An Introduction to
Statistics with Klong
Linear Regression
Here is another x/y set with values rounded to two decimal places.
We know that the variables are probably correlated, because of the
way in which the set has been created.
XY::(!30),'rndn(;2)'err(20;20;0.15*!30)
[[ 0 0.0 ] [ 1 4.27] [ 2 1.48] [ 3 4.26] [ 4 5.28]
[ 5 0.75] [ 6 0.9 ] [ 7 1.89] [ 8 5.27] [ 9 1.35]
[10 0.91] [11 1.65] [12 2.39] [13 4.3 ] [14 5.63]
[15 1.66] [16 0.64] [17 0.2 ] [18 4.46] [19 3.62]
[20 0.06] [21 7.27] [22 9.77] [23 2.43] [24 4.78]
[25 1.99] [26 3.9 ] [27 3.46] [28 7.73] [29 4.35]]

The scatter plot in fig.9 also suggests a correlation.
The lreg
function of nstat
can be used to compute the
slope
and
intercept
coefficients of a
regression line
through the x/y set:
lreg(XY)
[0.201581757508342594 1.21793548387096761]

The function returns a tuple containing the slope and intercept
values of the regression line, but these details do not have to be
memorized, because the lr
function will take care if it.
While lreg
fits a
model
to the data, lr
uses the model to
predict values of the Y
variable of the x/y set given values of the X variable. In fig.10,
lr
is used to plot the regression line through the set.
Its parameters are an
independent variable
and a linear regression model delivered by lreg
.
The nstat
module provides two methods for quantifying
the correlation between two variables, the
covariance
(cov
) and the normalized
correlation coefficient
(Pearson's r, cor
). They both expect each variable as a
separate data set:
cov(*'XY;{x@1}'XY)
15.1018333333333333
cor(*'XY;{x@1}'XY)
0.201581757508342604

Given a model, like
linear regression,
there are several ways to examine the quality of the predictions
made by the model. The nstat
module provides the following
of them: the
residual sum of squares
(RSS), the residual squared error (RSE), the
mean squared error
(MSE), and the
coefficient of determination
(r^{2}):
L::lreg(XY)
[0.201581757508342594 1.21793548387096761]
rss(XY;lr(;L))
304.98152685205783
rse(XY;lr(;L))
3.30033292071777115
mse(XY;lr(;L))
10.1660508950685943
r2(XY;lr(;L))
0.230445406440760124

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