import numpy as np
import pandas as pd
from sklearn.metrics import auc
from sklearn.utils.extmath import stable_cumsum
from sklearn.utils.validation import check_consistent_length
[docs]def uplift_curve(y_true, uplift, treatment):
"""Compute Uplift curve.
For computing the area under the Uplift Curve, see :func:`.uplift_auc_score`.
Args:
y_true (1d array-like): Correct (true) target values.
uplift (1d array-like): Predicted uplift, as returned by a model.
treatment (1d array-like): Treatment labels.
Returns:
array (shape = [>2]), array (shape = [>2]): Points on a curve.
See also:
:func:`.uplift_auc_score`: Compute normalized Area Under the Uplift curve from prediction scores.
:func:`.perfect_uplift_curve`: Compute the perfect Uplift curve.
:func:`.plot_uplift_curve`: Plot Uplift curves from predictions.
:func:`.qini_curve`: Compute Qini curve.
References:
Devriendt, F., Guns, T., & Verbeke, W. (2020). Learning to rank for uplift modeling. ArXiv, abs/2002.05897.
"""
# TODO: check the treatment is binary
y_true, uplift, treatment = np.array(y_true), np.array(uplift), np.array(treatment)
desc_score_indices = np.argsort(uplift, kind="mergesort")[::-1]
y_true, uplift, treatment = y_true[desc_score_indices], uplift[desc_score_indices], treatment[desc_score_indices]
y_true_ctrl, y_true_trmnt = y_true.copy(), y_true.copy()
y_true_ctrl[treatment == 1] = 0
y_true_trmnt[treatment == 0] = 0
distinct_value_indices = np.where(np.diff(uplift))[0]
threshold_indices = np.r_[distinct_value_indices, uplift.size - 1]
num_trmnt = stable_cumsum(treatment)[threshold_indices]
y_trmnt = stable_cumsum(y_true_trmnt)[threshold_indices]
num_all = threshold_indices + 1
num_ctrl = num_all - num_trmnt
y_ctrl = stable_cumsum(y_true_ctrl)[threshold_indices]
curve_values = (np.divide(y_trmnt, num_trmnt, out=np.zeros_like(y_trmnt), where=num_trmnt != 0) -
np.divide(y_ctrl, num_ctrl, out=np.zeros_like(y_ctrl), where=num_ctrl != 0)) * num_all
if num_all.size == 0 or curve_values[0] != 0 or num_all[0] != 0:
# Add an extra threshold position if necessary
# to make sure that the curve starts at (0, 0)
num_all = np.r_[0, num_all]
curve_values = np.r_[0, curve_values]
return num_all, curve_values
[docs]def perfect_uplift_curve(y_true, treatment):
"""Compute the perfect (optimum) Uplift curve.
This is a function, given points on a curve. For computing the
area under the Uplift Curve, see :func:`.uplift_auc_score`.
Args:
y_true (1d array-like): Correct (true) target values.
treatment (1d array-like): Treatment labels.
Returns:
array (shape = [>2]), array (shape = [>2]): Points on a curve.
See also:
:func:`.uplift_curve`: Compute the area under the Qini curve.
:func:`.uplift_auc_score`: Compute normalized Area Under the Uplift curve from prediction scores.
:func:`.plot_uplift_curve`: Plot Uplift curves from predictions.
"""
check_consistent_length(y_true, treatment)
y_true, treatment = np.array(y_true), np.array(treatment)
cr_num = np.sum((y_true == 1) & (treatment == 0)) # Control Responders
tn_num = np.sum((y_true == 0) & (treatment == 1)) # Treated Non-Responders
# express an ideal uplift curve through y_true and treatment
summand = y_true if cr_num > tn_num else treatment
perfect_uplift = 2 * (y_true == treatment) + summand
return uplift_curve(y_true, perfect_uplift, treatment)
[docs]def uplift_auc_score(y_true, uplift, treatment):
"""Compute normalized Area Under the Uplift Curve from prediction scores.
By computing the area under the Uplift curve, the curve information is summarized in one number.
For binary outcomes the ratio of the actual uplift gains curve above the diagonal to that of
the optimum Uplift Curve.
Args:
y_true (1d array-like): Correct (true) target values.
uplift (1d array-like): Predicted uplift, as returned by a model.
treatment (1d array-like): Treatment labels.
