A Curve Fitting Guide for the Busy Experimentalist

Curve fitting is an extremely useful analysis tool to describe the relationship between variables or discover a trend within noisy data. Here I’ll focus on a pragmatic introduction curve fitting: how to do it in Python, why can it fail and how do we interpret the results? Finally, I will also give a brief glimpse at the larger themes behind curve fitting, such as mathematical optimization, to the extent that I think is useful for the casual curve fitter.

Curve Fitting Made Easy with SciPy

We start by creating a noisy exponential decay function. The exponential decay function has two parameters: the time constant tau and the initial value at the beginning of the curve init. We’ll evenly sample from this function and add some white noise. We then use curve_fit to fit parameters to the data.

import numpy as np
import matplotlib.pyplot as plt
import scipy.optimize

# The exponential decay function
def exp_decay(x, tau, init):
    return init*np.e**(-x/tau)

# Parameters for the exp_decay function
real_tau = 30
real_init = 250

# Sample exp_decay function and add noise
np.random.seed(100)
dt=0.1
x = np.arange(0,100,dt)
noise=np.random.normal(scale=50, size=x.shape[0])
y = exp_decay(x, real_tau, real_init)
y_noisy = y + noise

# Use scipy.optimize.curve_fit to fit parameters to noisy data
popt, pcov = scipy.optimize.curve_fit(exp_decay, x, y_noisy)
fit_tau, fit_init = popt

# Sample exp_decay with optimized parameters
y_fit = exp_decay(x, opt_tau, opt_init)

fig, ax = plt.subplots(1)
ax.scatter(x, y_noisy,
           alpha=0.8,
           color= "#1b9e77",
           label="Exponential Decay + Noise")
ax.plot(x, y,
        color="#d95f02",
        label="Exponential Decay")
ax.plot(x, y_fit,
        color="#7570b3",
        label="Fit")
ax.set_xlabel("x")
ax.set_ylabel("y")
ax.legend()
ax.set_title("Curve Fit Exponential Decay")

Our fit parameters are almost identical to the actual parameters. We get 30.60 for fit_tau and 245.03 for fit_init both very close to the real values of 30 and 250. All we had to do was call scipy.optimize.curve_fit and pass it the function we want to fit, the x data and the y data. The function we are passing should have a certain structure. The first argument must be the input data. All other arguments are the parameters to be fit. From the call signature of def exp_decay(x, tau, init) we can see that x is the input data while tau and init are the parameters to be optimized such that the difference between the function output and y_noisy is minimal. Technically this can work for any number of parameters and any kind of function. It also works when the sampling is much more sparse. Below is a fit on 20 randomly chosen data points.

Of course the accuracy will decrease with the sampling. So why would this every fail? The most common failure mode in my opinion is bad initial parameters.

Choosing Good Initial Parameters

The initial parameters of a function are the starting parameters before being optimized. The initial parameters are very important because most optimization methods don’t just look for the best fit randomly. That would take too long. Instead, it starts with the initial parameters, changes them slightly and checks if the fit improves. When changing the parameters shows very little improvement, the fit is considered done. That makes it very easy for the method to stop with bad parameters if it stops in a local minimum or a saddle point. Let’s look at an example of a bad fit. We will change our tau to a negative number, which will result in exponential growth.

In this case fitting didn’t work. For a real_tau and real_init of -30 and 20 we get a fit_tau and fit_init of 885223976.9 and 106.4, both way off. So what happened? Although we never specified the initial parameters (p0), curve_fit chooses default parameters of 1 for both fit_tau and fit_init. Starting from 1, curve_fit never finds good parameters. So what happens if we choose better parameters? Looking at our exp_decay definition and the exponential growth in our noisy data, we know for sure that our tau has to be negative. Let’s see what happens when we choose a negative initial value of -5.

p0 = [-5, 1]
popt, pcov = scipy.optimize.curve_fit(exp_decay, x, y_noisy, p0=p0)
fit_tau, fit_init = popt
y_fit = exp_decay(x, fit_tau, fit_init)
fig, ax = plt.subplots(1)
ax.scatter(x, y_noisy,
           alpha=0.8,
           color= "#1b9e77",
           label="Exponential Decay + Noise")
ax.plot(x, y,
        color="#d95f02",
        label="Exponential Decay")
ax.plot(x, y_fit,
        color="#7570b3",
        label="Fit")
ax.set_xlabel("x")
ax.set_ylabel("y")
ax.legend()
ax.set_title("Curve Fit Exponential Growth Good Initials")

With an initial parameter of -5 for tau we get good parameters of -30.4 for tau and 20.6 for init (real values were -30 and 20). The key point is that initial conditions are extremely important because they can change the result we get. This is an extreme case, where the fit works almost perfectly for some initial parameters or completely fails for others. In more subtle cases different initial conditions might result in slightly better or worse fits that could still be relevant to our research question. But what does it mean for a fit to be better or worse? In our example we can always compare it to the actual function. In more realistic settings we can only compare our fit to the noisy data.

Interpreting Fitting Results

In most research setting we don’t know our exact parameters. If we did, we would not need to do fitting at all. So to compare the goodness of different parameters we need to compare our fit to the data. How do we calculate the error between our data and the prediction of the fit? There are many different measures but among the most simple ones is the sum of squared residuals (SSR).

def ssr(y, fy):
    """Sum of squared residuals"""
    return ((y - fy) ** 2).sum()

We take the difference between our data (y) and the output of our function given a parameter set (fy). We square that difference and sum it up. In fact this is what curve_fit optimizes. Its whole purpose is to find the parameters that give the smallest value of this function, the least square. The parameters that give the smallest SSR are considered the best fit. We saw that this process can fail, depending on the function and the initial parameters, but let’s assume for a moment it worked. If we found the smallest SSR, does that mean we found the perfect fit? Unfortunately not. What we found was a good estimate for the best fitting parameters given our function. There are probably other functions out there that can fit our data better. We can use the SSR to find better fitting functions in a process called cross-validation. Instead of comparing different parameters of the same function we compare different functions. However, if we increase the number of parameters we run into a problem called overfitting. I will not get into the details of overfitting here because it is beyond our scope.

