library(tidyverse)
## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
## ✔ dplyr 1.1.4 ✔ readr 2.1.5
## ✔ forcats 1.0.0 ✔ stringr 1.5.2
## ✔ ggplot2 4.0.0 ✔ tibble 3.3.0
## ✔ lubridate 1.9.4 ✔ tidyr 1.3.1
## ✔ purrr 1.1.0
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag() masks stats::lag()
## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors
library(ggpubr)
library(ggplot2)
set.seed(123)
# --- Simulated dataset with LST and NDVI ---
dat <- tibble(
group = rep(c("Urban", "NonUrban"), each = 30),
LST = c(rnorm(30, mean = 34.5, sd = 2), # higher in Urban
rnorm(30, mean = 30.0, sd = 2)),
NDVI = runif(60, 0.2, 0.8) # random vegetation index
)
# --- Frequentist model 1: t-test as lm() ---
fit_group <- lm(LST ~ group, data = dat)
summary(fit_group)
##
## Call:
## lm(formula = LST ~ group, data = dat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.8390 -1.1272 -0.2254 1.1182 3.9812
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 30.3567 0.3327 91.257 < 2e-16 ***
## groupUrban 4.0491 0.4704 8.607 5.93e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.822 on 58 degrees of freedom
## Multiple R-squared: 0.5609, Adjusted R-squared: 0.5533
## F-statistic: 74.08 on 1 and 58 DF, p-value: 5.935e-12
# Plot: LST by land use group
ggboxplot(dat, x = "group", y = "LST",
color = "group", palette = c("steelblue", "goldenrod"),
add = "jitter", shape = "group") +
theme_minimal() +
labs(title = "LST by Land Use Group",
x = "Land Use Group", y = "Land Surface Temperature (°C)")
# --- Frequentist model 2: regression LST ~ NDVI ---
fit_ndvi <- lm(LST ~ NDVI, data = dat)
summary(fit_ndvi)
##
## Call:
## lm(formula = LST ~ NDVI, data = dat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.3206 -2.1460 0.1752 1.9968 5.4897
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 31.180 1.187 26.27 <2e-16 ***
## NDVI 2.465 2.326 1.06 0.294
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.723 on 58 degrees of freedom
## Multiple R-squared: 0.019, Adjusted R-squared: 0.002087
## F-statistic: 1.123 on 1 and 58 DF, p-value: 0.2936
# Plot: regression of LST on NDVI
ggscatter(dat, x = "NDVI", y = "LST",
add = "reg.line", # Adds regression line
conf.int = TRUE, # Adds CI ribbon
cor.coef = TRUE, # Show correlation R
size = 3, alpha = 0.6,
shape = 21,
color = "steelblue",
fill = "skyblue") +
theme_minimal(base_size = 14) +
labs(title = "Regression of LST on NDVI",
x = "NDVI", y = "Land Surface Temperature (°C)")
# Define the helper function once
action_box <- function(title, items, border = "#e91e63", bg = "#fff5f7") {
html <- paste0(
'<div style="border-left:6px solid ', border,
';padding:12px 14px;background:', bg,
';border-radius:10px;margin:14px 0 6px 0">',
'<h4 style="margin:0 0 8px 0;color:', border, ';font-weight:bold;">', title, '</h4>',
'<ul style="margin:0;list-style-type:disc;color:black;">',
paste0("<li>", items, "</li>", collapse = ""),
"</ul></div>"
)
cat(html)
}
action_box(
title = "Action Items — Frequentist Models",
items = c(
"What is the coefficient? The coefficient for NDVI is 2.465",
"What is the p-value? p value for NDVI is 0.294",
"What does the CI mean? The CI is not shown",
"How would you describe the result if you were writing a results section? Do your best here. I mainly would like to see how you would present your results. We have not gone over this, so, feel free to use Google, etc for help. But, don’t blindly ask ChatGPT for it - I want to challenge you to start working on writing up results. We’ll come back to this as a class next week. A liner regression was used to look at whether NDVI predicts LST. The model estimated a positive, but not significant relationship"
),
border = "#2e7d32", bg = "#eefaf0"
)
Action Items — Frequentist Models
-
What is the coefficient? The coefficient for NDVI is 2.465
-
What is the p-value? p value for NDVI is 0.294
-
What does the CI mean? The CI is not shown
-
How would you describe the result if you were writing a results section?
