As a scientist I have the privilege to always learn and discover new things. I have recently started a new blog called Microbe Food, about single cell food production, you might be interested in. This leaves my programming blog mostly dormant for the moment but I hope the content is still useful.
SciPy features two different interfaces to solve differential equations: odeint and solve_ivp. The newer one is solve_ivp and it is recommended but odeint is still widespread, probably because of its simplicity. Here I will go through the difference between both with a focus on moving to the more modern solve_ivp interface. The primary advantage is that solve_ivp offers several methods for solving differential equations whereas odeint is restricted to one. We get started by setting up our system of differential equations and some parameters of the simulation.
UPDATE 07.02.2021: Note that I am focusing here on usage of the different interfaces and not on benchmarking accuracy or performance. Faruk Krecinic has contacted me and noted that assessing accuracy would require a system with a known solution as a benchmark. This is beyond the scope of this blog post. He also pointed out to me that the hmax parameter of odeint is important for the extent to which both interfaces give similar results. You can learn more about the parameters of odeint in the docs.
import numpy as np
from scipy.integrate import odeint, solve_ivp
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
def lorenz(t, state, sigma, beta, rho):
x, y, z = state
dx = sigma * (y - x)
dy = x * (rho - z) - y
dz = x * y - beta * z
return [dx, dy, dz]
sigma = 10.0
beta = 8.0 / 3.0
rho = 28.0
p = (sigma, beta, rho) # Parameters of the system
y0 = [1.0, 1.0, 1.0] # Initial state of the system
We will be using the Lorenz system. We can directly move on the solving the system with both odeint and solve_ivp.
Simulation results from odeint and solve_ivp. Note that the solve_ivp looks very different primarily because of the default temporal resolution that is applied. Changing the the temporal resolution and getting very similar results to odeint is easy and shown below.
The first thing that sticks out is that the solve_ivp solution is less smooth. That is because it is calculated at fewer time points, which in turn has to do with the difference between t_span and t. The odeint interface expects t, an array of time points for which we want to calculate the solution. The temporal resolution of the system is given by the interval between time points. The solve_ivp interface on the other hand expects t_span, a tuple that gives the start and end of the simulation interval. solve_ivp determines the temporal resolution by itself, depending on the integration method and the desired accuracy of the solution. We can confirm that the temporal resolution of solve_ivp is lower in this example by inspecting the output of both functions.
The t array has 4000 elements and therefore the result of odeint has 4000 rows, each row being a time point defined by t. The result of solve_ivp is different. It has its own time array as an attribute and it has 1989 elements. This tells us that solve_ivp indeed calculated fewer time points affection temporal resolution. So how can we increase the the number of time points in solve_ivp? There are three ways: 1. We can manually define the time points to integrate, similar to odeint. 2. We can decrease the error of the solution we are willing to tolerate. 3. We can change to a more accurate integration method. We will first change the integration method. The default integration method of solve_ivp is RK45 and we will compare it to the default method of odeint, which is LSODA.
Comparison between RK45 and LSODA integration methods of solve_ivp.
The LSODA method is already more accurate but we can make it even more accurate but it is still not as accurate as the solution we got from odeint. That is because we made odeint solve at even higher temporal resolution when we passed it t. To get the exact same result from solve_ivp we got from odeint, we must pass it the exact time points we want to solve with the t_eval parameter.
odeint and solve_ivp with identical integration method and temporal resolution.
Now both solutions have identical temporal resolution. But their solution is still not identical. I was unable to confirm why that is the case but I suspect very small floating point errors. The Lorenz attractor is a chaotic system and even small errors can make it diverge. The following plot shows the first variable of the system for odeint and solve_ivp from the above simulation. It confirms my suspicion that floating point accuracy is to blame but was unable to confirm the source.
Solutions from odeint and solve_ivp diverge even for identical temporal resolution and integration method.
There is one more way to make the system more smooth: decrease the tolerated error. We will not go into that here because it does not relate to the difference between odeint and solve_ivp. It can be used in both interfaces. There are some more subtle differences. For example, by default, odeint expects the first parameter of the problem to be the state and the second parameter to be time t. This is the other way around for solve_ivp. To make odeint accept a problem definition where time is the first parameter we set the parameter tfirst=True.
In summary, solve_ivp offers several integration methods while odeint only uses LSODA.
Differential equations are special because they don’t tell us the value of a variable straight up. Instead, they tell us by how much the variable will change with respect to the change of another variable. Usually that other variable is time. To numerically solve a system of differential equations we need to track the systems change over time starting at an initial state. This process is called numerical integration and there is a SciPy function for it called odeint. We will learn how to use this package by simulating the ‘hello world’ of differential equations: the Lorenz system.
Here is the first part of the code where we define the function that describes the dynamics of the system.
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint
from mpl_toolkits.mplot3d import Axes3D
def lorenz(state, t, sigma, beta, rho):
x, y, z = state
dx = sigma * (y - x)
dy = x * (rho - z) - y
dz = x * y - beta * z
return [dx, dy, dz]
We start with some imports. Of course we need NumPy and odeint is imported from scipy.integrat. Matplotlib will be used to plot the result of our simulation. After that we define the system of differential equations that defines our Lorenz system. It consists of three differential equations that we fit into one function called lorenz. This function needs a specific call signature (lorenz(state, t, sigma, beta, rho)) because we will later pass it to odeint which expects specific parameters in specific places. Most importantly, the first parameter must be the state of the system.The state of the Lorenz system is defined by three variables: x, y, z. Our state object has to be a sequence with an order that reflects this.
Inside the lorenz function, the first thing we do is to unpack the state into the three state variables. This is followed by the three differential equations that described the dynamic changes of the state variables. The fact that the variable t does not show up in any of these equations is a common point of confusion. The amount of change certainly depends on the amount of time. So why can we ignore t here? The answer is that our numerical integrator will keep track of t for us. For this particular system we could actually build a function that does not take the parameter t but I include it because it can be useful if you want to add discontinuities that depend on t.
