Remember, to use the numpy module, first it must be imported:
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import numpy as np
You can do a lot with the numpy module. Below is an example to jog your memory:
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np.linspace(0,10,11)
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Do you remember the Fibonacci sequence from yesterday's Lecture 1? Let's define our own function that will help us to write the Fibonacci sequence.
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def myFib(a,b):
return a+b
Remember loops too? Let's get the first 10 numbers in the Fibonacci sequence.
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fibLength = 10 #the length we want for our Fibonacci sequence
fibSeq = np.zeros(fibLength) #make a numpy array of 10 zeros
# Let's define the first 2 elements of the Fibonacci sequence
fibSeq[0] = 0
fibSeq[1] = 1
i = 2 #with the first 2 elements defined, we can calculate the rest of the sequence beginning with the 3rd element
while i-1 < fibLength-1:
nextFib = myFib(fibSeq[i-2],fibSeq[i-1])
fibSeq[i] = nextFib
i = i + 1
print(fibSeq)
There's your quick review of numpy and functions along with a while loop thrown in. Now we can move on to the content of Lecture 3.
In the previous lecture, you learned how to import the module numpy and how to use many of its associated functions. As you've seen, numpy gives us the ability to generate arrays of numbers using commands usch as np.linspace and others.
In addition to these commands, you can also use numpy to generate distributions of numbers. The two most frequently used distributions are the following:
np.random.randnp.random.randn
(notice the "n" that distinguishes the functions for generating normal vs. uniform distributions)
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import numpy as np
Let's generate a vector of length 5 populated with uniformly distributed random numbers. The function np.random.rand takes the array output size as an argument (in this case, 5).
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np.random.rand(5)
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Additionally, you are not limited to one-dimensional arrays! Let's make a 5x5, two-dimensional array:
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np.random.rand(5,5)
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Great, so now you have a handle on generating uniform distributions. Let's quickly look at one more type of distribution.
The equation for a Gaussian curve is the following:
$f(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{-(x-\mu)^2}{2\sigma^2}}$
where $\mu$ is the mean and $\sigma$ is the standard deviation.
Don't worry about memorizing this equation, but do know that it exists and that numbers can be randomly drawn from it.
In python, the command np.random.randn selects numbers from the "standard" normal distribution.
All this means is that, in the equation above, $\mu$ (mean) = 0 and $\sigma$ (standard deviation ) 1. randn takes the size of the output as an argument just like rand does.
Try running the cell below to see the numbers you get from a normal distribution.
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np.random.randn(5)
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So these numbers probably don't mean that much to you. Don't worry; they don't mean much to me either!
Instead of trying to derive meaning from a list of numbers, let's actually plot these outputs and see what they look like. This will allow us to determine whether or not these distributions actually look like what we are expecting. How do we do that? The answer is with histograms!
Histogram documentation: http://matplotlib.org/1.2.1/api/pyplot_api.html?highlight=hist#matplotlib.pyplot.hist
Understanding distributions is perhaps best done by plotting them in a histogram. Lucky for us, matplotlib makes that very simple for us.
To make a histogram, we use the command plt.hist, which takes -- at minimum -- a vector of values that we want to plot as a histogram. We can also specify the number of bins.
First things first: let's import matplotlib:
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import matplotlib.pyplot as plt
%matplotlib inline
Use what you learned above to define your variable X as a uniformly distributed array with 5000 elements.
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#your code here
X = np.random.rand(5000)
Now, let's use plt.hist to see what X looks like. First, run the cell below. Then, vary bins -- doing so will either increase or decrease the apparent effect of noise in your distribution.
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plt.hist(X, bins=20);
Nice job! Do you see why the "uniform distribution" is referred to as such?
In the cell below, generate a vector of length 5000, called X, from the normal (Gaussian) distribution and plot a histogram with 50 bins.
HINT: You will use a similar format as above when you defined and plotted a uniform distribution.
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#your code here
X = np.random.randn(5000)
plt.hist(X, bins=50);
Nice job! You just plotted a Gaussian distribution with mean of 0 and a standard deviation of 1.
As a reminder, this is considered the "standard" normal distribution, and it's not particularly interesting. We can transform the distribution given by np.random.randn (and make it more interesting!) using simple arithmetic.
Run the cell below to see. How is the code below different from the code you've already written?
