Quite a few months ago, I implemented decision tree regression, using recursion, from scratch, with Python. I based my implementation on some anonymous AI-generated code from a Google search. Bottom line: I refactored my old code but I still don’t like the use of recursion (when building the tree) so this blog post is really an exploration, not code that I’d use elsewhere. I will re-refactor to eliminate recursion when I get some free time. Put another way, the code in this post is purely an investigation — I have very little confidence in its correctness.
One recent morning before work, I reviewed that old decision tree regression code carefully, and noticed that even though the code is correct in the sense that it works properly, the code is ugly in the sense of readability (using arrays of Booleans), and good practice (using a value of None for an integer value to indicate a leaf node), and minor technical details (no shuffling column order in split() function). So I refactored my code.
For my updated implementation, I used a nested Node class which is defined inside of an outer MyDecisionTreeRegressor class. I prepended the “My” in front of the name so there wouldn’t be a name clash with the scikit-learn DecisionTreeRegressor module:
class MyDecisionTreeRegressor:
def __init__(self, max_depth=3, min_samples=2,
n_split_cols=-1, seed=0):
self.max_depth = max_depth
self.min_samples = min_samples # aka min_samples_split
self.n_split_cols = n_split_cols
self.root = None
self.rnd = np.random.RandomState(seed) # for split cols
# =========================================================
class Node:
def __init__(self, col_idx=-1, thresh=0.0,
left=None, right=None, value=0.0, is_leaf=True):
self.col_idx = col_idx
self.thresh = thresh
self.left = left
self.right = right
self.value = value
self.is_leaf = is_leaf # must be False for an in-node
# =========================================================
def best_split(self, X, y): . . .
def make_tree(self, X, y, depth=0): . . .
def fit(self, X, y): . . .
def predict_one(self, x, from_node): . . .
def predict(self, X): . . .
def explain(self, x){ . . .
The scikit library DecisionTreeRegressor module has a min_samples_split and a min_samples_leaf parameters to the constructor. The min_samples_split parameter is the fewest number of sample items in a Node for a split to be considered. The min_samples_leaf parameter is the fewest number of sample items in a resulting split. The interaction between the two parameters is kind of tricky, so I simplified and used just a min_samples which is the same as the scikit min_samples_split parameter.
All decision tree implementations have a max_depth parameter, but the meaning can be somewhat ambiguous because it depends on whether the root node is considered a level or not. For my implementation, if max_depth is 0, then the tree has just a root node and all predictions are the averaged of the target y values in the training data. If max_depth is 1, then the tree has a root node and a single left child and a single right child. If max_depth is 2, then the tree has at most 7 nodes. And so on. The scikit DecisionTreeRegressor works the same except that it requires max_depth to be greater than or equal to 1.
For my demo, I used a set of synthetic data that I generated using a neural network with random weights and biases. The data looks like:
-0.1660, 0.4406, -0.9998, -0.3953, -0.7065, 0.4840 0.0776, -0.1616, 0.3704, -0.5911, 0.7562, 0.1568 -0.9452, 0.3409, -0.1654, 0.1174, -0.7192, 0.8054 . . .
The first five values on each line are the predictors. The sixth value is the target to predict. All predictor values are between -1.0 and 1.0. Normalizing the predictor values is not necessary but is helpful when using the data with other regression techniques that require normalization (such as k-nearest neighbors regression). There are 200 items in the training data and 40 items in the test data.
