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Typo corrections in the documentation from Bing
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Keir Mierle
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@@ -17,11 +17,11 @@ Such optimization problems arise in almost every area of science and engineering
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Perhaps the simplest example of such a problem is the problem of Ordinary Linear Regression, where given observations $(x_1,y_1),\hdots, (x_k,y_k)$, we wish to find the line $y = mx + c$, that best explains $y$ as a function of $x$. One way to solve this problem is to find the solution to the following optimization problem
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\begin{equation}
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\arg\min_{m,c} \sum_{i=1}^k (y_i - m x_i - c)^2
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\arg\min_{m,c} \sum_{i=1}^k (y_i - m x_i - c)^2.
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\end{equation}
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With a little bit of calculus, this problem can be solved easily by hand. But what if, instead of a line we were interested in a more complicated relationship between $x$ and $y$, say for example $y = e^{mx + c}$. Then the optimization problem becomes
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\begin{equation}
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\arg\min_{m,c} \sum_{i=1}^k \left(y_i - e^{m x_i + c}\right)^2
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\arg\min_{m,c} \sum_{i=1}^k \left(y_i - e^{m x_i + c}\right)^2.
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\end{equation}
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This is a non-linear regression problem and solving it by hand is much more tedious. Ceres is designed to help you model and solve problems like this easily and efficiently.
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@@ -236,7 +236,7 @@ numeric errors and leads to slower convergence.
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\label{sec:tutorial:datafitting}
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The examples we have seen until now are simple optimization problems with no data. The original purpose of least squares and non-linear least squares analysis was fitting curves to data. It is only appropriate that we now consider an example of such a problem. Let us fit some data to the curve
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\begin{equation}
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y = e^{mx + c}
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y = e^{mx + c}.
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\end{equation}
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The full code and data for this example can be found in
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