The main point is that we must stay clear of misinterpretations of best fit. We are always fitting the parameters and not the function. If our fitting works, we get a good estimate for the best fitting parameters. But sometimes our fitting doesn’t work. This is because our fitting method did not converge to the minimum SSR and in the final chapter we will find out why that might happen in our example.

The Error Landscape of Exponential Decay

To understand why fitting can fail depending on the initial conditions we should consider the landscape of our sum of squared residuals (SSR). We will calculate it by assuming that we already know the init parameter, so we keep it constant. Then we calculate the SSR for many values of tau smaller than zero and many values for tau larger than zero. Plotting the SSR against the guessed tau will hopefully show us how the SSR looks around the ideal fit.

real_tau = -30.0
real_init = 20.0

noise=np.random.normal(scale=50, size=x.shape[0])
y = exp_decay(x, real_tau, real_init)
y_noisy = y + noise
dtau = 0.1
guess_tau_n = np.arange(-60, -4.9, dtau)
guess_tau_p = np.arange(1, 60, dtau)

# The SSR function
def ssr(y, fy):
    """Sum of squared residuals"""
    return ((y - fy) ** 2).sum()

loss_arr_n = [ssr(y_noisy, exp_decay(x, tau, real_init)) 
              for tau in guess_tau_n]
loss_arr_p = [ssr(y_noisy, exp_decay(x, tau, real_init))
              for tau in guess_tau_p]

"""Plotting"""
fig, ax = plt.subplots(1,2)
ax[0].scatter(guess_tau_n, loss_arr_n)
real_tau_loss = ssr(y_noisy, exp_decay(x, real_tau, real_init))
ax[0].scatter(real_tau, real_tau_loss, s=100)
ax[0].scatter(guess_tau_n[-1], loss_arr_n[-1], s=100)
ax[0].set_yscale("log")
ax[0].set_xlabel("Guessed Tau")
ax[0].set_ylabel("SSR Standard Log Scale")
ax[0].legend(("All Points", "Real Minimum", "-5 Initial Guess"))

ax[1].scatter(guess_tau_p, loss_arr_p)
ax[1].scatter(guess_tau_p[0], loss_arr_p[0], s=100)
ax[1].set_xlabel("Guessed Tau")
ax[1].set_ylabel("SSR")
ax[1].legend(("All Points", "1 Initial Guess"))

On the left we see the SSR landscape for tau smaller than 0. Here we see that towards zero, the error becomes extremely large (note the logarithmic y scale). This is because towards zero the exponential growth becomes ever faster. As we move to more negative values we find a minimum near -30 (orange), our real tau. This is the parameter curve_fit would find if it only optimized tau and started initially at -5 (green). The optimization method does not move to more negative values from -30 because there the SSR becomes worse, it increases.

On the right side we get a picture of why optimization failed when we started at 1. There is no local minimum. The SSR just keeps decreasing with larger values of tau. That is why the tau was so larger when fitting failed (885223976.9). If we set our initial parameter anywhere in this part of the SSR landscape, this is where tau will go. Now there are other optimization methods that can overcome bad initial parameters. But few are completely immune to this issue.

Easy to Learn Hard to Master.

Curve fitting is a very useful technique and it is really easy in Python with Scipy but there are some pitfalls. First of all, be aware of the initial values. They can lead to complete fitting failure or affect results in more subtle systematic ways. We should also remind ourselves that even with decent fitting results, there might be a more suitable function out there that can fit our data even better. In this particular example we always knew what the underlying function was. This is rarely the case in real research settings. Most of the time it is much more productive to think more deeply about possible underlying functions than finding more complicated fitting methods.

Finally, we barely scratched the surface here. Mathematical optimization is an entire field in itself and it is relevant to many areas such as statistics, machine learning, deep learning and many more. I tried to give the most pragmatic introduction to the topic here. If want to go deeper into the topic I recommend this Scipy lecture and of course the official Scipy documentation for optimization and root finding.

The Hodgkin-Huxley Neuron in Julia

The Hodgkin-Huxley model is one of the earliest mathematical descriptions of neural spiking. It was originally developed on data from the squid giant axon. Today, Hodgkin-Huxley like dynamics are also used to simulate the spiking of a variety of neuron types. I’ve recently written a script to simulate Hodgkin-Huxley dynamics in Julia. Here I am sharing that code and I will go through the most important elements. As I just started to learn Julia I will also mention some of the things I learned about Julia in the process.

using Plots
gr()

# Hyperparameters
tmin = 0.0
tmax = 1000.0
dt = 0.01
T = tmin:dt:tmax

# Parameters
gK = 35.0
gNa = 40.0
gL = 0.3
Cm = 1.0
EK = -77.0
ENa = 55.0
El = -65.0

# Potassium ion-channel rate functions
alpha_n(Vm) = (0.02 * (Vm - 25.0)) / (1.0 - exp((-1.0 * (Vm - 25.0)) / 9.0))
beta_n(Vm) = (-0.002 * (Vm - 25.0)) / (1.0 - exp((Vm - 25.0) / 9.0))

# Sodium ion-channel rate functions
alpha_m(Vm) = (0.182*(Vm + 35.0)) / (1.0 - exp((-1.0 * (Vm + 35.0)) / 9.0))
beta_m(Vm) = (-0.124 * (Vm + 35.0)) / (1.0 - exp((Vm + 35.0) / 9.0))

alpha_h(Vm) = 0.25 * exp((-1.0 * (Vm + 90.0)) / 12.0)
beta_h(Vm) = (0.25 * exp((Vm + 62.0) / 6.0)) / exp((Vm + 90.0) / 12.0)

# n, m & h steady-states
n_inf(Vm=0.0) = alpha_n(Vm) / (alpha_n(Vm) + beta_n(Vm))
m_inf(Vm=0.0) = alpha_m(Vm) / (alpha_m(Vm) + beta_m(Vm))
h_inf(Vm=0.0) = alpha_h(Vm) / (alpha_h(Vm) + beta_h(Vm))