The output of the from-scratch decision tree regression demo program is:
Begin decision tree regression scratch Python Loading synthetic train (200), test (40) data Done First three X predictors: [[-0.1660 0.4406 -0.9998 -0.3953 -0.7065] [ 0.0776 -0.1616 0.3704 -0.5911 0.7562] [-0.9452 0.3409 -0.1654 0.1174 -0.7192]] First three y targets: 0.4840 0.1568 0.8054 Creating tree model max_depth=6 min_samples=2 Accuracy train (within 0.10): 0.8150 Accuracy test (within 0.10): 0.4500 MSE train: 0.0004 MSE test: 0.0050 Predicting train_X[0] using scratch: [0.4909] IF column 0 > -0.2102 AND column 0 <= 0.3915 AND column 4 <= -0.2987 AND column 0 <= 0.0090 AND column 4 <= -0.6966 AND column 3 > -0.7393 AND THEN predicted = 0.4909 End demo ================================ Using scikit: Accuracy train (within 0.10): 0.8100 Accuracy test (within 0.10): 0.4000 MSE train: 0.0004 MSE test: 0.0057 Predicting train_X[0] using scikit: [0.4909]
The output of the from-scratch Python version and the scikit library version are not exactly the same because my from-scratch version uses maximization of explicit variance reduction to compute tree splits, but scikit uses minimization of quasi mean squared error, which is theoretically the same but can be slightly different because of the way in which scikit computes it. Direct reduction in variance is a bit more principled (in my opinion). The relevant scikit MSE variance reduction code is at github.com/scikit-learn/scikit-learn/blob/main/sklearn/tree/_criterion.pyx but I didn’t feel like spending the several hours needed to figure out what the library code is doing.
Additionally, my decision tree regressor version and the scikit version shuffle the order in which columns are searched when finding the split values. This prevents the first few columns from always being used when there are more than one splits that have the same variance reduction. But this introduces minor randomness into the tree models.
I implemented an accuracy() function that scores a predicted y value as correct if it’s within 0.10 of the true value. The accuracy of the decision tree model on the training data is good (81.50% — 163 out of 200 correct) but poor on the test data (45.00% — just 18 out of 40 correct). This is typical for decision trees because they usually overfit on the training data. And this is the reason why decision trees are almost never used by themselves. It’s far more common to combine many decision trees into an ensemble technique such as Bagging Tree, Random Forest, AdaBoost, or Gradient Boost.
For my updated demo, I added an explain() method that shows how an output is computed. Sometimes an explanation helps interpretability, but sometimes it doesn’t really help.
Decision trees are not plants. I’ve always been fascinated by movies and books with evil plants.
Left: “Baffling Mysteries” (#19, January 1954), cover artist (probably) George Roussos. This comic book ran 22 issues from 1951-1955.
Center: “Black Magic” (#19, December 1952), cover artist (definitely) Jack Kirby. This title/series ran from 1950-1961.
Right: “Adventures into the Unknown” (#48, October 1953, cover artist (probably) Ken Bald. This series ran from 1948-1967.
Demo program. Replace “lt” (less than), “gt”, “lte”, “gte” with Boolean operator symbols. (My blog editor chokes on symbols).
# decision_tree_regression_recursion_build.py
# common approach: pointers (nodes) and recursion (build)
# explicit variance reduction for splitting, aot MSE
import numpy as np
# ===========================================================
class MyDecisionTreeRegressor: # avoid scikit name collision
# if max_depth = 0, tree has just a root node with avg y
# if max_depth = 1, tree has at most 3 nodes (root, l, r)
# if max_depth = n, tree has at most 2^(n+1) - 1 nodes.
def __init__(self, max_depth=3, min_samples=2,
n_split_cols=-1, seed=0):
self.max_depth = max_depth
self.min_samples = min_samples # aka min_samples_split
self.n_split_cols = n_split_cols
self.root = None
self.rnd = np.random.RandomState(seed) # split col order
# ===============================================
class Node:
def __init__(self, col_idx=-1, thresh=0.0,
left=None, right=None, value=0.0, is_leaf=True):
self.col_idx = col_idx
self.thresh = thresh
self.left = left
self.right = right
