Functional Data Analysis of High-Frequency Household Energy Consumption Curves for Policy Evaluation
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1 Unponte2017: Mercati energetici e metodi quantitativi Università di Padova Padova, Italy October 12, 2017 Functional Data Analysis of High-Frequency Household Energy Consumption Curves for Policy Evaluation Simone Vantini MOX - Department of Mathematics, Politecnico di Milano, Milano, Italy.
2 From 1D-data to Functional data Functional Data Functions (i.e., curves, surfaces, trajectories, BW and RGB images, ) High-dimensional Data Multivariate Data Very long" vectors Vectors Univariate Data Numbers
3 Laser Welding of Metal Sheets Pini, A., Vantini, S., Colosimo, B. M., Grasso, M. (2017): Domain-Selective Functional ANOVA for Supervised Statistical Profile Monitoring of Signal Data, Journal of the Royal Statistical Society Series C (online).
4 Functional Two-population Test: a Simple Problem with Complex Data
5 Functional Two-population Test: a Simple Problem with Complex Data where Y 1 and Y 2 are L 2 -valued random functions Functional t-test:
6 Approaches to Null Hypothesis Significance Testing in FDA Functional data are not Gaussian In FDA sample sizes are intrinsically small Rejections are required to be located along the domain
7 Laser Welding of Metal Sheets 5 different locations [distance to the sheet boundaries] 3 gap values [distance between sheets] Plasma emission Laser reflection Metal emission Research question: Can we monitor the gap between the two sheets despite of the nuisance effect due to location by looking at the backward spectrum?
8 Modelling Framework: Two-way Functional Anova Two-way Functional ANOVA : common mean : gap effect (i=1,2,3) : location effect (j=1,,5) : independent errors (l=1,2,3) We want to test: If and in which wavelength intervals the gap effect is significant If and in which wavelength intervals the location effect is significant
9 Functional F-tests P-value functions Significant regions Gap Location
10 Estimation of Appliance Load Signatures Research question: Do household electric appliances have a specific load signature that can be identified over time in the various consumption curves? Fontana, M., Tavoni, M., Vantini, S. (2017+): Functional Data Analysis of High-Frequency Household Energy Consumption Curves for Policy Evaluation, MOX report, Dept. of Mathematics, Politecnico di Milano.
11 Data Data Family Administrative Data Profilation Data (425 Households) Data Type Client ID Display delivery date Municipality Display Version Presence of microgeneration devices Contractual power Number of People in the household Age and Sex Electric Appliances owned (34 items) Load Curves (1064 Households, ~600 days) Size of the household Sampled every 15 minutes
12 Functional-on-scalar Regression Model Electricity Consumption = f ( ) Appliances +ε
13 Functional-on-scalar Regression Model f = +ε(t)
14 Functional-on-scalar Regression Model = f + ε(t) Functional-on-scalarLinear Model y t ( = β +,-./01. t + β t I (2 + β t I (8 + ε t (
15 Bulding the Model Response: Electricity Consumption Curve Smoothing Fourier smoothing of the average consumption curve for each household K=3 K=11 K=31
16 Building the Model Regressors: Appliance Clustering Basic Appliances Blue Low Tech/High Consumption Appliances High Tech Appliances Red
17 Building the Model Regressors: Appliance Clustering Define a ownership index for each cluster k and each family f as I J I (< = 1? a N (0<, k in {B, H, L} < 0KL
18 Functional t-tests Baseline High Tech Low Tech β(t) P(t)
19 Energy Disaggregation
20 From Consumptions to Consumption Variations Functional-on-scalarLinear Model (Time Derivatives) Dy t ( = β +,-./01. (t) + β t I (2 + β t I (8 + ε t ( Consumption curves Consumption variation curves
21 Functional t-tests Baseline High Tech Low Tech β(t) P(t)
22 Many Other Applications of FDA
23 Some References Background: Ramsay, J. O., Silverman, B. W. (2005): Functional Data Analysis. New York: Springer-Verlag. General Methodology: Pini, A., & Vantini, S. (2017): Interval-wise testing for functional data. Journal of Nonparametric Statistics, 29(2), Specific Applications: Pini, A., Vantini, S., Colosimo, B. M., & Grasso, M. (2017): Domain-selective functional analysis of variance for supervised statistical profile monitoring of signal data. Journal of the Royal Statistical Society: Series C (Applied Statistics). Fontana, M., Tavoni, M., Vantini, S. (2017+): Functional Data Analysis of High-Frequency Household Energy Consumption Curves for Policy Evaluation, MOX report, Dept. of Mathematics, Politecnico di Milano Software: Pini, A, Vantini, S. (2016): fdatest, R package
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