# Conductances
GK(gK, n) = gK * (n ^ 4.0)
GNa(gNa, m) = gNa * (m ^ 3.0) * h
GL(gL) = gL

# Differential equations
function dv(Vm, GK, GNa, GL, Cm, EK, ENa, El, I, dt);
    ((I  - (GK * (v - EK)) - (GNa * (v - ENa)) - (GL * (v - El))) / Cm) * dt
end
dn(n, Vm, dt) = ((alpha_n(Vm) * (1.0 - n)) - (beta_n(Vm) * n)) * dt
dm(m, Vm, dt) = ((alpha_m(Vm) * (1.0 - m)) - (beta_m(Vm) * m)) * dt
dh(h, Vm, dt) = ((alpha_h(Vm) * (1.0 - h)) - (beta_h(Vm) * h)) * dt
I = T * 0.002

# Initial conditions and setup
v = -65
m = m_inf(v)
n = n_inf(v)
h = h_inf(v)

v_result = Array{Float64}(undef, length(T))
m_result = Array{Float64}(undef, length(T))
n_result = Array{Float64}(undef, length(T))
h_result = Array{Float64}(undef, length(T))

v_result[1] = v
m_result[1] = m
n_result[1] = n
h_result[1] = h


for t = 1:length(T)-1
    GKt = GK(gK, n)
    GNat = GNa(gNa, m)
    GLt = GL(gL)

    dvt = dv(v, GKt, GNat, GLt, Cm, EK, ENa, El, I[t], dt)
    dmt = dm(m, v, dt)
    dnt = dn(n, v, dt)
    dht = dh(h, v, dt)

    global v = v + dvt
    global m = m + dmt
    global n = n + dnt
    global h = h + dht

    v_result[t+1] = v
    m_result[t+1] = m
    n_result[t+1] = n
    h_result[t+1] = h
end

p1 = plot(T, v_result, xlabel="time (ms)", ylabel="voltage (mV)", legend=false, dpi=300)

The Parameters

There are seven parameters that define the Hodgkin-Huxley model. The maximum potassium, sodium and leak conductances gK, gNa and gL. Then there are the equilibrium potentials for potassium, sodium and leak, EK, ENa and El. Finally, there is Cm, the membrane capacitance. In a nutshell, the models comes down to calculating the fraction of sodium and potassium channels that are open at a time point. Together with the maximum conductance, the membrane potential and the equilibrium potential, the fraction of open channels tells us how much current is flowing at the time. The amount of current filtered by the membrane capacitance in turn tells us by how much the voltage changes. The fraction of open channels is given by n, m and h.

Differential Equations

We need to track the change of the voltage (v), potassium channel activation (n), sodium channel activation (m) and sodium channel inactivation (h). Let’s consider the potassium channels first. Active potassium channels can stay active or transition to the inactive state and inactive sodium channels. Since these potassium channels are voltage gated, the chance that they transition depends on the voltage. The function alpha_n gives the transition rate from inactive to active and beta_n gives the transition rate from active to inactive. For the sodium channels, the situation is almost identical. However, they can also be in a third state, that corresponds to depolarization induced inactivation.

Now that we are able to keep track of the states of our channels we can calculate the conductances. GK calculates the potassium conductance, GNa the sodium conductance and GL the leak conductance. Those conductances are then used in the dv function to calculate the currents based on the voltage. And that’s basically it. The integration method is simple forward euler inside the for loop.

Julia Notes From a Beginner

This is my very first Julia script so I have some thoughts. This is a completely non-optimized script and it is extremely fast, despite the for loop. From what I learned so far, for loops in Julia are known to be fast and they can allegedly outperform vectorized solutions. This is very different from Python, where we strictly avoid for loops, especially when concerned about performance.

For now I am confused by the scope and the use of the global keyword. Scope seems to operate similar to Python, where functions and for loops have seamless access to variables in the outer scope. Assigning variables on the other hand seems to be a problem inside the for loop, unless the global keyword is used.

Generally I am very happy with the Julia syntax. I think I could even code Python and Julia back to back. One major problem is of course indices starting at 1 but I get the difference in convention. I’m looking forward to my next script.

Managing Files and Directories with Python

We cannot always avoid the details of file management, especially when analyzing raw data. It can come in the form of multiple files distributed over multiple directories. Accessing those files and the directory system can be a critical aspect of raw data processing. Luckily, Python has very convenient methods to handle files and directories in the os module. Here we will look at functions to look inspect the contents of directories (os.listdir, os.walk) and functions that help us manipulate files and directoris (os.rename, os.remove, os.mkdir).

Directory Contents

One of the most important functions to manage files is os.listdir(directory). It returns a list of strings that contains to the names of all directories and files in path.

import os

directory = r"C:\Users\Daniel\tutorial"

dir_content = os.listdir(directory)
# ['images', 'test_one.txt', 'test_two.txt']

Now that we know what is in our directory, how do we find out which of these are files and which are subdirectories? You might be tempted to just check for a period (.) inside the string because it separates the file from its extension. This method is error prone, because directories could contain periods and files don’t necessarily have extensions. It is much better to use os.path.isfile() and os.path.isdir().

files = [x for x in dir_content if os.path.isfile(directory + os.sep + x)]
# ['test_one.txt', 'test_two.txt']

dirs = [x for x in dir_content if os.path.isdir(directory + os.sep + x)]
# ['images']

Now we know the contents of directory. We have two files and one directory. In case you were wondering about os.sep, that is the directory separator of the operating system. On my Window 10 that is the '\'. What if we need both files that are in our directory and those that are in sub-directories? This is the perfect case to use os.walk(). It gives a convenient way to loop through a directory and all its sub-directories.

for root, dirs, files in os.walk(directory):
    print(root)
    print(dirs)
    print(files)

# C:\Users\Daniel\tutorial
# ['images']
# ['file_1.txt', 'file_2.txt']
# C:\Users\Daniel\tutorial\images
# []
# ['plot_1.png', 'plot_2.png']

By default os.walk() goes from top down. The first name root tells us the full path of the directory we are currently at. Printing root tells us that we start at the top directory. While root is a string, both dirs and files are lists. They tell us for the current root, which files and directories are there. For the first directory we already found out on our own that the contents are. Two text files and a sub-directory. Our loop next goes to the sub-directory images. In there are no more sub-directories but two image files. If there were more sub-categories at any level (directory or directory\images), os.walk would go through all of them. Next we will find out how to create/move/rename/delete both files and directories.

