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I am rewriting the problem as $$\left. y = \dfrac{x}{C_0+C_1 x+C_2 x^2} \right\}\;(C_0+C_1 x+C_2 x^2)y = x$$ with the constants $C_0=C$, $C_1=-C D$ and $C_2=C D S^2$ and $x=M$, $y=R$ You want to find the coefficients $C_0$, $C_1$, and $C_2$ given 3 pairs of $(x_i,y_i)$ This can be treated as a 3×3 linear system when the three points are used in $C_0 y_i + ... 0 Honestly, it's just really easy to compute. While there are other methods that may give better answers in certain situations, Least Squares with a much simpler algorithm. It just involves constructing a matrix, where each element is a sum. Then we can use Row Reduction Echelon Form to find the coefficients. This simplicity means that it can be used ... 1 It's essentially due to Runge's Phenomenon for interpolating polynomials. Even without uncertainty, the behavior of higher order polynomials at their endpoints is very sensitive to the parameter values. Here is a link to a technical explanation that I will regurgitate here. It relates Runge's Phenomenon to regression. Below is a intuitive, but admittedly ... 0 I only recall a representation theorem in complex analysis (https://en.wikipedia.org/wiki/Weierstrass_factorization_theorem#The_Weierstrass_factorization_theorem) but without more details about the function, not much can be said. Note, however, that there is a bunch of specific method for nonlinear regression, e.g. ... 0 I asked for clarification because the way I interpret the question neither of the possible answers is correct. First, you cannot find correlation if either X or Y has all equal elements. In that case you would be trying to divide by a zero standard deviation. Second, I'm not sure what it means to say 'all other values take a constant value'. It seems to ... 0 If you have a data point that lacks either a value for the dependent or the independent variable, the data point does not carry any meaningful information and hence it is pretty much useless. For example, assume you are investigating the influence of education on income and in your sample you have a person who earns$100,000 per year, but you don’t know his ...

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begin quote If you know that $\sum_{i=1}^n e_i=0$, show that $\sum_{i=1}^n\epsilon_i=0$ where $e_i=Y_i-\hat{Y_i}$ and $\epsilon_i=Y_i-E[Y_i]$. I know that $$Y_i=B_0+B_1X_i+\epsilon_i$$ and $$E[Y_i]=B_0+B_1X_i$$ end quote Some assumptions are omitted here. Usually one says something like $Y_i = B_0+ B_1 X_i + \varepsilon_i$ where $B_0$, $B_1$, ...

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Reducing the order of the approximating curve would help reduce occurrence "sharp turns" but this affects the entire graph not just the start and end points.

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You can move $\Delta_{i}$ to the right-hand side and write $$Y_{i} = bX_{i} + (\varepsilon _{i} - \Delta_{i})$$. As long as $\Delta_{i}$ is independent and identically distributed (i.i.d) and uncorrelated with $X_{i}$, the OLS estimate of the $b$ will be BLUE, that is, the estimate of $b$ will be unbiased. So you don't have to modify the standard error of ...

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Recall that $$\operatorname{Cov}(X,Y) = \mathbb E[XY] - \mathbb E[X]\mathbb E[Y]$$ and $$\rho = \frac{\operatorname{Cov}(X,Y)}{\sqrt{\operatorname{Var}(X)}\sqrt{\operatorname{Var}(y)}}.$$ Now as $\mathbb E[X]=\mathbb E[Y] = 0$, we have $$\operatorname{Cov}(X,Y) = \mathbb E[XY],$$ and as $\operatorname{Var}(X)=\operatorname{Var}(Y)=1$, it follows that ...

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Hint: By Cauchy-Schwarz, we have $E\Big(\big|(X+Y)(X-Y)\big|\Big)\leq \sqrt{E\Big(|X+Y|^2\Big)\cdot E\Big(|X-Y|^2\Big)}$.

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If $d$ is the dummy variable, use the following regression model: $y = \beta_0 + \beta_1 d + \beta_2 x_1 + \beta_3(x_1 d) + \varepsilon$ Or your notation: $y \sim x_1 + d + d\cdot x_1$ $\beta_1$ corresponds to the difference (between the two groups defined by the dummy) in the intercepts while $\beta_3$ corresponds to the difference in the slopes.

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The fitting of your data1 (from github) with the method based on an integral equation is shown below. I will try to joint a paper "Triple exponential.docx" where one can find the method of fitting (In French, but the equations are lisible on other languages) Latter on, a new data set was proposed (Data7 from github), with small scatter. The results below ...

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You have usual regression model $$Y_i = bX_i + \varepsilon_i$$ but you can only measure $\tilde{Y}_i = Y_i+\delta_i$, with some measurement error $\delta_i$. Now the model becomes $$\tilde{Y}_i = bX_i + \varepsilon_i+\delta_i$$ and if $\varepsilon_i$ and $\delta_i$ are independent, the only thing that changes is that the variance of the error term ...

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I've used Claude's algorithm in a Fortran program to fit three and four exp term functions to data (on my github page): expFit

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A quick and dirty way which works. If $B,D,F$ are known, the problem reduces to a linear regression. So, make a three dimension grid and for each triplet compute the sum of squares until you find a minimum. For the best triplet, recompute $A,C,D$ and start the nonlinear regression. Because of symmetry, you must not compute all points. Suppose that you ...

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It means that the errors (presumably in different measurements) are independent random variables with jointly normal distribution.

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