Overview of Multi-step bootstrapping
Multi-step bootstrapping is a method used in the fields of statistics and machine learning, particularly for assessing uncertainty, and is an extension of the general bootstrapping method (a method for constructing a distribution of estimates by randomly re-extracting sample data) described in ‘Overview of ensemble learning, algorithms and implementation examples’. Multi-step bootstrapping has the following characteristics: 1.
1. multi-step resampling: whereas in normal bootstrapping, a single sample is randomly extracted from the original sample, in multi-step bootstrapping, resampling is performed over multiple steps. This allows the sequential nature of the data (e.g. time series data) to be taken into account.
2. accounting for temporal dependencies: for data such as time series data or autoregressive models, the next data point may be dependent on the previous data point Multi-step bootstrapping allows sampling while preserving these dependencies, helping to assess prediction accuracy and model uncertainty This can be useful for assessing the
3. improved accuracy: while the usual bootstrapping method obtains confidence intervals for estimation by randomly reproducing the distribution of the data, multi-step bootstrapping can obtain more reliable estimation results by sampling multiple times, making it a particularly effective approach for complex models and multi-step forecasting It is an effective approach for models.
4. application of the algorithm: multi-step bootstrapping is also often used in machine learning and is particularly effective for recursive predictive models and models where there is a time delay in the prediction process, which makes model training and evaluation more robust.
Relevant algorithms
The algorithms used in combination with multi-step bootstrapping are described below.
1. ARIMA (AutoRegressive Integrated Moving Average): ARIMA described in “Examples of implementations for general time series analysis using R and Python” is a general statistical model for analysing time series data that combines the autoregressive (AR), moving average (MA) and integral (I) components of the data. Multi-step bootstrapping is used to assess the predictive accuracy of ARIMA models and to obtain confidence intervals for future forecasts.
Relevance: through bootstrapping sampling in multi-step forecasting of time series data, it is used to account for uncertainty in the model and to assess forecasts over multiple steps.
2. Q-learning (reinforcement learning): multi-step bootstrapping is a reinforcement learning algorithm used in algorithms such as Q-learning described in “Overview of Q-Learning and Examples of Algorithms and Implementations” to help agents learn optimal behaviour as they interact with their environment, used to assess uncertainty and risk when agents make predictions over multiple steps.
Relevance: in reinforcement learning simulations, it helps to track how an agent’s behaviour changes over time and to assess the stability and efficiency of the model through computation over multiple steps.
3. Monte Carlo Simulation: a Monte Carlo simulation described in “Overview and Implementation of Markov Chain Monte Carlo Methods” is a stochastic method, an algorithm that finds a solution to a problem by simulation, especially for problems with high uncertainty or a large scope to be evaluated. Multi-step bootstrapping is combined with Monte Carlo simulation to generate multiple samples and increase confidence in the results.
Relevance: in Monte Carlo simulations, multi-step sampling of data is used to assess uncertainty in order to predict future scenarios.
4. bootstrap aggregating (bagging): bagging described in “Overview of ensemble learning and examples of algorithms and implementations” is a machine learning algorithm that trains a model from several different samples and averages their predictions to obtain a final prediction, while multi-step bootstrapping is a time series forecasts and complex models, with re-sampling at each step to ensure model stability.
Relevance: in time series data and risky forecasts, multiple forecasts can be generated using different samples and averaged together to obtain more reliable forecasts.
5. recursive partitioning: recursive partitioning is a type of decision tree algorithm described in “Overview of Decision Trees and Examples of Applications and Implementations” that iteratively partitions a dataset to obtain the best prediction, Multi-step bootstrapping is used when using recursive partitioning to make multi-step predictions and is used to assess the stability of the partitioning and the accuracy of the model.
Relevance: multiple resamplings can improve the prediction accuracy of decision tree models and stabilise time-consuming prediction tasks.
6. Kalman Filter: the Kalman filter described in “State Space Model with Clojure: Implementation of Kalman Filter” is an algorithm used to estimate the state of a dynamic system and is particularly effective when estimating from noisy data, Kalman filter prediction accuracy can be evaluated at multiple stages to improve the reliability of the system.