While t does not appear in the equations, sigma, beta & rho do. They are the parameters of the system and the system’s properties depend on them. We will set those parameters next.
These are the parameters Lorenz himself used and they are known to produce the type of dynamic that the Lorenz system is most known for: the Lorenz attractor. It is important that we store these parameters in a tuple in this exact order because of our functions structure. It must be a tuple rather than another type of collection because odeint expects it. Now that our parameters are defined, we will move on to define the initial values of the system. This is a critical part of solving differential equations. These equations tell us by how much the system state changes but they cannot tell us where to start.
y0 = [1.0, 1.0, 1.0]
Our system will start with all variables at 1.0. Now we can solve the system and plot the result.
We solve the system with a simple call to to odeint and we pass it the function that defines out system, the initial state, the time points t for which we want to solve the system and the parameters p. result is a two-dimensional array where the rows are the time points and the columns are the state variables at that those time points. And this is how we can solve differential equations with SciPy.
The deep feedforward neural network is the most simple network architecture. They are a great entry point to many deep learning concepts. They can also be pretty effective for many applications but they have been replaced by more specialized networks in most areas (for example recurrent neural networks or convolutional neural networks). Here we will build one with pyTorch and we will go from feature selection to training. The dataset is the Titanic challenge, where our model has to predict who survives the sinking of the Titanic. It is the introductory data science competition on Kaggle. We will start by looking at our data.
Missing Values and String to Numerical Conversion
First we need to load the data and find out what we are dealing with. It already comes split into training and test data, both being .csv files. We load both files and take a look at their general structure with .info().
For each passenger we have 11 features. In the test data, the 'Survived' column is missing, because our goal is to predict it. For the training data we know who survived and we will use this to predict who survived in the test data. Before we can do that, we need to take a closer look at our data. The first issue we encounter is missing values. In the training data are 891 passengers. However, the 'Age' column contains only 714 non-null values. The others are missing. We say they are not-a-number (NaN) and we need to take care of them. If we were to feed NaNs into our model it would completely destroy the calculations. One option is to remove the rows that contain NaNs. This would exclude a large number of rows. Probably more than we were comfortable with, considering that there will be other columns with NaNs. Additionally, the test data also have NaNs. Should we just give up for those test passengers and guess?
A better idea is to replace missing values with the column mean. Given no other information, the mean is the best guess. To be extra careful, we can take the column median, which is more resistance to outliers. Importantly, we will replace NaNs in the test data with the median from the training data. This is not strictly necessary for the Titanic toy example but it is really good practice if you ever design an important model that will be deployed on completely unknown data. Generally, we should use the test data exclusively to calculate our test accuracy. Nothing else. It should never touch the training data and we should act like we don’t even have access to it yet, until we get to the model testing. There are more complicated ways to replace NaNs but they go beyond this blog post. Let us replace NaNs with the median using the .fillna method. Because we need to do the same preprocessing to the training and test dataset we will loop through them. We will have to do the same for the "Fare" column. There is only one value missing and it is in the test data, but we must fill it.
train_test_datasets = [train, test]
median_age = train["Age"].median()
median_fare = train["Fare"].median()
for dataset in train_test_datasets:
dataset["Age"].fillna(median_age, inplace=True)
dataset["Fare"].fillna(median_fare, inplace=True)
That takes care of "Age" and “Fare” but we cannot apply the same strategy to "Cabin" or "Embarked" because they are nominal columns. Note that their dtype is object instead of a numeric types such as int64. Let us take a look at "Cabin" first.
train["Cabin"]
"""
PassengerId
1 NaN
2 C85
3 NaN
4 C123
5 NaN
887 NaN
888 B42
889 NaN
890 C148
891 NaN
Name: Cabin, Length: 891, dtype: object
"""
type(train["Cabin"][1])
# float
type(train["Cabin"][2])
# str
train["Cabin"].isna().sum()
# 687
Some rows contain strings, other contain floats. The number of NaNs is large at 687. I was tempted to simply delete this column because of the large number of NaNs. I decided to keep it because it might be important. I am no ship expert, but the location of the cabin might determine how accessible life boats are and thereby influence survival. Now there are several ways to proceed. I will go with a compromise between simplicity and retaining information. We will extract the first letter, creating a new nominal feature called “Cabin Letter”.
There are eight different cabin letters and the NaN problem persists. Because we are dealing with a nominal feature we can create a ninth category. These are the passengers where we don’t know their cabin number. We will do so as we convert the letters to numbers. For our network we will need to convert everything to floats anyway.
Now the NaNs are -1, which is fine for now. Later we will convert all these nominal variables to dummy variables anyway, but we will get to that. We could extract more information from "Cabin" but for this demonstration I will leave it here and delete "Cabin".
for dataset in train_test_datasets:
dataset.drop("Cabin", axis=1, inplace=True)
The final variable that suffers from NaNs is "Embarked". The data in it is less messy so we can go straight to getting the categorical codes.
for dataset in train_test_datasets:
dataset["Embarked"] = pd.Categorical(dataset["Embarked"])
dataset["Embarked"] = dataset["Embarked"].cat.codes
Now we succesfully removed all NaN values but some minor issues about our columns remain. "Name", "Sex" and "Ticket" do not have a numerical type. They are of type object. "Name" and "Ticket" probably don’t tell us a lot about survival, unless we would do some serious feature engineering on them. So we will simply drop them.
for dataset in train_test_datasets:
dataset.drop(["Name", "Ticket"], axis=1, inplace=True)
That just leaves us with "Sex". We can use .cat.codes as above to convert it to nubmers.
for dataset in train_test_datasets:
dataset["Sex"] = pd.Categorical(dataset["Sex"]).codes
Now let us take one more look at our datasets and make sure we took care of NaNs and everything is numeric.
