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Andrew Smith Denoising UK house prices 14th April 2010

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2 Regression on a graph Denoising UK house prices Discrete spatial processes Longitudinal data analysis Scatterplot smoothing Image analysis

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3 Outline UK house price data The need for regression Graph structure Regression on a graph Penalised regression Results and uses of algorithm

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4 Video available at

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5 Regression Seek a (simpler) pattern that explains observed data Model: Data=Signal+Noise Price year,town =True value year,town +σ year Z year,town

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6 No covariate values Euclidean distance inappropriate Missing values Challenges for existing methods

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7 Regression on a graph Consider observations Price year,town to be taken at the vertices of a graph Edges of the graph give an idea of closeness

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8 Consider observations Price year,town to be taken at the vertices of a graph Edges of the graph (neighbouring towns) give an idea of closeness Regression on a graph

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9 Penalised regression on a graph Need to penalise distance from data Penalty term: Sum (over all vertices) of squares of residuals

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10 Penalised regression on a graph Need to penalise roughness Penalty term: Sum (over all edges) of absolute differences

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11 Penalised regression on a graph Minimise: Distance from data + λ Roughness Computationally intensive, so use new algorithm

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12 Video available at

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13 Regional analysis The estimate identifies regions of constant value These regions change in a similar way through time

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14 Regional analysis The estimate identifies regions of constant value These regions change in a similar way through time They are not the same as government office regions

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15 Summary Detecting a smooth national trend in noisy UK house price data Use regression on a graph Penalise distance from data (vertices) and roughness (edges) arxiv.org/abs/ (Kovac & Smith 2009) All house price data courtesy of Halifax

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