Tutorial 3: Simultaneous fitting/regression

Week 3, Day 5: Network Causality

By Neuromatch Academy

Content creators: Ari Benjamin, Tony Liu, Konrad Kording

Content reviewers: Mike X Cohen, Madineh Sarvestani, Yoni Friedman, Ella Batty, Michael Waskom

Production editors: Gagana B, Spiros Chavlis

---

Tutorial objectives

Estimated timing of tutorial: 20 min

This is tutorial 3 on our day of examining causality. Below is the high level outline of what we’ll cover today, with the sections we will focus on in this notebook in bold:

1. Master definitions of causality

2. Understand that estimating causality is possible

3. Learn 4 different methods and understand when they fail

   1. perturbations

   2. correlations

   3. simultaneous fitting/regression

   4. instrumental variables

Notebook 3 objectives

In tutorial 2 we explored correlation as an approximation for causation and learned that correlation \(\neq\) causation for larger networks. However, computing correlations is a rather simple approach, and you may be wondering: will more sophisticated techniques allow us to better estimate causality? Can’t we control things?

Here we’ll use some common advanced (but controversial) methods that estimate causality from observational data. These methods rely on fitting a function to our data directly, instead of trying to use perturbations or correlations. Since we have the full closed-form equation of our system, we can try these methods and see how well they work in estimating causal connectivity when there are no perturbations. Specifically, we will:

• Learn about more advanced (but also controversial) techniques for estimating causality

  • conditional probabilities (regression)

• Explore limitations and failure modes

  • understand the problem of omitted variable bias

---

Setup

⚠ Experimental LLM-enhanced tutorial ⚠

This notebook includes Neuromatch’s experimental Chatify 🤖 functionality. The Chatify notebook extension adds support for a large language model-based “coding tutor” to the materials. The tutor provides automatically generated text to help explain any code cell in this notebook.

Note that using Chatify may cause breaking changes and/or provide incorrect or misleading information. If you wish to proceed by installing and enabling the Chatify extension, you should run the next two code blocks (hidden by default). If you do not want to use this experimental version of the Neuromatch materials, please use the stable materials instead.

To use the Chatify helper, insert the %%explain magic command at the start of any code cell and then run it (shift + enter) to access an interface for receiving LLM-based assitance. You can then select different options from the dropdown menus depending on what sort of assitance you want. To disable Chatify and run the code block as usual, simply delete the %%explain command and re-run the cell.

Note that, by default, all of Chatify’s responses are generated locally. This often takes several minutes per response. Once you click the “Submit request” button, just be patient– stuff is happening even if you can’t see it right away!

Thanks for giving Chatify a try! Love it? Hate it? Either way, we’d love to hear from you about your Chatify experience! Please consider filling out our brief survey to provide feedback and help us make Chatify more awesome!

Run the next two cells to install and configure Chatify…

%pip install -q davos
import davos
davos.config.suppress_stdout = True

Note: you may need to restart the kernel to use updated packages.

smuggle chatify      # pip: git+https://github.com/ContextLab/chatify.git
%load_ext chatify

Downloading and initializing model; this may take a few minutes...

llama.cpp: loading model from /home/runner/.cache/huggingface/hub/models--TheBloke--Llama-2-7B-Chat-GGML/snapshots/501a3c8182cd256a859888fff4e838c049d5d7f6/llama-2-7b-chat.ggmlv3.q5_1.bin
llama_model_load_internal: format     = ggjt v3 (latest)
llama_model_load_internal: n_vocab    = 32000
llama_model_load_internal: n_ctx      = 512
llama_model_load_internal: n_embd     = 4096
llama_model_load_internal: n_mult     = 256
llama_model_load_internal: n_head     = 32
llama_model_load_internal: n_layer    = 32
llama_model_load_internal: n_rot      = 128
llama_model_load_internal: freq_base  = 10000.0
llama_model_load_internal: freq_scale = 1
llama_model_load_internal: ftype      = 9 (mostly Q5_1)
llama_model_load_internal: n_ff       = 11008
llama_model_load_internal: model size = 7B
llama_model_load_internal: ggml ctx size =    0.08 MB
llama_model_load_internal: mem required  = 6390.60 MB (+ 1026.00 MB per state)

AVX = 1 | AVX2 = 1 | AVX512 = 1 | AVX512_VBMI = 0 | AVX512_VNNI = 0 | FMA = 1 | NEON = 0 | ARM_FMA = 0 | F16C = 1 | FP16_VA = 0 | WASM_SIMD = 0 | BLAS = 0 | SSE3 = 1 | VSX = 0 | llama_new_context_with_model: kv self size  =  256.00 MB