Returns:
float: Area Under the Uplift Curve.
See also:
:func:`.uplift_curve`: Compute Uplift curve.
:func:`.perfect_uplift_curve`: Compute the perfect (optimum) Uplift curve.
:func:`.plot_uplift_curve`: Plot Uplift curves from predictions.
:func:`.qini_auc_score`: Compute normalized Area Under the Qini Curve from prediction scores.
"""
check_consistent_length(y_true, uplift, treatment)
y_true, uplift, treatment = np.array(y_true), np.array(uplift), np.array(treatment)
x_actual, y_actual = uplift_curve(y_true, uplift, treatment)
x_perfect, y_perfect = perfect_uplift_curve(y_true, treatment)
x_baseline, y_baseline = np.array([0, x_perfect[-1]]), np.array([0, y_perfect[-1]])
auc_score_baseline = auc(x_baseline, y_baseline)
auc_score_perfect = auc(x_perfect, y_perfect) - auc_score_baseline
auc_score_actual = auc(x_actual, y_actual) - auc_score_baseline
return auc_score_actual / auc_score_perfect
[docs]def qini_curve(y_true, uplift, treatment):
"""Compute Qini curve.
For computing the area under the Qini Curve, see :func:`.qini_auc_score`.
Args:
y_true (1d array-like): Correct (true) target values.
uplift (1d array-like): Predicted uplift, as returned by a model.
treatment (1d array-like): Treatment labels.
Returns:
array (shape = [>2]), array (shape = [>2]): Points on a curve.
See also:
:func:`.uplift_curve`: Compute the area under the Qini curve.
:func:`.perfect_qini_curve`: Compute the perfect Qini curve.
:func:`.plot_qini_curves`: Plot Qini curves from predictions..
:func:`.uplift_curve`: Compute Uplift curve.
References:
Nicholas J Radcliffe. (2007). Using control groups to target on predicted lift:
Building and assessing uplift model. Direct Marketing Analytics Journal, (3):14–21, 2007.
Devriendt, F., Guns, T., & Verbeke, W. (2020). Learning to rank for uplift modeling. ArXiv, abs/2002.05897.
"""
# TODO: check the treatment is binary
y_true, uplift, treatment = np.array(y_true), np.array(uplift), np.array(treatment)
desc_score_indices = np.argsort(uplift, kind="mergesort")[::-1]
y_true = y_true[desc_score_indices]
treatment = treatment[desc_score_indices]
uplift = uplift[desc_score_indices]
y_true_ctrl, y_true_trmnt = y_true.copy(), y_true.copy()
y_true_ctrl[treatment == 1] = 0
y_true_trmnt[treatment == 0] = 0
distinct_value_indices = np.where(np.diff(uplift))[0]
threshold_indices = np.r_[distinct_value_indices, uplift.size - 1]
num_trmnt = stable_cumsum(treatment)[threshold_indices]
y_trmnt = stable_cumsum(y_true_trmnt)[threshold_indices]
num_all = threshold_indices + 1
num_ctrl = num_all - num_trmnt
y_ctrl = stable_cumsum(y_true_ctrl)[threshold_indices]
curve_values = y_trmnt - y_ctrl * np.divide(num_trmnt, num_ctrl, out=np.zeros_like(num_trmnt), where=num_ctrl != 0)
if num_all.size == 0 or curve_values[0] != 0 or num_all[0] != 0:
# Add an extra threshold position if necessary
# to make sure that the curve starts at (0, 0)
num_all = np.r_[0, num_all]
curve_values = np.r_[0, curve_values]
return num_all, curve_values
[docs]def perfect_qini_curve(y_true, treatment, negative_effect=True):
"""Compute the perfect (optimum) Qini curve.
For computing the area under the Qini Curve, see :func:`.qini_auc_score`.
Args:
y_true (1d array-like): Correct (true) target values.
treatment (1d array-like): Treatment labels.
negative_effect (bool): If True, optimum Qini Curve contains the negative effects
(negative uplift because of campaign). Otherwise, optimum Qini Curve will not
contain the negative effects.
Returns:
array (shape = [>2]), array (shape = [>2]): Points on a curve.
See also:
:func:`.qini_curve`: Compute Qini curve.
:func:`.qini_auc_score`: Compute the area under the Qini curve.
:func:`.plot_qini_curves`: Plot Qini curves from predictions..
