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Experimental data, calculating estimates of parameters

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PHY224
Introduction to Scientific Computing
In this assignment we focus on using experimental data, calculating estimates of parameters from the data,
and fitting the data with Python, numpy, and scipy. The example data used here is simulated for two
experiments – The speed of the Saturn V rocket on its journey to the Moon, and the dropping of a feather on
the moon to measure Lunar gravity.
1 Reading and plotting data
In an experiment to measure the speed of a Saturn V rocket on its way to the moon, the distance from the
Earth was recorded every 4 hours and saved into the file rocket.csv. Read the file into Python and plot the
data with uncertainties.
You should use numpy.loadtxt to load data from the file, and the plot should include errorbars, axes labels
and a title. Do not join the data with a line. Finally make sure you save your figure with a reasonable
filename.
2 Estimating the speed
The next step in finding the speed of the ship is to estimate the speed by taking the average value of
distance/time. Using the data in the rocket.csv file, calculate the mean and standard error of speed from
the data. You can ignore the uncertainties for this question.
Note that using mean and standard deviation in this way is useful for a quick estimate of your data, but it’s
not statistically robust because uncertainties are not considered.
3 Linear Regression
The distance vs. time equation is simple enough that we can analytically derive the best estimate of s0 and u
considering all data at once, given by the following equations:
uˆ =
qn
i=1(ti ≠ t
¯)(di ≠ ¯
d)
qn
i=1(ti ≠ t
¯)2
and
sˆ0 = ¯s ≠ uˆt.¯
where sˆ0 is the best estimate of variable s0, and s¯0 is the average value of the s0 data (and similarly for u).
Write a script to read the data from the rocket.csv file, and calculate the best estimates for u and s0 using
the above equations.
4 Plotting the prediction
With a prediction of the speed and starting position, we can add our prediction to the data plot. Write
a function that takes time, initial position, and speed, and predicts the location of the rocket (i.e. write
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PHY224 University of Toronto
s = s0 + ut as a Python function). Use this function to make a new plot with your measured data and a
predicted position generated for each measurement time. You can use your estimated values uˆ and sˆ0, or the
initial value of u¯ (and 0 km for the distance).
Your plot should now have the collection of data points with uncertainties, and a straight line showing the
predicted position. You will need to add a legend to your plot to identify the data and prediction clearly.
5 Characterizing the fit
Now that you have a set of parameters for the model function, you can determine how well the model fits the
data. One way to characterize the model fit is to calculate the ‰2
r metric, defined as
‰2
r = 1
N ≠ m
ÿ
N
i
(yi ≠ f(xi))2
‡2
i
.
‰2
r squared is made up of four parts
• yi ≠ f(xi) – The dierence between the measured data yi and the prediction with your model f(xi)
• ‡i – The uncertainty on each measurement’s dependent data (the errors on y).
• The sum of the squared-ratio of the previous two values. q
i
• 1
N≠m Is the denominator that scales ‰2
r to be around 1 for a good fit. In this term N is the number of
data points, and m is the number of parameters in your model (2 in this case).
Write a function to calculate ‰2
r, and use it to calculate ‰2
r for your best fitting parameters from the last
exercise.
6 Curve fitting
The linear regression equations you used above do not account for uncertainties, and the algebra gets more
complicated quickly as the model gets more complicated (e.g. when more terms are included). In addition,
the regression is linear and requires a model that is linear in the unknown parameters (ax + b is linear, axb is
not).
Instead, we are going to fit the measured data using a scipy function called curve_fit. curve_fit uses a
mathematical algorithm to fit (almost) any model to a dataset by making progressively better guesses at the
right answer and internally scoring each guess to make sure it improves the model fit. curve_fit comes from
the scipy.optimize library, and documentation can be found at https://docs.scipy.org/doc/scipy/reference/
generated/scipy.optimize.curve_fit.html
Typically, curve_fit will be called with the following code.
from scipy.optimize import curve_fit
popt, pcov = curve_fit(model_function, xdata, ydata,
sigma=yerrors, absolute_sigma=True,
p0=first_guess_parameters)
pstd = np.sqrt(np.diag(pcov))
In this code the variables are:
• model_function is the model of the data being measured.
• xdata are the independent data (a numpy array).
• ydata are the dependent data (a numpy array).
• yerrors are the errors on the dependent data (a numpy array).
• absolute_sigma=true is needed to tell curve_fit to treat your errors as absolute, not relative, values
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PHY224 University of Toronto
• first_guess_parameters are your first guess at each parameter in your model (a list, tuple, or numpy
array)
curve_fit then returns two variables, popt is a list of the optimized (i.e. best estimate) parameters. pcov is
the covariance matrix, whose diagonals are the variances of the optimized parameters. The square root of the
diagonal values are the standard deviations of each parameter independent of all other parameters (these are
the values you might use in error propagation). The pstd in the above code are the standard deviations.
Write the Python code to run curve_fit on the rocket dataset to find the best estimate of the starting
position and speed.
• Print out the estimates and their associated uncertainties (standard deviations).
• Print out the ‰2
r for the model with the new best estimate parameters.
• Plot the data with uncertainties, and plot the model using your linear regression parameters and the
curve_fit parameters as separate lines.
7 Feather Drop Experiment
Now that they are on the moon, one of the astronauts makes a measurement of the Lunar gravity by dropping
a feather from shoulder height, saving the data in feather.csv. This time the equation of motion includes
acceleration due to gravity
s = s0 + ut +
1
2
at2
Write a program that will
1. Load the time, position, and uncertainty data from the file feather.csv
2. Predict the position of the feather above the Lunar surface as a function of time given the initial
position s0, initial speed u, and constant acceleration a.
3. Fit the model to the data to find the optimal (i.e. best) parameters.
4. Print out the parameters fit by your model, including their uncertainties and units.
5. Plot the data including errorbars, and a prediction made by your model function using the best
parameters.
It is often helpful to provide curve_fit with an initial guess of the parameters. Assume that the astronaut is
between 1.5m and 2.0m tall, and Lunar gravity is around 1/7th of Earth’s gravity.
The code your write to answer this question will be a good starting point for the code your write
for all experiments in this course that involve fitting a model to data to derive a parameter
value.
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