Install and import feedback gadget

# @title Install and import feedback gadget

!pip3 install vibecheck datatops --quiet

from vibecheck import DatatopsContentReviewContainer
def content_review(notebook_section: str):
    return DatatopsContentReviewContainer(
        "",  # No text prompt
        notebook_section,
        {
            "url": "https://pmyvdlilci.execute-api.us-east-1.amazonaws.com/klab",
            "name": "neuromatch_cn",
            "user_key": "y1x3mpx5",
        },
    ).render()


feedback_prefix = "W3D5_T3"

# Imports
import numpy as np
import matplotlib.pyplot as plt

from sklearn.multioutput import MultiOutputRegressor
from sklearn.linear_model import Lasso

Figure Settings

# @title Figure Settings
import logging
logging.getLogger('matplotlib.font_manager').disabled = True
import ipywidgets as widgets  # interactive display
%config InlineBackend.figure_format = 'retina'
plt.style.use("https://raw.githubusercontent.com/NeuromatchAcademy/course-content/main/nma.mplstyle")

Plotting Functions

# @title Plotting Functions

def see_neurons(A, ax, ratio_observed=1, arrows=True):
  """
  Visualizes the connectivity matrix.

  Args:
      A (np.ndarray): the connectivity matrix of shape (n_neurons, n_neurons)
      ax (plt.axis): the matplotlib axis to display on

  Returns:
      Nothing, but visualizes A.
  """
  n = len(A)

  ax.set_aspect('equal')
  thetas = np.linspace(0, np.pi * 2, n, endpoint=False)
  x, y = np.cos(thetas), np.sin(thetas),
  if arrows:
    for i in range(n):
      for j in range(n):
        if A[i, j] > 0:
          ax.arrow(x[i], y[i], x[j] - x[i], y[j] - y[i], color='k',
                   head_width=.05, width = A[i, j] / 25, shape='right',
                   length_includes_head=True, alpha=.2)
  if ratio_observed < 1:
    nn = int(n * ratio_observed)
    ax.scatter(x[:nn], y[:nn], c='r', s=150, label='Observed')
    ax.scatter(x[nn:], y[nn:], c='b', s=150, label='Unobserved')
    ax.legend(fontsize=15)
  else:
    ax.scatter(x, y, c='k', s=150)
  ax.axis('off')


def plot_connectivity_matrix(A, ax=None):
  """Plot the (weighted) connectivity matrix A as a heatmap

    Args:
      A (ndarray): connectivity matrix (n_neurons by n_neurons)
      ax: axis on which to display connectivity matrix
  """
  if ax is None:
    ax = plt.gca()
  lim = np.abs(A).max()
  ax.imshow(A, vmin=-lim, vmax=lim, cmap="coolwarm")

Helper Functions

# @title Helper Functions


def sigmoid(x):
  """
  Compute sigmoid nonlinearity element-wise on x.

  Args:
    x (np.ndarray): the numpy data array we want to transform
  Returns
    (np.ndarray): x with sigmoid nonlinearity applied
  """
  return 1 / (1 + np.exp(-x))


def create_connectivity(n_neurons, random_state=42, p=0.9):
  """
  Generate our nxn causal connectivity matrix.

  Args:
    n_neurons (int): the number of neurons in our system.
    random_state (int): random seed for reproducibility

  Returns:
    A (np.ndarray): our 0.1 sparse connectivity matrix
  """
  np.random.seed(random_state)
  A_0 = np.random.choice([0, 1], size=(n_neurons, n_neurons), p=[p, 1 - p])

  # set the timescale of the dynamical system to about 100 steps
  _, s_vals, _ = np.linalg.svd(A_0)
  A = A_0 / (1.01 * s_vals[0])

  return A


def simulate_neurons(A, timesteps, random_state=42):
    """
    Simulates a dynamical system for the specified number of neurons and timesteps.

    Args:
      A (np.array): the connectivity matrix
      timesteps (int): the number of timesteps to simulate our system.
      random_state (int): random seed for reproducibility

    Returns:
      X has shape (n_neurons, timeteps).
    """
    np.random.seed(random_state)


    n_neurons = len(A)
    X = np.zeros((n_neurons, timesteps))

    for t in range(timesteps - 1):
      epsilon = np.random.multivariate_normal(np.zeros(n_neurons),
                                              np.eye(n_neurons))
      X[:, t + 1] = sigmoid(A.dot(X[:, t]) + epsilon)

      assert epsilon.shape == (n_neurons, )
    return X


def get_sys_corr(n_neurons, timesteps, random_state=42, neuron_idx=None):
  """
  A wrapper function for our correlation calculations between A and R.