"""
check_consistent_length(y_true, treatment)
n_samples = len(y_true)
y_true, treatment = np.array(y_true), np.array(treatment)
if not isinstance(negative_effect, bool):
raise TypeError(f'Negative_effects flag should be bool, got: {type(negative_effect)}')
# express an ideal uplift curve through y_true and treatment
if negative_effect:
x_perfect, y_perfect = qini_curve(
y_true, y_true * treatment - y_true * (1 - treatment), treatment
)
else:
ratio_random = (y_true[treatment == 1].sum() - len(y_true[treatment == 1]) *
y_true[treatment == 0].sum() / len(y_true[treatment == 0]))
x_perfect, y_perfect = np.array([0, ratio_random, n_samples]), np.array([0, ratio_random, ratio_random])
return x_perfect, y_perfect
[docs]def qini_auc_score(y_true, uplift, treatment, negative_effect=True):
"""Compute normalized Area Under the Qini curve (aka Qini coefficient) from prediction scores.
By computing the area under the Qini curve, the curve information is summarized in one number.
For binary outcomes the ratio of the actual uplift gains curve above the diagonal to that of
the optimum Qini curve.
Args:
y_true (1d array-like): Correct (true) target values.
uplift (1d array-like): Predicted uplift, as returned by a model.
treatment (1d array-like): Treatment labels.
negative_effect (bool): If True, optimum Qini Curve contains the negative effects
(negative uplift because of campaign). Otherwise, optimum Qini Curve will not contain the negative effects.
.. versionadded:: 0.2.0
Returns:
float: Qini coefficient.
See also:
:func:`.qini_curve`: Compute Qini curve.
:func:`.perfect_qini_curve`: Compute the perfect (optimum) Qini curve.
:func:`.plot_qini_curves`: Plot Qini curves from predictions..
:func:`.uplift_auc_score`: Compute normalized Area Under the Uplift curve from prediction scores.
References:
Nicholas J Radcliffe. (2007). Using control groups to target on predicted lift:
Building and assessing uplift model. Direct Marketing Analytics Journal, (3):14–21, 2007.
"""
# ToDO: Add Continuous Outcomes
check_consistent_length(y_true, uplift, treatment)
y_true, uplift, treatment = np.array(y_true), np.array(uplift), np.array(treatment)
if not isinstance(negative_effect, bool):
raise TypeError(f'Negative_effects flag should be bool, got: {type(negative_effect)}')
x_actual, y_actual = qini_curve(y_true, uplift, treatment)
x_perfect, y_perfect = perfect_qini_curve(y_true, treatment, negative_effect)
x_baseline, y_baseline = np.array([0, x_perfect[-1]]), np.array([0, y_perfect[-1]])
auc_score_baseline = auc(x_baseline, y_baseline)
auc_score_perfect = auc(x_perfect, y_perfect) - auc_score_baseline
auc_score_actual = auc(x_actual, y_actual) - auc_score_baseline
return auc_score_actual / auc_score_perfect
[docs]def uplift_at_k(y_true, uplift, treatment, strategy, k=0.3):
"""Compute uplift at first k observations by uplift of the total sample.
Args:
y_true (1d array-like): Correct (true) target values.
uplift (1d array-like): Predicted uplift, as returned by a model.
treatment (1d array-like): Treatment labels.
k (float or int): If float, should be between 0.0 and 1.0 and represent the proportion of the dataset
to include in the computation of uplift. If int, represents the absolute number of samples.
strategy (string, ['overall', 'by_group']): Determines the calculating strategy.
* ``'overall'``:
The first step is taking the first k observations of all test data ordered by uplift prediction
(overall both groups - control and treatment) and conversions in treatment and control groups
calculated only on them. Then the difference between these conversions is calculated.
* ``'by_group'``:
Separately calculates conversions in top k observations in each group (control and treatment)
sorted by uplift predictions. Then the difference between these conversions is calculated
.. versionchanged:: 0.1.0
* Add supporting absolute values for ``k`` parameter
* Add parameter ``strategy``
Returns:
float: Uplift score at first k observations of the total sample.
See also:
:func:`.uplift_auc_score`: Compute normalized Area Under the Uplift curve from prediction scores.
:func:`.qini_auc_score`: Compute normalized Area Under the Qini Curve from prediction scores.