self.value = value
self.is_leaf = is_leaf # must be False for an in-node
# ===============================================
def best_split(self, X, y):
# best split using explicit variance reduction
best_col_idx = -1 # indicates a bad split
best_thresh = 0.0
best_var_red = 0.0 # larger is better
n_rows, n_cols = X.shape
if len(y) == 0:
var_y = 0.0
else:
var_y = np.var(y) # baseline for variance reduction
rnd_cols = np.arange(n_cols)
self.rnd.shuffle(rnd_cols)
if self.n_split_cols != -1: # just use some cols
rnd_cols = rnd_cols[0:self.n_split_cols]
for j in range(len(rnd_cols)):
col_idx = rnd_cols[j]
examined_threshs = set()
for i in range(n_rows):
thresh = X[i][col_idx] # candidate threshold value
# if thresh value has already been checked, skip it
if thresh in examined_threshs == True:
continue
examined_threshs.add(thresh)
# get rows where x is lte, gt thresh
left_idxs = []
right_idxs = []
for r in range(len(X)):
if X[r][col_idx] "lte" thresh:
left_idxs.append(r)
else:
right_idxs.append(r)
if len(left_idxs) == 0 or \
len(right_idxs) == 0:
continue
left_wt = len(left_idxs) / n_rows
right_wt = len(right_idxs) / n_rows
left_y_vals = []
right_y_vals = []
for k in range(len(left_idxs)):
left_y_vals.append(y[left_idxs[k]])
for k in range(len(right_idxs)):
right_y_vals.append(y[right_idxs[k]])
if len(left_y_vals) == 0: # should never happen
left_var = 0.0
else:
left_var = left_wt * np.var(left_y_vals)
if len(right_y_vals) == 0:
right_var = 0.0
else:
right_var = right_wt * np.var(right_y_vals)
var_red = var_y - (left_var + right_var)
if var_red "gte" best_var_red:
best_col_idx = col_idx
best_thresh = thresh
best_var_red = var_red
return best_col_idx, best_thresh # -1 is bad/no split
# ---------------------------------------------------------
def make_tree(self, X, y, depth=0):
if depth == self.max_depth or len(y) "lt" self.min_samples:
return self.Node(value=np.mean(y), is_leaf=True)
col_idx, thresh = self.best_split(X, y)
# if feat_idx is None:
if col_idx == -1:
return self.Node(value=np.mean(y), is_leaf=True)
left_idxs = X[:, col_idx] "lte" thresh
right_idxs = ~left_idxs # "gt"
left_X = X[left_idxs]
left_y = y[left_idxs]
right_X = X[right_idxs]
right_y = y[right_idxs]
left = self.make_tree(left_X, left_y, depth+1)
right = self.make_tree(right_X, right_y, depth+1)
# next recursive call checks left and right
return self.Node(col_idx=col_idx,
thresh=thresh, left=left, right=right, is_leaf=False)
def fit(self, X, y):
self.root = self.make_tree(X, y)
def predict_one(self, x):
curr = self.root
while curr.is_leaf == False:
if x[curr.col_idx] "lte" curr.thresh:
curr = curr.left
else:
curr = curr.right
return curr.value
def predict(self, X):
result = np.zeros(len(X), dtype=np.float64)
for i in range(len(X)):
result[i] = self.predict_one(X[i])
return result
def explain(self, x):
print("\nIF ");
curr = self.root;
while curr.is_leaf == False:
print("column ", end="")
print(str(curr.col_idx) + " ", end="")
if x[curr.col_idx] "lte" curr.thresh:
print(" "lte" ", end="");
print("%8.4f AND " % curr.thresh)
curr = curr.left
else:
print(" "gt" ", end="")
print("%8.4f AND " % curr.thresh)
curr = curr.right;
print("THEN \npredicted = %0.4f " % curr.value)
# ===========================================================
# -----------------------------------------------------------
def accuracy(model, data_X, data_y, pct_close):
n = len(data_X)
n_correct = 0; n_wrong = 0
for i in range(n):
x = data_X[i].reshape(1,-1)
y = data_y[i]
y_pred = model.predict(x)
if np.abs(y - y_pred) "lt" np.abs(y * pct_close):
n_correct += 1
else:
n_wrong += 1
# print("Correct = " + str(n_correct))
# print("Wrong = " + str(n_wrong))
return n_correct / (n_correct + n_wrong)
# -----------------------------------------------------------
def MSE(model, data_X, data_y):
n = len(data_X)
sum = 0.0
for i in range(n):
x = data_X[i].reshape(1,-1)
y = data_y[i]
y_pred = model.predict(x)
sum += (y - y_pred) * (y - y_pred)
return sum / n
# -----------------------------------------------------------
def main():
print("\nBegin decision tree regression scratch Python ")
np.set_printoptions(precision=4, suppress=True,
floatmode='fixed')
np.random.seed(0) # not used this version
# 1. load data
print("\nLoading synthetic train (200), test (40) data ")
train_file = ".\\Data\\synthetic_train_200.txt"
