Resolution Tree algorithms have at all times fascinated me. They’re simple to implement and obtain good outcomes on numerous classification and regression duties. Mixed with boosting, determination bushes are nonetheless state-of-the-art in lots of functions.
Frameworks comparable to sklearn, Lightgbm, xgboost and catboost have achieved an excellent job till immediately. Nonetheless, previously few months, I’ve been lacking help for arrow datasets. Whereas lightgbm has not too long ago added help for that, it’s nonetheless lacking in most different frameworks. The arrow information format could possibly be an ideal match for determination bushes because it has a columnar construction optimized for environment friendly information processing. Pandas already added help for that and in addition polars makes use of the benefits.
Polars has proven some important efficiency benefits over most different information frameworks. It makes use of the info effectively and avoids copying the info unnecessarily. It additionally supplies a streaming engine that enables the processing of bigger information than reminiscence. This is the reason I made a decision to make use of polars as a backend for constructing a choice tree from scratch.
The purpose is to discover the benefits of utilizing polars for determination bushes by way of reminiscence and runtime. And, in fact, studying extra about polars, effectively defining expressions, and the streaming engine.
The code for the implementation may be discovered on this repository.
Code overview
To get a primary overview of the code, I’ll present the construction of the DecisionTreeClassifier first:
The primary essential factor may be seen within the imports. It was essential for me to maintain the import part clear and with as few dependencies as doable. This was profitable with solely having dependencies to polars, pickle, and typing.
The init methodology permits to outline if the polars streaming engine needs to be used. Additionally, the max_depth of the tree may be set right here. One other characteristic within the definition of categorical columns. These are dealt with otherwise than numerical options utilizing a goal encoding.
It’s doable to avoid wasting and cargo the choice tree mannequin. It’s represented as a nested dict and may be saved to disk as a pickled file.
The polars magic occurs within the match() and build_tree() strategies. These settle for each LazyFrames and DataFrames to have help for in-memory processing and streaming.
There are two prediction strategies accessible, predict() and predict_many(). The predict() methodology can be utilized on a small instance dimension, and the info must be offered as a dict. If we’ve an enormous check set, it’s extra environment friendly to make use of the predict_many() methodology. Right here, the info may be offered as a Polars Dataframe or LazyFrame.
import pickle
from typing import Iterable, Listing, Union
import polars as pl
class DecisionTreeClassifier:
def __init__(self, streaming=False, max_depth=None, categorical_columns=None):
...
def save_model(self, path: str) -> None:
...
def load_model(self, path: str) -> None:
...
def apply_categorical_mappings(self, information: Union[pl.DataFrame, pl.LazyFrame]) -> Union[pl.DataFrame, pl.LazyFrame]:
...
def match(self, information: Union[pl.DataFrame, pl.LazyFrame], target_name: str) -> None:
...
def predict_many(self, information: Union[pl.DataFrame, pl.LazyFrame]) -> Listing[Union[int, float]]:
...
def predict(self, information: Iterable[dict]):
...
def get_majority_class(self, df: Union[pl.DataFrame, pl.LazyFrame], target_name: str) -> str:
...
def _build_tree(
self,
information: Union[pl.DataFrame, pl.LazyFrame],
feature_names: record[str],
target_name: str,
unique_targets: record[int],
depth: int,
) -> dict:
...Becoming the tree
To coach the choice tree classifier, the match() methodology must be used.
def match(self, information: Union[pl.DataFrame, pl.LazyFrame], target_name: str) -> None:
"""
Match methodology to coach the choice tree.
:param information: Polars DataFrame or LazyFrame containing the coaching information.
:param target_name: Identify of the goal column
"""
columns = information.collect_schema().names()
feature_names = [col for col in columns if col != target_name]
# Shrink dtypes
information = information.choose(pl.all().shrink_dtype()).with_columns(
pl.col(target_name).solid(pl.UInt64).shrink_dtype().alias(target_name)
)
# Put together categorical columns with goal encoding
if self.categorical_columns:
categorical_mappings = {}
for categorical_column in self.categorical_columns:
categorical_mappings[categorical_column] = {
worth: index
for index, worth in enumerate(
information.lazy()
.group_by(categorical_column)
.agg(pl.col(target_name).imply().alias("avg"))
.type("avg")
.gather(streaming=self.streaming)[categorical_column]
)
}
self.categorical_mappings = categorical_mappings
information = self.apply_categorical_mappings(information)
unique_targets = information.choose(target_name).distinctive()
if isinstance(unique_targets, pl.LazyFrame):
unique_targets = unique_targets.gather(streaming=self.streaming)
unique_targets = unique_targets[target_name].to_list()
self.tree = self._build_tree(information, feature_names, target_name, unique_targets, depth=0)It receives a polars LazyFrame or DataFrame that accommodates all options and the goal column. To establish the goal column, the target_name must be offered.
Polars supplies a handy strategy to optimize the reminiscence utilization of the info.
information.choose(pl.all().shrink_dtype())With that, all columns are chosen and evaluated. It should convert the dtype to the smallest doable worth.
The explicit encoding
To encode categorical values, a goal encoding is used. For that, all cases of a categorical characteristic can be aggregated, and the typical goal worth can be calculated. Then, the cases are sorted by the typical goal worth, and a rank is assigned. This rank can be used because the illustration of the characteristic worth.
(
information.lazy()
.group_by(categorical_column)
.agg(pl.col(target_name).imply().alias("avg"))
.type("avg")
.gather(streaming=self.streaming)[categorical_column]
)Since it’s doable to offer polars DataFrames and LazyFrames, I take advantage of information.lazy() first. If the given information is a DataFrame, will probably be transformed to a LazyFrame. Whether it is already a LazyFrame, it solely returns self. With that trick, it’s doable to make sure that the info is processed in the identical approach for LazyFrames and DataFrames and that the gather() methodology can be utilized, which is barely accessible for LazyFrames.
