Critical Value Model of Sample Correlation Coefficient after Windowed Moving Average

What is it about?

This work concludes the steps for solving the confidence level $\xi$ based on sample correlation coefficient $|r|$: 1) Suppose window size is $k$, and take two independent samples ${\{x_i\},\{y_i\},}$ ${(i=1, ..., N)}$ through windowed moving average. And calculate reduction window size $k^{\prime}$ and freedom decrease factor $\bar{a}$ from formula \ref{eq9} and \ref{eq11} respectively; 2) Standardize the two samples in the step 1) so as to take them as Standardized Normal Distribution independently, and we have new samples ${\{x^{\prime}_i\},\{y^{\prime}_i\}(i=1, ..., N)}$; 3) If time lags $t$ is not considered, calculate sample correlation coefficient $|r|$ for ${\{x^{\prime}_i\}}$ and ${\{y^{\prime}_i\}}$; 4) When given confidence level $\xi$, we have slope $p_{1}$ and intercept $p_{2}$. Combine with freedom decrease factor $\bar{a}$ and formula \ref{eq12} to get parameter $b$ in order to obtain $t_{\alpha/2}(\bar{a}n+b)$. And then based on the formula \ref{eqa10} to own Model critical value $|r_{_\xi}|_{\rm MO}=\frac{t_{\alpha/2}(\bar{a}n+b)}{\sqrt{t_{\alpha/2}^2(\bar{a}n+b)+(\bar{a}n+b)}}$; 5) Repeat the step 4) to find out the confidence level $\xi$, which leads to the minimum of $||r|-|r_{\xi}|_{\rm{MO}}|$. Also, this $\xi$ is the confidence level based on sample correlation coefficient; 6) If considering time lags $t$, calculate the maximum of sample correlation coefficient $|r|_{\rm MAX}$ between ${\{x^{\prime}_i\}}$ and ${\{y^{\prime}_i\}}$. Then repeat the process from the step 4) to the step 5), and notice that the degree of freedom $n$ changes into $n-|t|$. Equations: \bar{a}=\frac{0.73\pm0.005}{k^{\prime}-(0.45\pm0.01)} b=p_{1}\cdot\ln(\bar{a})+p_{2}=p_{1}(\xi)\cdot\ln(\bar{a})+p_{2}(\xi) p_{1}=p_{1}(\xi)=-(0.2\pm0.01)\xi^{5.4\pm0.8}-(0.92\pm0.006) p_{2}=p_{2}(\xi)=(0.56\pm0.05)\xi^{9.2\pm1.5}-(0.78\pm0.01)

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Why is it important?

The critical value of correlation coefficient ${|r|_{\rm C}}$, which varies with the degree of windowed moving average $k$, freedom $n$, and confidence $\xi$, is widely accepted to judge the linear relationship between two independent samples $X$ and $Y$. In order to avoid referring to inconvenient critical value tables or using time-consuming Monte Carlo simulation (MC), this paper presents a mathematically unified model for calculating the critical value. In this paper, both independent samples $X$ and $Y$ follow the standard normal distribution. We analyze MC result ${|r|_{\rm MC}}$ for these samples $X$, $Y$ to establish the model parameters and then examine the model result ${|r|_{\rm MO}}$ for three different astronomical fields to calculate the confidence level $\xi$ of sample correlation coefficient ${|r|}$, which is compared with MC result ${|r|_{\rm MC}}$. The results show that the MC results ${|r|_{\rm MC}}$ and model results ${|r|_{\rm MO}}$ are in exceptionally good agreement, and the applicability of the model to obtain the confidence level $\xi$ is satisfactory when calculating the sample correlation coefficient ${|r|}$ for two independent samples $X$ and $Y$.

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