Relevance: model performance can be enhanced through multi-step sampling when dealing with time series prediction and noisy data.
7. Recurrent Neural Networks (RNNs) and LSTMs (Long Short-Term Memory): RNN described in “Overview of RNN and examples of algorithms and implementations” and LSTM described in “Overview of LSTM and Examples of Algorithms and Implementations“ are deep learning algorithms that deal with time series and sequential data, which learn dependencies over multiple steps, Multi-step bootstrapping is used in these models to assess the uncertainty of the prediction and can increase the accuracy of the model by going through multiple prediction steps.
Relevance: in time series forecasting, the forecasts generated by RNNs and LSTMs are resampled multiple times to estimate confidence intervals.
8. variational inference: variational inference descrtibed in “Overview of Variational Bayesian Learning and Various Implementations” is an efficient way to perform Bayesian inference, especially for complex probability distributions. multi-step bootstrapping re-samples the parameters of the model in the process of variational inference, used to evaluate predictions based on multiple assumptions.
Relevance: multi-step resampling is useful as a method to obtain optimal parameter estimates, in order to assess uncertainty over the parameters of the model.
These algorithms utilise multi-step bootstrapping to increase the accuracy of forecasts and the reliability of the model, and to help assess uncertainty and reduce risk.
implementation example
As an example of the implementation of multi-step bootstrapping, simple Python code is shown to evaluate the prediction accuracy for time series data. In this example, the ARIMA model is used to perform multi-step forecasting of time series data and Multi-step bootstrapping is applied to the forecast results.
The following code implements the ARIMA model using the statsmodels library and bootstrapping sampling to evaluate the confidence intervals of the forecasts.
Example implementation: the ARIMA model and Multi-step Bootstrapping
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from statsmodels.tsa.arima.model import ARIMA
from sklearn.utils import resample
# Sample time series data (actually use appropriate time series data)
np.random.seed(42)
data = np.cumsum(np.random.randn(100)) # random walk
# Building the ARIMA model.
def fit_arima_model(data):
model = ARIMA(data, order=(5, 1, 0)) # ARIMA(5,1,0) setting.
model_fit = model.fit()
return model_fit
# Implementation of Multi-step Bootstrapping
def multi_step_bootstrap(data, n_bootstrap, steps_ahead):
bootstrapped_predictions = []
# Bootstrap sampling and prediction
for _ in range(n_bootstrap):
boot_data = resample(data) # Data bootstrapping sample.
model_fit = fit_arima_model(boot_data)
forecast = model_fit.forecast(steps=steps_ahead) # Predicting n steps ahead
bootstrapped_predictions.append(forecast)
bootstrapped_predictions = np.array(bootstrapped_predictions)
return bootstrapped_predictions
# Parameter setting
n_bootstrap = 1000 # Number of bootstraps.
steps_ahead = 10 # Number of steps to be predicted
# Perform bootstrapping predictions.
bootstrapped_predictions = multi_step_bootstrap(data, n_bootstrap, steps_ahead)
# Calculation of means and confidence intervals for forecasts.
mean_forecast = bootstrapped_predictions.mean(axis=0)
lower_bound = np.percentile(bootstrapped_predictions, 2.5, axis=0)
upper_bound = np.percentile(bootstrapped_predictions, 97.5, axis=0)
# Plotting the results
plt.plot(data, label='Original Data')
plt.plot(np.arange(len(data), len(data) + steps_ahead), mean_forecast, label='Forecast', color='red')
plt.fill_between(np.arange(len(data), len(data) + steps_ahead), lower_bound, upper_bound, color='gray', alpha=0.5)
plt.title('ARIMA Forecast with Multi-step Bootstrapping')
plt.legend()
plt.show()Description.
- Data generation: data is sample time series data mimicking a random walk, which is replaced by real data in the actual problem.
- Applying the ARIMA model: the fit_arima_model function applies the ARIMA(5,1,0) model to train the time series data.