As an instance the result of the calculations within the completely different steps of the becoming course of, I apply it to a dataset for coronary heart illness prediction. It may be discovered on Kaggle and is printed beneath the Database Contents License.
Right here is an instance of the specific characteristic illustration for the glucose ranges:
┌──────┬──────┬──────────┐
│ rank ┆ gluc ┆ avg │
│ --- ┆ --- ┆ --- │
│ u32 ┆ i8 ┆ f64 │
╞══════╪══════╪══════════╡
│ 0 ┆ 1 ┆ 0.476139 │
│ 1 ┆ 2 ┆ 0.586319 │
│ 2 ┆ 3 ┆ 0.620972 │
└──────┴──────┴──────────┘For every of the glucose ranges, the chance of getting a coronary heart illness is calculated. That is sorted after which ranked so that every of the degrees is mapped to a rank worth.
Getting the goal values
Because the final a part of the match() methodology, the distinctive goal values are decided.
unique_targets = information.choose(target_name).distinctive()
if isinstance(unique_targets, pl.LazyFrame):
unique_targets = unique_targets.gather(streaming=self.streaming)
unique_targets = unique_targets[target_name].to_list()
self.tree = self._build_tree(information, feature_names, target_name, unique_targets, depth=0)This serves because the final preparation earlier than calling the _build_tree() methodology recursively.
Constructing the tree
After the info is ready within the match() methodology, the _build_tree() methodology known as. That is achieved recursively till a stopping criterion is met, e.g., the max depth of the tree is reached. The primary name is executed from the match() methodology with a depth of zero.
def _build_tree(
self,
information: Union[pl.DataFrame, pl.LazyFrame],
feature_names: record[str],
target_name: str,
unique_targets: record[int],
depth: int,
) -> dict:
"""
Builds the choice tree recursively.
If max_depth is reached, returns a leaf node with the bulk class.
In any other case, finds the most effective break up and creates inner nodes for left and proper kids.
:param information: The dataframe to guage.
:param feature_names: Identify of the characteristic columns.
:param target_name: Identify of the goal column.
:param unique_targets: distinctive goal values.
:param depth: The present depth of the tree.
:return: A dictionary representing the node.
"""
if self.max_depth shouldn't be None and depth >= self.max_depth:
return {"sort": "leaf", "worth": self.get_majority_class(information, target_name)}
# Make information lazy right here to keep away from that it's evaluated in every loop iteration.
information = information.lazy()
# Consider entropy per characteristic:
information_gain_dfs = []
for feature_name in feature_names:
feature_data = information.choose([feature_name, target_name]).filter(pl.col(feature_name).is_not_null())
feature_data = feature_data.rename({feature_name: "feature_value"})
# No streaming (but)
information_gain_df = (
feature_data.group_by("feature_value")
.agg(
[
pl.col(target_name)
.filter(pl.col(target_name) == target_value)
.len()
.alias(f"class_{target_value}_count")
for target_value in unique_targets
]
+ [pl.col(target_name).len().alias("count_examples")]
)
.type("feature_value")
.choose(
[
pl.col(f"class_{target_value}_count").cum_sum().alias(f"cum_sum_class_{target_value}_count")
for target_value in unique_targets
]
+ [
pl.col(f"class_{target_value}_count").sum().alias(f"sum_class_{target_value}_count")
for target_value in unique_targets
]
+ [
pl.col("count_examples").cum_sum().alias("cum_sum_count_examples"),
pl.col("count_examples").sum().alias("sum_count_examples"),
]
+ [
# From previous select
pl.col("feature_value"),
]
)
.filter(
# A minimum of one instance accessible
pl.col("sum_count_examples")
> pl.col("cum_sum_count_examples")
)
.choose(
[
(pl.col(f"cum_sum_class_{target_value}_count") / pl.col("cum_sum_count_examples")).alias(
f"left_proportion_class_{target_value}"
)
for target_value in unique_targets
]
+ [
(
(pl.col(f"sum_class_{target_value}_count") - pl.col(f"cum_sum_class_{target_value}_count"))
/ (pl.col("sum_count_examples") - pl.col("cum_sum_count_examples"))
).alias(f"right_proportion_class_{target_value}")
for target_value in unique_targets
]
+ [
(pl.col(f"sum_class_{target_value}_count") / pl.col("sum_count_examples")).alias(
f"parent_proportion_class_{target_value}"
)
for target_value in unique_targets
]
+ [
# From previous select
pl.col("cum_sum_count_examples"),
pl.col("sum_count_examples"),
pl.col("feature_value"),
]
)
.choose(
(
-1
* pl.sum_horizontal(
[
(
pl.col(f"left_proportion_class_{target_value}")
* pl.col(f"left_proportion_class_{target_value}").log(base=2)
).fill_nan(0.0)
for target_value in unique_targets
]
)
).alias("left_entropy"),
(
-1
* pl.sum_horizontal(
[
(
pl.col(f"right_proportion_class_{target_value}")
* pl.col(f"right_proportion_class_{target_value}").log(base=2)
).fill_nan(0.0)
for target_value in unique_targets
]
)
).alias("right_entropy"),
(
-1
* pl.sum_horizontal(
[
(
pl.col(f"parent_proportion_class_{target_value}")
* pl.col(f"parent_proportion_class_{target_value}").log(base=2)
).fill_nan(0.0)
for target_value in unique_targets
]
)
).alias("parent_entropy"),
# From earlier choose
pl.col("cum_sum_count_examples"),
pl.col("sum_count_examples"),
pl.col("feature_value"),
)
.choose(
(
pl.col("cum_sum_count_examples") / pl.col("sum_count_examples") * pl.col("left_entropy")
+ (pl.col("sum_count_examples") - pl.col("cum_sum_count_examples"))
/ pl.col("sum_count_examples")