"""
# ToDo: checker that treatment is binary and all groups is not empty
check_consistent_length(y_true, uplift, treatment)
y_true, uplift, treatment = np.array(y_true), np.array(uplift), np.array(treatment)
strategy_methods = ['overall', 'by_group']
if strategy not in strategy_methods:
raise ValueError(f'Uplift score supports only calculating methods in {strategy_methods},'
f' got {strategy}.'
)
n_samples = len(y_true)
order = np.argsort(uplift, kind='mergesort')[::-1]
_, treatment_counts = np.unique(treatment, return_counts=True)
n_samples_ctrl = treatment_counts[0]
n_samples_trmnt = treatment_counts[1]
k_type = np.asarray(k).dtype.kind
if (k_type == 'i' and (k >= n_samples or k <= 0)
or k_type == 'f' and (k <= 0 or k >= 1)):
raise ValueError(f'k={k} should be either positive and smaller'
f' than the number of samples {n_samples} or a float in the '
f'(0, 1) range')
if k_type not in ('i', 'f'):
raise ValueError(f'Invalid value for k: {k_type}')
if strategy == 'overall':
if k_type == 'f':
n_size = int(n_samples * k)
else:
n_size = k
# ToDo: _checker_ there are observations among two groups among first k
score_ctrl = y_true[order][:n_size][treatment[order][:n_size] == 0].mean()
score_trmnt = y_true[order][:n_size][treatment[order][:n_size] == 1].mean()
else: # strategy == 'by_group':
if k_type == 'f':
n_ctrl = int((treatment == 0).sum() * k)
n_trmnt = int((treatment == 1).sum() * k)
else:
n_ctrl = k
n_trmnt = k
if n_ctrl > n_samples_ctrl:
raise ValueError(f'With k={k}, the number of the first k observations'
' bigger than the number of samples'
f'in the control group: {n_samples_ctrl}'
)
if n_trmnt > n_samples_trmnt:
raise ValueError(f'With k={k}, the number of the first k observations'
' bigger than the number of samples'
f'in the treatment group: {n_samples_ctrl}'
)
score_ctrl = y_true[order][treatment[order] == 0][:n_ctrl].mean()
score_trmnt = y_true[order][treatment[order] == 1][:n_trmnt].mean()
return score_trmnt - score_ctrl
[docs]def response_rate_by_percentile(y_true, uplift, treatment, group, strategy='overall', bins=10):
"""Compute response rate (target mean in the control or treatment group) at each percentile.
Args:
y_true (1d array-like): Correct (true) target values.
uplift (1d array-like): Predicted uplift, as returned by a model.
treatment (1d array-like): Treatment labels.
group (string, ['treatment', 'control']): Group type for computing response rate: treatment or control.
* ``'treatment'``:
Values equal 1 in the treatment column.
* ``'control'``:
Values equal 0 in the treatment column.
strategy (string, ['overall', 'by_group']): Determines the calculating strategy. Default is 'overall'.
* ``'overall'``:
The first step is taking the first k observations of all test data ordered by uplift prediction
(overall both groups - control and treatment) and conversions in treatment and control groups
calculated only on them. Then the difference between these conversions is calculated.
* ``'by_group'``:
Separately calculates conversions in top k observations in each group (control and treatment)
sorted by uplift predictions. Then the difference between these conversions is calculated.
bins (int): Determines the number of bins (and relative percentile) in the data. Default is 10.
Returns:
array (shape = [>2]), array (shape = [>2]), array (shape = [>2]):
response rate at each percentile for control or treatment group,
variance of the response rate at each percentile,
group size at each percentile.
"""
group_types = ['treatment', 'control']
strategy_methods = ['overall', 'by_group']
n_samples = len(y_true)
check_consistent_length(y_true, uplift, treatment)
if group not in group_types:
raise ValueError(f'Response rate supports only group types in {group_types},'
f' got {group}.')
if strategy not in strategy_methods:
raise ValueError(f'Response rate supports only calculating methods in {strategy_methods},'
f' got {strategy}.')