Manipulating Files and Directories

Let’s say I want to rename the .txt files. I don’t like the numbering and would prefer them to have a leading zero in case they go into the double digits. We can use os.rename for this job.

directory = r"C:\Users\Daniel\tutorial"

dir_content = os.listdir(directory)

txt_f = [x for x in dir_content
         if os.path.isfile(directory + os.sep + x) and ".txt" in x]
# ['file_1.txt', 'file_2.txt']
for f_name in txt_f:
    f_name_split = f_name.split("_")
    num = f_name_split[1].split(".")[0]
    new_name = f_name_split[0] + "_" + num.zfill(2) + ".txt"
    os.rename(directory + os.sep + f_name,
              directory + os.sep + new_name)
    
os.listdir(directory)
['file_01.txt', 'file_02.txt', 'images']

Now that we renamed our files, let’s create another sub-directory for these .txt files. To create new directories we use os.mkdir().

os.listdir(directory)
# ['file_01.txt', 'file_02.txt', 'images']

os.mkdir(directory + os.sep + 'texts')

os.listdir(directory)
# ['file_01.txt', 'file_02.txt', 'images', 'texts']

Now we need to move the .txt files. There is no dedicated move function in the os module. Instead we use rename but instead of changing the name of the file, we change the path to the directory.

9directory = r"C:\Users\Daniel\tutorial"
dir_content = os.listdir(directory)
txt_f = [x for x in dir_content
         if os.path.isfile(directory + os.sep + x) and ".txt" in x]

for f in txt_f:
    old_path = directory + os.sep + f
    new_path = directory + os.sep + "texts" + os.sep + f
    os.rename(old_path, new_path)

os.listdir(directory)
# ['images', 'texts']

os.listdir(directory+os.sep+'texts')
# ['file_01.txt', 'file_02.txt']

Now our .txt files are in the ‘\texts’ sub-directory. Unfortunately there is no copy function in os. Instead we have to use another module called shutil. You can use a signature like this.

from shutil import copyfile
copyfile(source, destination)

Finally, to remove a file we simple use os.remove().

os.listdir(directory+os.sep+"texts")
# ['file_01.txt', 'file_02.txt']

os.remove(directory+os.sep+'texts'+os.sep+"file_01.txt")

os.listdir(directory+os.sep+"texts")
# ['file_02.txt']

And that’s it. You might have noticed that we did not cover how to create or read files. The os module is technically able to create and read files but in data science we usually depend on more high level interfaces to read files. For example, we might want to open a .csv file with pandas pd.read_csv. Using the lower level os functions will rarely be necessary. Thank you for reading and let me know if you have any questions.

In case you want to learn more about the os module, here are the os module docs.

Matplotlib and the Object-Oriented Interface

Matplotlib is a great data visualization library for Python and there are two ways of using it. The functional interface (also known as pyplot interface) allows us to interactively create simple plots. The object-oriented interface on the other hand gives us more control when we create figures that contain multiple plots. While having two interfaces gives us a lot of freedom, they also cause some confusion. The most common error is to use the functional interface when using the object-oriented would be much easier. For beginners it is now highly recommended to use the object-oriented interface under most circumstances because they have a tendency to overuse the functional one. I made that mistake myself for a long time. I started out with the functional interface and only knew that one for a long time. Here I will explain the difference between both, starting with the object-oriented interface. If you have never used it, now is probably the time to start.

Figures, Axes & Methods

When using the object-oriented interface, we create objects and do the plotting with their methods. Methods are the functions that come with the object. We create both a figure and an axes object with plt.subplots(1). Then we use the ax.plot() method from our axes object to create the plot. We also use two more methods, ax.set_xlabel() and ax.set_ylabel() to label our axes.

import matplotlib.pyplot as plt
import numpy as np

x = np.arange(0, 10, 0.1)
y = np.sin(np.pi * x) + x

fig, ax = plt.subplots(1)
ax.plot(x, y)
ax.set_xlabel("x")
ax.set_ylabel("y")

The big advantage is that we can very easily create multiple plots and we can very naturally keep track of where we are plotting what, because the method that does the plotting is associated with a specific axes object. In the next example we will plot on three different axes that we create all with plt.subplots(3).

x = np.arange(0,10,0.1)
ys = [np.sin(np.pi*x) + x,
      np.sin(np.pi*x) * x,
      np.sin(np.pi*x) / x]

fig, ax = plt.subplots(3)
ax[0].plot(x,ys[0])
ax[1].plot(x,ys[1])
ax[2].plot(x,ys[2])

ax[0].set_title("Addition")
ax[1].set_title("Multiplication")
ax[2].set_title("Division")

for a in ax:
    a.set_xlabel("x")
    a.set_ylabel("y")

When we create multiple axes objects, they are available to us through the ax array. We can index into them and we can also loop through all of them. We can take the above example even further and pack even the plotting into the for loop.

x = np.arange(0,10,0.1)
ys = [np.sin(np.pi*x) + x,
      np.sin(np.pi*x) * x,
      np.sin(np.pi*x) / x]

fig, ax = plt.subplots(3)
titles = ["Addition", "Multiplication", "Division"]
for idx, a in enumerate(ax):
    a.plot(x, ys[idx])
    a.set_title(titles[idx])
    a.set_xlabel("x")
    a.set_ylabel("y")

This code produces exactly the same three axes figure as above. There are other ways to use the object-oriented interface. For example, we can create an empty figure without axes using fig = plt.figure(). We can then create subplots in that figure with ax = fig.add_subplot(). This is exactly the same concept as always but instead of creating figure and axes at the same time, we use the figure method to create axes. If personally prefer fig, ax = plt.subplots() but fig.add_subplot() is slightly more flexible in the way it allows us to arrange the axes. For example, plt.subplots(x, y) allows us to create a figure with axes arranged in x rows and y columns. Using fig.add_subplot() we could create a column with 2 axes and another with 3 axes.

fig = plt.figure()
ax1 = fig.add_subplot(2,2,1)
ax2 = fig.add_subplot(2,2,3)
ax3 = fig.add_subplot(3,2,2)
ax4 = fig.add_subplot(3,2,4)
ax5 = fig.add_subplot(3,2,6)

Personally I prefer to avoid these arrangements, because things like tight_layout don’t work but it is doable and cannot be done with plt.subplots(). This concludes our overview of the object-oriented interface. Simply remember that you want to do your plotting through the methods of an axes object that you can create either with fig, ax = plt.subplots() or fig.add_subplot(). So what is different about the functional interface? Instead of plotting through axes methods, we do all our plotting through functions in the matplotlib.pyplot module.