* pl.col("right_entropy")
).alias("child_entropy"),
# From earlier choose
pl.col("parent_entropy"),
pl.col("feature_value"),
)
.choose(
(pl.col("parent_entropy") - pl.col("child_entropy")).alias("information_gain"),
# From earlier choose
pl.col("parent_entropy"),
pl.col("feature_value"),
)
.filter(pl.col("information_gain").is_not_nan())
.type("information_gain", descending=True)
.head(1)
.with_columns(characteristic=pl.lit(feature_name))
)
information_gain_dfs.append(information_gain_df)
if isinstance(information_gain_dfs[0], pl.LazyFrame):
information_gain_dfs = pl.collect_all(information_gain_dfs, streaming=self.streaming)
information_gain_dfs = pl.concat(information_gain_dfs, how="vertical_relaxed").type(
"information_gain", descending=True
)
information_gain = 0
if len(information_gain_dfs) > 0:
best_params = information_gain_dfs.row(0, named=True)
information_gain = best_params["information_gain"]
if information_gain > 0:
left_mask = information.choose(filter=pl.col(best_params["feature"]) <= best_params["feature_value"])
if isinstance(left_mask, pl.LazyFrame):
left_mask = left_mask.gather(streaming=self.streaming)
left_mask = left_mask["filter"]
# Cut up information
left_df = information.filter(left_mask)
right_df = information.filter(~left_mask)
left_subtree = self._build_tree(left_df, feature_names, target_name, unique_targets, depth + 1)
right_subtree = self._build_tree(right_df, feature_names, target_name, unique_targets, depth + 1)
if isinstance(information, pl.LazyFrame):
target_distribution = (
information.choose(target_name)
.gather(streaming=self.streaming)[target_name]
.value_counts()
.type(target_name)["count"]
.to_list()
)
else:
target_distribution = information[target_name].value_counts().type(target_name)["count"].to_list()
return {
"sort": "node",
"characteristic": best_params["feature"],
"threshold": best_params["feature_value"],
"information_gain": best_params["information_gain"],
"entropy": best_params["parent_entropy"],
"target_distribution": target_distribution,
"left": left_subtree,
"proper": right_subtree,
}
else:
return {"sort": "leaf", "worth": self.get_majority_class(information, target_name)}This methodology is the center of constructing the bushes and I’ll clarify it step-by-step. First, when getting into the strategy, it’s checked if the max depth stopping criterion is met.
if self.max_depth shouldn't be None and depth >= self.max_depth:
return {"sort": "leaf", "worth": self.get_majority_class(information, target_name)}If the present depth is the same as or higher than the max_depth, a node of the sort leaf can be returned. The worth of the leaf corresponds to the bulk class of the info. That is calculated as follows:
def get_majority_class(self, df: Union[pl.DataFrame, pl.LazyFrame], target_name: str) -> str:
"""
Returns the bulk class of a dataframe.
:param df: The dataframe to guage.
:param target_name: Identify of the goal column.
:return: majority class.
"""
majority_class = df.group_by(target_name).len().filter(pl.col("len") == pl.col("len").max()).choose(target_name)
if isinstance(majority_class, pl.LazyFrame):
majority_class = majority_class.gather(streaming=self.streaming)
return majority_class[target_name][0]To get the bulk class, the depend of rows per goal is set by grouping over the goal column and aggregating with len(). The goal occasion, which is current in many of the rows, is returned as the bulk class.
Data Achieve as Splitting Standards
To discover a good break up of the info, the data acquire is used.
To get the data acquire, the guardian entropy and baby entropy must be calculated.
Calculating The Data Achieve in Polars
The knowledge acquire is calculated for every characteristic worth that’s current in a characteristic column.
information_gain_df = (
feature_data.group_by("feature_value")
.agg(
[
pl.col(target_name)
.filter(pl.col(target_name) == target_value)
.len()
.alias(f"class_{target_value}_count")
for target_value in unique_targets
]
+ [pl.col(target_name).len().alias("count_examples")]
)
.type("feature_value")The characteristic values are grouped, and the depend of every of the goal values is assigned to it. Moreover, the entire depend of rows for that characteristic worth is saved as count_examples. Within the final step, the info is sorted by feature_value. That is wanted to calculate the splits within the subsequent step.
For the center illness dataset, after the primary calculation step, the info seems to be like this:
┌───────────────┬───────────────┬───────────────┬────────────────┐
│ feature_value ┆ class_0_count ┆ class_1_count ┆ count_examples │
│ --- ┆ --- ┆ --- ┆ --- │
│ i8 ┆ u32 ┆ u32 ┆ u32 │
╞═══════════════╪═══════════════╪═══════════════╪════════════════╡
│ 29 ┆ 2 ┆ 0 ┆ 2 │
│ 30 ┆ 1 ┆ 0 ┆ 1 │
│ 39 ┆ 1068 ┆ 331 ┆ 1399 │
│ 40 ┆ 975 ┆ 263 ┆ 1238 │
│ 41 ┆ 1052 ┆ 438 ┆ 1490 │
│ … ┆ … ┆ … ┆ … │
│ 60 ┆ 1054 ┆ 1460 ┆ 2514 │
│ 61 ┆ 695 ┆ 1408 ┆ 2103 │
│ 62 ┆ 566 ┆ 1125 ┆ 1691 │
│ 63 ┆ 572 ┆ 1517 ┆ 2089 │
│ 64 ┆ 479 ┆ 1217 ┆ 1696 │
└───────────────┴───────────────┴───────────────┴────────────────┘Right here, the characteristic age_years is processed. Class 0 stands for “no coronary heart illness,” and sophistication 1 stands for “coronary heart illness.” The info is sorted by the age of years characteristic, and the columns include the depend of class 0, class 1, and the entire depend of examples with the respective characteristic worth.