Everything looks in order. However, there is one more necessary preprocessing step before we can move to standardization. We need to convert categorical features to dummy variables. That means, each category in a column gets its own new column and a sample of that category will have a 1 there. Let’s see how that looks.
Convert Categories to Dummy Variables
We will use the .get_dummies function on all our categorical columns but first let us try it on the "Pclass" column.
Dummy variables make sense if we just stare at them long enough. What we are looking for is the conversion of each category into an individual column. train["Pclass"] has three categories, [1, 2, 3]. Therefore, we get three dummy variables. A row that has a 1 in "Pclass" will get a 1 in the first column and 0 in the other two. A row that has a 2 will get a 1 in the second column and 0 in the others. A row that has a 3 will get a 1 in the third column and 0 in the others. Now we will use the same function on all our categorical columns. Note that we will not use our usual loop trough train and test data because there is no way to generate the dummy variables inplace.
The code looks solid but something went wrong. We now have more columns in our train_dummies than in test_dummies. What happened? Let us look at the columns and see if we can spot the issue.
test_dummies is missing the following columns: "Embarked_-1“, "Embarked_0" and "Cabin Letter_7". Those are the three missing columns. Remember that "Survived" is purposely not in the test dataset. This happened because in the test data there were no rows that had missing values or a 0 in "Embarked" or a 7 in "Cabin Letter". This raises another issue. When we converted string categories to numbers with cat.codes, we may have converted letters differently in the test and training dataset. Luckily, we can convert strings directly to dummy variables. This is a great opportunity to take a step back and clean up our code. We will start from scratch.
import pandas as pd
# Load data
train = pd.read_csv("train.csv", index_col='PassengerId')
test = pd.read_csv("test.csv", index_col='PassengerId')
# Merge train and test for wrangling and preprocessing
train_test_datasets = [train, test]
"""Data wrangling"""
# Split cabin into letter and number
median_age = train["Age"].median()
median_fare = train["Fare"].median()
for idx, dataset in enumerate(train_test_datasets):
dataset["Age"].fillna(median_age, inplace=True)
dataset["Fare"].fillna(median_fare, inplace=True)
dataset["Cabin Letter"] = dataset["Cabin"].str.slice(0, 1)
dataset.drop("Cabin", axis=1, inplace=True)
#dataset["Embarked"] = dataset["Embarked"].cat.codes
dataset.drop(["Name", "Ticket"], axis=1, inplace=True)
categorical_cols = ["Pclass", "Sex", "Embarked", "Cabin Letter","SibSp"]
train_dummies = pd.get_dummies(train,
columns=categorical_cols,
prefix=categorical_cols,
dummy_na=True)
test_dummies = pd.get_dummies(test,
columns=categorical_cols,
prefix=categorical_cols,
dummy_na=True)
train_dummies.shape
# (891, 32)
test_dummies.shape
# (418, 30)
Now we have more columns because we added the dummy_na=True parameter. This gives us a "Sex_nan" column, although there were no NaNs in the original data. This is excessive but won’t be an issue for our network. Now we just need to add the one columns that is missing from test_dummies. It should contain only zeros, because there were no rows with that category in the test dataset.
import numpy as np
test_dummies["Cabin Letter_T"] = np.zeros(test_dummies.shape[0])
Now our dummy variables are in order and we can move on balancing the training data.
Balancing the Training Data
Balancing the training data will be important during learning. We want to train our model to predict survival. Imagine the extreme case, where we train a model only on passengers that survived. This model will be terrible, because it has no idea how a surviving passenger looks on paper. Now don’t worry, this training data has rows on both surviving and non-surviving passengers. However, a training data where one of the outcomes is more frequent can bias a model. Let us find out what’s the situation with this training data.
total_samples = train_dummies.shape[0] # Number of rows in DataFrame
number_surviving = (train_dummies['Survived'] == 1).sum() # Number of survivors
perc_survivors = (number_surviving / total_samples) * 100
# 38.38383838383838
In this training data 38% of passengers survived. There are no hard rules on how balanced data should be, but I am not happy with that number. If it was 45% I think it would be fine but we should do something about 38%. Because the number of survivors is lower than the non-survivors, we will randomly select as many non-survivors as there are survivors.
Now we have 342 survivors and 342 non-survivors. Perfectly balanced, as all things should be. Concatenating like this we must also remember to shuffle the rows with .sample(frac=1), otherwise we might run into problems later with unbalanced batches. Now that our data is balanced, we can move on to standardization.
Standardization
Both datasets are now free from missing values, categories are converted to dummy variables and our labels are balanced. Now we can move on to standardization, the process of bringing features to the same scale. We achieve this by subtracting the mean and dividing by the standard deviation. This has some advantages for neural networks. It can increase the learning rate and result in better fits. There are also disadvantages, as some features become harder to interpret because they lose their physical units. Because we rarely try to interpret neuronal networks, we almost always standardize our data. This is how it works with scikit-learn.
Originally the mean age is 29.23 and the standard deviation is 13.24. After standardization, the mean is near zero because we subtracted it and the standard deviation is near 1 because we divided by it. So our standardization worked. Now the absolute values of all features are on the scale of the standard normal distribution. We are now almost ready to create our actual network. But before, we will split our training data into a training and validation data. This will help us avoid overfitting.
Validation Data Split
Using train_test_split we get 547 training and 137 validation samples. test_size=0.1 specifies that 10% of samples should be in the test data, which in our case will be used for validation, as we already have another designated test data set. We are now ready to build our network an start training.
from sklearn.model_selection import train_test_split
X = train_scaled
X_train, X_validate, y_train, y_validate = train_test_split(X, y,
test_size=0.1)
X_train.shape
# (547, 31)
X_validate.shape
# (137, 31)
Building the Network with pyTorch
We start by converting our arrays to tensors. This is the data structure pyTorch will expect as input to the network later.