Do your best here. I mainly would like to see how you would present your
results. We have not gone over this, so, feel free to use Google, etc
for help. But, don’t blindly ask ChatGPT for it - I want to challenge
you to start working on writing up results. We’ll come back to this as a
class next week. A liner regression was used to look at whether NDVI
predicts LST. The model estimated a positive, but not significant
relationship
# Load required packages
library(brms) # for Bayesian regression
## Loading required package: Rcpp
## Loading 'brms' package (version 2.23.0). Useful instructions
## can be found by typing help('brms'). A more detailed introduction
## to the package is available through vignette('brms_overview').
##
## Attaching package: 'brms'
## The following object is masked from 'package:stats':
##
## ar
library(tidyverse) # for data handling
library(bayesplot) # for posterior checks
## This is bayesplot version 1.14.0
## - Online documentation and vignettes at mc-stan.org/bayesplot
## - bayesplot theme set to bayesplot::theme_default()
## * Does _not_ affect other ggplot2 plots
## * See ?bayesplot_theme_set for details on theme setting
##
## Attaching package: 'bayesplot'
## The following object is masked from 'package:brms':
##
## rhat
# Bayesian model: group ~ LST
bayes_group <- brm(
LST ~ group,
data = dat,
family = gaussian(),
prior = c(set_prior("normal(0, 5)", class = "b"),
set_prior("normal(0, 10)", class = "Intercept"),
set_prior("student_t(3, 0, 5)", class = "sigma")),
iter = 2000,
chains = 4,
warmup = 500,
seed = 123
)
## Compiling Stan program...
## WARNING: Rtools is required to build R packages, but is not currently installed.
##
## Please download and install the appropriate version of Rtools for 4.5.1 from
## https://cran.r-project.org/bin/windows/Rtools/.
## Trying to compile a simple C file
## Running "C:/PROGRA~1/R/R-45~1.1/bin/x64/Rcmd.exe" SHLIB foo.c
## using C compiler: 'gcc.exe (GCC) 14.2.0'
## gcc -I"C:/PROGRA~1/R/R-45~1.1/include" -DNDEBUG -I"C:/Users/orkne/AppData/Local/R/win-library/4.5/Rcpp/include/" -I"C:/Users/orkne/AppData/Local/R/win-library/4.5/RcppEigen/include/" -I"C:/Users/orkne/AppData/Local/R/win-library/4.5/RcppEigen/include/unsupported" -I"C:/Users/orkne/AppData/Local/R/win-library/4.5/BH/include" -I"C:/Users/orkne/AppData/Local/R/win-library/4.5/StanHeaders/include/src/" -I"C:/Users/orkne/AppData/Local/R/win-library/4.5/StanHeaders/include/" -I"C:/Users/orkne/AppData/Local/R/win-library/4.5/RcppParallel/include/" -DRCPP_PARALLEL_USE_TBB=1 -I"C:/Users/orkne/AppData/Local/R/win-library/4.5/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DUSE_STANC3 -DSTRICT_R_HEADERS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -D_HAS_AUTO_PTR_ETC=0 -include "C:/Users/orkne/AppData/Local/R/win-library/4.5/StanHeaders/include/stan/math/prim/fun/Eigen.hpp" -std=c++1y -I"C:/rtools45/x86_64-w64-mingw32.static.posix/include" -O2 -Wall -std=gnu2x -mfpmath=sse -msse2 -mstackrealign -c foo.c -o foo.o
## cc1.exe: warning: command-line option '-std=c++14' is valid for C++/ObjC++ but not for C
## In file included from C:/Users/orkne/AppData/Local/R/win-library/4.5/RcppEigen/include/Eigen/Core:19,
## from C:/Users/orkne/AppData/Local/R/win-library/4.5/RcppEigen/include/Eigen/Dense:1,
## from C:/Users/orkne/AppData/Local/R/win-library/4.5/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22,
## from <command-line>:
## C:/Users/orkne/AppData/Local/R/win-library/4.5/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: cmath: No such file or directory
