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The least squares method with Despacito

stats con chris
2022-03-27
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II. Methodology: The equation of a line

We define the linear equations of Despacito ($d$) and See You Again ($s$) as,

$$ \begin{align} y_{d} &= m_d x + b_d, \tag 9 \\ z_{s} &= m_s x + b_s. \tag{10} \end{align} $$

Considering the analytical expressions previously obtained we have,

$$\begin{align} m_d &= \frac{n \sum_i^n y_i x_i – \sum_i^n y_i \sum_i^n x_i}{n\sum_i^n x_i^2 – (\sum_i^n x_i)^2}, \tag{11} \\ b_{d} &=\frac{\sum_i^n y_i\sum_i^n x_i^2 – \sum_i^n y_i x_i \sum_i^n x_i }{n\sum_i^n x_i^2 – (\sum_i^n x_i)^2}. \tag{12} \end{align}$$

and

$$\begin{align} m_s &= \frac{n \sum_i^n z_i x_i – \sum_i^n z_i \sum_i^n x_i}{n\sum_i^n x_i^2 – (\sum_i^n x_i)^2}, \tag{13}\\ b_{s} &=\frac{\sum_i^n z_i\sum_i^n x_i^2 – \sum_i^n z_i x_i \sum_i^n x_i }{n\sum_i^n x_i^2 – (\sum_i^n x_i)^2}. \tag{14} \end{align}$$

To determine the numerical values of m and b we make use of the data given in Table I, refer to page 2. In this way we generate the following tables:

Table II: Despacito data.
Day ($x$)	Despacito ($y$)	$\lower 0.8em {y_ix_i}$	$x_i^2$
4	2310	9240	16
5	2328	11640	25
6	2348	14088	36
7	2369	16583	49
8	2393	19144	64
9	2415	21735	81
$\sum_i^6 x_i = 39$	$\sum_i^6 y_i =14163$	$\sum_i^6 y_i x_i =92430$	$\sum_i^6 x_i^2 = 271$

Table III: See You Again data.
Day ($x$)	See You Again ($z$)	$\lower 0.8em {z_ix_i}$	$x_i^2$
4	2872.8	11491.2	16
5	2875.9	14379.5	25
6	2879.0	17274.0	36
7	2882.2	20175.4	49
8	2885.7	23085.6	64
9	2889.0	26001.0	81
$\sum_i^6 x_i = 39$	$\sum_i^6 z_i =17284.6$	$\sum_i^6 z_i x_i =112406.7$	$\sum_i^6 x_i^2 = 271$

For Despacito, we replace the data given in Table II in Eqs. (11) and (12). For See You Again, we replace the data given in Table III in Eqs. (13) and (14). Then,

$$\begin{align} m_d &= \frac{6 \times 92430 – 14163\times 39}{6\times 271 – (39)^2} = 21.2, \tag{15} \\ b_{d} &=\frac{14163 \times 271 – 92430\times 39 }{6\times 271 – (39)^2} = 2222.9. \tag{16} \end{align}$$

and

$$\begin{align} m_s &= \frac{6 \times 112406.7 – 17284.6\times 39}{6\times 271 – (39)^2} = 3.2, \tag{17} \\ b_{s} &=\frac{17284.6 \times 271 – 112406.7\times 39 }{6\times 271 – (39)^2} = 2859.7. \tag{18} \end{align} $$

Finally, the linear equations are given by,

$$\begin{align} y_d &= 21.2 x + 2222.9, \tag{19} \\ z_s &= 3.2 x + 2859.7. \tag{20} \end{align}$$

The point where these lines intersect defines the moment when Despacito surpasses See You Again and becomes the most viewed video on YouTube. The resolution of this problem is described on the next page.

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