Statistical Analysis Course by Penn state University

  1. Welcome to STAT 505! | STAT 505 (psu.edu) Applied Multivariate Statistical Analysis
  2. Welcome to STAT 501! | STAT 501 (psu.edu) Regression Methods

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Visualizing Ellipse rotation and associated parametric equations

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  Diagonal matrix Ellipse axes are parallel to coordinate axes (no rotation). Zeros in off-diagonals means zero correlation. Same values in diagonals Ellipse is rotated 45 degrees if correlation (off-diagonal) is positive regardless of its magnitude.  Similarly -45 degrees if correlation is negative. Correlation matrix Correlation matrix has 1 in diagonals and values between -1 and 1 inclusive in off-diagonals. Ellipse size remains the same (always touches square of side 2 units).  Whether the correlation is positive or negative can be observed by the orientation of the ellipse. The amount of correlation can be interpreted by how thin the ellipse is. How It Works Below is a list of parametric equations starting from that of a general ellipse and modifying it step by step into a prediction ellipse, showing how different parts contribute at each step. Reference How to Draw Ellipse of Covariance Matrix (cookierobotics.com)
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Covariance Error Ellipse - sqrt of Eigen value as ellipse axis length

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  Axis-aligned confidence ellipses In general, the equation of an axis-aligned ellipse with a major axis of length   and a minor axis of length  , centered at the origin, is defined by the following equation: In our case, the length of the axes are defined by the standard deviations   and   of the data such that the equation of the error ellipse becomes: where s defines the scale of the ellipse and could be any arbitrary number (e.g. s=1).  The question is now how to choose  , such that the scale of the resulting ellipse represents a chosen confidence level (e.g. a 95% confidence level corresponds to s=5.991). The left hand side of equation ( 2 ) actually represents the sum of squares of independent normally distributed data samples, cov is zero.  The sum of squared Gaussian data points is known to be distributed according to a so called  Chi-Square distribution . A Chi-Square distribution is defined in terms of ‘degrees of fre...
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Random sample approach to rational subgroups (Xbar, R chart)

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A fundamental idea in the use of control charts is the collection of sample data according to what Shewhart called the rational subgroup concept. To illustrate this concept, suppose that  we are using an control chart to detect changes in the process mean . Then the rational subgroup concept means that subgroups or samples should be selected so that if assignable causes are present, the chance for differences between subgroups will be maximized, while the chance for differences due to these assignable causes within a subgroup will be minimized. The rational subgroup concept is very important. The  proper selection of samples requires careful consideration of the process, with the objective of obtaining as much useful information as possible from the control chart analysis . In random sample approach to rational subgroups, random sample (of defined number of units, here 5 units) is collected from all process output produced over the sampling interval. This method of rational su...
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