Statistics Symbols

This page is specifically for people in EGR 103 and represents a concordance of sorts among the lectures and the two textbooks with respect to different symbols for statistical quantities. It has been updated to reflect the use of Python in EGR 103.

In general, the index $$k$$ is used to mean the $$k$$th data point (out of $$N$$ data points, indexed 0 through $$N-1$$) and the index $$m$$ is used as the $$m$$th basis function of a general linear fit (out of $$M$$ basis functions, indexed 0 through $$M-1$$). The letter $$a$$ is used to represent the coefficient of a basis function and $$\phi(x)$$ is the basis function. That is to say, for a general linear fit with $M$ basis functions, the estimate of the $$k$$th dependent value is:

$$ \begin{align*} \hat{y}(x_k)&=\sum_{m=0}^{M-1}a_m\phi_m(x_k) \end{align*} $$

Symbols

The entries in the "Palm" column are taken from William J. Palm III's Introduction to Matlab 7 for Engineers, 2/e book, while those in the "Chapra" column are taken from Steven C. Chapra's Applied Numerical Methods with MATLAB for Engineers and Scientists, 2/e book. Entries in the "EGR 103" column, when not taken from Chapra or Palm, have been developed over the course of several years' of EGR 103 lectures.

\( \begin{array}{|c|c|c|c|}\hline \mbox{Quantity} & \mbox{Palm} & \mbox{Chapra} & \mbox{EGR 103}\\ \hline \mbox{Independent Data} & x & x & x \\ \hline \mbox{Dependent Data} & y & y & y \\ \hline \mbox{Individual Elements} & y_i & y_i & y_k \\ \hline \mbox{Mean Value} & \bar{y}=\frac{1}{n}\sum_{i=1}^ny_i & \bar{y}=\frac{\sum y_i}{n} & \bar{y}=\frac{\sum y_k}{N} \\ \hline \mbox{Sum of Squares of Data Residuals} & S=\sum_{i=1}^m\left(y_i-\bar{y}\right)^2 & S_t=\sum\left(y_i-\bar{y}\right)^2 & S_t = \sum_k\left(y_k-\bar{y}\right)^2 \\ \hline \mbox{Estimates (straight line)} & f(x_i) & a_0+a_1x_i& \hat{y}_k=p[0]x_k+p[1] \\ \hline \mbox{Estimates (general linear fit)} & f(x_i) & \hat{y}_i=\sum_{j=0}^ma_jz_{ji} & \hat{y}_k=\sum_{m=0}^{M-1}a_m\phi_m(x_k) \\ \hline \mbox{Sum of Squares of Estimate Residuals (straight line)} & J=\sum_{i=1}^m\left[f(x_i)-y_i\right]^2 & S_r=\sum\left(y_i-a_0-a_1x_i\right)^2 & S_r=\sum_k\left(y_k-\hat{y}_k\right)^2 \\ \hline \mbox{Sum of Squares of Estimate Residuals (general linear fit)} & \mbox{Not Used} & S_r=\sum_{i=1}^{n} \left(y_i-\hat{y}\right)^2 & S_r=\sum_k\left(y_k-\hat{y}_k\right)^2 \\ \hline \mbox{Coefficient of Determination} & r^2=1-\frac{J}{S} & r^2=\frac{S_t-S_r}{S_t} & r^2=\frac{S_t-S_r}{S_t}=1-\frac{S_r}{S_t} \\ \hline \mbox{Sample Standard Deviation} & \sigma=\sqrt{\frac{\sum_{i=1}^n(y_i-\bar{y})^2}{n-1}} & s_y=\sqrt{\frac{S_t}{n-1}} & s_y=\sqrt{\frac{S_t}{n-1}} \\ \hline \mbox{Coefficient of Variation} & \mbox{Not Used} & \mbox{c.v.}=\frac{s_y}{\bar{y}}*100\% & \mbox{c.v.}=\frac{s_y}{\bar{y}}*100\% \\ \hline \mbox{Standard Error of the Estimate (straight line)} & \mbox{Not Used} & s_{y/x} = \sqrt{\frac{S_r}{n-2}}& s_{y/x} = \sqrt{\frac{S_r}{n-2}} \\ \hline \mbox{Standard Error of the Estimate (general linear fit)} & \mbox{Not Used} & s_{y/x} = \sqrt{\frac{S_r}{n-(m+1)}}& s_{y/x} = \sqrt{\frac{S_r}{n-N}} \\ \hline \end{array} \)

Code

The following table will compare how to calculate various items in MATLAB and Python. The MATLAB version will use a row matrix:

a = [4 8 8 2 5 4]

while Python will use a list, a 1-D array and a data frame with the same contents:

import numpy as np
import pandas as pd
alist = [4, 8, 8, 2, 5, 4]
aarray = np.array(alist)
aframe = pd.DataFrame(alist)

Questions

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