One pyplot to Rule Them All

The functional interface works entirely through the pyplot module, which we import as plt by convention. In the example below we use it to create the exact same plot as in the beginning. We use plt to create the figure, do the plotting and label the axes.

import matplotlib.pyplot as plt
import numpy as np

x = np.arange(0,10,0.1)
y = np.sin(np.pi*x) + x

plt.figure()
plt.plot(x, y)
plt.xlabel("x")
plt.ylabel("y")

You might be wondering, why we don’t need to tell plt where to plot and which axes to label. It always works with the currently active figure or axes object. If there is no active figure, plt.plot() creates its own, including the axes. If a figure is already active, it creates an axes in that figure or plots into already existing axes. This make the functional interface less explicit and slightly less readable, especially for more complex figures. For the object-oriented interface, there is a specific object for any action, because a method must be called through an object. With the functional interface, it can be a guessing game where the plotting happens and we make ourselves highly dependent on the location in our script. The line we call plt.plot() on becomes crucial. Let’s recreate the three subplots example with the functional interface.

x = np.arange(0,10,0.1)
ys = [np.sin(np.pi*x) + x,
      np.sin(np.pi*x) * x,
      np.sin(np.pi*x) / x]

plt.figure()
plt.subplot(3, 1, 1)
plt.plot(x, ys[0])
plt.xlabel("x")
plt.ylabel("y")
plt.title("Addition")
plt.subplot(3, 1, 2)
plt.plot(x, ys[1])
plt.xlabel("x")
plt.ylabel("y")
plt.title("Multiplication")
plt.subplot(3, 1, 3)
plt.plot(x, ys[2])
plt.xlabel("x")
plt.ylabel("y")
plt.title("Division")

This one is much longer than the object-oriented code because we cannot label the axes in a for loop. To be fair, in this particular example we can put the entirety of our plotting and labeling into a for loop, but we have to put either everything or nothing into the loop. This is what I mean when I say the functional interface is less flexible.

x = np.arange(0,10,0.1)
ys = [np.sin(np.pi*x) + x,
      np.sin(np.pi*x) * x,
      np.sin(np.pi*x) / x]

plt.figure()
titles = ["Addition", "Multiplication", "Division"]
for idx, y in enumerate(ys):
    plt.subplot(3,1,idx+1)
    plt.plot(x, y)
    plt.xlabel("x")
    plt.ylabel("y")
    plt.title(titles[idx])

Both interfaces are very similar. You might have noticed that the methods in the object-oriented API have the form set_attribute. This is by design and follows from an object oriented convention, where methods that change attributes have a set prefix. Methods that don’t change attributes but create entirely new objects have an add prefix. For example add_subplot. Now that we have seen both APIs at work, why is the object-oriented API recommended?

Advantages of the Object-Oriented API

First of all, Matplotlib is internally object-oriented. The pyplot interface masks that fact in an effort to make the usage more MATLAB like by putting a functional layer on top. If we avoid plt and instead work with the object-oriented interface, our plotting becomes slightly faster. More importantly, the object-oriented interface is considered more readable and explicit. Both are very important when we write Python. Readability can be somewhat subjective but I hope the code could convince you that going through the plotting methods of an axes object makes it much more clear where we are plotting. We also get more flexibility to structure our code. Because plt depends on the order of plotting, we are constraint. With the object-oriented interface we can structure our code more clearly. We can for example split plotting, labeling and other tasks into their own code blocks.

In summary, I hope you will be able to use the object-oriented interface of Matplotlib now. Simply remember to create axes with fig, ax = plt.subplots() and then most of the work happens through the ax object. Finally, the object-oriented interface is recommended because it is more efficient, readable, explicit and flexible.

Image Segmentation with scikit-image

Image Segmentation is one of the most important steps in most imaging analysis pipelines. It separates between the background and the features of our images. It can also determine the number of distinct features and their location. Our ability to segment determines what we can analyze. We’ll look at a basic but complete segmentation pipeline with scikit-image. You can see the result in the title image where we segment four cells. First, we will need to threshold the image into a binary version where the background is 0 and the foreground is 1.

import numpy as np
import matplotlib.pyplot as plt
from skimage import io, filters, morphology, color

image = io.imread("example_image.tif")  # Load Image
threshold = filters.threshold_otsu(image)  # Calculate threshold
image_thresholded = image > threshold  # Apply threshold

# Show the results
fig, ax = plt.subplots(1, 2)
ax[0].imshow(image, 'gray')
ax[1].imshow(image_thresholded, 'gray')
ax[0].set_title("Intensity")
ax[1].set_title("Thresholded")

We calculate the threshold with the threshold_otsu function and apply it with a boolean operator. This threshold method works very well but there are two problems. First, there are very small particles that have nothing to do with our cell. To take care of those, we will apply morphological erosion. Second, there are holes in our cells. We will close those with morphological dilation.

# Apply 2 times erosion to get rid of background particles
n_erosion = 2
image_eroded = image_thresholded
for x in range(n_erosion):
    image_eroded = morphology.binary_erosion(image_eroded)

# Apply 14 times dilation to close holes
n_dilation = 14
image_dilated = image_eroded
for x in range(n_dilation):
    image_dilated = morphology.binary_dilation(image_dilated)

# Apply 4 times erosion to recover original size
n_erosion = 4
image_eroded_two = image_dilated
for x in range(n_erosion):
    image_eroded_two = morphology.binary_erosion(image_eroded_two)

fig, ax = plt.subplots(2,2)
ax[0,0].imshow(image_thresholded, 'gray')
ax[0,1].imshow(image_eroded, 'gray')
ax[1,0].imshow(image_dilated, 'gray')
ax[1,1].imshow(image_eroded_two, 'gray')
ax[0,0].set_title("Thresholded")
ax[0,1].set_title("Eroded 2x")
ax[1,0].set_title("Dilated 14x")
ax[1,1].set_title("Eroded 4x")

