# Additive model and multiplicative model

The simple formula for Regression Analysis is
Y = AX + B
This is an expression that represents a straight line with a slope of A and an intercept of B.

The structural equation is
Y = AX + B + E
E is an error, and the presence of this part expresses that the data is arranged in a form close to a straight line with variation. This formula is called the additive (addition) model.

## Two types of multiplicative models

There are two main types of explanations about multiplicative models in the world. The error term has a different shape.

### Multiplicative model exp(E)

Multiplicative model exp(E)
Y = B * (X^A) * exp(E)

### Multiplicative Model E

On this page, we refer to the "multiplicative model E" as the following equation:
Y = B * (X^A) * E

Multiplicative model E is often introduced as a model for time series analysis.

## Examples of multiplicative models

"E0.1" in the graph indicates that the number in parentheses is a normally distributed random number with a mean of 0 and a standard deviation of 0.1.

### Multiplicative model exp(E)

Here is an example of a multiplicative model exp(E). The model is similar to the multiplicative model E, which shows that the variation of Y increases as X increases, but also shows that Y increases overall.      ### Multiplicative Model E

Here is an example of a multiplicative model E. The model shows that as X grows, the variation in Y increases both positively and negatively.    In the graph above, there is no periodicity. If you want to deal with periodicity, it is easier to deal with it with an additive model, such as AR Model and Others.

## When to use multiplicative models

If you look up "Multiplicative model", you can find many articles that are introduced as a method of time series analysis.

If we think of X as a time series, the multiplicative model represents that the influence of uncertainty increases as time passes.

## Relationship between additive and multiplicative models exp(E)

Suddenly, the structural equation of the additive model is transformed as follows. If we do this deformation, we can see that the additive and multiplicative models are the same.

You can transform from a multiplicative model to an additive model by taking the logarithm for both sides of the multiplicative model.

The relationship between additive and multiplicative models E is similar.

## Differences between multiplicative and Proportional variance models

Multiplicative and Proportional variance models are similar, but in practice it is better to use them differently.

### Differences in use

Multiplicative models are well known as models for time series analysis. It seems to be adopted as a model to express that X is considered time and that uncertainty increases as time passes.

An example of a Proportional variance model is when the value of the measuring instrument is X. It seems to be suitable when dealing with the relationship between natural phenomena and their measurements.

### Difference between the multiplicative model exp(E) and the Proportional variance model

The multiplicative model exp(E) and the Proportional variance model can be almost unchanged at times and clearly different. The time that hardly changes is the lower case, the left is the multiplicative model exp(E), and the right is the Proportional variance model.  The simple form of the multiplicative model exp(E) is
Y = B * X * exp(E)

A simple form of a Proportional variance model is
Y = (B + E) * X

When E is small, the multiplicative model exp(E) is close to Y = B * X, and the Proportional variance model is close to
Y = B * X
and the two are similar.

When E is large, the difference is larger. On the left is the multiplicative model exp(E) and on the right is the Proportional variance model.  There are two points where the E is large and the difference appears. One is the difference between exp(E) and E, where E is normally distributed, and exp (E) has an extremely long base.

Another difference is the difference depending on the shape of the formula. The multiplicative model is a multiplication of three elements, and the Proportional variance model is a difference made by adding two multiplications, B and X, and E and X. In a multiplicative model, when E is large, the values of B and X are less relevant, and are determined by the influence of E. In the case of the Proportional variance model, even if E is large, the proportionality between X and Y remains.

When the variation becomes large, in the case of the multiplicative model exp(E), it feels like data sticking to the X axis (Y = 0). On the other hand, in the case of a Proportional variance model, the density is less dense near the X-axis. For Proportional variance models, the sloped line is also denser around it.

### Difference between multiplicative model E and Proportional variance model

The simple form of the multiplicative model E isbr> Y = B * X * E

In a Proportional variance model, if the slope term is 0, the multiplicative model E and the Proportional variance model are the same. On the left is the multiplicative model exp(E) and on the right is the Proportional variance model.  As the slope increases, the multiplicative model exp(E) and the Proportional variance model may be similar as described above, but the multiplicative model E does not have a slope like the Proportional variance model.

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