An Application of System Kinematics on a Stock Market

Definition of 30-market: Thirty stocks were chosen from Fidelity, and these thirty stocks in a given order make up  the 30-market.

Definition of 30-vectors: The thirty prices of these thirty stocks expresses the state of the 30-market.

Y=Y_(i)                                     （i=1,2,…30）,        (1-1）

Y’=Y_(i)/|Y|                               （i=1,2,…30）,        (1-2）

Where the |Y|=√∑(( Y_(i))²), （i=1,2,…30）,is the vector length, a scalar.

The market transition vectors (trends) are defined as the present over the previous states:

T_(i,k)=Y’_(i,k)/Y’_(i,k-1)                      （i=1,2,…30）        （2-1）

Where the second index indicates the time: k=present, k-1=previous.

Projection of the next states are expressed by a vector product of the present states and the trends:

P_(i,k+1)=Y_(i,k)*T_(i,k)                               （i=1,2,…30）        （2-2）

Where the P_(i,k+1) is the projection of the next states of the 30-market based on the existing information of the 30-market.

Bai-Jameson Filter connects the two sequential states:

Y_(i,k+1)=[Y_(i,k)*T_(i,k)* β +D_(i,k+1)* α]/（α+β）  （i=1,2,…30）    （3-1）

Where the D_(i,k+1) are the 30-observation vectors, the closing prices of the thirty stocks on time k+1. Thus, the expectation of the 30-market on k+1 are the weighted sum of the observations on k+1 and the projection from k. And, where α and βare the weights of D and P, respectively. Setting the two weights equal to α = β =0.5:

Y_(i,k+1)=[Y_(i,k)*T_(i,k)+D_(i,k+1)]/2         （i=1,2,…30）             （3-2）

The formula can be further extended into a time chain (3-3). This Bai-Jameson Chain uses history information to predict the future states in a manner of Fading Memory:

P_(i，k+1)= T_(i，k)(D_(i,k)+0.5*T_(i,k-1)(D_(i,k-1)+0.5*T_(i,k-2)(D_(i,k-2)+...+0.5*T₂(D₂+D₁))))

（i=1,2,…30）                                                      (3-3)              

Main Reference: T. Jay Bai, Trend Analysis, The Ethnic Publishing House, Beijing, 2006.

Translated from：http://blog.sciencenet.cn/blog-333331-768774.html

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