  Args:
    n_neurons (int): the number of neurons in our system.
    timesteps (int): the number of timesteps to simulate our system.
    random_state (int): seed for reproducibility
    neuron_idx (int): optionally provide a neuron idx to slice out

  Returns:
    A single float correlation value representing the similarity between A and R
  """

  A = create_connectivity(n_neurons, random_state)
  X = simulate_neurons(A, timesteps)

  R = correlation_for_all_neurons(X)

  return np.corrcoef(A.flatten(), R.flatten())[0, 1]


def correlation_for_all_neurons(X):
  """
  Computes the connectivity matrix for the all neurons using correlations

  Args:
    X: the matrix of activities

  Returns:
    estimated_connectivity (np.ndarray): estimated connectivity for the
                                         selected neuron, of shape (n_neurons,)
  """
  n_neurons = len(X)
  S = np.concatenate([X[:, 1:], X[:, :-1]], axis=0)
  R = np.corrcoef(S)[:n_neurons, n_neurons:]
  return R

The helper functions defined above are:

• sigmoid: computes sigmoid nonlinearity element-wise on input, from Tutorial 1

• create_connectivity: generates nxn causal connectivity matrix., from Tutorial 1

• simulate_neurons: simulates a dynamical system for the specified number of neurons and timesteps, from Tutorial 1

• get_sys_corr: a wrapper function for correlation calculations between A and R, from Tutorial 2

• correlation_for_all_neurons: computes the connectivity matrix for the all neurons using correlations, from Tutorial 2

---

Section 1: Regression: recovering connectivity by model fitting

Video 1: Regression approach

Submit your feedback

# @title Submit your feedback
content_review(f"{feedback_prefix}_Regression_approach_Video")

You may be familiar with the idea that correlation only implies causation when there are no hidden confounders. This aligns with our intuition that correlation only implies causality when no alternative variables could explain away a correlation.

A confounding example: Suppose you observe that people who sleep more do better in school. It’s a nice correlation. But what else could explain it? Maybe people who sleep more are richer, don’t work a second job, and have time to actually do homework. If you want to ask if sleep causes better grades, and want to answer that with correlations, you have to control for all possible confounds.

A confound is any variable that affects both the outcome and your original covariate. In our example, confounds are things that affect both sleep and grades.

Controlling for a confound: Confonds can be controlled for by adding them as covariates in a regression. But for your coefficients to be causal effects, you need three things:

1. All confounds are included as covariates

2. Your regression assumes the same mathematical form of how covariates relate to outcomes (linear, GLM, etc.)

3. No covariates are caused by both the treatment (original variable) and the outcome. These are colliders; we won’t introduce it today (but Google it on your own time! Colliders are very counterintuitive.)

In the real world it is very hard to guarantee these conditions are met. In the brain it’s even harder (as we can’t measure all neurons). Luckily today we simulated the system ourselves.

Video 2: Fitting a GLM

Submit your feedback

# @title Submit your feedback
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Recall that in our system each neuron effects every other via:

\[ \vec{x}_{t+1} = \sigma(A\vec{x}_t + \epsilon_t), \]

where \(\sigma\) is our sigmoid nonlinearity from before: \(\sigma(x) = \frac{1}{1 + e^{-x}}\)

Our system is a closed system, too, so there are no omitted variables. The regression coefficients should be the causal effect. Are they?

We will use a regression approach to estimate the causal influence of all neurons to neuron #1. Specifically, we will use linear regression to determine the \(A\) in:

(461)\[\begin{equation} \sigma^{-1}(\vec{x}_{t+1}) = A\vec{x}_t + \epsilon_t , \end{equation}\]

where \(\sigma^{-1}\) is the inverse sigmoid transformation, also sometimes referred to as the logit transformation: \(\sigma^{-1}(x) = \log(\frac{x}{1-x})\).

