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# Configure Jupyter so figures appear in the notebook
%matplotlib inline
# Configure Jupyter to display the assigned value after an assignment
%config InteractiveShell.ast_node_interactivity='last_expr_or_assign'
# import functions from the modsim library
from modsim import *
Here's a simple function that returns a value:
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def add_five(x):
return x + 5
And here's how we call it.
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y = add_five(3)
If you run a function on the last line of a cell, Jupyter displays the result:
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add_five(5)
But that can be a bad habit, because usually if you call a function and don't assign the result in a variable, the result gets discarded.
In the following example, Jupyter shows the second result, but the first result just disappears.
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add_five(3)
add_five(5)
When you call a function that returns a variable, it is generally a good idea to assign the result to a variable.
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y1 = add_five(3)
y2 = add_five(5)
print(y1, y2)
Exercise: Write a function called make_state that creates a State object with the state variables olin=10 and wellesley=2, and then returns the new State object.
Write a line of code that calls make_state and assigns the result to a variable named init.
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# Solution goes here
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# Solution goes here
Here's the code from the previous notebook.
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def step(state, p1, p2):
"""Simulate one minute of time.
state: bikeshare State object
p1: probability of an Olin->Wellesley customer arrival
p2: probability of a Wellesley->Olin customer arrival
"""
if flip(p1):
bike_to_wellesley(state)
if flip(p2):
bike_to_olin(state)
def bike_to_wellesley(state):
"""Move one bike from Olin to Wellesley.
state: bikeshare State object
"""
if state.olin == 0:
state.olin_empty += 1
return
state.olin -= 1
state.wellesley += 1
def bike_to_olin(state):
"""Move one bike from Wellesley to Olin.
state: bikeshare State object
"""
if state.wellesley == 0:
state.wellesley_empty += 1
return
state.wellesley -= 1
state.olin += 1
def decorate_bikeshare():
"""Add a title and label the axes."""
decorate(title='Olin-Wellesley Bikeshare',
xlabel='Time step (min)',
ylabel='Number of bikes')
Here's a modified version of run_simulation that creates a State object, runs the simulation, and returns the State object.
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def run_simulation(p1, p2, num_steps):
"""Simulate the given number of time steps.
p1: probability of an Olin->Wellesley customer arrival
p2: probability of a Wellesley->Olin customer arrival
num_steps: number of time steps
"""
state = State(olin=10, wellesley=2,
olin_empty=0, wellesley_empty=0)
for i in range(num_steps):
step(state, p1, p2)
return state
Now run_simulation doesn't plot anything:
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state = run_simulation(0.4, 0.2, 60)
But after the simulation, we can read the metrics from the State object.
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state.olin_empty
Now we can run simulations with different values for the parameters. When p1 is small, we probably don't run out of bikes at Olin.
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state = run_simulation(0.2, 0.2, 60)
state.olin_empty
When p1 is large, we probably do.
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state = run_simulation(0.6, 0.2, 60)
state.olin_empty
linspace creates a NumPy array of equally spaced numbers.
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p1_array = linspace(0, 1, 5)
We can use an array in a for loop, like this:
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for p1 in p1_array:
print(p1)
This will come in handy in the next section.
linspace is defined in modsim.py. You can get the documentation using help.
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help(linspace)
linspace is based on a NumPy function with the same name. Click here to read more about how to use it.
Exercise:
Use linspace to make an array of 10 equally spaced numbers from 1 to 10 (including both).
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# Solution goes here
Exercise: The modsim library provides a related function called linrange. You can view the documentation by running the following cell:
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help(linrange)
Use linrange to make an array of numbers from 1 to 11 with a step size of 2.
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# Solution goes here
p1_array contains a range of values for p1.
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p2 = 0.2
num_steps = 60
p1_array = linspace(0, 1, 11)
The following loop runs a simulation for each value of p1 in p1_array; after each simulation, it prints the number of unhappy customers at the Olin station:
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for p1 in p1_array:
state = run_simulation(p1, p2, num_steps)
print(p1, state.olin_empty)
Now we can do the same thing, but storing the results in a SweepSeries instead of printing them.
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sweep = SweepSeries()
for p1 in p1_array:
state = run_simulation(p1, p2, num_steps)
sweep[p1] = state.olin_empty
And then we can plot the results.
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plot(sweep, label='Olin')
decorate(title='Olin-Wellesley Bikeshare',
xlabel='Arrival rate at Olin (p1 in customers/min)',
ylabel='Number of unhappy customers')
Exercise: Wrap this code in a function named sweep_p1 that takes an array called p1_array as a parameter. It should create a new SweepSeries, run a simulation for each value of p1 in p1_array, store the results in the SweepSeries, and return the SweepSeries.
Use your function to plot the number of unhappy customers at Olin as a function of p1. Label the axes.
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# Solution goes here
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# Solution goes here
Exercise: Write a function called sweep_p2 that runs simulations with p1=0.5 and a range of values for p2. It should store the results in a SweepSeries and return the SweepSeries.
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# Solution goes here
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# Solution goes here
The following two exercises are a little more challenging. If you are comfortable with what you have learned so far, you should give them a try. If you feel like you have your hands full, you might want to skip them for now.
Exercise: Because our simulations are random, the results vary from one run to another, and the results of a parameter sweep tend to be noisy. We can get a clearer picture of the relationship between a parameter and a metric by running multiple simulations with the same parameter and taking the average of the results.
Write a function called run_multiple_simulations that takes as parameters p1, p2, num_steps, and num_runs.
num_runs specifies how many times it should call run_simulation.
After each run, it should store the total number of unhappy customers (at Olin or Wellesley) in a TimeSeries. At the end, it should return the TimeSeries.
Test your function with parameters
p1 = 0.3
p2 = 0.3
num_steps = 60
num_runs = 10Display the resulting TimeSeries and use the mean function provided by the TimeSeries object to compute the average number of unhappy customers (see Section 2.7).
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# Solution goes here
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# Solution goes here
Exercise: Continuting the previous exercise, use run_multiple_simulations to run simulations with a range of values for p1 and
p2 = 0.3
num_steps = 60
num_runs = 20Store the results in a SweepSeries, then plot the average number of unhappy customers as a function of p1. Label the axes.
What value of p1 minimizes the average number of unhappy customers?
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# Solution goes here
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# Solution goes here