Simple Linear Regression
Energy can be produced from wind using windmills. Choosing a site for a wind farm (i.e. the location of the windmills), however, can be a multi-million dollar gamble. If wind is inadequate at the site, then the energy produced over the lifetime of the wind farm can be much less than the cost of building the operation. Hence, accurate prediction of wind speed at a candidate site can be an important component in the decision to build or not to build. Because the energy produced is proportional to the square of the wind speed, even small errors in wind speed prediction can lead to significant impacts.
A potential approach for predicting wind speed at a candidate site is to use wind speed data from a nearby reference site. A reference site is a nearby location where the wind speed is already being monitored and should, theoretically, be similar to the candidate site. If the reference sites turn out to be good predictors of the candidate sites, leveraging the information from the reference site would allow windmill companies to estimate the wind speed at the candidate sites without going through a costly and lengthy data collection period.
The Windmill data set contains measurements of wind speed (in meters per second m/s) at candidate sites (CSpd) (column 1) and at corresponding reference sites (RSpd) (column 2) for 1,116 sites. Download the Windmill.txt file from Canvas (which can be found in Files -> Data Sets), and put it in the same folder as this quarto file.
The goal of this assignment is to predict whether the wind speed at the reference site would be a good measure of the wind speed at the candidate site. Since we have one explanatory variable (RSpd) and one response variable (CSpd), simple linear regression would be a good tool to see if the data is linearly related.
# <your code here>
wind = read_table("Windmill.txt")
── Column specification ────────────────────────────────────────────────────────
cols(
CSpd = col_double(),
RSpd = col_double()
)head(wind, 6)# A tibble: 6 × 2
CSpd RSpd
<dbl> <dbl>
1 6.9 5.97
2 7.1 7.22
3 7.8 7.94
4 6.9 6.02
5 5.5 6.16
6 3.1 1.77The response variable is CSpd since we are seeing if the reference sight is a good predictor of the candidate site.
RSpd.
# <your code here>
plot = ggplot(wind, mapping = aes(x = RSpd, y = CSpd)) +
geom_point() +
labs(
x = "Reference Site Wind Speed (m/s)",
y = "Candidate Site Wind Speed (m/s)",
title = "Wind Speed at Reference vs Candidate Sites"
)
plot
RSpd and CSpd appear to have a relatively strong positive linear correlation.
# <your code here>
cor(wind$RSpd, wind$CSpd)[1] 0.7555948The correlation coefficient was 0.7555948. This tells us that there is a strong positive linear assosiation.
\[\begin{align} \text{CSpd}_i &= \beta_0 + \beta_1 \times \text{RSpd}_i + \epsilon_i \\ \epsilon_i &\stackrel{iid}{\sim} N(0, \sigma^2) \end{align}\]
ggplot with geom_smooth, You can remove the standard error line with the option se = FALSE).# <your code here>
plot +
geom_smooth(method = "lm",
color = "red", se = FALSE)`geom_smooth()` using formula = 'y ~ x'
lm function. (c) Plot the model residuals against fitted values in a well-labeled scatterplot.# <your code here>
model = lm(CSpd ~ RSpd, data = wind)
model
Call:
lm(formula = CSpd ~ RSpd, data = wind)
Coefficients:
(Intercept) RSpd
3.1412 0.7557 wind = wind %>% mutate(
residuals = resid(model),
fitted.values = fitted(model)
)
ggplot(wind, aes(x = fitted.values, y = residuals)) +
geom_point() +
labs(
x = "Fitted Values",
y = "Residuals",
title = "Residuals vs Fitted Values"
)
\[\widehat{\text{Cand Speed}}_i = \hat{\beta}_0 + \hat{\beta}_1 \times \text{Ref Speed}_i.\]
\[\widehat{\text{Cand Speed}}_i = 3.14 + 0.76 \times \text{Ref Speed}_i.\]
For every 1 m/s increase in reference site’s wind speed, the candidate site’s wind speed is expected to increase by 0.76 m/s on average.
When the reference site wind speed is 0 m/s, the predicted wind speed at the candidate site is 3.14 m/s.
# <your code here>
new_data = data.frame(RSpd = 12)
predict(model, newdata = new_data) 1
12.21003 If we look at the original scatterplot, there isn’t any data past about 21–22 m/s for the reference site wind speed. Therefore, using the model to estimate the candidate site wind speed when the reference site wind speed is 25 m/s would be extrapolating and give an unreliable estimate.
# <your code here>
sigma = sigma(model)
sigma2 = sigma^2
sigma2[1] 6.082312X = model.matrix(model)
head(X) (Intercept) RSpd
1 1 5.9666
2 1 7.2176
3 1 7.9405
4 1 6.0174
5 1 6.1646
6 1 1.7687y = wind$CSpd
result = solve(t(X) %*% X) %*% t(X) %*% y
result [,1]
(Intercept) 3.1412324
RSpd 0.7557333This analysis was pretty fun; I enjoyed learning how to apply simple linear regression to a data set. My biggest takeaway was the new things I learned in R. After looking up a few things, it was really satisfying to see that I could get the same intercept and slope coefficients using linear algebra as with the linear model. I didn’t know how to do matrix inverses, matrix multiplication, or use the solve() function before.
The purpose of analyzing this data was to see if we could predict the wind speed at a candidate site based on measurements taken at a reference site. This is important because if we choose a candidate site with low wind speed, we could lose millions of dollars.
In our analysis, we found that there is a strong relationship between the reference site and candidate site wind speeds. This means that we can reasonably predict what a candidate site’s wind speed will be based on measurements taken at a nearby reference site, which will help us determine the best places to build wind farms.