This network takes our 31 input features and transforms them to 50 hidden units using a fully connected linear layer. This layer also learns a bias term by default. We then apply the rectifying linear unit to introduce some non-linearity. We then convert the 50 hidden units to one output unit, to which we apply the Sigmoid function. This means our deep neural network has one hidden layer with 50 units. The Sigmoid function makes sure that our output domain is between 0 and 1. This is important because we are making a binary classification of surviving (1) and non-surviving (0) passengers. Now we define our loos function and our learning method.
Our loss function is binary cross entropy loss, which works well for the binary classification problem we are facing. Our learning algorithm is Adam, a variety of gradient descent that features L2 regularization (weight_decay). Before we start training we will split our train data into 41 mini batches.
Training our network on the mini batches instead of the whole data makes learning quicker and can result in models that generalize better. Now we are ready to train the network.
n_epochs = 2000
loss_list = []
validate_loss_list = []
for epoch in range(n_epochs):
for batch_idx in range(n_batches):
optimizer.zero_grad()
outputs = model(train_features_batched[batch_idx].float())
loss = criterion(outputs.flatten().float(),
train_labels_batched[batch_idx].float())
loss.backward()
optimizer.step()
outputs = model(train_features.float())
validation_outputs = model(validation_features.float())
loss = criterion(outputs.flatten().float(),
train_labels.float())
validate_loss = criterion(validation_outputs.flatten().float(),
validation_labels.float())
loss_list.append(loss.item())
validate_loss_list.append(validate_loss)
print('Finished Training')
import matplotlib.pyplot as plt
plt.plot(loss_list, linewidth=3)
plt.plot(validate_loss_list, linewidth=3)
plt.legend(("Training Loss", "Validation Loss"))
plt.xlabel("Epoch")
plt.ylabel("BCE Loss")
Loss curves of model training. Yours will look different, as we did not seed the randomness during preprocessing or model initialization.
This code starts by creating empty lists to record the loss of both test and validation data. n_epochs tells us how many times we train the network on the entire data. The actual learning happens in the inner loop that exposes batch_idx. We train the network after each mini batch. The output is whatever the network current spits out for a given batch. Then we compare that output to the known labels, which is our loss. loss.backward() calculates the gradient of the loss with respect to the parameters. Finally, the actual learning happens when we call optimizer.step(), which adjust the parameters of our model according to the gradients. This is why pyTorch is handy, it takes very good care of the gradients. The loss we are plotting is calculated in the out loop, where the loss across the entire data instead of the mini batches is calculated. The rest is just plotting. So how did it go? Our network learns very quickly. After 250 epochs the validation loss already stops improving. Everything that comes afterwards is probably overfitting the training data. That means the training loss improves while the validation loss is stagnant or becomes worse. The last piece of code we have to write calculates our prediction for the test data and saves it in a way that we can upload it to Kaggle to find out how well we did.
I will upload three different predictions to Kaggle. One from a network that is untrained. To do so I will set n_epochs to 0. This amounts to guessing, because the network is randomly initialized. Then I will upload predictions from a model that stops after 200 epochs. This should be a well trained model. Then I will use a model that stopped after 2000 epochs. This should be slightly overfit. Let’s see how we did.
We got a 0.39952 accuracy score for the untrained model, meaning that 39.952 % of our predictions were correct. Pretty bad but expected from a guessing model. Our model trained on 200 epochs has a 0.70095 score. Much better and a good sign that our model actually learned something. Our model trained on 2000 epochs got a score of 0.74162. Not exactly what we would expect. This probably means that our validation data set was not representative for the test data set. I also read from other people that overfitting the test data improves test accuracy slightly in the Titanic example. How could we improve our model?
Where to go from here
I decided to use a neural network because I wanted to write about it but it is not the best model. A random forest classifier seems to do better here. The highest score without cheating is around 0.83, which is achieved with this kind of model. But what could we do to improve our above code. One great improvement would be to use another validation method. Our validation method sets a lot of data aside that the model is never trained on and we don’t have massive amounts of data to begin with. A cross-validation technique could help us us that data for training. A second improvement would be to use a more formal way to determine the epoch where we want to stop training. We just looked at it and picked nice looking spots in the loss curve. Early stopping is a technique that could help us find a more objective point to stop the training. Finally, we could learn more about the Titanic data and get more features out of the data. For example, we discarded the cabin number, keeping only the letter. However, the number could be very important. A good way to learn more about the Titanic data is to browse the Kaggle discussion board.
t-SNE (t-distributed stochastic neighbor embedding) is a popular dimensionality reduction technique. We often havedata where samples are characterized by n features. To reduce the dimensionality, t-SNE generates a lower number of features (typically two) that preserves the relationship between samples as good as possible. Here we will learn how to use the scikit-learn implementation of t-SNE and how it achieves dimensionality reduction step by step.
How to use t-SNE with scikit-learn
We will start by performing t-SNE on a part of the MNIST dataset. The MNIST dataset consists of images of hand drawn digits from 0 to 9. Accurately classifying each digit is a popular machine learning challenge. We can load the MNIST dataset with sklearn.
from sklearn.datasets import fetch_openml
import numpy as np
import matplotlib.pyplot as plt
# Load the MNIST data
X, y = fetch_openml('mnist_784', version=1, return_X_y=True, as_frame=False)
# Randomly select 1000 samples for performance reasons
np.random.seed(100)
subsample_idc = np.random.choice(X.shape[0], 1000, replace=False)
X = X[subsample_idc,:]
y = y[subsample_idc]
# Show two example images
fig, ax = plt.subplots(1,2)
ax[0].imshow(X[11,:].reshape(28,28), 'Greys')
ax[1].imshow(X[15,:].reshape(28,28), 'Greys')
ax[0].set_title("Label 3")
ax[1].set_title("Label 8")
Two example images from the MNIST dataset.
