Todd - Segment 19
To Calculate
1. Prove the assertion on lecture slide 5, namely that, for a multivariate normal distribution, the quantity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ({\mathbf x-\mathbf\mu})^T{\mathbf\Sigma}^{-1}({\mathbf x-\mathbf\mu})} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf x} is a random draw from the multivariate normal, is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi^2} distributed. |
By Cholesky factorization, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf \Sigma = \mathbf L \mathbf L^T }
. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf L \mathbf y = \mathbf x - \mathbf \mu}
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Then, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ({\mathbf x-\mathbf\mu})^T{\mathbf\Sigma}^{-1}({\mathbf x-\mathbf\mu}) = (\mathbf L \mathbf y)^T (\mathbf L \mathbf L^T)^{-1} (\mathbf L \mathbf y) = \mathbf L^T \mathbf y^T (\mathbf L^T)^{-1} \mathbf L^{-1} \mathbf L \mathbf y = \mathbf L^T (\mathbf L^T)^{-1} \mathbf L^{-1} \mathbf L \mathbf y^T \mathbf y = \mathbf y^T\mathbf y }
Note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf y^T\mathbf y = \sum_i{\mathbf y_i^2}}
by definition of vector multiplication. This expression is in the form of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi^2 = \sum_i{(\frac{\mathbf y_i - \mathbf \mu_i} {\sigma_i})^2}}
with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_i = 0, \sigma_i = 1}
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Therefore, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ({\mathbf x-\mathbf\mu})^T{\mathbf\Sigma}^{-1}({\mathbf x-\mathbf\mu})}
is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi^2}
distributed.
To Think About
1. Why are we so interested in t-values? Why do we square them? |
They are important because they tell us how far an observed value is from the expected value (ie: how far from the norm is this data point). We square them in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi^2} statistic because we are interested in capturing deviation from the mean, which is always positive. If we did not square them, a t-value could be negative, and the sum of several t-values could be smaller than they should be because negative values would cancel positive values.