# -0.1660,0.4406,-0.9998,-0.3953,-0.7065,0.4840
# 0.0776,-0.1616,0.3704,-0.5911,0.7562,0.1568
# -0.9452,0.3409,-0.1654,0.1174,-0.7192,0.8054
# . . .
train_X = np.loadtxt(train_file, comments="#",
usecols=[0,1,2,3,4],
delimiter=",", dtype=np.float64)
train_y = np.loadtxt(train_file, comments="#", usecols=5,
delimiter=",", dtype=np.float64)
test_file = ".\\Data\\synthetic_test_40.txt"
test_X = np.loadtxt(test_file, comments="#",
usecols=[0,1,2,3,4],
delimiter=",", dtype=np.float64)
test_y = np.loadtxt(test_file, comments="#", usecols=5,
delimiter=",", dtype=np.float64)
print("Done ")
print("\nFirst three X predictors: ")
print(train_X[0:3,:])
print("\nFirst three y targets: ")
for i in range(3):
print("%0.4f" % train_y[i])
md = 6 # max_depth
ms = 2 # min_samples to consider a split
print("\nCreating tree model max_depth=" + str(md) + \
" min_samples=" + str(ms))
tree = MyDecisionTreeRegressor(max_depth=md,
min_samples=ms)
tree.fit(train_X, train_y)
acc_train = accuracy(tree, train_X, train_y, 0.10)
print("\nAccuracy train (within 0.10): %0.4f " % acc_train)
acc_test = accuracy(tree, test_X, test_y, 0.10)
print("Accuracy test (within 0.10): %0.4f " % acc_test)
mse_train = MSE(tree, train_X, train_y)
print("\nMSE train: %0.4f " % mse_train)
mse_test = MSE(tree, test_X, test_y)
print("MSE test: %0.4f " % mse_test)
print("\nPredicting train_X[0] using scratch: ")
x = train_X[0].reshape(1,-1)
y_pred = tree.predict(x)
print(y_pred)
tree.explain(train_X[0])
print("\nEnd demo ")
print("\nUsing scikit: ")
from sklearn.tree import DecisionTreeRegressor
tree = DecisionTreeRegressor(max_depth=md,
min_samples_split=ms, random_state=0)
tree.fit(train_X, train_y)
acc_train = accuracy(tree, train_X, train_y, 0.10)
print("\nAccuracy train (within 0.10): %0.4f " % acc_train)
acc_test = accuracy(tree, test_X, test_y, 0.10)
print("Accuracy test (within 0.10): %0.4f " % acc_test)
mse_train = MSE(tree, train_X, train_y)
print("\nMSE train: %0.4f " % mse_train)
mse_test = MSE(tree, test_X, test_y)
print("MSE test: %0.4f " % mse_test)
print("\nPredicting train_X[0] using scikit: ")
y_pred = tree.predict(x)
print(y_pred)
if __name__ == "__main__":
main()
Training data:
# synthetic_train_200.txt # -0.1660, 0.4406, -0.9998, -0.3953, -0.7065, 0.4840 0.0776, -0.1616, 0.3704, -0.5911, 0.7562, 0.1568 -0.9452, 0.3409, -0.1654, 0.1174, -0.7192, 0.8054 0.9365, -0.3732, 0.3846, 0.7528, 0.7892, 0.1345 -0.8299, -0.9219, -0.6603, 0.7563, -0.8033, 0.7955 0.0663, 0.3838, -0.3690, 0.3730, 0.6693, 0.3206 -0.9634, 0.5003, 0.9777, 0.4963, -0.4391, 0.7377 -0.1042, 0.8172, -0.4128, -0.4244, -0.7399, 0.4801 -0.9613, 0.3577, -0.5767, -0.4689, -0.0169, 0.6861 -0.7065, 0.1786, 0.3995, -0.7953, -0.1719, 0.5569 0.3888, -0.1716, -0.9001, 0.0718, 0.3276, 0.2500 0.1731, 0.8068, -0.7251, -0.7214, 0.6148, 0.3297 -0.2046, -0.6693, 0.8550, -0.3045, 0.5016, 0.2129 0.2473, 0.5019, -0.3022, -0.4601, 0.7918, 0.2613 -0.1438, 0.9297, 0.3269, 0.2434, -0.7705, 0.5171 0.1568, -0.1837, -0.5259, 0.8068, 0.1474, 0.3307 -0.9943, 0.2343, -0.3467, 0.0541, 0.7719, 0.5581 0.2467, -0.9684, 0.8589, 0.3818, 0.9946, 0.1092 -0.6553, -0.7257, 0.8652, 0.3936, -0.8680, 0.7018 0.8460, 0.4230, -0.7515, -0.9602, -0.9476, 0.1996 -0.9434, -0.5076, 0.7201, 0.0777, 0.1056, 0.5664 0.9392, 0.1221, -0.9627, 0.6013, -0.5341, 0.1533 0.6142, -0.2243, 0.7271, 0.4942, 0.1125, 0.1661 0.4260, 0.1194, -0.9749, -0.8561, 0.9346, 0.2230 0.1362, -0.5934, -0.4953, 0.4877, -0.6091, 0.3810 0.6937, -0.5203, -0.0125, 0.2399, 0.6580, 0.1460 -0.6864, -0.9628, -0.8600, -0.0273, 0.2127, 0.5387 0.9772, 0.1595, -0.2397, 0.1019, 0.4907, 0.1611 0.3385, -0.4702, -0.8673, -0.2598, 0.2594, 0.2270 -0.8669, -0.4794, 0.6095, -0.6131, 0.2789, 0.4700 0.0493, 0.8496, -0.4734, -0.8681, 0.4701, 0.3516 0.8639, -0.9721, -0.5313, 0.2336, 0.8980, 0.1412 0.9004, 0.1133, 0.8312, 0.2831, -0.2200, 0.1782 0.0991, 0.8524, 0.8375, -0.2102, 0.9265, 0.2150 -0.6521, -0.7473, -0.7298, 0.0113, -0.9570, 0.7422 0.6190, -0.3105, 0.8802, 0.1640, 0.7577, 0.1056 0.6895, 0.8108, -0.0802, 0.0927, 0.5972, 0.2214 0.1982, -0.9689, 0.1870, -0.1326, 0.6147, 0.1310 -0.3695, 0.7858, 0.1557, -0.6320, 0.5759, 0.3773 -0.1596, 0.3581, 0.8372, -0.9992, 0.9535, 0.2071 -0.2468, 0.9476, 0.2094, 0.6577, 0.1494, 0.4132 0.1737, 0.5000, 0.7166, 0.5102, 0.3961, 0.2611 0.7290, -0.3546, 0.3416, -0.0983, -0.2358, 0.1332 -0.3652, 0.2438, -0.1395, 0.9476, 0.3556, 0.4170 -0.6029, -0.1466, -0.3133, 0.5953, 0.7600, 0.4334 -0.4596, -0.4953, 0.7098, 0.0554, 0.6043, 0.2775 0.1450, 0.4663, 0.0380, 0.5418, 0.1377, 0.2931 -0.8636, -0.2442, -0.8407, 0.9656, -0.6368, 0.7429 0.6237, 0.7499, 0.3768, 0.1390, -0.6781, 0.2185 -0.5499, 0.1850, -0.3755, 0.8326, 0.8193, 0.4399 -0.4858, -0.7782, -0.6141, -0.0008, 0.4572, 0.4197 0.7033, -0.1683, 0.2334, -0.5327, -0.7961, 0.1776 0.0317, -0.0457, -0.6947, 0.2436, 0.0880, 0.3345 0.5031, -0.5559, 0.0387, 0.5706, -0.9553, 0.3107 -0.3513, 0.7458, 0.6894, 