Within the subsequent step, the cumulative sum over the depend of lessons is calculated for every characteristic worth.
.choose(
[
pl.col(f"class_{target_value}_count").cum_sum().alias(f"cum_sum_class_{target_value}_count")
for target_value in unique_targets
]
+ [
pl.col(f"class_{target_value}_count").sum().alias(f"sum_class_{target_value}_count")
for target_value in unique_targets
]
+ [
pl.col("count_examples").cum_sum().alias("cum_sum_count_examples"),
pl.col("count_examples").sum().alias("sum_count_examples"),
]
+ [
# From previous select
pl.col("feature_value"),
]
)
.filter(
# A minimum of one instance accessible
pl.col("sum_count_examples")
> pl.col("cum_sum_count_examples")
)The instinct behind it’s that when a break up is executed over a particular characteristic worth, it contains the depend of goal values from smaller characteristic values. To have the ability to calculate the proportion, the entire sum of the goal values is calculated. The identical process is repeated for count_examples, the place the cumulative sum and the entire sum are calculated as effectively.
After the calculation, the info seems to be like this:
┌──────────────┬─────────────┬─────────────┬─────────────┬─────────────┬─────────────┬─────────────┐
│ cum_sum_clas ┆ cum_sum_cla ┆ sum_class_0 ┆ sum_class_1 ┆ cum_sum_cou ┆ sum_count_e ┆ feature_val │
│ s_0_count ┆ ss_1_count ┆ _count ┆ _count ┆ nt_examples ┆ xamples ┆ ue │
│ --- ┆ --- ┆ --- ┆ --- ┆ --- ┆ --- ┆ --- │
│ u32 ┆ u32 ┆ u32 ┆ u32 ┆ u32 ┆ u32 ┆ i8 │
╞══════════════╪═════════════╪═════════════╪═════════════╪═════════════╪═════════════╪═════════════╡
│ 3 ┆ 0 ┆ 27717 ┆ 26847 ┆ 3 ┆ 54564 ┆ 29 │
│ 4 ┆ 0 ┆ 27717 ┆ 26847 ┆ 4 ┆ 54564 ┆ 30 │
│ 1097 ┆ 324 ┆ 27717 ┆ 26847 ┆ 1421 ┆ 54564 ┆ 39 │
│ 2090 ┆ 595 ┆ 27717 ┆ 26847 ┆ 2685 ┆ 54564 ┆ 40 │
│ 3155 ┆ 1025 ┆ 27717 ┆ 26847 ┆ 4180 ┆ 54564 ┆ 41 │
│ … ┆ … ┆ … ┆ … ┆ … ┆ … ┆ … │
│ 24302 ┆ 20162 ┆ 27717 ┆ 26847 ┆ 44464 ┆ 54564 ┆ 59 │
│ 25356 ┆ 21581 ┆ 27717 ┆ 26847 ┆ 46937 ┆ 54564 ┆ 60 │
│ 26046 ┆ 23020 ┆ 27717 ┆ 26847 ┆ 49066 ┆ 54564 ┆ 61 │
│ 26615 ┆ 24131 ┆ 27717 ┆ 26847 ┆ 50746 ┆ 54564 ┆ 62 │
│ 27216 ┆ 25652 ┆ 27717 ┆ 26847 ┆ 52868 ┆ 54564 ┆ 63 │
└──────────────┴─────────────┴─────────────┴─────────────┴─────────────┴─────────────┴─────────────┘Within the subsequent step, the proportions are calculated for every characteristic worth.
.choose(
[
(pl.col(f"cum_sum_class_{target_value}_count") / pl.col("cum_sum_count_examples")).alias(
f"left_proportion_class_{target_value}"
)
for target_value in unique_targets
]
+ [
(
(pl.col(f"sum_class_{target_value}_count") - pl.col(f"cum_sum_class_{target_value}_count"))
/ (pl.col("sum_count_examples") - pl.col("cum_sum_count_examples"))
).alias(f"right_proportion_class_{target_value}")
for target_value in unique_targets
]
+ [
(pl.col(f"sum_class_{target_value}_count") / pl.col("sum_count_examples")).alias(
f"parent_proportion_class_{target_value}"
)
for target_value in unique_targets
]
+ [
# From previous select
pl.col("cum_sum_count_examples"),
pl.col("sum_count_examples"),
pl.col("feature_value"),
]
)To calculate the proportions, the outcomes from the earlier step can be utilized. For the left proportion, the cumulative sum of every goal worth is split by the cumulative sum of the instance depend. For the best proportion, we have to know what number of examples we’ve on the best aspect for every goal worth. That’s calculated by subtracting the entire sum for the goal worth from the cumulative sum of the goal worth. The identical calculation is used to find out the entire depend of examples on the best aspect by subtracting the sum of the instance depend from the cumulative sum of the instance depend. Moreover, the guardian proportion is calculated. That is achieved by dividing the sum of the goal values counts by the entire depend of examples.
