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cs:localization:kalman:introduction:start [2017/01/16 23:39] Brian Moore First revision complete |
cs:localization:kalman:introduction:start [2017/01/31 02:06] (current) Brian Moore [Section 1 - Linear Least Squares Estimation] |
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| In more formal terms, for some $m$ measurements $Y$ that are linear functions of a system with $n$ unknown states $X$ where $m>=n$. Such systems are said to be //over-determined,// whereby it is impossible to choose values of $X$ that will satisfy every measurement perfectly, and thus a compromise of values of $X$ is chosen that minimizes the total sum of the squares of the error between each measurement | In more formal terms, for some $m$ measurements $Y$ that are linear functions of a system with $n$ unknown states $X$ where $m>=n$. Such systems are said to be //over-determined,// whereby it is impossible to choose values of $X$ that will satisfy every measurement perfectly, and thus a compromise of values of $X$ is chosen that minimizes the total sum of the squares of the error between each measurement | ||
| - | $$ X_{est} = \text{arg}\,\min\limits_{\beta}\sum\|y-X\beta||^2 $$ | + | $$ X_{est} = \text{arg}\,\min\limits_{X}\sum\|y-\beta X||^2 $$ |
| Given the matrix format: | Given the matrix format: | ||
| Line 206: | Line 206: | ||
| \end{bmatrix} \\ | \end{bmatrix} \\ | ||
| - | X_{est} = (\beta'\beta)^{-1}\beta Y | + | X_{est} = (\beta'\beta)^{-1}\beta' Y |
| $$ | $$ | ||