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mu = 5 #the mean of the distribution
sigma = 3 #the standard deviation
Y = sigma * np.random.randn(5000) + mu
plt.hist([X,Y],bins=50,histtype='step');
Before moving onto the next section, vary the values of mu and sigma in the above code to see how your histogram changes. You should find that changing mu (the mean) affects the center of the distribution while changing sigma (the standard deviation) affects the width of the distribution.
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#write your observations here
For simplicity's sake, we've used plt.hist without generating any return variables. Remember that plt.hist takes in your data (X) and the number of bins, and it makes histograms from it. In the process, plt.hist generates variables that you can store; we just haven't thus far. Run the cell below to see -- it should replot the Gaussian from above while also generating the output variables.
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N,bins, = plt.hist(X, bins=50)
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_
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Something that might be useful to you is that you can make use of variables outputted by plt.hist -- particularly bins and N.
The bins array returned by plt.hist is longer (by one element) than the actual number of bins. Why? Because the bins array contains all the edges of the bins. For example, if you have 2 bins, you will have 3 edges. Does this make sense?
So you can generate these outputs, but what can you do with them? You can average consecutive elements from the bins output to get, in a sense, a location of the center of a bin. Let's call it bin_avg. Then you can plot the number of observations in that bin (N) against the bin location (bin_avg).
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bin_avg = (bins[1:]+bins[:-1])/2
plt.plot(bin_avg, N, 'r*')
plt.show()
The plot above (red stars) should look like it overlays the histogram plot above it. If that's what you see, nice job! If not, let your instructor and/or TAs know before moving onto the next section.
If you ever want to check that your distributions are giving you what you expect, you can use numpy to calculate the mean and standard deviation of your distribution. Let's do this for X, our Gaussian distribution, and print the results.
Run the cell below. Are your mean and standard deviation what you expect them to be?
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mean = np.mean(X)
std = np.std(X)
print('mean: '+ repr(mean) )
print('standard deviation: ' + repr(std))
So you've learned how to generate distributions of numbers, plot them, and generate statistics on them. This is a great starting point, but let's try working with some real data!
Hope you're excited -- we're about to get our hands on some real data! Let's import a list of fluorescence lifetimes in nanoseconds from Nitrogen-Vacancy defects in diamond.
(While it is not at all necessary to understand the physics behind this, know that this is indeed real data! You can read more about it at http://www.nature.com/articles/ncomms11820 if you are so inclined. This data is from Fig. 6a).
Do you remember learning how to import data in yesterday's Lecture 2? The command you want to use is np.loadtxt. The data we'll be working with is called LifetimeData.txt, and it's located in the Data folder.
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lifetimes = np.loadtxt('Data/LifetimeData.txt')
Next, plot a histogram of this data set (play around with the number of bins, too).
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#your code here
N,bins,patches = plt.hist(lifetimes,bins=10)
Now, calculate and print the mean and standard deviation of this distribution.
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#your code here
mean = np.mean(lifetimes)
std = np.std(lifetimes)
print("mean: "+repr(mean))
print("standard deviation: "+repr(std))
Nice job! Now that you're used to working with real data, we're going to try to fit some more real data to known functions to gain a better understanding of that data.
In this section, we're going to introduce you to the Python module known as scipy (short for Scientific Python).
scipy allows you to perform a range of functions such as numerical integration and optimization. In particular, it's useful for data analysis, which we shall see shortly. In particular, we will do curve fitting using curve_fit from scipy.optimize.
Curve fitting documentation: https://docs.scipy.org/doc/scipy-0.18.1/reference/generated/scipy.optimize.curve_fit.html
In this section, you will learn how to use curve fitting on simulated data. Next will be real data!
First, let's load the modules.
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import numpy as np
from scipy.optimize import curve_fit
import matplotlib.pyplot as plt
%matplotlib inline
We will show you an example, and then you get to try it out for yourself!
We start by creating an equally-spaced numpy array x consisting of 100 numbers from -5 to 5. Try it out yourself below.
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# your code here
x = np.linspace(-5,5,100)
Next, we will define a function $f(x) = \frac 1 3x^2+3$ that will square the elements in x and add an offset. Call this function f, and implement it below.
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# your code here
def f(x):
return 1/3*x**2 + 3
Now set y equal to f called with the numpy array x as the input.
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y = f(x)
Now we will add some noise to the array y using the np.random.rand() function and store it in a new variable called y_noisy.
