Difference between revisions of "Worksheets/Week9"

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=== Part 1 ===
=== Part 1 ===


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<html><div data-datacamp-exercise data-lang="r" data-height="auto">
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.
    <code data-type="sample-code">
center = 36
range = 24


rw.x <- c(24, 48)
coded.x <- (rw.x - center)/(0.5*range)
y0 <- c(28, 63)


model.0 <- lm(y0 ~ coded.x)
<html><div data-datacamp-exercise data-lang="r" data-height="500">
summary(model.0)
    <code data-type="sample-code">
 
# What is the interpretation of the
# * slope?
# * intercept
# * why are R2 and SE where 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_bw() +
  theme(axis.text=element_text(size=18), legend.position = "none") +
  theme(axis.title=element_text(face="bold", size=14))
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: 54 and 55. So about
# 10 units difference.
 
# 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
 
 
# Try fitting a linear model now through all the 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)
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
 
 
 
# A a quadratic component through all the 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.quad <- lm(y1 ~ coded.x + I(coded.x^2))
summary(model.1.quad)
 
# Show the quadratic fit between -2 and +2 in coded units
# 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
x.test <- data.frame(coded.x=coded.x)
rw.x <- c(24, 48, 36, 36, 60)
coded.x <- (rw.x - center)/(0.5*range)
predict(model.1.quad, newdata=x.test)  # predicts ~54
 
# 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:
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
#------------
 
#------------
# Reset our frame of reference; keep our range the same (24 hours)
# -1: 36
# +1: 60
#  0: 48 hours
center = 48
range = 24
 
rw.x <- c(48, 36, 36, 60)
y3  <- c(63, 55, 54, 66)
 
# Rebuild the model, and start the plots again
model.3 <- lm(y3 ~ rw.x + I(1/(rw.x)))
summary(model.3)
 
x.min = 35
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_bw() +
  #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, 0.1))
plot_data$y <- predict(model.3, newdata=plot_data)
p <- p + geom_line(data=plot_data, color="blue", size=1)
p
 
# Run experiment at +2 [72 hours; use 75 hours - just a little more]: predict it first
rw.x <- c(48, 36, 36, 60, 75)
coded.x <- (rw.x - center)/(0.5*range)
x.test <- data.frame(coded.x=coded.x)
predict(model.3, newdata=x.test)
 
# Expect a predicted value of 66 in the output. Actual: 79. So about 12 units difference.
# Update model and plot
rw.x <- c(48, 36, 36, 60, 75)
y4  <- c(63, 55, 54, 66, 79)
 
# Rebuild the model, and start the plots again
model.4 <- lm(y4 ~ rw.x + I(rw.x^2))
summary(model.4)
 
 
# The models are not working well for us. Let's try at around 90 hours.
# Expect an outcome of around 85. Got a value of 76. Stabilized? Try again
# at 95 hours
# Point 7: Try 90 hours. Stabilizing?                    : 76
# Point 8. Try 95 hours. Seems to confirm stabilization  : 81
# Point 9: Overshoot(?) of 105                            : 72
 
rw.x <- c(48, 36, 36, 60, 75, 90, 95, 105)
y5  <- c(63, 55, 54, 66, 79, 76, 81, 72)
 
# Rebuild the model, and start the plots again
model.5 <- lm(y5 ~ rw.x + I(rw.x^2))
summary(model.5)
 
 
raw_data <- data.frame(rw.x = rw.x, y = y5)
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
 
 
# Rebuild the model with a differt structure, and start the plots again
model.6 <- lm(y5 ~ rw.x + I(1/sqrt(rw.x)))
summary(model.6)


plot_data <- data.frame(rw.x = seq(x.min, x.max, 0.1))
plot_data$y <- predict(model.6, newdata=plot_data)
p <- p + geom_line(data=plot_data, color="purple", size=1)
p


     </code>
     </code>
</div></html>
</div></html>

Latest revision as of 04:51, 6 May 2019

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.


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