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.
It provides a somewhat predictive model to allow companies to see a reference site and have a decent estimate of how different the wind speed will be at the corresponding site.
Wind <- read_table("Windmill.txt")
── Column specification ────────────────────────────────────────────────────────
cols(
CSpd = col_double(),
RSpd = col_double()
)print(Wind)# A tibble: 1,116 × 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 1.77
7 6.8 4.65
8 11.4 10.9
9 12.9 11.7
10 13.5 13.0
# ℹ 1,106 more rowsThe corresponding site wind speeds
The reference site wind speeds
ggplot(Wind, aes(x=RSpd, y=CSpd))+
geom_point()+
labs (x = 'Reference site wind speeds',
y= 'corresponding site wind speeds ') 
The relationship between RSpd and CSpd seems to have a decently strong positive linear relationship.
cor(Wind$RSpd, Wind$CSpd)[1] 0.7555948This shows a strong positive correlation between these two variables. As the wind speed of RSpd goes up CSpd tends to go up as well.
Wind speed at corresponding site = y intercept + Slope * Wind speed at reference site + ( Wind speed at corresponding site - predicted wind speed at corresponding site), when the errors are independantlly and identically distributed with Normal distribution centered around a mean of 0 and have consistent variability.
\[\begin{align} corr speed_i &= yintercept + Slope \times Refspeed_i + (\widehat{\text{corr speed}}_i -corr speed_1) \\ \epsilon_i &\stackrel{iid}{\sim} N(0, \sigma^2) \end{align} \] when the errors are interdependently and identically distributed with Normal distribution centered around a mean of 0 and have consistent variability.
ggplot with geom_smooth, You can remove the standard error line with the option se = FALSE).ggplot(Wind, aes(x=RSpd, y=CSpd))+
geom_point()+
labs (x = 'Reference site wind speeds',
y= 'corresponding site wind speeds ') +
geom_smooth(method = lm, se=FALSE) `geom_smooth()` using formula = 'y ~ x'
lm function. (c) Plot the model fitted values (x-axis) against the residuals (y-axis) in a well-labeled scatter plot.#creating lm dataframe
model1<-lm(CSpd ~ RSpd, data = Wind)
summary(model1)
Call:
lm(formula = CSpd ~ RSpd, data = Wind)
Residuals:
Min 1Q Median 3Q Max
-7.7877 -1.5864 -0.1994 1.4403 9.1738
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.14123 0.16958 18.52 <2e-16 ***
RSpd 0.75573 0.01963 38.50 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.466 on 1114 degrees of freedom
Multiple R-squared: 0.5709, Adjusted R-squared: 0.5705
F-statistic: 1482 on 1 and 1114 DF, p-value: < 2.2e-16#creating new colomns for values and residuals
Wind$residuals <- model1$residuals
Wind$fits <- model1$fitted.values
#Creating scatter plot
ggplot(Wind, aes(x=fits, y=residuals))+
geom_point()+
labs (x = 'fitted values',
y= 'model residuals ') +
geom_smooth(method = lm, se=FALSE) `geom_smooth()` using formula = 'y ~ x'
\[\widehat{\text{Cand Speed}}_i = {\text {3.14 + 0.76 *}} {\text{ Ref Speed}}_i\]
< your response here. Note that you can write math in R markdown by surrounding the math in dollar signs. >
When winds speeds go up 1 meter per second in the reference sites wind speeds on average go up 0.76 meters per second in the corresponding sites
When wind speeds are at 0 meters per second at reference sites wind speeds tend to be on average 3.14 meters a second at corresponding sites.
3.14+0.76 * 12 [1] 12.26Because 25m/s is out side of our models range which would means we would be extrapolating
sum(model1$residuals^2)/model1$df.residual [1] 6.082312I am starting to see how these linear models can be good starting place for solving a problem. I also feel like I am personally learning just what different notations mean and I feel like I am starting to recognize patterns and terms.
The purpose of this analysis is to see how well the reference sites are able to predict wind speeds at corresponding sites. This is so when someone wants to invest in building windmills in a particular area that we can show with at least some certainty, with the data we can easily get from reference sites, that they haven’t made a bad investment. What we have found is that reference sites do tend to predict wind speeds at corresponding sites fairly accurately and that a certain degree of variability can be accounted for any future investments.