Important question: What value for the array size should we pass into this function?
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# your code here
y_noisy = y + np.random.rand(100)
Let's see what the y values look like now
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plt.plot(x,y_noisy)
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It seems like there's still a rough parabolic shape, so let's see if we can recover the original y values without any noise.
We can treat this y_noisy as data values that we want to fit with a parabolic funciton. To do this, we first need to define the general form of a quadratic function:
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def quadratic(x,a,b,c):
return a*x**2 + b*x + c
Then, we want to find the optimal values of a, b, and c that will give a function that fits best to y_noisy.
We do this using the curve_fit function in the following way:
curve_fit(f,xdata,ydata)
where f is the model we're fitting to (quadratic in this case).
This function will return the optimal values for a, b, and c in a list. Try it out!
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optimal_values, _ = curve_fit(quadratic,x,y_noisy)
a = optimal_values[0]
b = optimal_values[1]
c = optimal_values[2]
print(a, b, c)
Now that we have the fitted parameters, let's use quadratic to plot the fitted parabola alongside the noisy y values.
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y_fitted = quadratic(x,a,b,c)
plt.plot(x,y_fitted)
plt.plot(x,y_noisy)
And we can also compare y_fitted to the original y values without any noise:
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plt.plot(x,y_fitted)
plt.plot(x,y)
Not a bad job for your first fit function!
In this section, you will visualize real data and plot a best-fit function to model the underlying physics.
You just used curve_fit above to fit simulated data to a linear function. Using that code as your guide, combined with the steps below, you will use curve_fit to fit your real data to a non-linear function that you define. This exercise will combine most of what you've learned so far!
Here is the basic outline on how to use curve_fit. As this is the last section, you will mostly be on your own. Try your best with new skills you have learned here and feel free to ask for help!
1) Load in your x and y data. You will be using "photopeak.txt", which is in the folder Data.
HINT 1: When you load your data, I recommend making use of the usecols and unpack argument.
HINT 2: Make sure the arrays are the same length!
2) Plot this data to see what it looks like. Determine the function your data most resembles.
3) Define the function to which your data will be fit.
4) PART A: Use curve_fit and point the output to popt and pcov. These are the fitted parameters (popt) and their estimated errors (pcov).
4) PART B - OPTIONAL (only do this if you get through all the other steps): Input a guess (p0) and bounds (bounds) into curve_fit. For p0, I would suggest [0.5, 0.1, 5].
5) Pass the popt parameters into the function you've defined to create the model fit.
6) Plot your data and your fitted function.
7) Pat yourself on the back!
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# Step 1: Import the data
# Step 2: Plot the data to see what it looks like
xData,yData = np.loadtxt('Data\photopeak.txt', usecols=(0,1), unpack=True)
plt.plot(xData,yData,'*')
plt.title('xData and yData')
So you've imported your data and plotted it. It should look similar to the figure below. Run the next cell to see.
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from IPython.display import display, Image
display(Image(filename='Data\photopeak.png'))
What type of function would you say this is? Think back to the distributions we've learned about today. Any ideas?
Based on what you think, define your function below.
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# Step 3: Define your function here
def myGaussian(Xvals,A,mu,sigma):
return (A/np.sqrt(2*np.pi*sigma**2))*np.exp(-((Xvals-mu)**2/(2*sigma**2)))
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# Step 3.5: SANITY CHECK! Use this step as a way to check that the function you defined above is mathematically correct.
mu = 0.66 #the mean of the distribution
sigma = 0.04 #the standard deviation
A = 10;
Xvals = np.linspace(0.50,0.80,100)
Yvals = A*myGaussian(Xvals,A,mu,sigma)
plt.plot(Xvals,Yvals)
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# Step 4: Use curve_fit to generate your output parameters
popt, pcov = curve_fit(myGaussian, xData, yData, p0=[0.5, 0.1, 5])
#perr = np.sqrt(np.diag(pcov))
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# Step 5: Generate your model fit
xFit = np.linspace(min(xData),max(xData),100) #give this
line_fit = myGaussian(xFit, *popt)
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# Step 6: Plot the best fit function and the scatter plot of data
plt.plot(xData, yData, 'r*')
plt.plot(xFit, line_fit)
There you have it! You've completed advanced curve fitting. Congratulations -- give yourself a pat on the back (Step 7)!