- Multi-step Bootstrapping: the multi_step_bootstrap function resamples the data using bootstrap sampling and applies the ARIMA model to each sample to predict steps_ahead steps. This is repeated a specified number of times (n_bootstrap) to obtain multiple predictions.
- Calculate confidence intervals: calculate confidence intervals (lower_bound and upper_bound) using the mean (mean_forecast) of the predictions obtained from 1000 bootstraps and the 2.5th and 97.5th percentiles.
- Plotting the results: matplotlib is used to plot the original time series data, the forecast results and their confidence intervals.
Output results: this implementation displays the values predicted by the ARIMA model and their confidence intervals, together with the original data. The confidence intervals are obtained through multiple bootstrapped forecasts and visually show the uncertainty of the predicted values.
Application examples
Specific applications of multi-step bootstrapping include.
1. stock price forecasting (time series forecasting)
- PROBLEM: Stock prices are a typical time-series forecasting problem, where future stock prices are predicted from past data, and investors need to take into account the uncertainty of future stock prices when making investment decisions. Multi-step bootstrapping is used to assess this uncertainty.
- METHODS:
- Use historical stock price data to train an ARIMA model, for example.
- Multi-step bootstrapping is used to generate bootstrap samples of historical stock price data and predict future stock prices based on each sample.
- The distribution of predicted share prices (mean and confidence interval) is used to assess the uncertainty of future share prices.
- Application: e.g. predict future share prices one month ahead using one year of past share price data for a given company. Calculating confidence intervals in forecasts helps investors to understand worst- and best-case scenarios and to manage risk.
- Benefits: multi-step bootstrapping allows uncertainty and risk in forecast outcomes to be explicitly stated, which can enhance risk management when developing investment strategies. For example, it provides an indication of the range of possible upswings or downswings of 10% in relation to a stock price forecast, allowing asset allocation according to risk.
2. demand forecasting in manufacturing (Supply Chain Management)
- PROBLEM: Companies forecast demand for their products, plan production and manage inventories. Demand forecasting involves uncertainty, which needs to be assessed, especially in multi-step demand forecasting.
- METHODS:
- Based on historical sales data, demand forecasting models are built using ARIMA models and Exponential Smoothing.
- Multi-step bootstrapping is applied to randomly sample from the historical data and forecast future demand for each sample.
- Based on the multiple forecasts obtained, the mean and confidence intervals for future demand are calculated.
- Application: e.g. a manufacturing company using monthly sales data for the last five years to forecast demand for the next six months, Multi-step bootstrapping can be used to account for forecast errors and obtain multiple demand scenarios.
- Benefits: multi-step bootstrapping allows the range of forecast error for future demand to be defined, enabling inventory management to prevent overproduction and shortages. Confidence intervals are provided so that risky scenarios can be planned for in advance.
3. energy consumption forecasting (Smart Grids)
- PROBLEM: In infrastructure systems such as smart grids, which are also discussed in ‘Electricity storage technology, smart grids and GNNs’, it is important to predict the energy consumption of households and businesses. Several factors influence the forecasting of energy consumption, such as seasonal variations and contingencies, and multi-step bootstrapping can be used to assess the uncertainty of the forecasted values.
- METHODS:
- Build a time-series forecasting model (e.g. ARIMA) based on historical energy consumption data (e.g. one year of consumption data).
- Generate samples from the historical data using multi-step bootstrapping to forecast future consumption.
- Based on the predictions of each sample, calculate the mean and confidence interval of future energy consumption.
- Application: for example, if household-specific energy consumption data for the past year in a city is used to forecast consumption for the next three months, Multi-step bootstrapping can provide confidence intervals that indicate within which range future consumption will converge.
- Benefits: confidence intervals allow energy companies to estimate energy requirements more accurately in order to avoid over- or under-supply, and to plan precautionarily for changes in consumption patterns.
4. climate forecasting (Weather Forecasting)
- PROBLEM: Climate models have a high degree of uncertainty when forecasting long-term weather and temperature. Multi-step bootstrapping is an effective approach to improve the accuracy of future temperature and precipitation forecasts in weather forecasting.