## 679 | #include <cmath>
## | ^~~~~~~
## compilation terminated.
## make: *** [C:/PROGRA~1/R/R-45~1.1/etc/x64/Makeconf:289: foo.o] Error 1
## WARNING: Rtools is required to build R packages, but is not currently installed.
##
## Please download and install the appropriate version of Rtools for 4.5.1 from
## https://cran.r-project.org/bin/windows/Rtools/.
## Start sampling
##
## SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
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summary(bayes_group)
## Family: gaussian
## Links: mu = identity
## Formula: LST ~ group
## Data: dat (Number of observations: 60)
## Draws: 4 chains, each with iter = 2000; warmup = 500; thin = 1;
## total post-warmup draws = 6000
##
## Regression Coefficients:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 30.35 0.34 29.69 31.01 1.00 6217 4133
## groupUrban 4.02 0.47 3.10 4.95 1.00 5714 4047
##
## Further Distributional Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 1.86 0.18 1.55 2.25 1.00 5287 4325
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
bayes_R2(bayes_group)
## Estimate Est.Error Q2.5 Q97.5
## R2 0.5488601 0.06017632 0.411371 0.6421825
pp_check(bayes_group, ndraws = 100)
# Bayesian model: NDVI ~ LST
bayes_reg <- brm(
LST ~ NDVI,
data = dat,
family = gaussian(),
prior = c(set_prior("normal(0, 5)", class = "b"),
set_prior("normal(0, 10)", class = "Intercept"),
set_prior("student_t(3, 0, 5)", class = "sigma")),
iter = 2000,
chains = 4,
warmup = 500,
seed = 123
)
## Compiling Stan program...
## WARNING: Rtools is required to build R packages, but is not currently installed.
##
## Please download and install the appropriate version of Rtools for 4.5.1 from
## https://cran.r-project.org/bin/windows/Rtools/.
## Trying to compile a simple C file
## Running "C:/PROGRA~1/R/R-45~1.1/bin/x64/Rcmd.exe" SHLIB foo.c
## using C compiler: 'gcc.exe (GCC) 14.2.0'
## gcc -I"C:/PROGRA~1/R/R-45~1.1/include" -DNDEBUG -I"C:/Users/orkne/AppData/Local/R/win-library/4.5/Rcpp/include/" -I"C:/Users/orkne/AppData/Local/R/win-library/4.5/RcppEigen/include/" -I"C:/Users/orkne/AppData/Local/R/win-library/4.5/RcppEigen/include/unsupported" -I"C:/Users/orkne/AppData/Local/R/win-library/4.5/BH/include" -I"C:/Users/orkne/AppData/Local/R/win-library/4.5/StanHeaders/include/src/" -I"C:/Users/orkne/AppData/Local/R/win-library/4.5/StanHeaders/include/" -I"C:/Users/orkne/AppData/Local/R/win-library/4.5/RcppParallel/include/" -DRCPP_PARALLEL_USE_TBB=1 -I"C:/Users/orkne/AppData/Local/R/win-library/4.5/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DUSE_STANC3 -DSTRICT_R_HEADERS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -D_HAS_AUTO_PTR_ETC=0 -include "C:/Users/orkne/AppData/Local/R/win-library/4.5/StanHeaders/include/stan/math/prim/fun/Eigen.hpp" -std=c++1y -I"C:/rtools45/x86_64-w64-mingw32.static.posix/include" -O2 -Wall -std=gnu2x -mfpmath=sse -msse2 -mstackrealign -c foo.c -o foo.o
## cc1.exe: warning: command-line option '-std=c++14' is valid for C++/ObjC++ but not for C
## In file included from C:/Users/orkne/AppData/Local/R/win-library/4.5/RcppEigen/include/Eigen/Core:19,
## from C:/Users/orkne/AppData/Local/R/win-library/4.5/RcppEigen/include/Eigen/Dense:1,
## from C:/Users/orkne/AppData/Local/R/win-library/4.5/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22,
## from <command-line>:
## C:/Users/orkne/AppData/Local/R/win-library/4.5/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: cmath: No such file or directory