Let \(W\) be the \(\vec{x}_t\) values, up to the second-to-last timestep \(T-1\):

(462)\[\begin{equation} W = \begin{bmatrix} \mid & \mid & ... & \mid \\ \vec{x}_0 & \vec{x}_1 & ... & \vec{x}_{T-1} \\ \mid & \mid & ... & \mid \end{bmatrix}_{n \times (T-1)} \end{equation}\]

Let \(Y\) be the \(\vec{x}_{t+1}\) values for a selected neuron, indexed by \(i\), starting from the second timestep up to the last timestep \(T\):

(463)\[\begin{equation} Y = \begin{bmatrix} x_{i,1} & x_{i,2} & ... & x_{i, T} \\ \end{bmatrix}_{1 \times (T-1)} \end{equation}\]

You will then fit the following model:

(464)\[\begin{equation} \sigma^{-1}(Y^T) = W^\top V \end{equation}\]

where \(V\) is the \(n \times 1\) coefficient matrix of this regression, which will be the estimated connectivity matrix between the selected neuron and the rest of the neurons.

Review: As you learned in Week 1, lasso a.k.a. \(L_1\) regularization causes the coefficients to be sparse, containing mostly zeros. Think about why we want this here.

Coding Exercise 1: Use linear regression plus lasso to estimate causal connectivities

You will now create a function to fit the above regression model and V. We will then call this function to examine how close the regression vs the correlation is to true causality.

Code:

You’ll notice that we’ve transposed both \(Y\) and \(W\) here and in the code we’ve already provided below. Why is that?

This is because the machine learning models provided in scikit-learn expect the rows of the input data to be the observations, while the columns are the variables. We have that inverted in our definitions of \(Y\) and \(W\), with the timesteps of our system (the observations) as the columns. So we transpose both matrices to make the matrix orientation correct for scikit-learn.

• Because of the abstraction provided by scikit-learn, fitting this regression will just be a call to initialize the Lasso() estimator and a call to the fit() function

• Use the following hyperparameters for the Lasso estimator:

  • alpha = 0.01

  • fit_intercept = False

• How do we obtain \(V\) from the fitted model?

We will use the helper function logit.

# Set parameters
n_neurons = 50  # the size of our system
timesteps = 10000  # the number of timesteps to take
random_state = 42
neuron_idx = 1

# Set up system and simulate
A = create_connectivity(n_neurons, random_state)
X = simulate_neurons(A, timesteps)

Execute this cell to enable helper function logit

# @markdown Execute this cell to enable helper function `logit`

def logit(x):
  """
  Applies the logit (inverse sigmoid) transformation

  Args:
    x (np.ndarray): the numpy data array we want to transform
  Returns
    (np.ndarray): x with logit nonlinearity applied
  """
  return np.log(x/(1-x))

def get_regression_estimate(X, neuron_idx):
  """
  Estimates the connectivity matrix using lasso regression.

  Args:
    X (np.ndarray): our simulated system of shape (n_neurons, timesteps)
    neuron_idx (int):  a neuron index to compute connectivity for

  Returns:
    V (np.ndarray): estimated connectivity matrix of shape (n_neurons, n_neurons).
                    if neuron_idx is specified, V is of shape (n_neurons,).
  """
  # Extract Y and W as defined above
  W = X[:, :-1].transpose()
  Y = X[[neuron_idx], 1:].transpose()

  # Apply inverse sigmoid transformation
  Y = logit(Y)

  ############################################################################
  ## TODO: Insert your code here to fit a regressor with Lasso. Lasso captures
  ## our assumption that most connections are precisely 0.
  ## Fill in function and remove
  raise NotImplementedError("Please complete the regression exercise")
  ############################################################################

  # Initialize regression model with no intercept and alpha=0.01
  regression = ...

  # Fit regression to the data
  regression.fit(...)

  V = regression.coef_

  return V


# Estimate causality with regression
V = get_regression_estimate(X, neuron_idx)

print(f"Regression: correlation of estimated with true connectivity: {np.corrcoef(A[neuron_idx, :], V)[1, 0]:.3f}")
print(f"Lagged correlation of estimated with true connectivity: {get_sys_corr(n_neurons, timesteps, random_state, neuron_idx=neuron_idx):.3f}")

You should find that:

Regression: correlation of estimated connectivity with true connectivity: 0.865
Lagged correlation of estimated connectivity with true connectivity: 0.703

Click for solution

Submit your feedback

# @title Submit your feedback
content_review(f"{feedback_prefix}_Linear_regression_with_lasso_to_estimate_causal_connectivities_Exercise")

We can see from these numbers that multiple regression is better than simple correlation for estimating connectivity.

---

Section 2: Partially Observed Systems

Estimated timing to here from start of tutorial: 10 min

If we are unable to observe the entire system, omitted variable bias becomes a problem. If we don’t have access to all the neurons, and so therefore can’t control them, can we still estimate the causal effect accurately?