By default, the MNIST data we fetch comes with 70000 images. We randomly select 1000 of those to make this demonstration faster. Each image consists of 784 pixels and they come as a flat one dimensional array. To display them as an image we reshape them into a 28×28 matrix. The images are in X and their labels in y.
X.shape
# (1000, 784)
# 1000 Samples with 784 features
y.shape
# (1000,)
# 1000 labels
np.unique(y)
# array(['0', '1', '2', '3', '4', '5', '6', '7', '8', '9'], dtype=object)
# The 10 classes of the images
Using t-SNE on this data is shockingly easy thanks to scikit-learn. We simply import the TSNE class, pass it our data and then fit.
from sklearn.manifold import TSNE
import pandas as pd
import seaborn as sns
# We want to get TSNE embedding with 2 dimensions
n_components = 2
tsne = TSNE(n_components)
tsne_result = tsne.fit_transform(X)
tsne_result.shape
# (1000, 2)
# Two dimensions for each of our images
# Plot the result of our TSNE with the label color coded
# A lot of the stuff here is about making the plot look pretty and not TSNE
tsne_result_df = pd.DataFrame({'tsne_1': tsne_result[:,0], 'tsne_2': tsne_result[:,1], 'label': y})
fig, ax = plt.subplots(1)
sns.scatterplot(x='tsne_1', y='tsne_2', hue='label', data=tsne_result_df, ax=ax,s=120)
lim = (tsne_result.min()-5, tsne_result.max()+5)
ax.set_xlim(lim)
ax.set_ylim(lim)
ax.set_aspect('equal')
ax.legend(bbox_to_anchor=(1.05, 1), loc=2, borderaxespad=0.0)
t-SNE example on MNIST subsample.
Considering that we did not specify any parameters except n_components, this looks pretty good. Before we dive into the parameters, we will go through t-SNE step by step and take some looks under the hood of the scikit-learn implementation.
t-SNE step by step
The Distance Matrix
The first step of t-SNE is to calculate the distance matrix. In our t-SNE embedding above, each sample is described by two features. In the actual data, each point is described by 728 features (the pixels). Plotting data with that many features is impossible and that is the whole point of dimensionality reduction. However, even at 728 features, each point is a certain distance apart from every other point. There are many different distance metrics that make sense but probably the most straightforward one is the euclidean distance.
Definition of euclidean distance for two features. Image credit licensed under CC BY 4.0 by Kmhkmh.
The above definition of euclidean distance for two features extends to n features (p1,p2,p2,…,pn). Once again we can use scikit-learn to calculate the euclidean distance matrix. Because a distance matrix of the unsorted samples doesn’t look like much, we also calculate it after sorting the samples by label.
When the samples are sorted by label, squareish patterns emerge in the distance matrix. White means smaller euclidean distances. This is reassuring. After all, we would expect the drawing of a one to be more similar to other drawings of a one than to a drawing of a zero. That’s what the squares near the white diagonal represent. t-SNE tries to roughly preserve the distances between samples but it does not work on the raw distances. It works on the joint probabilities, bringing us to our second step.
Joint Probabilities
The distance matrix tells us how far samples are apart. The joint probabilities tell us how likely it is that samples choose each others as “neighbors”. The two are of course related, because nearby samples should have a higher chance to be neighbors than further apart samples. The t-distribution is defined by its mean, the degrees of freedom and the scale (σ). For our t-SNE purposes we set the mean to 0 (at 0, samples are exactly at the same place). The degrees of freedom are set to the number of components minus one. That’s one in our case, since we want two components. The last free parameter is sigma and it is important because it determines how wide the tails of the distribution are.
t-distributions for sigma of 1 and 2
This is where the perplexity parameter comes in. The user chooses a perplexity value (recommended values are between 5 and 50) and based on the perplexity, t-SNE then chooses sigmas that satisfy that perplexity. To understand what this means, consider the first row of our distance matrix. It tells us the distance of our first point to each other point and as we transform that row with the t-distribution we get our own distribution P. The perplexity is defined 2H(P) where H is the Shannon entropy. Different values for sigma will results in different distributions, which differ in entropy and therefore differ in perplexity.
Distributions of joint probabilities for different values of sigma.
Above we can see what happens to the joint probability distribution as we increase sigma. With increasing sigma the entropy increases and so does the perplexity. t-SNE performs a binary search for the sigma that produces the perplexity specified by the user. This means that the perplexity controls the chance of far away points to be chosen as neighbors. Therefor, perplexity is commonly interpreted as a measure for the number of samples neigbors. The default value for perplexity is 30 in the sklearn implementation of t-SNE. Instead of implementing our own binary search we will take a shortcut to calculating the joint probabilities. We will use sklearns internal function to do it.
from sklearn.manifold import _t_sne
perplexity = 30 # Same as the default perplexity
p = _t_sne._joint_probabilities(distances=distance_matrix,
desired_perplexity = perplexity,
verbose=False)
As a small note, our joint probabilities are no longer a matrix. You may have noticed that the distance matrix is symmetric along one of its diagonals and the diagonal is all zeros. So we only keep the upper triangular of the matrix in a flat array p. That’s all we need to move from joint probabilities to the next step.
Optimize Embedding with Gradient Descent
Now that we have the joint probabilities from our high dimensional data, we want to generate a low dimensional embedding with just two features that preserves the joint probabilities as good as possible. First we need to initialize our low dimensional embedding. By default, sklearn will use a random initialization so that’s what we will use. Once we initialized our embedding, we will optimize it using gradient descent. This optimization is at the core of t-SNE and we will be done afterwards. To achieve a good embedding, t-SNE optimizes the Kullback-Leibler divergence between the joint probabilites of the data and their embedding. It is a measure for the similarity of two distributions. The sklearn TSNE class comes with its own implementation of the Kullback-Leibler divergence and all we have to do is pass it to the _gradient_descent function with the initial embedding and the joint probabilities of the data.