0.0769, 0.7332, 0.3170 0.2205, 0.5992, -0.9309, 0.5405, 0.4635, 0.3532 -0.4806, -0.4859, 0.2646, -0.3094, 0.5932, 0.3202 0.9809, -0.3995, -0.7140, 0.8026, 0.0831, 0.1600 0.9495, 0.2732, 0.9878, 0.0921, 0.0529, 0.1289 -0.9476, -0.6792, 0.4913, -0.9392, -0.2669, 0.5966 0.7247, 0.3854, 0.3819, -0.6227, -0.1162, 0.1550 -0.5922, -0.5045, -0.4757, 0.5003, -0.0860, 0.5863 -0.8861, 0.0170, -0.5761, 0.5972, -0.4053, 0.7301 0.6877, -0.2380, 0.4997, 0.0223, 0.0819, 0.1404 0.9189, 0.6079, -0.9354, 0.4188, -0.0700, 0.1907 -0.1428, -0.7820, 0.2676, 0.6059, 0.3936, 0.2790 0.5324, -0.3151, 0.6917, -0.1425, 0.6480, 0.1071 -0.8432, -0.9633, -0.8666, -0.0828, -0.7733, 0.7784 -0.9444, 0.5097, -0.2103, 0.4939, -0.0952, 0.6787 -0.0520, 0.6063, -0.1952, 0.8094, -0.9259, 0.4836 0.5477, -0.7487, 0.2370, -0.9793, 0.0773, 0.1241 0.2450, 0.8116, 0.9799, 0.4222, 0.4636, 0.2355 0.8186, -0.1983, -0.5003, -0.6531, -0.7611, 0.1511 -0.4714, 0.6382, -0.3788, 0.9648, -0.4667, 0.5950 0.0673, -0.3711, 0.8215, -0.2669, -0.1328, 0.2677 -0.9381, 0.4338, 0.7820, -0.9454, 0.0441, 0.5518 -0.3480, 0.7190, 0.1170, 0.3805, -0.0943, 0.4724 -0.9813, 0.1535, -0.3771, 0.0345, 0.8328, 0.5438 -0.1471, -0.5052, -0.2574, 0.8637, 0.8737, 0.3042 -0.5454, -0.3712, -0.6505, 0.2142, -0.1728, 0.5783 0.6327, -0.6297, 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-0.1628, 0.2002 0.0115, -0.6209, 0.9300, -0.4116, -0.7931, 0.4052 -0.7114, -0.9718, 0.4319, 0.1290, 0.5892, 0.3661 0.3915, 0.5557, -0.1870, 0.2955, -0.6404, 0.2954 -0.3564, -0.6548, -0.1827, -0.5172, -0.1862, 0.4622 0.2392, -0.4959, 0.5857, -0.1341, -0.2850, 0.2470 -0.3394, 0.3947, -0.4627, 0.6166, -0.4094, 0.5325 0.7107, 0.7768, -0.6312, 0.1707, 0.7964, 0.2757 -0.1078, 0.8437, -0.4420, 0.2177, 0.3649, 0.4028 -0.3139, 0.5595, -0.6505, -0.3161, -0.7108, 0.5546 0.4335, 0.3986, 0.3770, -0.4932, 0.3847, 0.1810 -0.2562, -0.2894, -0.8847, 0.2633, 0.4146, 0.4036 0.2272, 0.2966, -0.6601, -0.7011, 0.0284, 0.2778 -0.0743, -0.1421, -0.0054, -0.6770, -0.3151, 0.3597 -0.4762, 0.6891, 0.6007, -0.1467, 0.2140, 0.4266 -0.4061, 0.7193, 0.3432, 0.2669, -0.7505, 0.6147 -0.0588, 0.9731, 0.8966, 0.2902, -0.6966, 0.4955 -0.0627, -0.1439, 0.1985, 0.6999, 0.5022, 0.3077 0.1587, 0.8494, -0.8705, 0.9827, -0.8940, 0.4263 -0.7850, 0.2473, -0.9040, -0.4308, -0.8779, 0.7199 0.4070, 0.3369, -0.2428, -0.6236, 0.4940, 0.2215 -0.0242, 0.0513, -0.9430, 0.2885, -0.2987, 0.3947 -0.5416, -0.1322, -0.2351, -0.0604, 0.9590, 0.3683 