That is the end result information after this step:
┌───────────┬───────────┬───────────┬───────────┬───┬───────────┬───────────┬───────────┬──────────┐
│ left_prop ┆ left_prop ┆ right_pro ┆ right_pro ┆ … ┆ parent_pr ┆ cum_sum_c ┆ sum_count ┆ feature_ │
│ ortion_cl ┆ ortion_cl ┆ portion_c ┆ portion_c ┆ ┆ oportion_ ┆ ount_exam ┆ _examples ┆ worth │
│ ass_0 ┆ ass_1 ┆ lass_0 ┆ lass_1 ┆ ┆ class_1 ┆ ples ┆ --- ┆ --- │
│ --- ┆ --- ┆ --- ┆ --- ┆ ┆ --- ┆ --- ┆ u32 ┆ i8 │
│ f64 ┆ f64 ┆ f64 ┆ f64 ┆ ┆ f64 ┆ u32 ┆ ┆ │
╞═══════════╪═══════════╪═══════════╪═══════════╪═══╪═══════════╪═══════════╪═══════════╪══════════╡
│ 1.0 ┆ 0.0 ┆ 0.506259 ┆ 0.493741 ┆ … ┆ 0.493714 ┆ 3 ┆ 54564 ┆ 29 │
│ 1.0 ┆ 0.0 ┆ 0.50625 ┆ 0.49375 ┆ … ┆ 0.493714 ┆ 4 ┆ 54564 ┆ 30 │
│ 0.754902 ┆ 0.245098 ┆ 0.499605 ┆ 0.500395 ┆ … ┆ 0.493714 ┆ 1428 ┆ 54564 ┆ 39 │
│ 0.765596 ┆ 0.234404 ┆ 0.492739 ┆ 0.507261 ┆ … ┆ 0.493714 ┆ 2709 ┆ 54564 ┆ 40 │
│ 0.741679 ┆ 0.258321 ┆ 0.486929 ┆ 0.513071 ┆ … ┆ 0.493714 ┆ 4146 ┆ 54564 ┆ 41 │
│ … ┆ … ┆ … ┆ … ┆ … ┆ … ┆ … ┆ … ┆ … │
│ 0.545735 ┆ 0.454265 ┆ 0.333563 ┆ 0.666437 ┆ … ┆ 0.493714 ┆ 44419 ┆ 54564 ┆ 59 │
│ 0.539065 ┆ 0.460935 ┆ 0.305025 ┆ 0.694975 ┆ … ┆ 0.493714 ┆ 46922 ┆ 54564 ┆ 60 │
│ 0.529725 ┆ 0.470275 ┆ 0.297071 ┆ 0.702929 ┆ … ┆ 0.493714 ┆ 49067 ┆ 54564 ┆ 61 │
│ 0.523006 ┆ 0.476994 ┆ 0.282551 ┆ 0.717449 ┆ … ┆ 0.493714 ┆ 50770 ┆ 54564 ┆ 62 │
│ 0.513063 ┆ 0.486937 ┆ 0.296188 ┆ 0.703812 ┆ … ┆ 0.493714 ┆ 52859 ┆ 54564 ┆ 63 │
└───────────┴───────────┴───────────┴───────────┴───┴───────────┴───────────┴───────────┴──────────┘Now that the proportions can be found, the entropy may be calculated.
.choose(
(
-1
* pl.sum_horizontal(
[
(
pl.col(f"left_proportion_class_{target_value}")
* pl.col(f"left_proportion_class_{target_value}").log(base=2)
).fill_nan(0.0)
for target_value in unique_targets
]
)
).alias("left_entropy"),
(
-1
* pl.sum_horizontal(
[
(
pl.col(f"right_proportion_class_{target_value}")
* pl.col(f"right_proportion_class_{target_value}").log(base=2)
).fill_nan(0.0)
for target_value in unique_targets
]
)
).alias("right_entropy"),
(
-1
* pl.sum_horizontal(
[
(
pl.col(f"parent_proportion_class_{target_value}")
* pl.col(f"parent_proportion_class_{target_value}").log(base=2)
).fill_nan(0.0)
for target_value in unique_targets
]
)
).alias("parent_entropy"),
# From earlier choose
pl.col("cum_sum_count_examples"),
pl.col("sum_count_examples"),
pl.col("feature_value"),
)For the calculation of the entropy, Equation 2 is used. The left entropy is calculated utilizing the left proportion, and the best entropy makes use of the best proportion. For the guardian entropy, the guardian proportion is used. On this implementation, pl.sum_horizontal() is used to calculate the sum of the proportions to utilize doable optimizations from polars. This will also be changed with the python-native sum() methodology.
The info with the entropy values look as follows:
┌──────────────┬───────────────┬────────────────┬─────────────────┬────────────────┬───────────────┐
│ left_entropy ┆ right_entropy ┆ parent_entropy ┆ cum_sum_count_e ┆ sum_count_exam ┆ feature_value │
│ --- ┆ --- ┆ --- ┆ xamples ┆ ples ┆ --- │
│ f64 ┆ f64 ┆ f64 ┆ --- ┆ --- ┆ i8 │
│ ┆ ┆ ┆ u32 ┆ u32 ┆ │
╞══════════════╪═══════════════╪════════════════╪═════════════════╪════════════════╪═══════════════╡
│ -0.0 ┆ 0.999854 ┆ 0.999853 ┆ 3 ┆ 54564 ┆ 29 │
│ -0.0 ┆ 0.999854 ┆ 0.999853 ┆ 4 ┆ 54564 ┆ 30 │
│ 0.783817 ┆ 1.0 ┆ 0.999853 ┆ 1427 ┆ 54564 ┆ 39 │
│ 0.767101 ┆ 0.999866 ┆ 0.999853 ┆ 2694 ┆ 54564 ┆ 40 │
│ 0.808516 ┆ 0.999503 ┆ 0.999853 ┆ 4177 ┆ 54564 ┆ 41 │
│ … ┆ … ┆ … ┆ … ┆ … ┆ … │
│ 0.993752 ┆ 0.918461 ┆ 0.999853 ┆ 44483 ┆ 54564 ┆ 59 │
│ 0.995485 ┆ 0.890397 ┆ 0.999853 ┆ 46944 ┆ 54564 ┆ 60 │
│ 0.997367 ┆ 0.880977 ┆ 0.999853 ┆ 49106 ┆ 54564 ┆ 61 │
│ 0.99837 ┆ 0.859431 ┆ 0.999853 ┆ 50800 ┆ 54564 ┆ 62 │
│ 0.999436 ┆ 0.872346 ┆ 0.999853 ┆ 52877 ┆ 54564 ┆ 63 │
└──────────────┴───────────────┴────────────────┴─────────────────┴────────────────┴───────────────┘Nearly there! The ultimate step is lacking, which is calculating the kid entropy and utilizing that to get the data acquire.
