Intelligent Database Systems Lab 國立雲林科技大學 National Yunlin University of Science and Technology Local linear correlation analysis with the SOM Advisor : Dr. Hsu Presenter : ching-wen Hong Author : Antonio Piras, Alain Germond

Intelligent Database Systems Lab N.Y.U.S.T. I. M. 2 Outline  Motivation  Objection  Linear correlation  Local linear correlation  The computing of local linear correlation in SOM  Application to electrical load forecasting  Conclusion  My opinion

Intelligent Database Systems Lab N.Y.U.S.T. I. M. 3 Motivation  When two variables are not linearly correlated, it does not mean that they are not linearly correlated in the some local region. For example, temperature and electrical energy consumption are not linearly correlated, but the linear correlation is negative in winter and positive in summer. So the linear correlation coefficient is not a good tool in some application.

Intelligent Database Systems Lab N.Y.U.S.T. I. M. 4 Objection  This paper uses a local linear correlation to select relevant input variables for non-linear regression. The method is an extension to the concept of SOM and linear correlation.  The method is based on the SOM which allows to compute and analysis the local linear correlation between variables in neighbour subspace.

Intelligent Database Systems Lab N.Y.U.S.T. I. M. 5 Linear correlation  Consider two variables x and y, and x, y variables with zero mean.  Data set D={(x(t),y(t)) | i=1,…,N}

Intelligent Database Systems Lab N.Y.U.S.T. I. M. 6 Linear correlation  -1≤r xy ≤1  r xy =0 ~ x, y are no linear correlation  r xy =1 ~ x, y are a perfect positive linear correlation  r xy =-1 ~ x, y are a perfect negative linear correlation

Intelligent Database Systems Lab N.Y.U.S.T. I. M. 7 Local linear correlation  When two variables are not linearly correlated, it does not mean that they are non-linear correlated. It could happen that in some regions of the definition space the variables are correlated and in some other regions not.  For example, temperature and electrical energy consumption are not linearly correlated, but the linear correlation is negative in winter and positive in summer.

Intelligent Database Systems Lab N.Y.U.S.T. I. M. 8 The computing of local linear correlation in SOM  A vector quantization algorithm which divides the manifold where v=(x,y) in S subspaces.  The m i is the centres of the subspaces v i, i=1,..,s  We can compute m i and local linear correlation r xy,i by the classical Kohonen learning rule.

Intelligent Database Systems Lab N.Y.U.S.T. I. M. 9 The classical Kohonen learning rule.

Intelligent Database Systems Lab N.Y.U.S.T. I. M. 10 Measuring local linear correlations  The local linear correlation r xy,i can be viewed as a weighted measure of neighbouring correlations.  We now need to sum up the local linear correlations with a single index to discover quickly the important variables.  The measuring index of local linear correlation:RMSLC S is the number of clusters, r,i is the local correlation between x and y in the cluster v i,i=1,…,s

Intelligent Database Systems Lab N.Y.U.S.T. I. M. 11 Measuring local linear correlations

Intelligent Database Systems Lab N.Y.U.S.T. I. M. 12 Application to electrical load forecasting  The input variables: 1.the max and min temperatures of the forecast day and of the day before of three recording points:TMIN1,TMAX1, TMIN2,TMAX2, TMIN3,TMAX3, and TMIN1(-1),TMAX1 (-1), TMIN2 (-1),TMAX2 (-1), TMIN3 (-1),TMAX3 (-1) 2.The load of the forecast day, of the 10 day before and 1 day:Y,Y(-10),Y(-1)

Intelligent Database Systems Lab N.Y.U.S.T. I. M. 13

Intelligent Database Systems Lab N.Y.U.S.T. I. M. 14 The training of closed Kohonen ring  One network with 12 units has been trained with a set composed of 2yr historical data (700 patterns).  1.we obtain the mean and the standard deviation of the input vector in each cluster.2.the covariance and the correlation coefficient between the load Y and the rest of variables in the input vector according to the Eqs.(4)-(7).

Intelligent Database Systems Lab N.Y.U.S.T. I. M. 15 The training of closed Kohonen ring is terminated  Spring: neurons  Summer: neurons 1-2  Autumn: neurons  Winter: neurons 7-8-9

Intelligent Database Systems Lab N.Y.U.S.T. I. M. 16  A clear negative correlation in winter (r=-0.8)  A insignificant positive correlation in summer (r=0.2)  Y(-1) is significantly positive correlated with the load of today.  Y(-10) is not correlated with the load of today.  The TMIN3 was more important than TMAX3.

Intelligent Database Systems Lab N.Y.U.S.T. I. M. 17  A clear negative correlation in winter (r=-0.6)  A insignificant negative correlation in summer (r=-0.2)  Y(-1) is significantly positive correlated with the load of today.  Y(-10) is not correlated with the load of today.  The TMAX3 was more important than TMIN3.

Intelligent Database Systems Lab N.Y.U.S.T. I. M. 18 Quantify the importance of each variables (for 1 p.m. data, networks of 1and 12 )  We knew that the temperature is an important explanatory variable, but this fact cannot be shown with the simple linear correlation coefficient computed over a year of observations.

Intelligent Database Systems Lab N.Y.U.S.T. I. M. 19 Conclusion  In the paper we presented a method to the linear correlation between variables in neighbour subspaces based on SOM.  The visualisation of the local linear correlation computed at each unit of SOM allows to understand the varying dependency between variables.  Future works will focus on the application the method to theoric in put distributions.

Intelligent Database Systems Lab N.Y.U.S.T. I. M. 20 My opinion  1.We can present a method to the multiple linear correlation( 複相關 ) between variables in neighbour subspaces based on SOM.  2. 先以 SOM 分群後, 在再依各群去計算相關係 數, 其結果不知與本文結果有何差異？