0.1055, 0.7783, -0.2901, -0.5090, 0.8220, 0.2984 -0.9129, 0.9015, 0.1128, -0.2473, 0.9901, 0.4776 -0.9378, 0.1424, -0.6391, 0.2619, 0.9618, 0.5368 0.7498, -0.0963, 0.4169, 0.5549, -0.0103, 0.1614 -0.2612, -0.7156, 0.4538, -0.0460, -0.1022, 0.3717 0.7720, 0.0552, -0.1818, -0.4622, -0.8560, 0.1685 -0.4177, 0.0070, 0.9319, -0.7812, 0.3461, 0.3052 -0.0001, 0.5542, -0.7128, -0.8336, -0.2016, 0.3803 0.5356, -0.4194, -0.5662, -0.9666, -0.2027, 0.1776 -0.2378, 0.3187, -0.8582, -0.6948, -0.9668, 0.5474 -0.1947, -0.3579, 0.1158, 0.9869, 0.6690, 0.2992 0.3992, 0.8365, -0.9205, -0.8593, -0.0520, 0.3154 -0.0209, 0.0793, 0.7905, -0.1067, 0.7541, 0.1864 -0.4928, -0.4524, -0.3433, 0.0951, -0.5597, 0.6261 -0.8118, 0.7404, -0.5263, -0.2280, 0.1431, 0.6349 0.0516, -0.8480, 0.7483, 0.9023, 0.6250, 0.1959 -0.3212, 0.1093, 0.9488, -0.3766, 0.3376, 0.2735 -0.3481, 0.5490, -0.3484, 0.7797, 0.5034, 0.4379 -0.5785, -0.9170, -0.3563, -0.9258, 0.3877, 0.4121 0.3407, -0.1391, 0.5356, 0.0720, -0.9203, 0.3458 -0.3287, -0.8954, 0.2102, 0.0241, 0.2349, 0.3247 -0.1353, 0.6954, -0.0919, -0.9692, 0.7461, 0.3338 0.9036, -0.8982, -0.5299, -0.8733, -0.1567, 0.1187 0.7277, -0.8368, -0.0538, -0.7489, 0.5458, 0.0830 0.9049, 0.8878, 0.2279, 0.9470, -0.3103, 0.2194 0.7957, -0.1308, -0.5284, 0.8817, 0.3684, 0.2172 0.4647, -0.4931, 0.2010, 0.6292, -0.8918, 0.3371 -0.7390, 0.6849, 0.2367, 0.0626, -0.5034, 0.7039 -0.1567, -0.8711, 0.7940, -0.5932, 0.6525, 0.1710 0.7635, -0.0265, 0.1969, 0.0545, 0.2496, 0.1445 0.7675, 0.1354, -0.7698, -0.5460, 0.1920, 0.1728 -0.5211, -0.7372, -0.6763, 0.6897, 0.2044, 0.5217 0.1913, 0.1980, 0.2314, -0.8816, 0.5006, 0.1998 0.8964, 0.0694, -0.6149, 0.5059, -0.9854, 0.1825 0.1767, 0.7104, 0.2093, 0.6452, 0.7590, 0.2832 -0.3580, -0.7541, 0.4426, -0.1193, -0.7465, 0.5657 -0.5996, 0.5766, -0.9758, -0.3933, -0.9572, 0.6800 0.9950, 0.1641, -0.4132, 0.8579, 0.0142, 0.2003 -0.4717, -0.3894, -0.2567, -0.5111, 0.1691, 0.4266 0.3917, -0.8561, 0.9422, 0.5061, 0.6123, 0.1212 -0.0366, -0.1087, 0.3449, -0.1025, 0.4086, 0.2475 0.3633, 0.3943, 0.2372, -0.6980, 0.5216, 0.1925 -0.5325, -0.6466, -0.2178, -0.3589, 0.6310, 0.3568 0.2271, 0.5200, -0.1447, -0.8011, -0.7699, 0.3128 0.6415, 0.1993, 0.3777, -0.0178, -0.8237, 0.2181 -0.5298, -0.0768, -0.6028, -0.9490, 0.4588, 0.4356 0.6870, -0.1431, 0.7294, 0.3141, 0.1621, 0.1632 -0.5985, 0.0591, 0.7889, -0.3900, 0.7419, 0.2945 0.3661, 0.7984, -0.8486, 0.7572, -0.6183, 0.3449 0.6995, 0.3342, -0.3113, -0.6972, 