[BUG] Fix incorrect undifferencing in ARIMA

aeon/forecasting/_arima.py

@@ -12,6 +12,7 @@
 from aeon.forecasting._extract_paras import _extract_arma_params
 from aeon.forecasting._nelder_mead import nelder_mead
 from aeon.forecasting.base import BaseForecaster
+from aeon.transformations.series._diff import _undifference
 
 
 class ARIMA(BaseForecaster):
@@ -123,19 +124,9 @@ def _fit(self, y, exog=None):
         # parameters
         differenced_forecast = self.fitted_values_[-1]
 
-        if self.d == 0:
-            forecast_value = differenced_forecast
-        elif self.d == 1:
-            forecast_value = differenced_forecast + self._series[-1]
-        else:  # for d > 1, iteratively undifference
-            forecast_value = differenced_forecast
-            last_vals = self._series[-self.d :]
-            for _ in range(self.d):
-                forecast_value += last_vals[-1] - last_vals[-2]
-                # Shift values to avoid appending to list (efficient)
-                last_vals = np.roll(last_vals, -1)
-                last_vals[-1] = forecast_value  # Extract the parameter values
-        self.forecast_ = forecast_value
+        self.forecast_ = _undifference(
+            np.array([differenced_forecast]), self._series[-self.d :]
+        )[0]
         if self.use_constant:
             self.c_ = formatted_params[0][0]
         self.phi_ = formatted_params[1][: self.p]
@@ -196,12 +187,7 @@ def _predict(self, y, exog=None):
         forecast_diff = c + ar_forecast + ma_forecast
 
         # Undifference the forecast
-        if d == 0:
-            return forecast_diff
-        elif d == 1:
-            return forecast_diff + y[-1]
-        else:
-            return forecast_diff + np.sum(y[-d:])
+        return _undifference(np.array([forecast_diff]), self._series[-self.d :])[0]
 
     def _forecast(self, y, exog=None):
         """Forecast one ahead for time series y."""
@@ -240,17 +226,7 @@ def iterative_forecast(self, y, prediction_horizon):
 
         # Correct differencing using forecast values
         y_forecast_diff = forecast_series[n : n + h]
-        d = self.d
-        if d == 0:
-            return y_forecast_diff
-        else:  # Correct undifferencing
-            # Start with last d values from original y
-            undiff = list(self._series[-d:])
-            for i in range(h):
-                # Take the last d values and sum them
-                reconstructed = y_forecast_diff[i] + sum(undiff[-d:])
-                undiff.append(reconstructed)
-            return np.array(undiff[d:])
+        return _undifference(np.array([y_forecast_diff]), self._series[-self.d :])
 
 
 @njit(cache=True, fastmath=True)

aeon/testing/testing_config.py

@@ -62,6 +62,9 @@
     # Unknown issue not producing the same results
     "RDSTRegressor": ["check_regressor_against_expected_results"],
     "RISTRegressor": ["check_regressor_against_expected_results"],
+    # Requires y to be passed in invers_transform,
+    # but this is not currently enabled/supported
+    "DifferenceTransformer": ["check_transform_inverse_transform_equivalent"],
 }
 
 # Exclude specific tests for estimators here only when numba is disabled

aeon/transformations/series/_diff.py

@@ -1,8 +1,11 @@
+"""Difference Transformer."""
+
 import numpy as np
+from numba import njit
 
 from aeon.transformations.series.base import BaseSeriesTransformer
 
-__maintainer__ = ["TinaJin0228"]
+__maintainer__ = ["TinaJin0228", "alexbanwell1"]
 __all__ = ["DifferenceTransformer"]
 
 
@@ -59,6 +62,7 @@ class DifferenceTransformer(BaseSeriesTransformer):
 
