doc: add doc strings

Author: pipipyau

StatTools/filters/symbolic_kalman.py

@@ -1,3 +1,5 @@
+"""Symbolic Kalman filter."""
+
 from sympy import (
     Function,
     Matrix,
@@ -6,14 +8,16 @@
 )
 
 
-def nth_order_derivative(n, k):
+def nth_order_derivative(n: int, k):
+    """Get the nth order derivative of x(k)."""
     x = Function("x")
     if n <= 1:
         return x(k) - x(k - 1)
     return nth_order_derivative(n - 1, k) - nth_order_derivative(n - 1, k - 1)
 
 
-def get_all_coeffs(expr, n):
+def get_all_coeffs(expr, n: int) -> dict:
+    """Get all coefficients of the expression."""
     x = Function("x")
     k = symbols("k")
     coeffs = {}
@@ -24,16 +28,17 @@ def get_all_coeffs(expr, n):
     return coeffs
 
 
-def Fmij_formula(n, i, j):
+def Fmij_formula(n: int, i: int, j: int):
+    """Get the (i,j) element of the matrix."""
     k = symbols("k")
     m = n - 1
 
-    # denominator знаменатель
+    # denominator
     expr = nth_order_derivative(j, k)
     coeffs = get_all_coeffs(expr, j)
     den = coeffs[j]
 
-    # numerator числитель
+    # numerator
     a = symbols(f"a_{j + 1}")
     num = -a
     for l in range(0, m - j):
@@ -49,7 +54,8 @@ def Fmij_formula(n, i, j):
     return simplify(num / den)
 
 
-def get_sympy_filter_matrix(n):
+def get_sympy_filter_matrix(n: int) -> Matrix:
+    """Get filter matrix."""
     data = []
     for i in range(n):
         row = []
@@ -60,7 +66,8 @@ def get_sympy_filter_matrix(n):
     return Matrix(data)
 
 
-def refine_filter_matrix(filter_matrix: Matrix, n: int, ar_filter: list):
+def refine_filter_matrix(filter_matrix: Matrix, n: int, ar_filter: list) -> Matrix:
+    """Refine filter matrix using AR coefficients."""
     filter_matrix_refined = filter_matrix.copy()
     for i in range(1, n + 1):
         filter_matrix_refined = filter_matrix_refined.subs({f"a_{i}": ar_filter[i]})