# Create the initial embedding
n_samples = X.shape[0]
n_components = 2
X_embedded = 1e-4 * np.random.randn(n_samples,
n_components).astype(np.float32)
embedding_init = X_embedded.ravel() # Flatten the two dimensional array to 1D
# kl_kwargs defines the arguments that are passed down to _kl_divergence
kl_kwargs = {'P': p,
'degrees_of_freedom': 1,
'n_samples': 1000,
'n_components':2}
# Perform gradient descent
embedding_done = _t_sne._gradient_descent(_t_sne._kl_divergence,
embedding_init,
0,
1000,
kwargs=kl_kwargs)
# Get first and second TSNE components into a 2D array
tsne_result = embedding_done[0].reshape(1000,2)
# Convert do DataFrame and plot
tsne_result_df = pd.DataFrame({'tsne_1': tsne_result[:,0],
'tsne_2': tsne_result[:,1],
'label': y})
fig, ax = plt.subplots(1)
sns.scatterplot(x='tsne_1', y='tsne_2', hue='label', data=tsne_result_df, ax=ax,s=120)
lim = (tsne_result.min()-5, tsne_result.max()+5)
ax.set_xlim(lim)
ax.set_ylim(lim)
ax.set_aspect('equal')
ax.legend(bbox_to_anchor=(1.05, 1), loc=2, borderaxespad=0.0)
And that’s it. It doesn’t look identical to the t-SNE we did above because we did not seed the initialization of the and the gradient descent. In fact, your results will look slightly different if you follow this guide. The sign of success you are looking for are similarly well defined clusters. We could have gone a bit deeper here and there. For example we could have written our own implementations of the Kullback-Leibler divergence or gradient descent. I’ll leave that for another time. Here are some useful links if you want to dig deeper into t-SNE or its sklearn implementation.
A DataFrame is a spreadsheet like data structure. We can think of it as a collection of rows and columns. This row-column structure is useful for many different kinds of data. The most widely used DataFrame implementation in Python is from the Pandas package. First we will learn how to create DataFrames. We will also learn how to do some basic data analysis with them. Finally, we will compare the DataFrame to the ndarray data structure and learn why data frames are useful in other packages such as Seaborn.
How to Create a DataFrame
There two major ways to create a DataFrame. We can directly call DataFrame() and pass it data in a dictionary, list or array. Alternatively we can use several functions to load data from a file directly into a DataFrame. While it is very common in data science to load data from file, there are also many occasions where we need to create DataFrame from other data structures. We will first learn how to create a DataFrame from a dictionary.
In our dictionary the keys are used as the column names. The data under each key then becomes the column. The row names are defined separately by passing a collection to the index parameter of DataFrame. We can get column and row names with the columns and index attributes.
These names are useful because they give us a descriptive way of indexing into columns and rows. If we use indexing syntax on the DataFrame, we can get individual columns.
In that case there are no dictionary keys that could be use to infer the column names. This means we need to pass the column_names to the columns parameter. Mostly anything that structures our data in a two-dimensional way can be used to create a DataFrame. Next we will learn about functions that allow us to load different file types as a DataFrame.
Loading Files as a DataFrame
The list of file types Pandas can read and write is rather long and you can find it here. I only want to cover the most commonly used .csv file here. They have the particular advantage that they can also be read by humans, because they are essentially text files. They are also widely supported by a variety of languages and programs. First, let’s create our file. Because it is a text file, we can write a literal string to file.
text_file = open("example.csv", "w")
text_file.write(""",Frequency,Location,Cell Type
C1,20,2,Interneuron
C2,50,3,Interneuron
C3,8,1,Pyramidal""")
text_file.close()
In this file columns are separated by commas and rows are separated by new lines. This is what .csv means, it stands for comma-separated values. To load this file into a DataFrame we need to pass the file name and which column contains the row names. Pandas assumes by default that the first row contains the column names.
There are many more parameters we can specify for read_csv in case we have a file that is structured differently. In fact we can load files that have a value delimiter other than the comma, by specifying the delimiter parameter.
We specify '-' as the delimiter and and it also works. Although the function is called read_csv it is not strictly bound to comma separated values. We can also skip rows, columns and specify many more options you can learn about from the documentation. For well structured .csv files however, we need very few arguments as shown above. Next we will learn how to do basic calculations with the DataFrame.
Basic Math with DataFrame
A variety of functions such as df.mean(), df.median() and df.std() are available to do basic statistics on our DataFrame. By default they all return values per column. That is because columns are assumed to contain our variables (or features) and each row contains a sample.
One big advantage of the column is that within a column the data type is clearly defined. Within a row on the other hand different data types can exist. In our case we have two numeric types and a string. When we call these statistical methods, numeric types are ignored. In our case that is 'Cell Type'. Technically we can also use the axis parameter to calculate these statistics for each sample but this is not always useful and has to again ignore one of the columns.
Often times these operations make more sense for individual columns. As explained above we can use indexing to get individual columns and we can even assign new results to an existing or new column.
If you are familiar with NumPy, most of these DataFrame operations will seem very familiar because they mostly work like array operations. Because Pandas builds on NumPy, most NumPy functions (for example np.sin) work on numeric columns. I don’t want to go deeper and instead move on to visualizing DataFrames with Seaborn.
Seaborn for Data Visualization
Seaborn is a high-level data visualization package that builds on Matplotlib. It does not necessarily require a DataFrame. It can work with other data structures such as ndarray but it is particularly convenient with DataFrame. First, let us get a more interesting data set. Luckily Seaborn comes with some nice example data sets and they conveniently load into Pandas DataFrame.
The Iris data set contains information about different species of iris plants. It contains 150 samples and 5 features. The 'species' feature tells us what species a particular sample belongs to. The names of those columns are very useful when we structure our plots in Seaborn. Let’s first try a basic bar graph.