Erosion turns any pixel black that is contact with another black pixel. This is how erosion can get rid of small particles. In our case we need to apply erosion twice. Once those particles disappeared, we can use dilation to close the holes in our cells. To close all the holes, we have to slightly over dilate, which makes the cells slightly bigger than they actually are. To recover the original morphology we apply some more erosions. Here is an example that shows how erosion and dilation work in detail. It also illustrates what being “in contact” with another pixel means by default.

cross = np.array([[0,0,0,0,0], [0,0,1,0,0], [0,1,1,1,0], 
                [0,0,1,0,0], [0,0,0,0,0]], dtype=np.uint8)
cross_eroded = morphology.binary_erosion(cross)
cross_dilated = morphology.binary_dilation(cross)
fig, ax = plt.subplots(1,3)
ax[0].imshow(cross, 'gray')
ax[1].imshow(cross_eroded, 'gray')
ax[2].imshow(cross_dilated, 'gray')
ax[0].set_title("Cross")
ax[1].set_title("Cross Eroded")
ax[2].set_title("Cross Dilated")

Now we are essentially done segmenting foreground and background. But we also want to assign distinct labels to our objects.

labels = morphology.label(image_eroded_two)
labels_rgb = color.label2rgb(labels,
                             colors=['greenyellow', 'green',
                                     'yellow', 'yellowgreen'],
                             bg_label=0)
image.shape
# (342, 382)
labels.shape
# (342, 382)
fig, ax = plt.subplots(2,2)
ax[0,0].imshow(labels==1, 'gray')
ax[0,1].imshow(labels==2, 'gray')
ax[1,0].imshow(labels==3, 'gray')
ax[1,1].imshow(labels_rgb)
ax[0,0].set_title("label == 1")
ax[0,1].set_title("label == 2")
ax[1,0].set_title("label == 3")
ax[1,1].set_title("All labels RGB")

We use morphology.label to generate a label for each connected feature. This returns an array that has the same shape as our original image but the pixels are no longer zero or one. The background is zero but each feature gets its own integer. All pixels belonging to the first label are equal to 1, pixels of the second label equal to 2 and so on. To visualize those labels all in one image, we call color.label2rgb to get color representations for each label in RGB space. And that’s it.

Segmentation is crucial for image analysis and I hope this tutorial got you on a good way to do your own segmentation with scikit-image. This pipeline is not perfect but illustrates the concept well. There are many more functions in the morphology module to filter binary images, but they all come down to a sequence of erosions and dilations. If you want to adapt this approach for your own images, I would recommend to play around with the number of erosions and dilations. Let me know how it worked for you.

Contrast Adjustment with scikit-image

Contrast is one of the most important properties of an image and contrast adjustment is one of the easiest things we can do to make our images look better. There are many ways to adjust the contrast and with most of them we have to be careful because they can artificially change our images. The title image shows three basic methods of contrast adjustment and how they affect the image histogram. We will cover all three methods here but first let us consider what it means for an image to have low contrast.

Each pixel in an image has an intensity value. In an 8-bit image the values can be between 0 and 255. Contrast issues occur, when the largest intensity value in our image is smaller than 255 or the smallest value is larger than 0. The reason is that we are not using the entire range of possible values, making the image overall darker than it could be. So what about the contrast of our example image? We can use .min() and .max() to find maximum and minimum intensity.

import numpy as np
import matplotlib.pyplot as plt
from skimage import io, exposure, data

image = io.imread("example_image.tif")
image.max()
# 52
image.min()
# 1

Our maximum of 52 is very far away from 255, which explains why our image is so dark. On the minimum we are almost perfect. To correct the contrast we can user the exposure module which gives us the function rescale_intensity.

image_minmax_scaled = exposure.rescale_intensity(image)
image_minmax_scaled.max()
# 255
image_minmax_scaled.min()
# 0

Now both the minimum and the maximum are optimized. All pixels that were equal to the original minimum are now 0 and all pixels equal to the maximum are now 255. But what happens to the values in the middle? Let’s look at a smaller example.

arr = np.array([2, 40, 100, 205, 250], dtype=np.uint8)

arr_rescaled = exposure.rescale_intensity(arr)
# array([  0,  39, 100, 208, 255], dtype=uint8)

As expected, the minimum 2 became 0 and the maximum 250 became 255. In the middle, 40 became smaller, nothing happened to 100 and 205 became larger. We will look at each step to find out how we got there.

arr = np.array([2, 40, 100, 205, 250], dtype=np.uint8)
min, max = arr.min(), arr.max()
arr_subtracted = arr - min  # Subtract the minimum
# array([  0,  38,  98, 203, 248], dtype=uint8)
arr_divided = arr_subtracted / (max - min)  # Divide by new max
# array([0.        , 0.15322581, 0.39516129, 0.81854839, 1.        ])
arr_multiplied = arr_divided * 255  # Multiply by dtype max
# array([  0.        ,  39.07258065, 100.76612903, 208.72983871,
#        255.        ])
# Convert dtype to original uint8
arr_rescaled = np.asarray(arr_multiplied, dtype=arr.dtype)
# array([  0,  39, 100, 208, 255], dtype=uint8)

We can get there in four simple steps. Subtract the minimum, divide by the maximum of the new subtracted array, multiply by the maximum value of the data type and finally convert back to the original data type. This works well if we want to rescale by minimum and maximum of the image but sometimes we need to use different values. Especially the maximum can be easily dominated by noise. For all we know, it could be just one pixel that is not necessarily representative for the entire image. As you can see from the histogram in the title image, very few pixels are near the maximum. This means we can use percentile rescaling with little information loss.

percentiles = np.percentile(image, (0.5, 99.5))
# array([ 1., 28.])
scaled = exposure.rescale_intensity(image,
                                    in_range=tuple(percentiles))

This time we scale from 1 to 28, which means that all values above or equal to 28 become the new maximum 255. As we chose the 99.5 percentile, this affects roughly 5% of the upper pixels. You can see the consequence in the image on the right. The image becomes brighter but it also means that we lose information. Pixels that were distinguishable now look exactly the same, because they are all 255. It is up to you if you can afford to lose those features. If you do quantitative image analysis you should perform rescaling with caution and always look out for information loss.

