the small size of the dataset. In a future study, the data volume will need to be significantly expanded to over 5000, to cover additional conditions. Finally, other data-driven models or more sophisticated algorithms will need to be analyzed to remove the fast simulation condition, such as machine learning (ML) or deep neural network models. The current results suggest a solid underlaying pattern between building inputs and the annual daylighting performance with the currently available data. Therefore, a larger dataset and more advanced ML models could help to improve the daylighting prediction and assist architects in daylighting design.