## 679 | #include <cmath>
## | ^~~~~~~
## compilation terminated.
## make: *** [C:/PROGRA~1/R/R-45~1.1/etc/x64/Makeconf:289: foo.o] Error 1
## WARNING: Rtools is required to build R packages, but is not currently installed.
##
## Please download and install the appropriate version of Rtools for 4.5.1 from
## https://cran.r-project.org/bin/windows/Rtools/.
## Start sampling
##
## SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
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summary(bayes_reg)
## Family: gaussian
## Links: mu = identity
## Formula: LST ~ NDVI
## Data: dat (Number of observations: 60)
## Draws: 4 chains, each with iter = 2000; warmup = 500; thin = 1;
## total post-warmup draws = 6000
##
## Regression Coefficients:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 31.35 1.11 29.17 33.56 1.00 5195 4115
## NDVI 2.03 2.14 -2.24 6.25 1.00 5365 3902
##
## Further Distributional Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 2.77 0.27 2.31 3.36 1.00 5265 4168
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
bayes_R2(bayes_reg)
## Estimate Est.Error Q2.5 Q97.5
## R2 0.0254301 0.02912638 3.293696e-05 0.1066629
pp_check(bayes_reg, ndraws = 100)
# Define the helper function once
action_box <- function(title, items, border = "#e91e63", bg = "#fff5f7") {
html <- paste0(
'<div style="border-left:6px solid ', border,
';padding:12px 14px;background:', bg,
';border-radius:10px;margin:14px 0 6px 0">',
'<h4 style="margin:0 0 8px 0;color:', border, ';font-weight:bold;">', title, '</h4>',
'<ul style="margin:0;list-style-type:disc;color:black;">',
paste0("<li>", items, "</li>", collapse = ""),
"</ul></div>"
)
cat(html)
}
action_box(
title = "Action Items — Bayesian Models",
items = c(
"Interpretation: What are the coefficient values? Intercept = 31.35, NDVI = 2.03",
"What does the credible interval say? [-2.24, 6.25] crosses 0 so no clear effect",
"How is this different from the lm() output? Positive estimate, but Bayesian focuses on uncertainty instead of p-values",
"Are the models significant (in a Bayesian sense?). How do you tell? Not significant because CI includes 0",
"How would you describe your results if you were writing them up? NDVI shows a weak, uncertain effect on LST"
),
border = "#FF6EC7", bg = "#ffe6f5"
)
## <div style="border-left:6px solid #FF6EC7;padding:12px 14px;background:#ffe6f5;border-radius:10px;margin:14px 0 6px 0"><h4 style="margin:0 0 8px 0;color:#FF6EC7;font-weight:bold;">Action Items — Bayesian Models</h4><ul style="margin:0;list-style-type:disc;color:black;"><li>Interpretation: What are the coefficient values? Intercept = 31.35, NDVI = 2.03</li><li>What does the credible interval say? [-2.24, 6.25] crosses 0 so no clear effect</li><li>How is this different from the lm() output? Positive estimate, but Bayesian focuses on uncertainty instead of p-values</li><li>Are the models significant (in a Bayesian sense?). How do you tell? Not significant because CI includes 0</li><li>How would you describe your results if you were writing them up? NDVI shows a weak, uncertain effect on LST</li></ul></div>