if not isinstance(bins, int) or bins <= 0:
raise ValueError(f'Bins should be positive integer. Invalid value bins: {bins}')
if bins >= n_samples:
raise ValueError(f'Number of bins = {bins} should be smaller than the length of y_true {n_samples}')
y_true, uplift, treatment = np.array(y_true), np.array(uplift), np.array(treatment)
order = np.argsort(uplift, kind='mergesort')[::-1]
trmnt_flag = 1 if group == 'treatment' else 0
if strategy == 'overall':
y_true_bin = np.array_split(y_true[order], bins)
trmnt_bin = np.array_split(treatment[order], bins)
group_size = np.array([len(y[trmnt == trmnt_flag]) for y, trmnt in zip(y_true_bin, trmnt_bin)])
response_rate = np.array([np.mean(y[trmnt == trmnt_flag]) for y, trmnt in zip(y_true_bin, trmnt_bin)])
else: # strategy == 'by_group'
y_bin = np.array_split(y_true[order][treatment[order] == trmnt_flag], bins)
group_size = np.array([len(y) for y in y_bin])
response_rate = np.array([np.mean(y) for y in y_bin])
variance = np.multiply(response_rate, np.divide((1 - response_rate), group_size))
return response_rate, variance, group_size
[docs]def weighted_average_uplift(y_true, uplift, treatment, strategy='overall', bins=10):
"""Weighted average uplift.
It is an average of uplift by percentile.
Weights are sizes of the treatment group by percentile.
Args:
y_true (1d array-like): Correct (true) target values.
uplift (1d array-like): Predicted uplift, as returned by a model.
treatment (1d array-like): Treatment labels.
strategy (string, ['overall', 'by_group']): Determines the calculating strategy. Default is 'overall'.
* ``'overall'``:
The first step is taking the first k observations of all test data ordered by uplift prediction
(overall both groups - control and treatment) and conversions in treatment and control groups
calculated only on them. Then the difference between these conversions is calculated.
* ``'by_group'``:
Separately calculates conversions in top k observations in each group (control and treatment)
sorted by uplift predictions. Then the difference between these conversions is calculated
bins (int): Determines the number of bins (and the relative percentile) in the data. Default is 10.
Returns:
float: Weighted average uplift.
"""
strategy_methods = ['overall', 'by_group']
n_samples = len(y_true)
check_consistent_length(y_true, uplift, treatment)
if strategy not in strategy_methods:
raise ValueError(f'Response rate supports only calculating methods in {strategy_methods},'
f' got {strategy}.')
if not isinstance(bins, int) or bins <= 0:
raise ValueError(f'Bins should be positive integer.'
f' Invalid value bins: {bins}')
if bins >= n_samples:
raise ValueError(f'Number of bins = {bins} should be smaller than the length of y_true {n_samples}')
response_rate_trmnt, variance_trmnt, n_trmnt = response_rate_by_percentile(
y_true, uplift, treatment, group='treatment', strategy=strategy, bins=bins)
response_rate_ctrl, variance_ctrl, n_ctrl = response_rate_by_percentile(
y_true, uplift, treatment, group='control', strategy=strategy, bins=bins)
uplift_scores = response_rate_trmnt - response_rate_ctrl
weighted_avg_uplift = np.dot(n_trmnt, uplift_scores) / np.sum(n_trmnt)
return weighted_avg_uplift
[docs]def uplift_by_percentile(y_true, uplift, treatment, strategy='overall', bins=10, std=False, total=False):
"""Compute metrics: uplift, group size, group response rate, standard deviation at each percentile.
Metrics in columns and percentiles in rows of pandas DataFrame:
- ``n_treatment``, ``n_control`` - group sizes.
- ``response_rate_treatment``, ``response_rate_control`` - group response rates.
- ``uplift`` - treatment response rate substract control response rate.
- ``std_treatment``, ``std_control`` - (optional) response rates standard deviation.
- ``std_uplift`` - (optional) uplift standard deviation.
Args:
y_true (1d array-like): Correct (true) target values.
uplift (1d array-like): Predicted uplift, as returned by a model.
treatment (1d array-like): Treatment labels.
strategy (string, ['overall', 'by_group']): Determines the calculating strategy. Default is 'overall'.
* ``'overall'``:
The first step is taking the first k observations of all test data ordered by uplift prediction
(overall both groups - control and treatment) and conversions in treatment and control groups
calculated only on them. Then the difference between these conversions is calculated.
* ``'by_group'``:
Separately calculates conversions in top k observations in each group (control and treatment)
sorted by uplift predictions. Then the difference between these conversions is calculated
std (bool): If True, add columns with the uplift standard deviation and the response rate standard deviation.