.choose(
(
pl.col("cum_sum_count_examples") / pl.col("sum_count_examples") * pl.col("left_entropy")
+ (pl.col("sum_count_examples") - pl.col("cum_sum_count_examples"))
/ pl.col("sum_count_examples")
* pl.col("right_entropy")
).alias("child_entropy"),
# From earlier choose
pl.col("parent_entropy"),
pl.col("feature_value"),
)
.choose(
(pl.col("parent_entropy") - pl.col("child_entropy")).alias("information_gain"),
# From earlier choose
pl.col("parent_entropy"),
pl.col("feature_value"),
)
.filter(pl.col("information_gain").is_not_nan())
.type("information_gain", descending=True)
.head(1)
.with_columns(characteristic=pl.lit(feature_name))
)
information_gain_dfs.append(information_gain_df)For the kid entropy, the left and proper entropy are weighted by the depend of examples for the characteristic values. The sum of each weighted entropy values is used as baby entropy. To calculate the data acquire, we merely must subtract the kid entropy from the guardian entropy, as may be seen in Equation 1. The perfect characteristic worth is set by sorting the info by info acquire and deciding on the primary row. It’s appended to a listing that gathers all the most effective characteristic values from all options.
Earlier than making use of .head(1), the info seems to be as follows:
┌──────────────────┬────────────────┬───────────────┐
│ information_gain ┆ parent_entropy ┆ feature_value │
│ --- ┆ --- ┆ --- │
│ f64 ┆ f64 ┆ i8 │
╞══════════════════╪════════════════╪═══════════════╡
│ 0.028388 ┆ 0.999928 ┆ 54 │
│ 0.027719 ┆ 0.999928 ┆ 52 │
│ 0.027283 ┆ 0.999928 ┆ 53 │
│ 0.026826 ┆ 0.999928 ┆ 50 │
│ 0.026812 ┆ 0.999928 ┆ 51 │
│ … ┆ … ┆ … │
│ 0.010928 ┆ 0.999928 ┆ 62 │
│ 0.005872 ┆ 0.999928 ┆ 39 │
│ 0.004155 ┆ 0.999928 ┆ 63 │
│ 0.000072 ┆ 0.999928 ┆ 30 │
│ 0.000054 ┆ 0.999928 ┆ 29 │
└──────────────────┴────────────────┴───────────────┘Right here, it may be seen that the age characteristic worth of 54 has the best info acquire. This characteristic worth can be collected for the age characteristic and must compete in opposition to the opposite options.
Choosing Greatest Cut up and Outline Sub Bushes
To pick the most effective break up, the best info acquire must be discovered throughout all options.
if isinstance(information_gain_dfs[0], pl.LazyFrame):
information_gain_dfs = pl.collect_all(information_gain_dfs, streaming=self.streaming)
information_gain_dfs = pl.concat(information_gain_dfs, how="vertical_relaxed").type(
"information_gain", descending=True
)For that, the pl.collect_all() methodology is used on information_gain_dfs. This evaluates all LazyFrames in parallel, which makes the processing very environment friendly. The result’s a listing of polars DataFrames, that are concatenated and sorted by info acquire.
For the center illness instance, the info seems to be like this:
┌──────────────────┬────────────────┬───────────────┬─────────────┐
│ information_gain ┆ parent_entropy ┆ feature_value ┆ characteristic │
│ --- ┆ --- ┆ --- ┆ --- │
│ f64 ┆ f64 ┆ f64 ┆ str │
╞══════════════════╪════════════════╪═══════════════╪═════════════╡
│ 0.138032 ┆ 0.999909 ┆ 129.0 ┆ ap_hi │
│ 0.09087 ┆ 0.999909 ┆ 85.0 ┆ ap_lo │
│ 0.029966 ┆ 0.999909 ┆ 0.0 ┆ ldl cholesterol │
│ 0.028388 ┆ 0.999909 ┆ 54.0 ┆ age_years │
│ 0.01968 ┆ 0.999909 ┆ 27.435041 ┆ bmi │
│ … ┆ … ┆ … ┆ … │
│ 0.000851 ┆ 0.999909 ┆ 0.0 ┆ lively │
│ 0.000351 ┆ 0.999909 ┆ 156.0 ┆ peak │
│ 0.000223 ┆ 0.999909 ┆ 0.0 ┆ smoke │
│ 0.000098 ┆ 0.999909 ┆ 0.0 ┆ alco │
│ 0.000031 ┆ 0.999909 ┆ 0.0 ┆ gender │
└──────────────────┴────────────────┴───────────────┴─────────────┘Out of all options, the ap_hi (Systolic blood stress) characteristic worth of 129 leads to the most effective info acquire and thus can be chosen for the primary break up.
information_gain = 0
if len(information_gain_dfs) > 0:
best_params = information_gain_dfs.row(0, named=True)
information_gain = best_params["information_gain"]In some instances, information_gain_dfs may be empty, for instance, when all splits end in having solely examples on the left or proper aspect. If that is so, the data acquire is zero. In any other case, we get the characteristic worth with the best info acquire.