We use sns.barplot and we have to pass our DataFrame to the data parameter. Then for x and y we define which column name should appear there. We put 'species' on the x-axis so that is how data is aggregated inside the bars. Setosa, versicolor and virginica are the different species. The sns.set() function defines multiple parameters of Seaborn and forces a certain style on the plots that I personally prefer. Bar graphs have grown out of fashion and for good reason. They are not very informative about the distribution of their underlying values. I prefer the violin plot to get a better idea of the distribution.
We even get a small box plot within the violin plot for free. Seaborn works its magic through the DataFrame column names. This makes plotting more convenient but also makes our code more descriptive than it would be with pure NumPy. Our code literally tells us, that 'species' will be on the x-axis.
Summary
We learned that we can create a DataFrame from a dictionary or another kind of collection. The most important features are the column and row names. Columns organize features and rows organize samples by convention. We can also load files into a DataFrame. For example we can use read_csv to load .csv or other text based files. We can also use methods like df.mean() to get basic statistics of our DataFrame. Finally, Seaborn is very useful to visualize a DataFrame.
Creating great looking animations in Julia is shockingly easy thanks for the Plots package and some macro magic. Here we will learn how to turn data into high quality animations. We will learn about the @animate macro, frames and the gif function.
Two Steps to Animations
To create animations we simply generate frames with the @animate macro and then generate a file with the gif function. Both are part of the Plots package from Julia, so we have to start with using Plots to make that package available. Next create the data we will plot, which is a simple sine wave. Then we generate the frame with the @animate macro and animate them with the gif function. This is the code and the resulting animation.
using Plots
x = collect(1:0.1:30)
y = sin.(x)
df = 2
anim = @animate for i = 1:df:length(x)
plot(x[1:i], y[1:i], legend=false)
end
gif(anim, "tutorial_anim_fps30.gif", fps = 30)
Macros and Meta-Programming
The @animate macro deserves some extra attention, because it looks like magic. Macros are related to a concept called meta-programming. In Julia, all code is a data structure that can be manipulated in a similar way to all other data structures. This effectively means that we can write code that manipulates our code. That’s what a macro is, a function that modifies code. In our case, the code being modified is the for loop behind our @animate macro. It is modified in a way that it catches the frame at the end of each loop iteration and saves it into anim. We can create code that does the same job ourselves.
anim = Plots.Animation()
for i = 1:df:length(x)
plot(x[1:i], y[1:i], legend=false)
Plots.frame(anim)
end
gif(anim, "tutorial_anim_fps30.gif", fps = 30)
We use the Plots.Animation() function to create our animation object where we will store our frames. During the for loop we then call Plots.frame(anim) to store the frame after each iteration in our anim object. These are the essential steps that the @animate macro takes care of. If you want to learn what the macro does in detail you can call @macroexpand on it.
@macroexpand @animate for i = 1:df:length(x)
plot(x[1:i], y[1:i], legend=false)
end
There is another macro that is even more convenient. The @gif macro. It saves us from having to call gif() on our anim object.
@gif for i = 1:df:length(x)
plot(x[1:i], y[1:i], legend=false)
end
This directly displays the animation if interactive Julia is available for it. The downside of this is that we do not save the animation to disk and we do not have an anim object available to do more animations later. It is most useful to quickly troubleshoot animations interactively.
Beyond plot()
The @animate macro supports animations of anything that can be plotted with Plots. For example, we can animate a heatmap.
anim = @animate for i = 1:100
mat = rand(0:100, 32, 32)
heatmap(mat, clim=(0,255))
end
gif(anim, "tutorial_heatmap_anim.gif", fps = 10)
The frame that is being caught is the state of the active figure at the end of the for loop. The for loop itself gives us a great deal of control, how many frames we want to create. For example in the previous examples, I skipped frames with the df variable.
Every value in Julia has a type. Like other popular languages such as Python, Julia is dynamically typed. That means, we don’t need to explicitly define the type of a value when we create it. However, types are particularly important in Julia because we can use explicit type definitions to speed up calculations. Here we will get an introduction into the type system of Julia.
Dynamic Typing
When we assign a value in Julia we don’t need to specify its type. The type is guessed based on the given value. Let’s create an integer, a float and a string.
We use typeof() to find out what type a value has. To create an integer, we assign a number without decimal. To create a float, we simply attach a decimal place. For a string we need quotation marks around our value. There is another way to more explicitly define the type. Our float value is a Float64 (it uses 64 bits). What if we wanted a Float32?
We use the convert(Type, value) function to explicitly convert 3.0 to Float32. If we want to ensure that a value is of a certain type, we can use the double colon syntax (::). It evaluates normally, if the value has the specified type but throws an error, if it has a different type.
This type assertion syntax is frequently used when defining functions. A function is a piece of code that takes input arguments and processes them to produce output values. We will learn more about functions later. Because Julia is dynamically typed, we could write our functions in a way that they work the same regardless of input types. However, defining functions in a way that is specific to the input types can be good for performance and Julia makes heavy use of that fact. For a given function, multiple methods might exist. Each method is responsible for a given set of input parameters. For example, we can inspect all the different methods of the mathematical cos by calling the methods function on it.