     _tags = {
         "capability:multivariate": True,
+        "capability:inverse_transform": True,
         "X_inner_type": "np.ndarray",
         "fit_is_empty": True,
     }
@@ -90,3 +94,96 @@ def _transform(self, X, y=None):
         Xt = diff_X
 
         return Xt
+
+    def _inverse_transform(self, X, y=None):
+        """
+        Inverse transform to reconstruct the original time series.
+
+        Parameters
+        ----------
+        X : Time series to inverse transform. With shape (n_channels, n_timepoints).
+        y : ignored argument for interface compatibility
+
+        Returns
+        -------
+        Xt : np.ndarray
+            Reconstructed original time series.
+        """
+        if y is None or y.shape[1] < self.order:
+            raise ValueError(
+                f"Inverse transformm requires first {self.order} original \
+                  data values supplied as y, but inverse_transform called with y=None"
+            )
+        if y.shape[0] != X.shape[0]:
+            raise ValueError(
+                f"y must have the same number of channels as X. "
+                f"Got X.shape[0]={X.shape[0]}, y.shape[0]={y.shape[0]}"
+            )
+        X = np.array(X, dtype=np.float64)
+        y = np.array(y, dtype=np.float64)
+        initial_values = y[:, : self.order]
+
+        return np.array(
+            [
+                _undifference(diff_X, initial_value)
+                for diff_X, initial_value in zip(X, initial_values)
+            ]
+        )
+
+
+@njit(cache=True, fastmath=True)
+def _comb(n, k):
+    """
+    Calculate the binomial coefficient C(n, k) = n! / (k! * (n - k)!).
+
+    Parameters
+    ----------
+    n : int
+        The total number of items.
+    k : int
+        The number of items to choose.
+
+    Returns
+    -------
+    int
+        The binomial coefficient C(n, k).
+    """
+    if k < 0 or k > n:
+        return 0
+    if k > n - k:
+        k = n - k  # Take advantage of symmetry
+    c = 1
+    for i in range(k):
+        c = c * (n - i) // (i + 1)
+    return c
+
+
+@njit(cache=True, fastmath=True)
+def _undifference(diff, initial_values):
+    """
+    Reconstruct original time series from an n-th order differenced series.
+
+    Parameters
+    ----------
+    diff : array-like
+        n-th order differenced series of length N - n
+    initial_values : array-like
+        The first n values of the original series before differencing (length n)
+
+    Returns
+    -------
+    original : np.ndarray
+        Reconstructed original series of length N
+    """
+    n = len(initial_values)
+    kernel = np.array(
+        [(-1) ** (k + 1) * _comb(n, k) for k in range(1, n + 1)],
+        dtype=initial_values.dtype,
+    )
+    original = np.empty((n + len(diff)), dtype=initial_values.dtype)
+    original[:n] = initial_values
+
+    for i, d in enumerate(diff):
+        original[n + i] = np.dot(kernel, original[i : n + i][::-1]) + d
+
+    return original

aeon/transformations/series/tests/test_diff.py

@@ -12,26 +12,41 @@ def test_diff():
     dt1 = DifferenceTransformer(order=1)
     Xt1 = dt1.fit_transform(X)
     expected1 = np.array([[3.0, 5.0, 7.0, 9.0, 11.0]])
+    X_hat1 = dt1.inverse_transform(Xt1, X)
 
     np.testing.assert_allclose(
         Xt1, expected1, equal_nan=True, err_msg="Value mismatch for order 1"
     )
+    np.testing.assert_allclose(
+        X_hat1, X, equal_nan=True, err_msg="Inverse transform failed for order 1"
+    )
 
     dt2 = DifferenceTransformer(order=2)
     Xt2 = dt2.fit_transform(X)
     expected2 = np.array([[2.0, 2.0, 2.0, 2.0]])
+    X_hat2 = dt2.inverse_transform(Xt2, X)
 
     np.testing.assert_allclose(
         Xt2, expected2, equal_nan=True, err_msg="Value mismatch for order 2"
     )
+    np.testing.assert_allclose(
+        X_hat2, X, equal_nan=True, err_msg="Inverse transform failed for order 2"
+    )
 
     Y = np.array([[1, 2, 3, 4], [5, 3, 1, 8]])
 
     Yt1 = dt1.fit_transform(Y)
     expected3 = np.array([[1, 1, 1], [-2, -2, 7]])
+    Y_hat1 = dt1.inverse_transform(Yt1, Y)
     np.testing.assert_allclose(
         Yt1,
         expected3,
         equal_nan=True,
         err_msg="Value mismatch for order 1,multivariate",
     )
+    np.testing.assert_allclose(
+        Y_hat1,
+        Y,
+        equal_nan=True,
+        err_msg="Inverse transform failed for order 1,multivariate",
+    )