Worksheets/Week9
Part 1
The aim is to find the duration [hours] of the experiments to maximize the yield of a biological product. If you operate the reactor too long some side reactions start to occur which consume your product. If you operate for too short a time, then you miss out on the opportunity of creating more product.
# Copy/paste the code piece by piece.
# Or run it piece by piece in RStudio.
# Start small: with just 2 experiments
center = 36
range = 24
rw.x <- c(24, 48)
coded.x <- (rw.x - center)/(0.5*range)
y0 <- c(28, 63)
# Coded -1 : 24 hours
# Coded +1 : 48 hours
model.0 <- lm(y0 ~ coded.x)
summary(model.0)
# What is the interpretation of the
# * slope?
# * intercept
# * why are R2 and SE the values they are?
# Basic plot of everything so far:
raw_data <- data.frame(coded.x = coded.x, y = y0)
library(ggplot2)
p <- ggplot(data=raw_data, aes(x=coded.x, y=y)) +
geom_point(size=5) +
xlab("Coded value for x_A") +
scale_x_continuous(breaks=seq(-2, 5, 1)) +
ylab("Outcome variable") +
scale_y_continuous(breaks=seq(0, 150, 10)) +
theme(axis.text=element_text(size=18), legend.position = "none") +
theme(axis.title=element_text(face="bold", size=14))
p
# Add the linear fit: through the 2 points
plot_data <- data.frame(coded.x = seq(-2, +5, 0.1))
plot_data$y <- predict(model.0, newdata=plot_data)
p <- p + geom_line(data=plot_data, color="blue", size=1)
p
# Run experiment at center point: predict it first
rw.x <- c(24, 48, 36)
coded.x <- (rw.x - center)/(0.5*range)
x.test <- data.frame(coded.x=coded.x)
predict(model.0, newdata=x.test)
# Expect a predicted value of 45.5 in the output. Actual: 55.
# Run a second experiment at the center, to get a feeling for spread. 54
# Similar values, and about 10 units difference from linear model.
# Try fitting a linear model now through all the available data points:
rw.x <- c(24, 48, 36, 36)
y1 <- c(28, 63, 55, 54)
coded.x <- (rw.x - center)/(0.5*range)
model.1 <- lm(y1 ~ coded.x)
summary(model.1)
# Show the linear fit with the extra data point (center points): green line
raw_data <- data.frame(coded.x = coded.x, y = y1)
plot_data$y <- predict(model.1, newdata=plot_data)
p <- p + geom_point(aes(x=coded.x, y=y, color='darkgreen', size=5), data=raw_data)
p <- p + geom_line(data=plot_data, color="darkgreen", size=1)
p
# Seems like a quadratic model through all the data points could approximate it:
rw.x <- c(24, 48, 36, 36)
y1 <- c(28, 63, 55, 54)
coded.x <- (rw.x - center)/(0.5*range)
model.1.quad <- lm(y1 ~ coded.x + I(coded.x^2))
summary(model.1.quad)
# Show the quadratic fit between -2 and +3 in coded units: red line
# In real-world units this corresonds to _____ and _____
plot_data <- data.frame(coded.x = seq(-2, +3, 0.1))
plot_data$y <- predict(model.1.quad, newdata=plot_data)
p <- p + geom_line(data=plot_data, color="red", size=1)
p
#------------
# Try a new point at +2: that is RW = coded * 0.5 * range + center; x = 60 hours
rw.x <- c(24, 48, 36, 36, 60)
coded.x <- (rw.x - center)/(0.5*range)
x.test <- data.frame(coded.x=coded.x)
predict(model.1.quad, newdata=x.test) # predicts 53.5
# Acutal y = 66. Therefore, our model isn't so good. Improve it with the new point
rw.x <- c(24, 48, 36, 36, 60)
y2 <- c(28, 63, 55, 54, 66)
coded.x <- (rw.x - center)/(0.5*range)
model.2.quad <- lm(y2 ~ coded.x + I(coded.x^2))
summary(model.2.quad)
# Plot it again: purple
raw_data <- data.frame(coded.x = coded.x, y = y2)
plot_data <- data.frame(coded.x = seq(-2, +3, 0.1))
plot_data$y <- predict(model.2.quad, newdata=plot_data)
p <- p + geom_point(aes(x=coded.x, y=y, color='purple', size=5), data=raw_data)
p <- p + geom_line(data=plot_data, color="purple", size=1)