0.2707, 0.1712 0.2565, 0.9126, 0.1798, -0.6043, -0.1413, 0.2893 -0.3265, 0.9839, -0.2395, 0.9854, 0.0376, 0.4770 0.2690, -0.1722, 0.9818, 0.8599, -0.7015, 0.3954 -0.2102, -0.0768, 0.1219, 0.5607, -0.0256, 0.3949 0.8216, -0.9555, 0.6422, -0.6231, 0.3715, 0.0801 -0.2896, 0.9484, -0.7545, -0.6249, 0.7789, 0.4370 -0.9985, -0.5448, -0.7092, -0.5931, 0.7926, 0.5402
Test data:
# synthetic_test_40.txt # 0.7462, 0.4006, -0.0590, 0.6543, -0.0083, 0.1935 0.8495, -0.2260, -0.0142, -0.4911, 0.7699, 0.1078 -0.2335, -0.4049, 0.4352, -0.6183, -0.7636, 0.5088 0.1810, -0.5142, 0.2465, 0.2767, -0.3449, 0.3136 -0.8650, 0.7611, -0.0801, 0.5277, -0.4922, 0.7140 -0.2358, -0.7466, -0.5115, -0.8413, -0.3943, 0.4533 0.4834, 0.2300, 0.3448, -0.9832, 0.3568, 0.1360 -0.6502, -0.6300, 0.6885, 0.9652, 0.8275, 0.3046 -0.3053, 0.5604, 0.0929, 0.6329, -0.0325, 0.4756 -0.7995, 0.0740, -0.2680, 0.2086, 0.9176, 0.4565 -0.2144, -0.2141, 0.5813, 0.2902, -0.2122, 0.4119 -0.7278, -0.0987, -0.3312, -0.5641, 0.8515, 0.4438 0.3793, 0.1976, 0.4933, 0.0839, 0.4011, 0.1905 -0.8568, 0.9573, -0.5272, 0.3212, -0.8207, 0.7415 -0.5785, 0.0056, -0.7901, -0.2223, 0.0760, 0.5551 0.0735, -0.2188, 0.3925, 0.3570, 0.3746, 0.2191 0.1230, -0.2838, 0.2262, 0.8715, 0.1938, 0.2878 0.4792, -0.9248, 0.5295, 0.0366, -0.9894, 0.3149 -0.4456, 0.0697, 0.5359, -0.8938, 0.0981, 0.3879 0.8629, -0.8505, -0.4464, 0.8385, 0.5300, 0.1769 0.1995, 0.6659, 0.7921, 0.9454, 0.9970, 0.2330 -0.0249, -0.3066, -0.2927, -0.4923, 0.8220, 0.2437 0.4513, -0.9481, -0.0770, -0.4374, -0.9421, 0.2879 -0.3405, 0.5931, -0.3507, -0.3842, 0.8562, 0.3987 0.9538, 0.0471, 0.9039, 0.7760, 0.0361, 0.1706 -0.0887, 0.2104, 0.9808, 0.5478, -0.3314, 0.4128 -0.8220, -0.6302, 0.0537, -0.1658, 0.6013, 0.4306 -0.4123, -0.2880, 0.9074, -0.0461, -0.4435, 0.5144 0.0060, 0.2867, -0.7775, 0.5161, 0.7039, 0.3599 -0.7968, -0.5484, 0.9426, -0.4308, 0.8148, 0.2979 0.7811, 0.8450, -0.6877, 0.7594, 0.2640, 0.2362 -0.6802, -0.1113, -0.8325, -0.6694, -0.6056, 0.6544 0.3821, 0.1476, 0.7466, -0.5107, 0.2592, 0.1648 0.7265, 0.9683, -0.9803, -0.4943, -0.5523, 0.2454 -0.9049, -0.9797, -0.0196, -0.9090, -0.4433, 0.6447 -0.4607, 0.1811, -0.2389, 0.4050, -0.0078, 0.5229 0.2664, -0.2932, -0.4259, -0.7336, 0.8742, 0.1834 -0.4507, 0.1029, -0.6294, -0.1158, -0.6294, 0.6081 0.8948, -0.0124, 0.9278, 0.2899, -0.0314, 0.1534 -0.1323, -0.8813, -0.0146, -0.0697, 0.6135, 0.2386
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