Video 3: Omitted variable bias

Submit your feedback

# @title Submit your feedback
content_review(f"{feedback_prefix}_Omitted_variable_bias_Video")

Video correction: the labels “connectivity from”/”connectivity to” are swapped in the video but fixed in the figures/demos below

We first visualize different subsets of the connectivity matrix when we observe 75% of the neurons vs 25%.

Recall the meaning of entries in our connectivity matrix: \(A[i,j] = 1\) means a connectivity from neuron \(i\) to neuron \(j\) with strength \(1\).

Execute this cell to visualize subsets of connectivity matrix

# @markdown Execute this cell to visualize subsets of connectivity matrix

# Run this cell to visualize the subsets of variables we observe
n_neurons = 25
A = create_connectivity(n_neurons)

fig, axs = plt.subplots(2, 2, figsize=(10, 10))
ratio_observed = [0.75, 0.25]  # the proportion of neurons observed in our system

for i, ratio in enumerate(ratio_observed):
  sel_idx = int(n_neurons * ratio)

  offset = np.zeros((n_neurons, n_neurons))
  axs[i, 1].title.set_text(f"{int(ratio * 100)}% neurons observed")
  offset[:sel_idx, :sel_idx] =  1 + A[:sel_idx, :sel_idx]
  im = axs[i, 1].imshow(offset, cmap="coolwarm", vmin=0, vmax=A.max() + 1)
  axs[i, 1].set_xlabel("Connectivity from")
  axs[i, 1].set_ylabel("Connectivity to")
  plt.colorbar(im, ax=axs[i, 1], fraction=0.046, pad=0.04)
  see_neurons(A, axs[i, 0], ratio)

plt.suptitle("Visualizing subsets of the connectivity matrix", y = 1.05)
plt.show()

Interactive Demo 3: Regression performance as a function of the number of observed neurons

We will first change the number of observed neurons in the network and inspect the resulting estimates of connectivity in this interactive demo. How does the estimated connectivity differ?

Execute this cell to get helper functions get_regression_estimate_full_connectivity and get_regression_corr_full_connectivity

# @markdown Execute this cell to get helper functions `get_regression_estimate_full_connectivity` and `get_regression_corr_full_connectivity`

def get_regression_estimate_full_connectivity(X):
  """
  Estimates the connectivity matrix using lasso regression.

  Args:
    X (np.ndarray): our simulated system of shape (n_neurons, timesteps)
    neuron_idx (int): optionally provide a neuron idx to compute connectivity for
  Returns:
    V (np.ndarray): estimated connectivity matrix of shape (n_neurons, n_neurons).
                    if neuron_idx is specified, V is of shape (n_neurons,).
  """
  n_neurons = X.shape[0]

  # Extract Y and W as defined above
  W = X[:, :-1].transpose()
  Y = X[:, 1:].transpose()

  # apply inverse sigmoid transformation
  Y = logit(Y)

  # fit multioutput regression
  reg = MultiOutputRegressor(Lasso(fit_intercept=False,
                                    alpha=0.01, max_iter=250 ), n_jobs=-1)
  reg.fit(W, Y)

  V = np.zeros((n_neurons, n_neurons))
  for i, estimator in enumerate(reg.estimators_):
    V[i, :] = estimator.coef_

  return V


def get_regression_corr_full_connectivity(n_neurons, A, X, observed_ratio, regression_args):
    """
    A wrapper function for our correlation calculations between A and the V estimated
    from regression.

    Args:
      n_neurons (int): number of neurons
      A (np.ndarray): connectivity matrix
      X (np.ndarray): dynamical system
      observed_ratio (float): the proportion of n_neurons observed, must be betweem 0 and 1.
      regression_args (dict): dictionary of lasso regression arguments and hyperparameters

    Returns:
      A single float correlation value representing the similarity between A and R
    """
    assert (observed_ratio > 0) and (observed_ratio <= 1)

    sel_idx = np.clip(int(n_neurons*observed_ratio), 1, n_neurons)

    sel_X = X[:sel_idx, :]
    sel_A = A[:sel_idx, :sel_idx]

    sel_V = get_regression_estimate_full_connectivity(sel_X)
    return np.corrcoef(sel_A.flatten(), sel_V.flatten())[1, 0], sel_V

Execute this cell to enable demo. the plots will take a few seconds to update after moving the slider.

