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The ability to return an instance of a strict subclass of int is deprecated, and may be removed in a future version of Python.__int__ returned non-int (type %.200s)%.200s does not export expected C %.8s %.200sC %.8s %.200s.%.200s has wrong signature (expected %.500s, got %.500s)Interpreter change detected - this module can only be loaded into one interpreter per process.unbound method %.200S() needs an argument__annotations__ must be set to a dict object__kwdefaults__ must be set to a dict objectchanges to cyfunction.__kwdefaults__ will not currently affect the values used in function calls__defaults__ must be set to a tuple objectchanges to cyfunction.__defaults__ will not currently affect the values used in function calls__qualname__ must be set to a string object__name__ must be set to a string objectFailed to import '%.20s.decompress' - cannot initialise module strings. String compression was configured with the C macro 'CYTHON_COMPRESS_STRINGS=%d'.%.200s.%.200s is not a type object%.200s.%.200s size changed, may indicate binary incompatibility. Expected %zd from C header, got %zd from PyObjectAcquisition count is %d (line %d)invalid vtable found for imported typemultiple bases have vtable conflict: '%.200s' and '%.200s'Unexpected format string character: '%c' while calling a Python objectNULL result without error in PyObject_CallBuffer dtype mismatch, expected %s%s%s but got %sBuffer dtype mismatch, expected '%s' but got %s in '%s.%s'Expected a dimension of size %zu, got %zuExpected %d dimensions, got %dPython does not define a standard format string size for long double ('g')..Buffer dtype mismatch; next field is at offset %zd but %zd expectedBig-endian buffer not supported on little-endian compilerBuffer acquisition: Expected '{' after 'T'Cannot handle repeated arrays in format stringDoes not understand character buffer dtype format string ('%c')Expected a dimension of size %zu, got %dExpected a comma in format string, got '%c'Expected %d dimension(s), got %dUnexpected end of format string, expected ')'%.200s() takes %.8s %zd positional argument%.1s (%zd given)sklearn.tree._criterion.RegressionCriterion.__reduce__sklearn.tree._criterion.MAE.__reduce__sklearn.tree._criterion.ClassificationCriterion.updatesklearn.tree._criterion.RegressionCriterion.updatesklearn.tree._criterion.Criterion.__setstate__sklearn.tree._criterion.Criterion.__getstate__sklearn.tree._criterion.MAE.initsklearn.tree._criterion.RegressionCriterion.initsklearn.tree._criterion.ClassificationCriterion.initsklearn.tree._criterion.MAE.reverse_resetsklearn.tree._criterion.__pyx_unpickle_Criterion__set_statecannot pass None into a C function argument that is declared 'not None'sklearn.tree._criterion.Criterion.__setstate_cython__sklearn.tree._criterion.Criterion.__reduce_cython__Buffer has wrong number of dimensions (expected %d, got %d)Item size of buffer (%zd byte%s) does not match size of '%s' (%zd byte%s)sklearn.tree._criterion.ClassificationCriterion.__cinit__sklearn.tree._criterion.ClassificationCriterion.__reduce__sklearn.tree._criterion.__pyx_unpickle_CriterionIncompatible checksums (0x%x vs (0x%x, 0x%x, 0x%x) = (%s))sklearn.tree._criterion._py_precompute_absolute_errors'%.200s' object is unsliceablesklearn.tree._criterion.RegressionCriterion.__cinit__sklearn.tree._criterion.MAE.__cinit__Module '_criterion' has already been imported. Re-initialisation is not supported.compile time Python version %d.%d of module '%.100s' %s runtime version %d.%dUnable to initialize pickling for %.200s_ARRAY_API is not PyCapsule objectmodule compiled against ABI version 0x%x but this version of numpy is 0x%xmodule was compiled against NumPy C-API version 0x%x (NumPy 1.23) but the running NumPy has C-API version 0x%x. Check the section C-API incompatibility at the Troubleshooting ImportError section at https://numpy.org/devdocs/user/troubleshooting-importerror.html#c-api-incompatibility for indications on how to solve this problem.FATAL: module compiled as unknown endianFATAL: module compiled as little endian, but detected different endianness at runtime../../../tmp/build-env-m273bhe1/lib/python3.11/site-packages/numpy/__init__.cython-30.pxd_cython_3_2_5.cython_function_or_method_cython_3_2_5._common_types_metatypesklearn.tree._criterion.PoissonHalf Poisson deviance as impurity criterion. Poisson deviance = 2/n * sum(y_true * log(y_true/y_pred) + y_pred - y_true) Note that the deviance is >= 0, and since we have `y_pred = mean(y_true)` at the leaves, one always has `sum(y_pred - y_true) = 0`. It remains the implemented impurity (factor 2 is skipped): 1/n * sum(y_true * log(y_true/y_pred) Mean absolute error impurity criterion. It has almost nothing in common with other regression criterions so it doesn't inherit from RegressionCriterion. MAE = (1 / n)*(\sum_i |y_i - p_i|), where y_i is the true value and p_i is the predicted value. In a decision tree, that prediction is the (weighted) median of the targets in the node. How this implementation works ----------------------------- This class precomputes in `reset`, for the current node, the absolute-error values and corresponding medians for all potential split positions: every p in [start, end). For that: - We first compute the rank of each samples node-local sorted order of target values. `self.ranks[p]` gives the rank of sample p. - While iterating the segment of indices (p in [start, end)), we * "activate" one sample at a time at its rank within a prefix sum tree, the `WeightedFenwickTree`: `tree.add(rank, y, weight)` The tree maintains cumulative sums of weights and of `weight * y` * search for the half total weight in the tree: `tree.search(current_total_weight / 2)`. This allows us to retrieve/compute: * the current weighted median value * the absolute-error contribution via the standard pinball-loss identity: AE = (wy_right - median * w_right) + (median * w_left - wy_left) - We perform two such passes: * one forward from `start` to `end - 1` to fill `left_abs_errors[p]` and `left_medians[p]` for left children. * one backward from `end - 1` down to `start` to fill `right_abs_errors[p]` and `right_medians[p]` for right children. Complexity: time complexity is O(n log n), indeed: - computing ranks is based on sorting: O(n log n) - add and search operations in the Fenwick tree are O(log n). => the forward and backward passes are O(n log n). How the other methods use the precomputations -------------------------------------------- - `reset` performs the precomputation described above. It also stores the node weighted median per output in `node_medians` (prediction value of the node). - `update(new_pos)` only updates `weighted_n_left` and `weighted_n_right`; no recomputation of errors is needed. - `children_impurity` reads the precomputed absolute errors at `left_abs_errors[pos - 1]` and `right_abs_errors[pos]` and scales them by the corresponding child weights and `n_outputs` to report the impurity of each child. - `middle_value` and `check_monotonicity` use the precomputed `left_medians[pos - 1]` and `right_medians[pos]` to derive the mid-point value and to validate monotonic constraints when enabled. - Missing values are not supported for MAE: `init_missing` raises. For a complementary, in-depth discussion of the mathematics and design choices, see the external report: https://github.com/cakedev0/fast-mae-split/blob/main/report.ipynb Mean squared error impurity criterion. MSE = var_left + var_right Gini Index impurity criterion. This handles cases where the target is a classification taking values 0, 1, ... K-2, K-1. If node m represents a region Rm with Nm observations, then let count_k = 1/ Nm \sum_{x_i in Rm} I(yi = k) be the proportion of class k observations in node m. The Gini Index is then defined as: index = \sum_{k=0}^{K-1} count_k (1 - count_k) = 1 - \sum_{k=0}^{K-1} count_k ** 2 sklearn.tree._criterion.EntropyCross Entropy impurity criterion. This handles cases where the target is a classification taking values 0, 1, ... K-2, K-1. If node m represents a region Rm with Nm observations, then let count_k = 1 / Nm \sum_{x_i in Rm} I(yi = k) be the proportion of class k observations in node m. The cross-entropy is then defined as cross-entropy = -\sum_{k=0}^{K-1} count_k log(count_k) sklearn.tree._criterion.RegressionCriterionAbstract regression criterion. This handles cases where the target is a continuous value, and is evaluated by computing the variance of the target values left and right of the split point. The computation takes linear time with `n_samples` by using :: var = \sum_i^n (y_i - y_bar) ** 2 = (\sum_i^n y_i ** 2) - n_samples * y_bar ** 2 sklearn.tree._criterion.ClassificationCriterionAbstract criterion for classification.sklearn.tree._criterion.CriterionInterface for impurity criteria. 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