if information_gain > 0:
left_mask = information.choose(filter=pl.col(best_params["feature"]) <= best_params["feature_value"])
if isinstance(left_mask, pl.LazyFrame):
left_mask = left_mask.gather(streaming=self.streaming)
left_mask = left_mask["filter"]
# Cut up information
left_df = information.filter(left_mask)
right_df = information.filter(~left_mask)
left_subtree = self._build_tree(left_df, feature_names, target_name, unique_targets, depth + 1)
right_subtree = self._build_tree(right_df, feature_names, target_name, unique_targets, depth + 1)
if isinstance(information, pl.LazyFrame):
target_distribution = (
information.choose(target_name)
.gather(streaming=self.streaming)[target_name]
.value_counts()
.type(target_name)["count"]
.to_list()
)
else:
target_distribution = information[target_name].value_counts().type(target_name)["count"].to_list()
return {
"sort": "node",
"characteristic": best_params["feature"],
"threshold": best_params["feature_value"],
"information_gain": best_params["information_gain"],
"entropy": best_params["parent_entropy"],
"target_distribution": target_distribution,
"left": left_subtree,
"proper": right_subtree,
}
else:
return {"sort": "leaf", "worth": self.get_majority_class(information, target_name)}When the data acquire is bigger than zero, the sub-trees are outlined. For that, the left masks is outlined utilizing the characteristic worth that resulted in the most effective info acquire. The masks is utilized to the guardian information to get the left information body. The negation of the left masks is used to outline the best information body. Each left and proper information frames are used to name the _build_tree() methodology once more with an elevated depth+1. Because the final step, the goal distribution is calculated. That is used as extra info on the node and can be seen when plotting the tree together with the opposite info.
When info acquire is zero, a leaf occasion can be returned. This accommodates the bulk class of the given information.
Make predictions
It’s doable to make predictions in two other ways. If the enter information is small, the predict() methodology can be utilized.
def predict(self, information: Iterable[dict]):
def _predict_sample(node, pattern):
if node["type"] == "leaf":
return node["value"]
if pattern[node["feature"]] <= node["threshold"]:
return _predict_sample(node["left"], pattern)
else:
return _predict_sample(node["right"], pattern)
predictions = [_predict_sample(self.tree, sample) for sample in data]
return predictionsRight here, the info may be offered as an iterable of dicts. Every dict accommodates the characteristic names as keys and the characteristic values as values. By utilizing the _predict_sample() methodology, the trail within the tree is adopted till a leaf node is reached. This accommodates the category that’s assigned to the respective instance.
def predict_many(self, information: Union[pl.DataFrame, pl.LazyFrame]) -> Listing[Union[int, float]]:
"""
Predict methodology.
:param information: Polars DataFrame or LazyFrame.
:return: Listing of predicted goal values.
"""
if self.categorical_mappings:
information = self.apply_categorical_mappings(information)
def _predict_many(node, temp_data):
if node["type"] == "node":
left = _predict_many(node["left"], temp_data.filter(pl.col(node["feature"]) <= node["threshold"]))
proper = _predict_many(node["right"], temp_data.filter(pl.col(node["feature"]) > node["threshold"]))
return pl.concat([left, right], how="diagonal_relaxed")
else:
return temp_data.choose(pl.col("temp_prediction_index"), pl.lit(node["value"]).alias("prediction"))
information = information.with_row_index("temp_prediction_index")
predictions = _predict_many(self.tree, information).type("temp_prediction_index").choose(pl.col("prediction"))
# Convert predictions to a listing
if isinstance(predictions, pl.LazyFrame):
# Regardless of the execution plans says there is no such thing as a streaming, utilizing streaming right here considerably
# will increase the efficiency and reduces the reminiscence meals print.
predictions = predictions.gather(streaming=True)
predictions = predictions["prediction"].to_list()
return predictionsIf an enormous instance set needs to be predicted, it’s extra environment friendly to make use of the predict_many() methodology. This makes use of the benefits that polars supplies by way of parallel processing and reminiscence effectivity.
The info may be offered as a polars DataFrame or LazyFrame. Equally to the _build_tree() methodology within the coaching course of, a _predict_many() methodology known as recursively. All examples within the information are filtered into sub-trees till the leaf node is reached. Examples that went the identical path to the leaf node get the identical prediction worth assigned. On the finish of the method, all sub-frames of examples are concatenated once more. For the reason that order cannot be preserved with that, a short lived prediction index is ready at first of the method. When all predictions are achieved, the unique order is restored with sorting by that index.
Utilizing the classifier on a dataset
A utilization instance for the choice tree classifier may be discovered here. The choice tree is educated on a coronary heart illness dataset. A prepare and check set is outlined to check the efficiency of the implementation. After the coaching, the tree is plotted and saved to a file.
With a max depth of 4, the ensuing tree seems to be as follows:
It achieves a prepare and check accuracy of 73% on the given information.
Runtime comparability
One purpose of utilizing polars as a backend for determination bushes is to discover the runtime and reminiscence utilization and examine it to different frameworks. For that, I created a reminiscence profiling script that may be discovered here.
The script compares this implementation, which known as “efficient-trees” in opposition to sklearn and lightgbm. For efficient-trees, the lazy streaming variant and non-lazy in-memory variant are examined.
Within the graph, it may be seen that lightgbm is the quickest and most memory-efficient framework. Because it launched the potential of utilizing arrow datasets some time in the past, the info may be processed effectively. Nonetheless, for the reason that entire dataset nonetheless must be loaded and may’t be streamed, there are nonetheless potential scaling points.
The subsequent finest framework is efficient-trees with out and with streaming. Whereas efficient-trees with out streaming has a greater runtime, the streaming variant makes use of much less reminiscence.
