# Method

## Begin

In the field of my study, the method of times series forecasting is so similar, almost every method is based on the trend and seasonal, but it is clear that not all times series follow these rules. Some times series come from the real volatility, and it doesn’t have the obvious features of trend and seasonal. In specific, this kind of series is usually defined as chaos series according to the chaos theory.

But how to forecast this kind of series? The jerk equation is the simplest method to do this by chaos theory.

## Methodology

The jerk equation defined the three basic concepts.

First concept is velocity, it’s the rate of change in time of the price, it could be showed in this equation.

$$ v = \frac{\mathrm{d}x}{\mathrm{d}t} $$

The second concept is the acceleration, it means the rate of change of velocity, or the second derivative of the price. $$ a = \frac{\mathrm{d}v}{\mathrm{d}t} = \frac{\mathrm{d}^2 x}{\mathrm{d}t^2} $$

Then the third concept and the most important concept is **Jerk**, it means the rate of change of acceleration, or the third derivative of the price.

$$ j = \frac{\mathrm{d}a}{\mathrm{d}t} = \frac{\mathrm{d}^2 v}{\mathrm{d}t^2} = \frac{\mathrm{d}^3 x}{\mathrm{d}t^3} $$

After the $v$, $a$, $j$, are defined by the price. The differential equations form could be showed as below. $$ J(x^{\prime \prime \prime},x^{\prime \prime},x^{\prime},x) = 0 $$

This equation is called **jerk equations**, this kind of equation is the minimal setting for solutions showing chaotic behavior. Thus,the approach to model price movement will be model its behavior with the equation of the form.
$$
A\frac{\mathrm{d}^3 x}{\mathrm{d}t^3} + B\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} + C\frac{\mathrm{d} x}{\mathrm{d}t} + Dx +E = 0
$$

We could convert this equation to the form below.

$$ A x^{\prime \prime \prime} + B x^{\prime \prime} + C x^{\prime} + D x +E =0 $$ In this equation, the A, B, C, D and E are the real numbers and is the parameters we need to estimate.

When E = 0, this differential equation could be simplified as below. $$ A\lambda^3 + B\lambda^2 + C\lambda + D = 0 $$

To solve this equation, we could get the $x(t)$ as this. $$ x(t) = C_1 e^{\lambda_1 t} + C_2 e^{\lambda_2 t} + C_3 e^{\lambda_3 t} $$

The jerk equation could become this form. $$ x_0 = C_1 +C_2+ C_3 \\ v_0 = C_1\lambda_1 + C_2\lambda_2 + C_3\lambda_3 \\ a_0 = C_1\lambda_1^2 + C_2\lambda_2^2 + C_3\lambda_3^2 \\ j_0 = C_1\lambda_1^3 + C_2\lambda_2^3 + C_3\lambda_3^3 $$

To give the time change, we do the discretization of this equation. $$ x^{\prime} \approx \frac{x_{i+1}-x_i}{\Delta t} $$

$$ x^{\prime \prime} \approx \frac{x_{i+1}-2x_i+x_{i-1}}{\Delta t^2} $$

$$ x^{\prime \prime \prime} \approx \frac{x_{i+1}- 3x_i+3x_{i-1}-x_{i-1}}{\Delta t^3} $$

Then we could regard this system as a linear system and solve this equations by a least-squares method.

$$ A\vec{x} = \vec{b} $$

Then we could get the estimate parameters easily. After doing the estimation, we should consider the predictor method of these equations.

We could convert the jerk equation to this form.

$$ A(x_{i+1}-3x_i+3x_{i-1}-x_{i-2})+B(x_{i+1}-2x_i+x_{i-1})+C(x_{i+1}-x_i)+Dx_i+E = 0 $$

The price next day could be approximated by following equation, with the linear system.

$$ x_{i+1} = \frac{-E+Ax_{i-2}-x_{i-1}(3A+B)-x_i(-3A-2B-C+D)}{A+B+C} $$

At last, we could push the days by this equation and try to do the prediction through this equation. It’s a good method of doing prediction.