Plotting 2D Vectors with Matplotlib

Vectors are extremely important in linear algebra and beyond. One of the most common visual representations of a vector is the arrow. Here we will learn how to plot vectors with Matplotlib. The title image shows two vectors and their sum. As a first step we will plot the vectors originating at 0, shown below.

import matplotlib.pyplot as plt
import numpy as np

vectors = np.array(([2, 0], [3, 2]))
vector_addition = vectors[0] + vectors[1]
vectors = np.append(vectors, vector_addition[None,:], axis=0)

tail = [0, 0]
fig, ax = plt.subplots(1)
ax.quiver(*tail,
           vectors[:, 0],
           vectors[:, 1],
           scale=1,
           scale_units='xy',
           angles = 'xy',
           color=['g', 'r', 'k'])

ax.set_xlim((-1, vectors[:,0].max()+1))
ax.set_ylim((-1, vectors[:,1].max()+1))

We have two vectors stored in our vectors array. Those are [2, 0] and [3, 2]. Both in order of [x, y] as you can see from the image. We can perform vector addition between the two by simply adding vectors[0] + vectors[1]. Then we use np.append so we have all three vectors in the same array. Now we define the origin in tail, because we will want the tail of the arrow to be located at [0, 0]. Then we create the figure and axes to plot in with plt.subplots(). The plotting itself can be done with one call to the ax.quiver method. But it is quite the call, with a lot of parameters so let’s go through it.

First, we need to define the origin, so we pass *tail. Why the asterisk? ax.quiver really takes two parameters for the origin, X and Y. The asterisk causes [0, 0] to be unpacked into those two parameters. Next, we pass the x coordinates (vectors[:, 0]) and then the y coordinates (vectors[:, 1]) of our vectors. The next three parameters scale, scale_units and angles are necessary to make the arrow length match the actual numbers. By default, the arrows are scaled, based on the average of all plotted vectors. We get rid of that kind of scaling. Try removing some of those to get a better idea of what I mean. Finally, we pass three colors, one for each arrow.

So what do we need to plot the head to tail aligned vectors as in the title image? We just need to pass the vectors where the origin is the other vector.

ax.quiver(vectors[1::-1,0],
          vectors[1::-1,1],
          vectors[:2,0],
          vectors[:2,1],
          scale=1,
          scale_units='xy',
          angles = 'xy',
          color=['g', 'r'])

This is simple because it is the same quiver method but it is complicated because of the indexing syntax. Now, we no longer unpack *tail. Instead we pass x and y origins separately. In vectors[1::-1,0] the 0 gets the x coordinates. -1 inverts the array. If we would not invert, each vector would be it’s own origin. The 1 skips the first vector, which is the summed vector because we inverted. vectors[1::-1,1] gives us the y coordiantes. Finally we just need to skip the summed vector when we pass x and y magnitudes. The rest is the same.

So that’s it. Unfortunately, ax.quiver only works for 2D vectors. It also isn’t specifically made to present vectors that have a common origin. Its main use case is to plot vector fields. This is why some of the plotting here feels clunky. There is also ax.arrow which is more straightforward but only creates one arrow per method call. I hope this post was helpful for you. Let me know if you have other ways to plot vectors.

Analyzing Image Histograms with scikit-image

An image says more than a thousand words but histograms are also very important. Digital images are made of pixels and each of them has a value. A histogram tells us how many pixels of the image have a certain value. The title plot shows Chelsea the cat and the histograms for each color channel. Here is the code that generated the figure.

import numpy as np
import skimage
import matplotlib.pyplot as plt

image = skimage.data.chelsea()
image_red, image_green, image_blue = image[:,:,0], image[:,:,1], image[:,:,2]

fig, ax = plt.subplots(2,3)
ax[0,0].imshow(image_red, cmap='gray')
ax[0,1].imshow(image_green, cmap='gray')
ax[0,2].imshow(image_blue, cmap='gray')

bins = np.arange(-0.5, 255+1,1)
ax[1,0].hist(image_red.flatten(), bins = bins, color='r')
ax[1,1].hist(image_green.flatten(), bins=bins, color='g')
ax[1,2].hist(image_blue.flatten(), bins=bins, color='b')

Because Chelsea is part of the scikit-image example data, we can simply load it with skimage.data.chelsea(). With image.shape we can find out that our image has three dimensions. The first two are y and x coordinates whereas the third one represents the colors red, green and blue (RGB). We split the colors into their own variables before visualizing each of them as a grayscale image and below it we plot the histogram. Here is a short version of the above code with some slightly advanced Python features.

fig, ax = plt.subplots(2,3)
bins = np.arange(-0.5, 255+1,1)
for ci, c in enumerate('rgb'):
    ax[0,ci].imshow(image[:,:,ci], cmap='gray')
    ax[1,ci].hist(image[:,:,ci].flatten(), bins = bins, color=c)

We can see from the histogram and the grayscale image that Chelsea is slightly more red than blue or green. But how can we get more quantitative information out of the histogram? We can use np.histogram and the usual numpy functions to learn more about the properties of our histograms.

hist_red = np.histogram(image_red.flatten(), bins=bins)
hist_red[0].argmax()
# 156

The np.histogram function gives us a tuple, where the first entry are the counts and the second entry are the bin edges. This is the reason we have to index into hist_red to call .argmax() on the correct array. .argmax() tells us that the peak of the histogram is at bin 156. This means that most pixels have an intensity value of 156. The peak can be deceiving, especially when the distribution is skewed or multi-modal but for this tutorial we will accept it as a first pass. Let’s see how the other channels look.

hist_red = np.histogram(image_red.flatten(), bins=bins)
green = np.histogram(image_green.flatten(), bins=bins)
hist_blue = np.histogram(image_blue.flatten(), bins=bins)

print(hist_red[0].argmax(),
      hist_green[0].argmax(),
      hist_blue[0].argmax())

# 156 116 97

As our eyes suspected, the green and blue channel have peaks at smaller intensity values than the red channel. This confirms our suspicion that Chelsea probably is a red cat. I hope this tutorial has been helpful to get you started with scikit-image. We learned that RGB images come in an array of shape (y, x, c), where c is the color channel. We can use plt.hist() to calculate and plot the histogram and np.hist() to calculate the histogram without plotting.