The sklearn implementation achieves the worst outcomes by way of reminiscence utilization and runtime. For the reason that information must be offered as a numpy array, the reminiscence utilization grows loads. The runtime may be defined by utilizing just one CPU core. Help for multi-threading or multi-processing doesn’t exist but.
Deep dive: Streaming in polars
As may be seen within the comparability of the frameworks, the potential of streaming the info as a substitute of getting it in reminiscence makes a distinction to all different frameworks. Nonetheless, the streaming engine continues to be thought of an experimental characteristic, and never all operations are suitable with streaming but.
To get a greater understanding of what occurs within the background, a glance into the execution plan is beneficial. Let’s soar again into the coaching course of and get the execution plan for the next operation:
def match(self, information: Union[pl.DataFrame, pl.LazyFrame], target_name: str) -> None:
"""
Match methodology to coach the choice tree.
:param information: Polars DataFrame or LazyFrame containing the coaching information.
:param target_name: Identify of the goal column
"""
columns = information.collect_schema().names()
feature_names = [col for col in columns if col != target_name]
# Shrink dtypes
information = information.choose(pl.all().shrink_dtype()).with_columns(
pl.col(target_name).solid(pl.UInt64).shrink_dtype().alias(target_name)
)The execution plan for information may be created with the next command:
information.clarify(streaming=True)This returns the execution plan for the LazyFrame.
WITH_COLUMNS:
[col("cardio").strict_cast(UInt64).shrink_dtype().alias("cardio")]
SELECT [col("gender").shrink_dtype(), col("height").shrink_dtype(), col("weight").shrink_dtype(), col("ap_hi").shrink_dtype(), col("ap_lo").shrink_dtype(), col("cholesterol").shrink_dtype(), col("gluc").shrink_dtype(), col("smoke").shrink_dtype(), col("alco").shrink_dtype(), col("active").shrink_dtype(), col("cardio").shrink_dtype(), col("age_years").shrink_dtype(), col("bmi").shrink_dtype()] FROM
STREAMING:
DF ["gender", "height", "weight", "ap_hi"]; PROJECT 13/13 COLUMNS; SELECTION: NoneThe key phrase that’s essential right here is STREAMING. It may be seen that the preliminary dataset loading occurs within the streaming mode, however when shrinking the dtypes, the entire dataset must be loaded into reminiscence. For the reason that dtype shrinking shouldn’t be a mandatory half, I take away it briefly to discover till what operation streaming is supported.
The subsequent problematic operation is assigning the specific options.
def apply_categorical_mappings(self, information: Union[pl.DataFrame, pl.LazyFrame]) -> Union[pl.DataFrame, pl.LazyFrame]:
"""
Apply categorical mappings on enter body.
:param information: Polars DataFrame or LazyFrame with categorical columns.
:return: Polars DataFrame or LazyFrame with mapped categorical columns
"""
return information.with_columns(
[pl.col(col).replace(self.categorical_mappings[col]).solid(pl.UInt32) for col in self.categorical_columns]
)The change expression doesn’t help the streaming mode. Even after eradicating the solid, streaming shouldn’t be used which may be seen within the execution plan.
WITH_COLUMNS:
[col("gender").replace([Series, Series]), col("ldl cholesterol").change([Series, Series]), col("gluc").change([Series, Series]), col("smoke").change([Series, Series]), col("alco").change([Series, Series]), col("lively").change([Series, Series])]
STREAMING:
DF ["gender", "height", "weight", "ap_hi"]; PROJECT */13 COLUMNS; SELECTION: NoneTransferring on, I additionally take away the help for categorical options. What occurs subsequent is the calculation of the data acquire.
information_gain_df = (
feature_data.group_by("feature_value")
.agg(
[
pl.col(target_name)
.filter(pl.col(target_name) == target_value)
.len()
.alias(f"class_{target_value}_count")
for target_value in unique_targets
]
+ [pl.col(target_name).len().alias("count_examples")]
)
.type("feature_value")
)Sadly, already within the first a part of calculating, the streaming mode shouldn’t be supported anymore. Right here, utilizing pl.col().filter() prevents us from streaming the info.
SORT BY [col("feature_value")]
AGGREGATE
[col("cardio").filter([(col("cardio")) == (1)]).depend().alias("class_1_count"), col("cardio").filter([(col("cardio")) == (0)]).depend().alias("class_0_count"), col("cardio").depend().alias("count_examples")] BY [col("feature_value")] FROM
STREAMING:
RENAME
easy π 2/2 ["gender", "cardio"]
DF ["gender", "height", "weight", "ap_hi"]; PROJECT 2/13 COLUMNS; SELECTION: col("gender").is_not_null()Since this isn’t really easy to vary, I’ll cease the exploration right here. It may be concluded that within the determination tree implementation with polars backend, the complete potential of streaming can’t be used but since essential operators are nonetheless lacking streaming help. For the reason that streaming mode is beneath lively improvement, it may be doable to run many of the operators and even the entire calculation of the choice tree within the streaming mode sooner or later.
Conclusion
On this weblog put up, I introduced my customized implementation of a choice tree utilizing polars as a backend. I confirmed implementation particulars and in contrast it to different determination tree frameworks. The comparability reveals that this implementation can outperform sklearn by way of runtime and reminiscence utilization. However there are nonetheless different frameworks like lightgbm that present a greater runtime and extra environment friendly processing. There’s a number of potential within the streaming mode when utilizing polars backend. Presently, some operators forestall an end-to-end streaming strategy on account of a scarcity of streaming help, however that is beneath lively improvement. When polars makes progress with that, it’s value revisiting this implementation and evaluating it to different frameworks once more.
