1. Predicting Energy Consumption in Buildings using Multiple Linear Regression
Introduction
Linear regression is used to model energy consumption in buildings. The variables of interest included in the model are temperature, insulation and number of bedrooms. Linear regression can also be used:
To see how well the independent (explanatory, x, or predictor) variables explain the dependent (response, y, or outcome) variable.
To identify which subsets from many independent variables is most effective for estimating the dependent variable.

2. Assumptions are: All observations should be independent.
Your data should not suffer from multicollinearity. That is the independent variables should not be highly related.
Residual from model fit should follow a normal distribution.
Each of the independent (explanatory, x, or predictor) variables should have a linear relationship with the dependent (response, y, or outcome) variable. It is always a good idea to check this assumption using a matrix scatterplot.

3. Simple Linear Regression and Multiple Linear Regression Models
simple regression model can be represented mathematically as:
𝒚= 𝒃 𝟎 + 𝒃 𝟏 ∗ 𝒙 𝟏 +𝜺
If there are two or more predictors it is a multiple regression model represented as: 
𝒚= 𝒃 𝟎 + 𝒃 𝟏 ∗ 𝒙 𝟏 + 𝒃 𝟐 ∗ 𝒙 𝟐 +…+𝜺
y is the response variable and 𝑥 1 , 𝑥 2 are the predictor variables and 𝜀 is the error term; 𝑏 0 ,𝑏 1 , 𝑏 2 are regression coefficients.

4. The Data File
𝐸𝑛𝑒𝑟𝑔𝑦 𝐶𝑜𝑛𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛= 𝑏 0 + 𝑏 1 ∗𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑒+ 𝑏 2 ∗𝑖𝑛𝑠𝑢𝑙𝑎𝑡𝑖𝑜𝑛+ 𝑏 3 ∗𝑏𝑒𝑑𝑟𝑜𝑜𝑚+𝜀

5. Assumption 4 using Matrix Plot

6. Running the Linear Regression Procedure
Analyze -> Regression -> Linear….
Dependent Variable: Oil Consumed in January (gallons) [oilconsu]
Independent: Average Outside Temperature in Degree Fahrenheit [tempf], Insulation in inches [insu] and number of bedrooms [bedrooms]
Statistics…
Descriptives
Make sure that Estimates and Model fit are selected.
Select Collinearity diagnostics (this will assist us in checking assumption 2)
Plots…
Select Histogram and Normal probability plot (this will assist us in checking assumption 3)
Click OK to generate the output.

7. The completed dialogue box is shown below:
These steps will generate lots of output. We will now examine the relevant part of the output and attempt to interpret it.

8. Descriptive Statistics
The average energy consumption of all the 15 buildings in the study is gallons with a standard deviation of

9. Correlation

10. Correlation best reported as

11. Model Summary
R2 is interpreted as the proportion of the total variation in energy consumption accounted for by the three independent variables (temperature, insulation and bedrooms). In other words temperature, insulation and bedrooms “explains” 97.8% of the variability of energy consumption.

12. ANOVA (Analysis of Variance) Table
Null (H0): 𝑏 1 , 𝑏 2 𝑏 3 = 0, i.e. there is no regression versus Alternative (H1): 𝑏 1 , 𝑏 2 𝑏 3  0, i.e. there is some regression.
Here F = and P = (< 0.05). Hence, we reject H0 at the 5% significance level and conclude that energy consumption is linearly dependent on temperature, insulation and bedrooms.

13. Coefficients Table
𝑬𝒏𝒆𝒓𝒈𝒚 𝑪𝒐𝒏𝒔𝒖𝒎𝒑𝒕𝒊𝒐𝒏=𝟑𝟕𝟏.𝟐−𝟑.𝟓∗𝒕𝒆𝒎𝒑𝒆𝒓𝒂𝒕𝒖𝒓𝒆−𝟏𝟕.𝟕∗𝒊𝒏𝒔𝒖𝒍𝒂𝒕𝒊𝒐𝒏+𝟒𝟖.𝟏∗𝒃𝒆𝒅𝒓𝒐𝒐𝒎

14. Assumption 3: Histogram and Normal P-P Plot

15. Restriction and Validation
Predictions using the generated model would have to be limited to the range of values for temperature, insulation and number of bedrooms used for building the model.
For the model to widely adopted, it has to be validated. The model can be validated using one of the following methods:
By collecting new data and comparing the model prediction to the actual observation.
Or if you had a large data set, to divide the data set into two. Use one to build the model and the other to validate the model. You may have to use the correlation procedure to compare model prediction to actual energy consumption

16. Summary
Linear regression indicates that temperature, insulation and bedrooms are significant predictors of energy consumption [F = and P = (< 0.05)]. In other words temperature insulation and bedrooms “explains” 97.2% of the variability of energy consumption. All the predictors are significant at the 5% level [temperature t=-4.266, p=0.001 (<0.05); insulation t= p=0.001 (<0.05) and bedrooms t=2.445 p=0.033 (<0.05)]. Standardised beta coefficients indicate that temperature contributes the most (beta =0.563) followed by insulation (beta=0.404) and then bedrooms (beta=0.327). The data was suitable for linear regression as it satisfied all the assumptions needed for the linear regression model.

17. Questions