methods(cos)
"""
# 12 methods for generic function cos:
cos(x::BigFloat) in Base.MPFR at mpfr.jl:744
cos(::Missing) in Base.Math at math.jl:1167
cos(a::Complex{Float16}) in Base.Math at math.jl:1115
cos(a::Float16) in Base.Math at math.jl:1114
cos(z::Complex{T}) where T in Base at complex.jl:823
cos(x::T) where T<:Union{Float32, Float64} in Base.Math at special/trig.jl:100
cos(x::Real) in Base.Math at special/trig.jl:124
cos(A::LinearAlgebra.Hermitian{#s661,S} where S<:(AbstractArray{#s662,2} where #s662<:#s661) where #s661<:Complex) in LinearAlgebra at C:\Users\Daniel\AppData\Local\Programs\Julia\Julia-1.4.2\share\julia\stdlib\v1.4\LinearAlgebra\src\symmetric.jl:907
cos(A::Union{LinearAlgebra.Hermitian{#s662,S}, LinearAlgebra.Symmetric{#s662,S}} where S where #s662<:Real) in LinearAlgebra at C:\Users\Daniel\AppData\Local\Programs\Julia\Julia-1.4.2\share\julia\stdlib\v1.4\LinearAlgebra\src\symmetric.jl:903
cos(D::LinearAlgebra.Diagonal) in LinearAlgebra at C:\Users\Daniel\AppData\Local\Programs\Julia\Julia-1.4.2\share\julia\stdlib\v1.4\LinearAlgebra\src\diagonal.jl:561
cos(A::AbstractArray{#s662,2} where #s662<:Real) in LinearAlgebra at C:\Users\Daniel\AppData\Local\Programs\Julia\Julia-1.4.2\share\julia\stdlib\v1.4\LinearAlgebra\src\dense.jl:793
cos(A::AbstractArray{#s662,2} where #s662<:Complex) in LinearAlgebra at C:\Users\Daniel\AppData\Local\Programs\Julia\Julia-1.4.2\share\julia\stdlib\v1.4\LinearAlgebra\src\dense.jl:800
"""
The cos function has 12 different methods. The double colon type assertion syntax checks the input type. Which method is used can depend on the type of all inputs. This is called multiple dispatch. Multiple inputs determine which method is dispatched. We will learn more about multiple dispatch in another post. Since arrays are central to scientific computing, we next look at the type system and arrays.
Array Types
There are several different ways to define arrays. One way is to use literal numbers enclosed by square brackets. In that case, the type is inferred in the same way as above, where we used literal numbers.
For an array, typeof() tells us not only the type of the array elements but also how many dimensions the array has. One important feature of the array is that all elements must have the same type. So what happens when we create an array from literals with different types? The type is determined by the most complex type.
We see that only one float value makes the entire array Float64. However, when one of the elements is a string, the type becomes Any, rather than String. That is because Julia does not convert numbers to strings. We get an error if we call convert(String, 3). If the most complex type is the Float64, Julia tries to convert all other values to that type. This works in the case where the other values are integers, because convert(Float64, 3) works. If other values cannot be converted to the most complex type all values take on the Any type. This means that values in the array can be of different types. They could literally be anything. This defeats the purpose of an array so we generally try to avoid it. There are other array creation methods besides the literal way. We can call a variety of methods that allow us to create arrays and they usually allow us to specify the type.
If we don’t specify the type we want, the zeros() function default to Float64. There are other functions such as ones(), giving an array of ones, or rand(), giving an array of random numbers. The type of the output can be specified for each.
That is it for our basic introduction into types. You can find a more complete introduction to Julia types in the official documentation. In summary, we don’t need to explicitly specify the type of values but sometimes it might help to make math more efficient. We will learn in later posts more about performance and efficiency in Julia.
To get us started with Julia we cover three basics. Arithmetic operators, name assignment.
Arithmetic operators
The standard arithmetic operators are addition (+), subtraction (-), multiplication (*), division (/), power (^) and remainder (%). They work as expected and the only one that is different for the Python crowd is power. That one is Matlab consistent. The normal precedence of operations applies. First power. Then multiplication and division. Then remainder. Finally addition and subtraction. Parentheses can be used to change the order of operation.
1 + 3 * 2
# 7
(1 + 3) * 2
# 8
2 * 3 ^ 2
# 18
In those examples, both sides of the operator are scalars. It gets a little more interesting when at least one of them is a vector or a matrix. Not all of the above operations are defined between vectors and scalars. Only division and multiplication are defined. We create a vector using square brackets ([]) with the elements separated by commas.
The other operations are not defined and throw an error.
[3, 1, 4] ^ 2
"""
MethodError: no method matching ^(::Array{Int64,1}, ::Int64)
Closest candidates are:
^(!Matched::Float16, ::Integer) at math.jl:885
^(!Matched::Regex, ::Integer) at regex.jl:712
^(!Matched::Missing, ::Integer) at missing.jl:155
...
Stacktrace:
[1] macro expansion at .\none:0 [inlined]
[2] literal_pow(::typeof(^), ::Array{Int64,1}, ::Val{2}) at .\none:0
[3] top-level scope at In[52]:1
"""
The reasons for this design choice have to do with Julias focus on linear algebra and are not important here. If we want this operation to work in an element-wise manner, we have to force it explicitly. We can do so using the dot (.). This way we can explicitly force every operator to be applied element-wise.
Now that we have our arithmetic operators, let’s move on to name assignment so we can store the results of our operations.
Name Assignment
We assign names to values with the = operator using the syntax name = value. Once a name is assigned, we can use the name instead of the value in operations.
Once a name is assigned to a value we can reassign that same name to a different value without problem.
result = 2 + 2
result = 10
# 10
If we want to assign a name that is not supposed to be reassigned, we can use the const keyword. If we try to reassign a constant name we get an error.
const a = 8.3144621
a = 3
"""
invalid redefinition of constant a
Stacktrace:
[1] top-level scope at In[73]:2
"""
There are a few rules about the names we can assign. Generally, Unicode characters (UTF-8) are allowed. This means we can do something like this.
δt = 0.0001
Here we are using the special character delta. If you want to quickly generate such a character, many Julia environments allow you to do this by typing \delta and hitting tab. I recommend using these sparingly, as they might confuse people transitioning from other languages that don’t allow unicode names. On the other hand they might be useful to make your code style more mathy. Not allowed as names are built-in keywords that have special meaning and trying to assign them will result in an error.
if = 3
# syntax: unexpected "="
If you are interested in more details about variables and name assignments you can take a look at the official documentation. In the next blog post we will take a look at the type system of Julia.
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.
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).
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 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.
Data, Science, Energy 