# @markdown Execute this cell to enable demo. the plots will take a few seconds to update after moving the slider.

n_neurons = 50
A = create_connectivity(n_neurons, random_state=42)
X = simulate_neurons(A, 4000, random_state=42)

reg_args = {
    "fit_intercept": False,
    "alpha": 0.001
}

@widgets.interact(n_observed=widgets.IntSlider(min=5, max=45, step=5,
                                               continuous_update=False))
def plot_observed(n_observed):
  to_neuron = 0
  fig, axs = plt.subplots(1, 3, figsize=(15, 5))
  sel_idx = n_observed
  ratio = (n_observed) / n_neurons
  offset = np.zeros((n_neurons, n_neurons))
  axs[0].title.set_text(f"{int(ratio * 100)}% neurons observed")
  offset[:sel_idx, :sel_idx] =  1 + A[:sel_idx, :sel_idx]
  im = axs[1].imshow(offset, cmap="coolwarm", vmin=0, vmax=A.max() + 1)
  plt.colorbar(im, ax=axs[1], fraction=0.046, pad=0.04)

  see_neurons(A,axs[0], ratio, False)
  corr, R = get_regression_corr_full_connectivity(n_neurons, A, X,
                                                  ratio, reg_args)
  big_R = np.zeros(A.shape)
  big_R[:sel_idx, :sel_idx] =  1 + R
  im = axs[2].imshow(big_R, cmap="coolwarm", vmin=0, vmax=A.max() + 1)
  plt.colorbar(im, ax=axs[2],fraction=0.046, pad=0.04)
  c = 'w' if n_observed<(n_neurons-3) else 'k'
  axs[2].text(0, n_observed + 3, f"Correlation: {corr:.2f}", color=c, size=15)
  axs[1].title.set_text("True connectivity")
  axs[1].set_xlabel("Connectivity from")
  axs[1].set_ylabel("Connectivity to")
  axs[2].title.set_text("Estimated connectivity")
  axs[2].set_xlabel("Connectivity from")
  plt.show()

Next, we will inspect a plot of the correlation between true and estimated connectivity matrices vs the percent of neurons observed over multiple trials. What is the relationship that you see between performance and the number of neurons observed?

Note: the cell below will take about 25-30 seconds to run.

Plot correlation vs. subsampling

# @markdown Plot correlation vs. subsampling
import warnings
warnings.filterwarnings('ignore')

# we'll simulate many systems for various ratios of observed neurons
n_neurons = 50
timesteps = 5000
ratio_observed = [1, 0.75, 0.5, .25, .12]  # the proportion of neurons observed in our system
n_trials = 3  # run it this many times to get variability in our results

reg_args = {
    "fit_intercept": False,
    "alpha": 0.001
}

corr_data = np.zeros((n_trials, len(ratio_observed)))
for trial in range(n_trials):

  A = create_connectivity(n_neurons, random_state=trial)
  X = simulate_neurons(A, timesteps)
  print(f"simulating trial {trial + 1} of {n_trials}")


  for j, ratio in enumerate(ratio_observed):
    result,_ = get_regression_corr_full_connectivity(n_neurons, A, X,
                                                     ratio, reg_args)
    corr_data[trial, j] = result

corr_mean = np.nanmean(corr_data, axis=0)
corr_std = np.nanstd(corr_data, axis=0)

plt.plot(np.asarray(ratio_observed) * 100, corr_mean)
plt.fill_between(np.asarray(ratio_observed) * 100,
                 corr_mean - corr_std, corr_mean + corr_std,
                 alpha=.2)
plt.xlim([100, 10])
plt.xlabel("Percent of neurons observed")
plt.ylabel("connectivity matrices correlation")
plt.title("Performance of regression as a function of the number of neurons observed")
plt.show()

simulating trial 1 of 3

/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 7.035e-01, tolerance: 6.170e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.194e+00, tolerance: 5.991e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 7.444e+01, tolerance: 1.037e+00
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.527e+00, tolerance: 7.297e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 5.612e+01, tolerance: 8.502e-01
  model = cd_fast.enet_coordinate_descent(

/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 9.205e+00, tolerance: 9.716e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.571e+00, tolerance: 7.069e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 9.267e-01, tolerance: 6.659e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.111e+00, tolerance: 5.913e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.784e+00, tolerance: 7.325e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 9.089e-01, tolerance: 6.599e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.105e+00, tolerance: 6.815e-01
  model = cd_fast.enet_coordinate_descent(

/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 2.171e+00, tolerance: 7.299e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.078e+00, tolerance: 6.658e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 4.060e+01, tolerance: 7.243e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.791e+00, tolerance: 7.903e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.440e+00, tolerance: 7.260e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 2.302e+00, tolerance: 7.114e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.018e+00, tolerance: 6.731e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.636e+00, tolerance: 7.693e-01
  model = cd_fast.enet_coordinate_descent(

