Create 04_workshop_solutions.R

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+# workshop_4_solutions.R
+# Dr. Fraser
+
+# In today's workshop, let's practice our many, many ways 
+# of using failure functions to analyze system reliability! 
+
+# See workshop_4_solutions.R for solutions (but only after class!)
+
+###################################
+# 0. Load Packages
+###################################
+library(dplyr)
+library(ggplot2)
+library(mosaicCalc)
+
+###################################
+# 1. Making Functions
+###################################
+
+# You learned to make Functions in Week 3. Let's practice!
+# https://timothyfraser.com/sigma/skill-functions-in-r.html
+
+# Nintendo is product testing its next Switch console.
+# 1000 enthusiastic children received a console in the mail,
+# played video games for days on end, and 
+# parents called in when the console eventually broke.
+
+# On average, 1 console failed every 1200 hours (50 days).
+
+# So, we say the constant failure rate lambda was 1/1200 hours
+
+# Using your notes from Workshop 4, let's write a few functions using lambda!
+
+
+### Example Function l() (for lambda)
+
+# The exponential distribution has one parameter
+# lambda = the rate of failure
+# 1 / mean time to failure
+
+
+# units = how many pokeballs failed
+# hours = how many hours they were used
+
+# mu = hours / units ( mean hours to failure)
+# lambda = 1 / mu
+# every hour, how many pokeballs do we expect to fail?
+
+l = function(units, hours){ units / hours }
+# We have an empirical failure rate of 0.2 pokeballs per hour
+l(units = 200, hours = 1000)
+
+# Try it out
+l(units = 1, hours = 1200)
+# We can also, respecting the order of units and hours, write:
+l(1, 1200)
+
+
+
+
+
+
+# Note: In r, for exponential distribution,
+# we write e^something as exp(something)
+
+# Q1. Let's write our probability density function d(t)
+# PDF loooks like: d(t) = lambda * e^(-lambda*t)
+d = function(t, lambda){  lambda * exp(-1*lambda*t) }
+
+# PDF f(t) is always d(t) in R
+d(t = 100, lambda = 0.01)
+dexp(x = 100, rate = 0.01)
+
+# Failure F(t)
+f = function(t, lambda){ 1 - exp(-1*lambda*t) }
+# cumulative probability of failure by 100 hours?
+f(t = 100, lambda = 0.01)
+pexp(q = 100, rate = 0.01)
+# Or, you could integrate the PDF to produce the equivalent function!!
+f2 = mosaicCalc::antiD(tilde = d(t, lambda) ~ t)
+f2(t = 100, lambda = 0.01)
+
+
+# Reliability R(t)
+f = function(t, lambda){ 1 - exp(-1*lambda*t) }
+r = function(t, lambda){ 1 - f(t, lambda) }
+# EXCEPT
+
+r = function(t, lambda){ 
+  # Tell my funciton r(t, lambda) that f(t, lambda) means this:
+  f = function(t, lambda){ 1 - exp(-1*lambda*t) }
+  # calcualte it!
+  1 - f(t, lambda)
+  # output = 1 - f(t, lambda);  return(output)
+}
+
+r(t = 100, lambda = 0.01)
+
+
+lambda_a = .01
+lambda_b = .005
+
+dat = tibble(
+  t = 1:1000,
+  prob_a = r(t = t, lambda = lambda_a),
+  prob_b = r(t = t, lambda = lambda_b)
+)
+
+ggplot() +
+  geom_area(data = dat, mapping = aes(x = t, y = prob_b),
+            alpha = 0.5, fill = "darksalmon") +
+  geom_area(data = dat, mapping = aes(x = t, y = prob_a),
+            alpha = 0.75, fill = "seagreen") +
+  labs(x = "Hours to Failure", y = "Reliability (%)")
+
+# Failure Rate Function
+z = function(t, lambda){ dexp(t, lambda) / (1 - pexp(t, lambda) )}
+z(t = 0.5, lambda = 0.01)
+
+
+# Q2. Let's write our failure function f(t)
+f = function(t, lambda){1 - exp(-1*lambda * t)}
+
+# Q3. Let's write our reliability function r(t)
+r = function(t, lambda){exp(-1 * lambda * t)}
+
+# Q4. Let's write our failure rate z(t) (hazard rate)
+z = function(t, lambda){
+  # density function / aka change in failure function
+  lambda * exp(-1 * lambda * t) /
+    # reliability function
+    exp(-1*lambda*t)
+}
+
+
+# Q5. Test d(t), f(t), r(t), and z(t)
+# over a span of 150 days (1 to 3600 hours)
+# and visualize each with hist()
+
+d(t = 1:3600, lambda = 1/1200) %>% hist()
+
+f(t = 1:3600, lambda = 1/1200) %>% hist()
+
+r(t = 1:3600, lambda = 1/1200) %>% hist()
+
+z(t = 1:3600, lambda = 1/1200) %>% hist()
+
+# What do you notice?
+
+##################################
+# Clear environment
+rm(list = ls())
+
+
+####################################
+# 2. Higher level Functions
+####################################
+
+# Sometimes, we make functions that USE functions inside them.
+# Just make sure you either 
+# (1) put function A before function B, or 
+# (2) put function A INSIDE function B before using it
+
+
+# We can re-write r(t) USING our function f(t)
+f = function(t, lambda){ 1 - exp(-1*lambda*t) }
+r = function(t, lambda){ 1 - f(t, lambda)   }
+
+# Or we can embed f(t) in r(t)
+r = function(t, lambda){ 
+  # Write f(t)
+  f = function(t, lambda){ 1 - exp(-1*lambda*t) }
+  # Then calculate and return r(t)
+  1 - f(t, lambda)   
+}
+
+# We can use embedded functions to create super functions, like h(t) and afr(t)