p
#------------
#------------
# Normally at this point I would reset the frame of reference;
# keep our range the same (24 hours)
# -1: 36 hours [prior coded "0" now becomes "-1]
# 0: 48 hours [prior coded "+1" now becomes "0"]
# +1: 60 hours [prior coded "+2" now becomes "+1"]
center = 48
range = 24 # unchanged, but could also have been chosen to be smaller or bigger
# At this point I will also switch the model type back to real-world units.
# For 2 reasons: there is only 1 variable that is interesting (duration), so
# coded units don't matter; secondly the model types we will use cannot be
# built when the coded x-value is negative.
rw.x <- c(24, 48, 36, 36, 60)
y3 <- c(28, 63, 55, 54, 66)
# Rebuild the model, and start the plots again. Try different model type
model.3 <- lm(y3 ~ I(1/(rw.x)))
summary(model.3)
x.min = 20
x.max = 105
# Basic plot of everything so far:
raw_data <- data.frame(rw.x = rw.x, y = y3)
p <- ggplot(data=raw_data, aes(x=rw.x, y=y)) +
geom_point(size=5) +
xlab("Real-world values: time") +
scale_x_continuous(breaks=seq(x.min, x.max,5)) +
ylab("Outcome variable") +
scale_y_continuous(breaks=seq(0, 150, 10)) +
#theme(axis.text=element_text(size=18), legend.position = "none") +
theme(axis.title=element_text(face="bold", size=14)) +
expand_limits(x = c(x.min, x.max)) +
expand_limits(y = c(50, 80))
p
plot_data <- data.frame(rw.x = seq(x.min, x.max, 0.5))
plot_data$y <- predict(model.3, newdata=plot_data)
p <- p + geom_line(data=plot_data, color="blue", size=1)
p
# Run experiment at use 75 hours - just a little more]: predict it first
rw.x <- c(24, 48, 36, 36, 60, 75)
x.test <- data.frame(rw.x=rw.x)
predict(model.3, newdata=x.test)
# Expect a predicted value of 74.3 in the output. # Actual result: 79.
# So about 5 units difference. Not too bad.
rw.x <- c(24, 48, 36, 36, 60, 75)
y4 <- c(28, 63, 55, 54, 66, 79)
# Rebuild the model, and start the plots again. Different model type: x + x^2
model.4 <- lm(y4 ~ I(1/(rw.x)))
summary(model.4)
# Plot it again: purple
raw_data <- data.frame(rw.x = rw.x, y = y4)
plot_data <- data.frame(rw.x = seq(x.min, x.max, 0.5))
plot_data$y <- predict(model.4, newdata=plot_data)
p <- p + geom_point(aes(x=rw.x, y=y, color='purple', size=5), data=raw_data)
p <- p + geom_line(data=plot_data, color="purple", size=1)
p
# -------
# Let's try at around 90 hours. Expect an outcome of around 80.
# Got a value of 76 instead. Stabilized?
rw.x <- c(24, 48, 36, 36, 60, 75, 90)
y5 <- c(28, 63, 55, 54, 66, 79, 76)
# Rebuild the model, and start the plots again. Different model type: x + x^2
model.5 <- lm(y5 ~ I(1/(rw.x)))
summary(model.5)
# Plot it again: green
raw_data <- data.frame(rw.x = rw.x, y = y5)
plot_data <- data.frame(rw.x = seq(x.min, x.max, 0.5))
plot_data$y <- predict(model.5, newdata=plot_data)
p <- p + geom_point(aes(x=rw.x, y=y, color='green', size=5), data=raw_data)
p <- p + geom_line(data=plot_data, color="green", size=1)
p
# Try adding few more points:
# Point 8. Try 95 hours. Seems to confirm stabilization. Predict 79; actual: 81
# Point 9: Overshoot(?) to 105 hour to see decline. Predict 81; actual: 72
# Build a model with all values now
rw.x <- c(48, 36, 36, 60, 75, 90, 95, 105)
y6 <- c(63, 55, 54, 66, 79, 76, 81, 72)
# Rebuild the model, and start the plots again
model.6 <- lm(y6 ~ I(1/(rw.x)))
summary(model.6)
raw_data <- data.frame(rw.x = rw.x, y = y6)
plot_data <- data.frame(rw.x = seq(x.min, x.max, 0.1))
plot_data$y <- predict(model.5, newdata=plot_data)
p <- p + geom_point(aes(x=rw.x, y=y, color='red', size=5), data=raw_data)
p <- p + geom_line(data=plot_data, color="red", size=1)
p
# The current model structure does not allow for decrease/stabilization.
# Rebuild it with a different structure
# Try various ones: x + 1/sqrt(x) SE = 4.11
# Try various ones: x + 1/x SE = 4.24
# Try various ones: x + log(x) SE = 3.98
# Try various ones: x + x^2 SE = 3.41
model.6.revised <- lm(y6 ~ rw.x + I((rw.x)^2)) #
summary(model.6.revised)
x.min = 35
x.max = 105
raw_data <- data.frame(rw.x = rw.x, y = y6)
p <- ggplot(data=raw_data, aes(x=rw.x, y=y)) +
geom_point(size=5) +
xlab("Real-world values: time") +
scale_x_continuous(breaks=seq(x.min, x.max,5)) +
ylab("Outcome variable") +
scale_y_continuous(breaks=seq(0, 150, 10)) +
theme(axis.title=element_text(face="bold", size=14)) +
expand_limits(x = c(x.min, x.max)) +
expand_limits(y = c(50, 80))
p
plot_data <- data.frame(rw.x = seq(x.min, x.max, 0.5))
plot_data$y <- predict(model.6.revised, newdata=plot_data)
p <- p + geom_line(data=plot_data, color="orange", size=1)
p