/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.467e+00, tolerance: 7.440e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.721e+00, tolerance: 6.617e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 2.472e+00, tolerance: 7.109e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.154e+00, tolerance: 6.479e-01
  model = cd_fast.enet_coordinate_descent(

/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.633e+00, tolerance: 7.605e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 8.637e-01, tolerance: 6.640e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 2.259e+01, tolerance: 8.183e-01
  model = cd_fast.enet_coordinate_descent(

/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 7.204e-01, tolerance: 6.170e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 8.021e-01, tolerance: 5.991e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.172e+00, tolerance: 7.297e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 4.223e+00, tolerance: 1.037e+00
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 6.555e+01, tolerance: 8.502e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.365e+00, tolerance: 7.069e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 4.059e+00, tolerance: 9.716e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.058e+00, tolerance: 6.659e-01
  model = cd_fast.enet_coordinate_descent(

/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 2.163e+00, tolerance: 7.325e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.122e+00, tolerance: 6.599e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.291e+00, tolerance: 6.815e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 2.268e+00, tolerance: 7.299e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.162e+00, tolerance: 6.658e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 2.813e+01, tolerance: 7.243e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.818e+00, tolerance: 7.903e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 8.262e-01, tolerance: 7.260e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.352e+00, tolerance: 7.114e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.152e+00, tolerance: 6.731e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 2.719e+00, tolerance: 7.693e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 9.390e-01, tolerance: 7.440e-01
  model = cd_fast.enet_coordinate_descent(

/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 9.340e-01, tolerance: 6.522e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.202e+00, tolerance: 6.617e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 3.547e+00, tolerance: 7.109e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 8.700e-01, tolerance: 6.170e-01
  model = cd_fast.enet_coordinate_descent(

/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 2.419e+00, tolerance: 7.297e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 4.904e+00, tolerance: 1.037e+00
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 3.428e+00, tolerance: 8.502e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.687e+00, tolerance: 7.069e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 3.013e+00, tolerance: 9.716e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.223e+00, tolerance: 6.659e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.523e+00, tolerance: 6.815e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.665e+00, tolerance: 6.658e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.683e+00, tolerance: 7.903e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.236e+00, tolerance: 7.114e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 2.632e+00, tolerance: 7.299e-01
  model = cd_fast.enet_coordinate_descent(

/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.873e+00, tolerance: 1.037e+00
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 2.514e+00, tolerance: 8.502e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.991e+00, tolerance: 9.716e-01
  model = cd_fast.enet_coordinate_descent(

simulating trial 2 of 3

/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 8.023e-01, tolerance: 6.741e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.731e+00, tolerance: 7.361e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 2.002e+01, tolerance: 8.649e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.238e+00, tolerance: 6.406e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 8.129e-01, tolerance: 6.262e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 9.260e-01, tolerance: 6.778e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.617e+00, tolerance: 6.656e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 2.394e+00, tolerance: 9.632e-01
  model = cd_fast.enet_coordinate_descent(

/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 9.720e-01, tolerance: 5.902e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.908e+00, tolerance: 7.739e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.209e+00, tolerance: 6.617e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.312e+01, tolerance: 8.605e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 2.495e+00, tolerance: 7.708e-01
  model = cd_fast.enet_coordinate_descent(

/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.896e+00, tolerance: 7.932e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 3.575e+01, tolerance: 8.446e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 8.333e+01, tolerance: 1.165e+00
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 2.991e+00, tolerance: 7.790e-01
  model = cd_fast.enet_coordinate_descent(

/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 8.719e-01, tolerance: 5.862e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 6.998e-01, tolerance: 6.702e-01
  model = cd_fast.enet_coordinate_descent(

/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 7.094e-01, tolerance: 6.741e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.758e+00, tolerance: 7.361e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 5.846e+00, tolerance: 8.649e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 7.478e-01, tolerance: 6.406e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.533e+00, tolerance: 6.778e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.073e+00, tolerance: 6.656e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 3.776e+00, tolerance: 9.632e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 9.124e-01, tolerance: 6.149e-01
  model = cd_fast.enet_coordinate_descent(

/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 7.977e-01, tolerance: 6.159e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.171e+00, tolerance: 7.739e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 6.217e-01, tolerance: 5.514e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.102e+00, tolerance: 6.617e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.723e+00, tolerance: 7.708e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 4.001e+00, tolerance: 8.605e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 2.556e+00, tolerance: 7.932e-01
  model = cd_fast.enet_coordinate_descent(

