maths: add autocorrelation function

linear_algebra/src/polynom_for_points.py

@@ -12,27 +12,31 @@ def points_to_polynomial(coordinates: list[list[int]]) -> str:
         ...
     ValueError: The program cannot work out a fitting polynomial.
     >>> points_to_polynomial([[1, 0], [2, 0], [3, 0]])
-    'f(x)=x^2*0.0+x^1*-0.0+x^0*0.0'
+    'f(x)=x^2*0.0+x^1*0.0+x^0*0.0'
     >>> points_to_polynomial([[1, 1], [2, 1], [3, 1]])
-    'f(x)=x^2*0.0+x^1*-0.0+x^0*1.0'
+    'f(x)=x^2*0.0+x^1*0.0+x^0*1.0'
     >>> points_to_polynomial([[1, 3], [2, 3], [3, 3]])
-    'f(x)=x^2*0.0+x^1*-0.0+x^0*3.0'
+    'f(x)=x^2*0.0+x^1*0.0+x^0*3.0'
     >>> points_to_polynomial([[1, 1], [2, 2], [3, 3]])
-    'f(x)=x^2*0.0+x^1*1.0+x^0*0.0'
+    'f(x)=x^2*4.9343245538895844e-17+x^1*1.0+x^0*0.0'
     >>> points_to_polynomial([[1, 1], [2, 4], [3, 9]])
-    'f(x)=x^2*1.0+x^1*-0.0+x^0*0.0'
+    'f(x)=x^2*1.0+x^1*0.0+x^0*0.0'
     >>> points_to_polynomial([[1, 3], [2, 6], [3, 11]])
-    'f(x)=x^2*1.0+x^1*-0.0+x^0*2.0'
+    'f(x)=x^2*0.9999999999999996+x^1*9.992007221626407e-16+x^0*1.9999999999999993'
     >>> points_to_polynomial([[1, -3], [2, -6], [3, -11]])
-    'f(x)=x^2*-1.0+x^1*-0.0+x^0*-2.0'
+    'f(x)=x^2*-0.9999999999999996+x^1*-9.992007221626407e-16+x^0*-1.9999999999999993'
     >>> points_to_polynomial([[1, 5], [2, 2], [3, 9]])
-    'f(x)=x^2*5.0+x^1*-18.0+x^0*18.0'
+    'f(x)=x^2*5.0+x^1*-18.000000000000004+x^0*18.000000000000004'
     >>> points_to_polynomial([[1, 1], [1, 2], [1, 3]])
     'x=1'
     >>> points_to_polynomial([[1, 1], [2, 2], [2, 2]])
     Traceback (most recent call last):
         ...
     ValueError: The program cannot work out a fitting polynomial.
+    >>> points_to_polynomial([[0, 1], [1, 2], [2, 5]])
+    'f(x)=x^2*1.0+x^1*0.0+x^0*1.0'
+    >>> points_to_polynomial([[0, 0], [1, 1], [2, 4]])
+    'f(x)=x^2*1.0+x^1*0.0+x^0*0.0'
     """
     if len(coordinates) == 0 or not all(len(pair) == 2 for pair in coordinates):
         raise ValueError("The program cannot work out a fitting polynomial.")
@@ -62,6 +66,11 @@ def points_to_polynomial(coordinates: list[list[int]]) -> str:
     vector: list[float] = [coordinates[count_of_line][1] for count_of_line in range(x)]
 
     for count in range(x):
+        # Partial pivoting: swap in the row with the largest absolute pivot value
+        max_row = max(range(count, x), key=lambda r: abs(matrix[r][count]))
+        matrix[count], matrix[max_row] = matrix[max_row], matrix[count]
+        vector[count], vector[max_row] = vector[max_row], vector[count]
+
         for number in range(x):
             if count == number:
                 continue

maths/autocorrelation.py

@@ -0,0 +1,50 @@
+"""
+Autocorrelation measures the correlation of a signal with a delayed
+copy of itself. It is widely used in time series analysis, signal
+processing, and statistics.
+
+Reference: https://en.wikipedia.org/wiki/Autocorrelation
+"""
+
+
+def autocorrelation(data: list[float], lag: int) -> float:
+    """
+    Calculate the autocorrelation of a time series at a given lag.
+
+    :param data: A list of numerical values representing the time series.
+    :param lag: The number of time steps to shift the series.
+    :return: The autocorrelation coefficient at the given lag.
+
+    >>> round(autocorrelation([1, 2, 3, 4, 5], 1), 4)
+    0.4
+    >>> round(autocorrelation([1, 2, 3, 4, 5], 0), 4)
+    1.0
+    >>> autocorrelation([1, 2, 3], 5)
+    Traceback (most recent call last):
+        ...
+    ValueError: Lag must be less than the length of the data.
+    """
+    if lag >= len(data):
+        raise ValueError("Lag must be less than the length of the data.")
+
+    n = len(data)
+    mean = sum(data) / n
+    variance = sum((x - mean) ** 2 for x in data) / n
+
+    if variance == 0:
+        raise ValueError("Variance of data is zero, autocorrelation undefined.")
+
+    covariance = (
+        sum((data[i] - mean) * (data[i - lag] - mean) for i in range(lag, n)) / n
+    )
+
+    return covariance / variance
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()
+    data = [1, 2, 3, 4, 5, 4, 3, 2, 1]
+    for lag in range(5):
+        print(f"Lag {lag}: {autocorrelation(data, lag):.4f}")