# How the regression method affects the smoothness of the ’regression curve’ when we used them as a smoothing method?

The following Quizzes are the rough　translation (with minor modification) of Quizzes No.08 of the exam of the "2019's semi-first grade of Japan Statistical Society Certificate (JSSC)" (See (ref 1) ).

The correct answers by the JSSC  are as described below (See the column named "【The correct answers】." ). However, I cannot imagine how to achieve the correct answer.

My Question:

• (My Question 0) Why can we smooth the graph of Fig 1 by using (Formula 1)or (Formulas 2) ? Detail of this question is as described in the column named 【 Detail of My Question 0】.
• (My Question 1) How to achieve the correct answer to the following Quizzes? : How the regression method affects the smoothness of the ’regression curve’ when we used them as a smoothing method?

Detail of the (My Question 1) are:

【 Detail of My Question 0】
The following graphs in both Fig 2 and Figs 3 show the relationship between the time (month) and the count of high energy-charged particles. How can we get the relationship between time (month) and the number of particles from this regression method? Curves, shown in ① to ④ of Figs 2 (and Fig 3) show the relationship between time (month) v.s number of particles. Is this the correct understanding that:

• In the following (Quiz 1) and (Quiz 2), "the number of particles (y)" is an explanatory variable, not an objective variable?
• In the case of (Quiz 1) below, $${\beta}_{i}$$ that minimizes (Equation 1) considered as $${y}_{i}$$?
• Although it was not specified in the original question sentence of following Quiz 1 and Quiz 2, do $${y}_{i}$$ have to be arranged in chronological order? (Chronological order means Manners that increase $$i$$ as the time goes on.)

If so, what are the advantages compared to the elementally smoothing method such as the moving average method?

Quizzes
The following graph shows the count of the High energy-charged particles during the period from October 2000 to October 2018. Total 217 data $${y}_{i}$$ (i=1,2,..., 217; These data are quoted at secondhand from the OLU) are plotted in this graph. Answer the following (Quiz 1) and (Quiz 2).

Fig.1

(Quiz 1) To recognize the characteristics of data, smoothing is performed by the Fused Lasso method using the following equation, and we set λ =500. (Hereinafter, the following formula is referred to as (Formula 1). )

Then, which one is the most appropriate graph as a result of above-mentioned smooshing? Select the best answer from the following ① to ④.

Fig.2

(Quiz 2) Refer to the abovementioned total 217 data $${y}_{i}$$, smoothing was performed using a method different from Quiz 1. The result of this smoothing is shown in the following graph.

Figs.3

Corresponding to the the graph above (See Figs.3), which of the following formulas, ① to ④ is most suitable as a smoothing method?(Hereinafter, the following formulas are referred to as (Formula 2-①), ...,and (Formula 2-④) , respectively.)Select the best answer from the following ① to ④.

The correct answers are opened to the public by the JSSC (Ref.1). The correct answer of Quiz(1) is ④, and the correct answer of (Quiz 2) is ④. (So the correct answer of both Quizzes are ④.)

References:
(Ref1)Question No 8 of the exam of the "2019's semi-first grade of Japan Statistical Society Certificate(Written in Japanese.)

P.S. I'm not very good at English, so I'm sorry if I have some impolite or unclear expressions.

One characteristic of the LASSO penalty of the form $$+\sum_i |\alpha_i|$$ is that the solution vector $$(\alpha_1, \ldots, \alpha_n)$$ tends to have lots of zeros. You can find explanations of this behavior in many places.
Applying this statement to the first question implies the Fused LASSO encourages many of the $$|\beta_{i+1} - \beta_i|$$ to be zero. Which plot has many $$i$$ such that $$\beta_{i+1} - \beta_i = 0$$?
For the second question, the graph seems to be piecewise linear. Can you use the same reasoning to see why (4) encourages piecewise linear solutions? (Hint: if the $$(\beta_1, \ldots, \beta_n)$$ form a line, what is $$\beta_{i+2} - 2 \beta_{i+1} + \beta_i$$?)
• Let me clarify my understanding: For the case of (Quiz 1), ${\beta}_{i}$ that minimizes (Equation 1) considered as ${y}_{i}$? So $({t}_{i},{\beta}_{i})$ are plotted on Fig.2 and the penalty-term acts like a moving average? ( Please refer to the updated【My Question】column.) If so, I think I am reaching an understanding. – Blue Various Aug 22 at 10:13
• @BlueVarious The scatter plots are of the points $(i, y_i)$, whereas the line plots are of $(i, \hat{\beta}_i)$ where $\hat{\beta}$ is the solution to the optimization problem. – angryavian Aug 23 at 4:38