/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.035e+00, tolerance: 6.173e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 2.325e+01, tolerance: 8.446e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 7.232e+01, tolerance: 1.165e+00
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 6.535e-01, tolerance: 6.413e-01
  model = cd_fast.enet_coordinate_descent(

/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.339e+00, tolerance: 7.361e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 4.226e+00, tolerance: 8.649e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.151e+00, tolerance: 6.262e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.598e+00, tolerance: 6.778e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 6.910e-01, tolerance: 6.656e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 3.940e+00, tolerance: 9.632e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 6.720e-01, tolerance: 6.253e-01
  model = cd_fast.enet_coordinate_descent(

/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 8.304e-01, tolerance: 7.739e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.768e+00, tolerance: 8.649e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 2.679e+00, tolerance: 9.632e-01
  model = cd_fast.enet_coordinate_descent(

simulating trial 3 of 3

/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 7.731e-01, tolerance: 6.676e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.265e+00, tolerance: 6.124e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.043e+00, tolerance: 6.539e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 6.497e+00, tolerance: 8.507e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 2.578e+01, tolerance: 7.448e-01
  model = cd_fast.enet_coordinate_descent(

/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 4.344e+00, tolerance: 8.328e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 2.512e+00, tolerance: 9.272e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.905e+01, tolerance: 7.096e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.355e+01, tolerance: 1.073e+00
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 8.221e-01, tolerance: 6.339e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.189e+01, tolerance: 8.865e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 8.500e-01, tolerance: 6.317e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.015e+00, tolerance: 7.284e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 6.883e+01, tolerance: 1.060e+00
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.905e+00, tolerance: 7.797e-01
  model = cd_fast.enet_coordinate_descent(

/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 6.462e+00, tolerance: 7.751e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 7.263e-01, tolerance: 6.171e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 2.120e+00, tolerance: 7.435e-01
  model = cd_fast.enet_coordinate_descent(

/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 7.577e-01, tolerance: 5.916e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 7.973e+00, tolerance: 7.764e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 2.432e+00, tolerance: 7.365e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 2.278e+00, tolerance: 6.878e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.042e+00, tolerance: 7.395e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 7.468e-01, tolerance: 6.390e-01
  model = cd_fast.enet_coordinate_descent(

/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 2.049e+00, tolerance: 6.547e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 8.916e-01, tolerance: 6.178e-01
  model = cd_fast.enet_coordinate_descent(

/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 8.255e-01, tolerance: 6.124e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 7.204e-01, tolerance: 6.539e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 2.674e+00, tolerance: 8.507e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 7.252e+00, tolerance: 7.448e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 3.965e+00, tolerance: 8.328e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 2.364e+00, tolerance: 9.272e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 2.032e+00, tolerance: 7.096e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 6.873e+00, tolerance: 1.073e+00
  model = cd_fast.enet_coordinate_descent(

/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.029e+00, tolerance: 6.339e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.714e+00, tolerance: 8.865e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.713e+00, tolerance: 7.284e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 5.834e+01, tolerance: 1.060e+00
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.367e+00, tolerance: 7.797e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.837e+00, tolerance: 7.751e-01
  model = cd_fast.enet_coordinate_descent(

/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 2.221e+00, tolerance: 7.435e-01
  model = cd_fast.enet_coordinate_descent(

/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 9.202e-01, tolerance: 6.539e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 3.476e+00, tolerance: 8.507e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.863e+00, tolerance: 7.448e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 2.303e+00, tolerance: 9.272e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 3.737e+00, tolerance: 8.328e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 4.466e+00, tolerance: 1.073e+00
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.130e+00, tolerance: 7.096e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 5.684e+00, tolerance: 1.060e+00
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 1.415e+00, tolerance: 7.751e-01
  model = cd_fast.enet_coordinate_descent(
/opt/hostedtoolcache/Python/3.9.17/x64/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 3.224e+00, tolerance: 8.865e-01
  model = cd_fast.enet_coordinate_descent(

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# @title Submit your feedback
content_review(f"{feedback_prefix}_Regression_performance_as_a_function_of_the_number_of_observed_neurons_Interactive_Demo")

---

Summary

Estimated timing of tutorial: 20 min

Video 4: Summary

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# @title Submit your feedback
content_review(f"{feedback_prefix}_Summary")

In this tutorial, we learned:

1. To use regression for estimating causality

2. The problem of omitted variable bias, and how it arises in practice