Default is False.
total (bool): If True, add the last row with the total values. Default is False.
The total uplift is a weighted average uplift. See :func:`.weighted_average_uplift`.
The total response rate is a response rate on the full data amount.
bins (int): Determines the number of bins (and the relative percentile) in the data. Default is 10.
Returns:
pandas.DataFrame: DataFrame where metrics are by columns and percentiles are by rows.
"""
strategy_methods = ['overall', 'by_group']
n_samples = len(y_true)
check_consistent_length(y_true, uplift, treatment)
if strategy not in strategy_methods:
raise ValueError(f'Response rate supports only calculating methods in {strategy_methods},'
f' got {strategy}.')
if not isinstance(total, bool):
raise ValueError(f'Flag total should be bool: True or False.'
f' Invalid value total: {total}')
if not isinstance(std, bool):
raise ValueError(f'Flag std should be bool: True or False.'
f' Invalid value std: {std}')
if not isinstance(bins, int) or bins <= 0:
raise ValueError(f'Bins should be positive integer.'
f' Invalid value bins: {bins}')
if bins >= n_samples:
raise ValueError(f'Number of bins = {bins} should be smaller than the length of y_true {n_samples}')
y_true, uplift, treatment = np.array(y_true), np.array(uplift), np.array(treatment)
response_rate_trmnt, variance_trmnt, n_trmnt = response_rate_by_percentile(
y_true, uplift, treatment, group='treatment', strategy=strategy, bins=bins)
response_rate_ctrl, variance_ctrl, n_ctrl = response_rate_by_percentile(
y_true, uplift, treatment, group='control', strategy=strategy, bins=bins)
uplift_scores = response_rate_trmnt - response_rate_ctrl
uplift_variance = variance_trmnt + variance_ctrl
percentiles = [round(p * 100 / bins, 1) for p in range(1, bins + 1)]
df = pd.DataFrame({
'percentile': percentiles,
'n_treatment': n_trmnt,
'n_control': n_ctrl,
'response_rate_treatment': response_rate_trmnt,
'response_rate_control': response_rate_ctrl,
'uplift': uplift_scores
})
if total:
response_rate_trmnt_total, variance_trmnt_total, n_trmnt_total = response_rate_by_percentile(
y_true, uplift, treatment, strategy=strategy, group='treatment', bins=1)
response_rate_ctrl_total, variance_ctrl_total, n_ctrl_total = response_rate_by_percentile(
y_true, uplift, treatment, strategy=strategy, group='control', bins=1)
weighted_avg_uplift = 1 / n_trmnt_total * np.dot(n_trmnt, uplift_scores)
df.loc[-1, :] = ['total', n_trmnt_total, n_ctrl_total, response_rate_trmnt_total,
response_rate_ctrl_total, weighted_avg_uplift]
if std:
std_treatment = np.sqrt(variance_trmnt)
std_control = np.sqrt(variance_ctrl)
std_uplift = np.sqrt(uplift_variance)
if total:
std_treatment = np.append(std_treatment, np.sum(std_treatment))
std_control = np.append(std_control, np.sum(std_control))
std_uplift = np.append(std_uplift, np.sum(std_uplift))
df.loc[:, 'std_treatment'] = std_treatment
df.loc[:, 'std_control'] = std_control
df.loc[:, 'std_uplift'] = std_uplift
df = df \
.set_index('percentile', drop=True, inplace=False) \
.astype({'n_treatment': 'int32', 'n_control': 'int32'})
return df
[docs]def treatment_balance_curve(uplift, treatment, winsize):
"""Compute the treatment balance curve: proportion of treatment group in the ordered predictions.
Args:
uplift (1d array-like): Predicted uplift, as returned by a model.
treatment (1d array-like): Treatment labels.
winsize(int): Size of the sliding window for calculating the balance between treatment and control.
Returns:
array (shape = [>2]), array (shape = [>2]): Points on a curve.
"""
uplift, treatment = np.array(uplift), np.array(treatment)
desc_score_indices = np.argsort(uplift, kind="mergesort")[::-1]
treatment = treatment[desc_score_indices]
balance = np.convolve(treatment, np.ones(winsize), 'valid') / winsize
idx = np.linspace(1, 100, len(balance))
return idx, balance