makes some changes to the rank-deficient part

docs/tutorials/plot_06_calcium_imaging.py

@@ -28,24 +28,6 @@
 nap.nap_config.suppress_conversion_warnings = True
 warnings.filterwarnings("ignore", category=UserWarning, message="The feature matrix is not of dtype")
 
-# %%
-# Later in the tutorial, we will compute the rank of the GLM feature matrix.
-# To ensure the computation accounts for the model's intercept term,
-# we can add a constant column of ones before calculating the rank.
-# Below is a utility function for adding the intercept column.
-
-def add_intercept(X):
-    """Add an intercept term to design matrix.
-
-    Convert matrix to float64, drops nans and add intercept term.
-    """
-    # convert to float64 for rank computation precision
-    X = np.asarray(X, dtype=np.float64)
-    # drop nans
-    X = X[nmo.tree_utils.get_valid_multitree(X)]
-    return np.hstack([np.ones((X.shape[0], 1)), X])
-
-
 
 
 # %%
@@ -201,48 +183,41 @@ def add_intercept(X):
 X = basis.compute_features(head_direction, Y[:, selected_neurons])
 
 # %%
-# A design matrix $X$ including a CyclicBSpline and an intercept term is rank deficient, i.e. one can find non-zero
-# coefficients $\mathbf{w} \neq \mathbf{0}$ such that $X \cdot \mathbf{w} = \mathbf{0}$.
-
-print(f"Number of features: {X.shape[1] + 1}")  # num coefficients + intercept
-print(f"Matrix rank: {np.linalg.matrix_rank(add_intercept(X))}")
-
-# %%
-# By setting,
-
-w = np.ones((X.shape[1] + 1))
-w[0] = -1
-w[1 + heading_basis.n_basis_funcs:] = 0
-
-
-# %%
-# We have that,
-
-np.max(np.abs(np.dot(add_intercept(X), w)))
-
-# %%
-# This implies that there will be infinite different parameters that results in the same firing rate,
-# or equivalently there will be infinite many equivalent solutions to an un-regularized GLM.
-
-# define some random coefficients
-coef = np.random.randn(X.shape[1] + 1)
-
-# the firing rate is softplus([1, X] * coef)
-# adding w to the coefficients does not change the output rate.
-firing_rate = jax.nn.softplus(np.dot(add_intercept(X), coef))
-firing_rate_2 = jax.nn.softplus(np.dot(add_intercept(X), coef + w))
-
-# check that the rate match
-np.allclose(firing_rate, firing_rate_2)
-
-# %%
-# We can avoid this issue by dropping linearly dependent columns in the design matrix.
-
-X, idx = apply_identifiability_constraints_by_basis_component(basis, X)
-
-print(f"Number of features: {X.shape[1] + 1}")  # drops one column
-print(f"Matrix rank: {np.linalg.matrix_rank(add_intercept(X))}")
-
+#
+# Before we use this design matrix to fit the population, we need to take a brief detour
+# into linear algebra. Depending on your design matrix is constructed, it is likely to
+# be rank-deficient, in which case it has a null space. Practically, that means that
+# there are infinitely many different sets of parameters that predict the same firing
+# rate. If you want to interpret your parameters, this is bad!
+#
+# While this multiplicity of solutions is always a potential issue when fitting models,
+# it is particularly relevant when using basis objects in nemos, as many of our basis
+# sets completely tile the input space (i.e., summing across all $n$ basis functions
+# returns 1 everywhere), which, when combined with the intercept term always present in
+# the GLM (i.e., the base firing rate), will give you a rank-deficient matrix.
+#
+# We thus recommend that you always check the rank of your design matrix and provide
+# some tools to drop the linearly-dependent columns, if necessary, which will guarantee
+# that your design matrix is full rank and thus that there is one unique solution.
+#
+# !!! tip "Linear Algebra"
+#
+#     To read more about matrix rank, see
+#     [Wikipedia](https://en.wikipedia.org/wiki/Rank_(linear_algebra)#Main_definitions).
+#     Gil Strang's Linear Algebra course, [available for free
+#     online](https://web.mit.edu/18.06/www/), and [NeuroMatch
+#     Academy](https://compneuro.neuromatch.io/tutorials/W0D3_LinearAlgebra/student/W0D3_Tutorial2.html)
+#     are also great resources.
+#
+# In this case, we are using the CyclicBSpline basis functions, which uniformly tile and
+# thus will result in a rank-deficient matrix. Therefore, we will use a utility function
+# to drop a column from the matrix and make it full-rank:
+
+# The number of features is the number of columns plus one (for the intercept)
+print(f"Number of features in the rank-deficient design matrix: {X.shape[1] + 1}")
+X, _ = apply_identifiability_constraints_by_basis_component(basis, X)
+# We have dropped one column
+print(f"Number of features in the full-rank design matrix: {X.shape[1] + 1}")
 
 # %%
 # ## Train & test set