# Principal Component Density Estimation for Scenario Generation Using Normalizing Flows

arXiv: https://arxiv.org/abs/2104.10410 · HTML: https://mlsanity.com/paper/2104.10410/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 62 references that are in the map.
- First posted on arXiv 21 Apr 2021.

## Builds on (most cited first, 15 of 15)

- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Wasserstein GAN](https://mlsanity.com/paper/1701.07875/) (2017)
- [Flows for simultaneous manifold learning and density estimation](https://mlsanity.com/paper/2003.13913/) (2020)
- [TensorFlow Distributions](https://mlsanity.com/paper/1711.10604/) (2017)
- [Understanding and Mitigating Exploding Inverses in Invertible Neural Networks](https://mlsanity.com/paper/2006.09347/) (2020)
- [Normalizing Flows on Riemannian Manifolds](https://mlsanity.com/paper/1611.02304/) (2016)
- [Model-Free Renewable Scenario Generation Using Generative Adversarial Networks](https://mlsanity.com/paper/1707.09676/) (2017)
- [Day-ahead electricity price forecasting with high-dimensional structures: Univariate vs. multivariate modeling frameworks](https://mlsanity.com/paper/1805.06649/) (2018)
- [Scenario Forecasting of Residential Load Profiles](https://mlsanity.com/paper/1906.07373/) (2019)
- [Bayesian Renewables Scenario Generation via Deep Generative Networks](https://mlsanity.com/paper/1802.00868/) (2018)
- [An Unsupervised Deep Learning Approach for Scenario Forecasts](https://mlsanity.com/paper/1711.02247/) (2017)
- [Forecasting Spatio-Temporal Renewable Scenarios: a Deep Generative Approach](https://mlsanity.com/paper/1903.05274/) (2019)
- [Generative Adversarial Networks for Operational Scenario Planning of Renewable Energy Farms: A Study on Wind and Photovoltaic](https://mlsanity.com/paper/1906.00662/) (2019)

## Built on by (most cited first, 10 of 10)

- [Canonical normalizing flows for manifold learning](https://mlsanity.com/paper/2310.12743/) (2023)
- [Multivariate Probabilistic Forecasting of Intraday Electricity Prices using Normalizing Flows](https://mlsanity.com/paper/2205.13826/) (2022)
- [Modular Flows: Differential Molecular Generation](https://mlsanity.com/paper/2210.06032/) (2022)
- [Nonlinear Isometric Manifold Learning for Injective Normalizing Flows](https://mlsanity.com/paper/2203.03934/) (2022)
- [Principal Manifold Flows](https://mlsanity.com/paper/2202.07037/) (2022)
- [Normalizing Flow-based Day-Ahead Wind Power Scenario Generation for Profitable and Reliable Delivery Commitments by Wind Farm Operators](https://mlsanity.com/paper/2204.02242/) (2022)
- [Multivariate Scenario Generation of Day-Ahead Electricity Prices using Normalizing Flows](https://mlsanity.com/paper/2311.14033/) (2023)
- [Least Squares and Marginal Log-Likelihood Model Predictive Control using Normalizing Flows](https://mlsanity.com/paper/2409.17632/) (2024)
- [Random Projection Flows for Efficient Manifold Density Estimation](https://mlsanity.com/paper/2509.25228/) (2025)
- [Data-Driven Conditional Flexibility Index](https://mlsanity.com/paper/2601.16028/) (2026)

## Abstract (arXiv)

Neural networks-based learning of the distribution of non-dispatchable renewable electricity generation from sources such as photovoltaics (PV) and wind as well as load demands has recently gained attention. Normalizing flow density models are particularly well suited for this task due to the training through direct log-likelihood maximization. However, research from the field of image generation has shown that standard normalizing flows can only learn smeared-out versions of manifold distributions. Previous works on normalizing flow-based scenario generation do not address this issue, and the smeared-out distributions result in the sampling of noisy time series. In this paper, we exploit the isometry of the principal component analysis (PCA), which sets up the normalizing flow in a lower-dimensional space while maintaining the direct and computationally efficient likelihood maximization. We train the resulting principal component flow (PCF) on data of PV and wind power generation as well as load demand in Germany in the years 2013 to 2015. The results of this investigation show that the PCF preserves critical features of the original distributions, such as the probability density and frequency behavior of the time series. The application of the PCF is, however, not limited to renewable power generation but rather extends to any data set, time series, or otherwise, which can be efficiently reduced using PCA.