{"id":5424,"date":"2021-09-13T06:00:00","date_gmt":"2021-09-12T21:00:00","guid":{"rendered":"https:\/\/skimura.com\/?p=5424"},"modified":"2022-01-23T10:57:45","modified_gmt":"2022-01-23T01:57:45","slug":"%e7%b5%8c%e5%96%b6%e8%80%85%e3%81%8c%e7%9b%ae%e7%b7%9a%e3%82%92%e4%b8%8a%e3%81%92%e3%82%8b%e3%80%81%e3%81%a8%e3%81%84%e3%81%86%e6%95%b0%e7%90%86%e7%9a%84%e6%84%8f%e5%91%b3","status":"publish","type":"post","link":"https:\/\/skimura.com\/en\/%e7%b5%8c%e5%96%b6%e8%80%85%e3%81%8c%e7%9b%ae%e7%b7%9a%e3%82%92%e4%b8%8a%e3%81%92%e3%82%8b%e3%80%81%e3%81%a8%e3%81%84%e3%81%86%e6%95%b0%e7%90%86%e7%9a%84%e6%84%8f%e5%91%b3\/","title":{"rendered":"The Mathematical Meaning of “Keep Raising Your Eyes!”"},"content":{"rendered":"\n

Today, I’d like to share with you that the formula that I introduced in my blogs of September 3, 2021, “A Super Simple Formula to Dispel Fears of Plummeting Sales under COVID-19<\/a><\/em><\/strong><\/span>” and September 11, 2021, “Understanding Continuity is Power with a Formula<\/a><\/span><\/em><\/strong>” actually hides another essential secret.<\/p>\n

Let’s start with the formula.<\/p>\n

$$x_{t}=\u03b1x_{t-1}+\u00b5+\u03b5_{t}\\tag{1}$$<\/p>\n

xt<\/sub><\/em> is the sales at time t<\/em>, xt-1<\/sub><\/em> is the previous sales, \u03b1<\/em> is the sales coefficient, \u00b5<\/em> is a constant, and \u03b5t<\/sub><\/em> is the error term. This simple equation taught us that sales that dropped due to COVID-19 will always return to normal, and conversely, sales that temporarily increased rapidly will return to normal if no action is taken afterwards. In each of these cases, we have looked at the time series change in sales xt<\/sub><\/em> by stimulating the error \u03b5t<\/sub><\/em> in equation (1) to negative or positive values.<\/p>\n

Equation (1), when expanded based on the rule, leads to equation (2), which is<\/p>\n

$$x_{t}=\u03b1x_{t-1}+\u00b5+\u03b5_{t}=\u03b1^tx_{0}+\u00b5(1+\u03b1+\u03b1^2+\u03b1^3+ \u30fb\u30fb\u30fb+\u03b1^t)+(\u03b1^{t-1}\u03b5_{1}+\u03b1^{t-2}\u03b5_{2}+\u03b1^{t-3}\u03b5_{3}+\u30fb\u30fb\u30fb+\u03b5_{t})\\tag{2}$$<\/p>\n

First term: $\u03b1^tx_{0}$ approaches zero when $\u03b1<1$.<\/p>\n

Second term: $\u00b5(1+\u03b1+\u03b1^2+\u03b1^3+ … +\u03b1^t)$ converges to a certain value when $\u03b1<1$.<\/p>\n

The third term: $\u03b1^{t-1}\u03b5_{1}+\u03b1^{t-2}\u03b5_{2}+\u03b1^{t-3}\u03b5_{3}+…+\u03b5_{t}$ is a collection of errors with coefficients and is normally distributed<\/p>\n

For example, a COVID-19 is a large negative stimulus to part of the third term, while a temporary surge in sales is a large positive stimulus. In both cases, when the stimulus passes, the situation returns to its normal steady state. Now, if we look closely at equation (2), we can see that it is not only the third term (error term) that raises the level of sales xt<\/sub><\/em>. Pay attention to the second term, \u00b5<\/em>. Depending on whether this \u00b5<\/em> is large or small, the value of convergence will vary greatly. In fact, it can be said that this term has a more essential impact on sales. In other words, this \u00b5<\/em> is the manager’s perspective.<\/p>\n

Now, let’s start the calculation. Fig. 1<\/em> shows the time series change in xt<\/sub><\/em> when \u00b5 is doubled to 20<\/em> after t=10<\/em>.<\/p>\n

Fig. 1<\/em><\/p>\n

\"\"<\/p>\n

Let x0<\/sub>=100, \u03b1=0.9, and \u03b5t<\/sub>=N(0,5) be a normal distribution.<\/em>
The time series change in sales xt<\/sub> when the value of \u00b5 is increased from 10 to 20 after t=10\u00a0<\/em><\/h5>\n

\u00a0<\/p>\n

Indeed, when the manager doubles the value of \u00b5<\/em>, the sales xt<\/sub><\/em> also doubles. Now, suppose further that aggressive advertising activities were carried out from t=10<\/em> to 13<\/em> and continued thereafter (Fig. 2<\/em>).<\/p>\n

Fig. 2<\/em><\/p>\n

\"\"<\/p>\n

x0<\/sub>=100\u3001\u03b1=0.9\u3001\u03b50\uff5e10<\/sub>=N(0,5) , <\/em>\u03b511\uff5e<\/sub>=N(10,5) <\/em><\/h5>\n
\u03b510<\/sub>=50, \u03b511<\/sub>=40, \u03b512<\/sub>=30, \u03b513<\/sub>=20<\/em><\/h5>\n
\u00b5 is 10 until t=9, and 20 after t=14<\/em><\/h5>\n

\u00a0<\/p>\n

The sales tripled due to the “combined effect” of the continuous advertising activities!<\/p>\n

\u00a0<\/p>\n

This concludes our three-part series on time series analysis<\/em>. Even though equation (1) is very simple in form, it has taught us many things such as:<\/p>\n