Animations with Matplotlib

Anything that can be plotted with Matplotlib can also be animated. This is especially useful when data changes over time. Animations allow us to see the dynamics in our data, which is nearly impossible with most static plots. Here we will learn how to animate with Matplotlib by producing this traveling wave animation.

This is the code to make the animation. It creates the traveling wave, defines two functions that handle the animation and creates the animation with the FuncAnimation class. Let’s take it step by step.

import numpy as np
from matplotlib.animation import FuncAnimation
import matplotlib.pyplot as plt

# Create the traveling wave
def wave(x, t, wavelength, speed):
    return np.sin((2*np.pi)*(x-speed*t)/wavelength)

x = np.arange(0,4,0.01)[np.newaxis,:]
t = np.arange(0,2,0.01)[:,np.newaxis]
wavelength = 1
speed = 1
yt = wave(x, t, wavelength, speed)  # shape is [t,y]

# Create the figure and axes to animate
fig, ax = plt.subplots(1)
# init_func() is called at the beginning of the animation
def init_func():
    ax.clear()

# update_plot() is called between frames
def update_plot(i):
    ax.clear()
    ax.plot(x[0,:], yt[i,:], color='k')

# Create animation
anim = FuncAnimation(fig,
                     update_plot,
                     frames=np.arange(0, len(t[:,0])),
                     init_func=init_func)

# Save animation
anim.save('traveling_wave.mp4',
          dpi=150,
          fps=30,
          writer='ffmpeg')

On the first three lines we import NumPy, Matplotlib and most importantly the FuncAnimation class. It will take the center stage in our code as it will create the animation later on by combining all the parts we need. On lines 5-13 we create the traveling wave. I don’t want to go into too much detail, as it is just a toy example for the animation. The important part is that we get the array yt, which defines the wave at each time point. So yt[0] contains the wave at t0 , yt[1] at t1 and so on. This is important, since we will be iterating over time during the animation. If you want to learn more about the traveling wave, you can change wavelength, speed and play around with the wave() function.

Now that we have our wave, we can start preparing the animation. We create a the figure and the axes we want to use with plt.subplots(1). Then we create a the init_func(). This one will be called whenever the animation starts or repeats. In this particular example it is pretty useless. I include it here because it is a useful feature for more complex animations.

Now we get to update_plot(), the heart of our animation. This function updates our figure between frames. It determines what we see on each frame. It is the most important function and it is shockingly simple. The parameter i is an integer that defines what frame we are at. We use that integer as an index into the first dimension of yt. We plot the wave as it looks at t=i. Importantly, we must clean up our axes with ax.clear(). If we would forget about clearing, our plot would quickly become all black, filled with waves.

Now FuncAnimation is where it all comes together. We pass it fig, update_plot and init_func. We also pass frames, those are the values that i will take on during the animation. Technically, this gets the animation going in your interactive Python console but most of the time we want to save our animation. We do that by calling anim.save(). We pass it the file name as a string, the resolution in dpi, the frames per second and finally the writer class used for generating the animation. Not all writers work for all file formats. I prefer .mp4 with the ffmpeg writer. If there are issues with saving, the most common problem is that the writer we are trying to use is not installed. If you want to find out if the ffmpeg writer is available on your machine, you can type matplotlib.animation.FFMpegWriter().isAvailable(). It returns True if the writer is available and False otherwise. If you are using Anaconda you can install the codec from here.

This wraps up our tutorial. This particular example is very simple, but anything that can be plotted can also be animated. I hope you are now on your way to create your own animations. I will leave you with a more involved animation I created.

Filtering Data with SciPy

Time series data may contain signals at many different frequencies. Sharp increases or decreases have a high frequency. Slow increases or decreases have a low frequency. Filtering allows us to take different frequency components out of the data.

Signal filtering is a science on its own and I’ll focus on the practical aspects here and stick to two filter types: butterworth and Chebyshev type I. Each of those filters can be used for different purposes. We can use them as low pass, high pass, band pass or notch filters. Low pass filters leave low frequencies alone but attack high frequencies. High pass filters leave high frequencies alone but attach low frequencies. The title image shows an example of low and high pass filters used on the same data. Band pass filters leave a specific frequency band alone and attack all other frequencies. Notch filters attack a specific frequency band, leaving the rest alone. Let’s look at an example.

import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
from scipy.signal import butter, cheby1, filtfilt

data = np.load("example_data.npy")

order = 3
Wn = 4000  # in Hz
btype = 'lowpass'
fs = 50000  # in Hz

b, a = butter(order, Wn, btype, fs = fs)
data_butter = filtfilt(b, a, data)

This is a butterworth lowpass filter with a cutoff frequency of 4000Hz (Wn). That means, signals below 4000Hz are is the pass band. They are largely left alone. Signals above 4000Hz are in the stop band, they are diminished. fs is the sampling frequency of the data. If the units are Hz, it tells us how many data points are recorded during one second. filtfilt is the function that does the actual filtering on the data, based on the filter (b, a) that was designed previously. Filtering is not a perfect process. Filters have what is called roll-off at the critical 4000Hz frequency.

Ideally, we would like a filter response that falls down straight. Anything in the pass band is untouched, anything in the stop band is shutdown the same way. As you can see, our actual filter does no live up to the ideal. It already slightly attenuates signal that is part of the pass band and it falls much slower in the stop band. If we need a steeper roll off, we can increase the order of our filter.

Some filter types have steeper roll off than others. For example, the Chebyshev type I filter achieves steeper roll off by tolerating some ripple in the pass band.

This can lead to distortions in the data depending on the size of the ripple. The Chebyshev type I filter takes an argument rp that defines the amount of pass band ripple that is tolerated in units of dB. The more ripple we tolerate, the steeper the roll off will be. Here you can see how large ripple causes oscillations in the data.

Generally, the butterworth filter is sufficient for most situations and is safer because it does not ripple. I hope this post helped you filtering your own data. If you want to learn more, check out the SciPy signal docs. Both the butter and cheby1 filter are there with many, many more.