- METHODS:
- Climate models are trained based on historical weather data.
- Using multi-step bootstrapping, different weather scenarios are sampled and forecasts are made for each of them.
- Based on the results of multiple forecasts, the mean and confidence intervals of the forecasts are calculated.
- Application: e.g. to forecast temperatures for the next year using average monthly temperature data for the last 20 years, Multi-step bootstrapping is used to obtain forecast results based on multiple scenarios to show the range of future temperature variability.
- Benefits: multi-step bootstrapping allows forecast results based on different scenarios to be obtained and allows planning for the possibility of extreme weather events (e.g. extreme weather events).
Multi-step bootstrapping is used in practice in various sectors such as finance, manufacturing, energy and weather forecasting to clarify uncertainties in forecast models, and is a very useful method for increasing the reliability of forecasts and managing risk by generating multiple scenarios. The results can be.
reference book
References relevant to multi-step bootstrapping are listed below.
1. statistics and bootstrapping methods
– The Bootstrap Methods and Their Application (An Introduction to the Bootstrap)
– Author(s): Brad Efron, Robert Tibshirani
– Abstract: A pioneering book on bootstrapping methods. It details the theoretical background of bootstrapping and its practical applications. Particularly useful for learning the foundations of bootstrapping methods in multi-stage forecasting.
– Areas of application: statistical analysis, assessment of forecasting accuracy.
– Bootstrap Methods: A Guide for Practitioners and Researchers
– Authors: A. C. Davison, D. V. Hinkley
– Abstract: This book provides an in-depth look at the practical implementation of bootstrapping and how to evaluate it. It explains how bootstrapping is applied to multi-stage forecasting and simulation.
– Areas of application: statistical methods, simulation
2. time series analysis and forecasting
– ‘Time Series Analysis and Its Applications: With R Examples’
– Author(s): Shumway, Robert H., Stoffer, David S.
– Abstract: Provides a practical approach to the analysis of time series data, including ARIMA models and forecasting methods, as well as a section on bootstrapping applications.
– Areas of application: time series forecasting, financial data, energy consumption forecasting.
– ‘Applied Time Series Analysis with R’
– Authors: Wayne A. Woodward, Henry L. Gray, Alan C. Elliott
– Abstract: A book for learning to analyse time series data using R, including ARIMA, outlier detection and other methods related to multistage forecasting.
– Areas of application: time series analysis, forecasting
3. monte carlo methods and simulation
– Monte Carlo Methods in Financial Engineering
– Author: Paul Glasserman
– Abstract: Details the application of Monte Carlo methods in financial engineering. In particular, the reader will learn how to combine bootstrapping and Monte Carlo methods.
– Areas of application: financial modelling, risk management
– Simulation Modelling and Analysis
– Author: Averill M. Law
– Abstract: A comprehensive guide to simulation modelling. It describes statistical methods in models for system simulation and multi-stage forecasting.
– Areas of application: simulation, forecasting
4. data science and predictive modelling
– Hands-On Time Series Analysis with R: Build Effective Time Series Models in R using the Most Popular Packages
– Author: Rami Krispin
– Abstract: Describes how to perform practical time series analysis and forecasting using R. Particular reference is made to how to evaluate predictive models and how to utilise bootstrapping to calculate confidence intervals.
– Areas of application: time series analysis, data science, machine learning
– Data Science for Business: What You Need to Know about Data Mining and Data-Analytic Thinking
– Author(s): Foster Provost, Tom Fawcett
– Abstract: Focuses on how data science can be used to solve business problems. It also discusses the use of bootstrapping in predictive modelling and risk assessment.
– Areas of application: data science, business forecasting
5. machine learning and prediction
– ‘Hands-On Machine Learning with Scikit-Learn, Keras, and TensorFlow: Concepts, Tools, and Techniques to Build Intelligent Systems’
– Author: Aurélien Géron
– Abstract: A practical introduction to machine learning. Techniques for improving model evaluation and prediction accuracy, such as bootstrapping and cross-validation, are covered.
– Areas of application: machine learning, model evaluation
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