ISSUE

Induce autocorrelation of this equation:

X_n= λY_n+(1-λ) X_(n-1)+2/3(Y_(n+1)-Y_(n-1))

Note: Generally the two arrays are expressed as (Y1,Y2), not (X,Y) to avoid confusion with X = independent and Y = dependent in the linear (non-time series) regression. See Frechet (1951). The question really is: "What is the joint probability of the correlations of two arrays (Y1,Y2)?" The answer can be found in Frechet's explicit correlation induction.

SMOOTHED DATA

In autocorrelation analysis smoothed data is equal to the weighted average of the data. Smoothed data (d*) may be given as:

(1)   d* = 0.25di-1 + 0.50di + 0.25di+1

… where d* = smoothed data, di-1 = last observation, di = current observation and di+1 = future observation. Under this structure the coefficients (0.25, 0.50 and 0.25) are known as filter factors. Note that ¼ or 0.25 is used for both past and future; it means that under this notation, the past and future have equal weight; whereas the present is weighted 0.50 or ½---this approach seems to offer a balanced fit.

However, in the question, the proposed equation: X_n= λY_n+(1-λ) X_(n-1)+2/3(Y_(n+1)-Y_(n-1)) appears to have different notation---a closer look tells us, it is the same as equation (1) with the exception of the introduction of lambda, and the introduction of a lag for X. The true form of X_n= λY_n+(1-λ) X_(n-1)+2/3(Y_(n+1)-Y_(n-1)) is ….

There are two extreme solutions to the equation. Assume that lambda = 1, then:

Xn = (1)Yn + (1 – 1)Xn-1 + (2/3)Yn+1

(2)   Xn = Yn + (2/3)Yn+1

The interpretation of this solution: Xn = Yn + (2/3)Yn+1 may be clear by putting the notation in the d* format, thus:

(3)   d* = Xn – Yn + (2/3)Yn+1

It means that the smoothed data is comprised of the present value of X and the present value of Y weighted at unspecified proportion of L + (1 + L) and 0.67 of future value of Y.

The second extreme solution is, say, let lambda = 0, then:

Xn = (0)Yn + (1 – 0)Xn-1 + (2/3)Yn+1

(4)  Xn = Xn-1 + (2/3)Yn+1

Putting solution 2 in the d* format, the following is obtained:

d* = Xn-1 – Xn + (2/3)Yn+1

It means that the smoothed data depends on the past value of X and less the present value of X weighted at unspecified proportion of L + (1 + L) and 0.67 of future value of Y.

Since the equation X_n= λY_n+(1-λ) X_(n-1)+2/3(Y_(n+1)-Y_(n-1)) allocates 67% weight to the future, the allocation seems not to follow the balanced fit in d*. Can it be possible? Yes, it depends on data and the assumption made by the analysis. These terms in the equation: 2/3(Y_(n+1)-Y_(n-1)) contains a correction factor, i.e. although the heavier weight is given to the future, but it is off set by Yn-1 or past data of Y.

The use of X and Y here is interesting. Generally, we use only X (univariate). But the proposed equation uses Y also. Note that elsewhere the notation is (Y1, Y2). This two arrays analysis is called joint probability determination.

LAMBDA FOR EXPLICIT CORRELATION INDUCTION

Lambda (L) was treated by Frechet in 1951 when he characterized the bound of joint probability of Y1 and Y2. The bound of the joint probability was given as:

(5)   H-(y1,y2) = max{F1(y1) + F2(y2) – 1, 0}

and

(6)   H+(y1,y2) = min{F1(y1) + F2(y2) – 1, 0}

The line of reasoning above (two extreme scenario) was consistent with Frechet’s method. In rechet F1(y1) is the cumulative distribution of Y1 and F2(y2) is the cdf for y2. Therefore, H-(y1,y2) and H+(y1,y2) are the minimum and maximum-correlation joint cdfs of (Y1,Y2). Frechet thus offered:

(7)   H-(y1,y2)< H(y1,y2) < H+(y1,y2)

For all (y1,y2) and all possible joint distribution of H(y1,y2). In order to characterize the composite of the two joint distribution (Y1,Y2), Frechet introduced lambda thus:

(8)   LH-(y1,y2) + (1-L)H+(y1,y2)

… where 0 < L 0, and 1, b < 1.

The lambda (L) is the weight. This lambda weight: L and (1 – L) is called “composition probabilities.” The distribution involved is called "composite distribution." See Nelsen (1987).

REFERENCES

(1) Frechet, M. 1951. Sur les tableaux de correlation dont les marges sont donnes.

Annales de l'Universite de Lyon, Section A, 14, 53-77.

(2) Nelsen, R. B. 1987. Discrete Bivariate Distributions with Given Marginals and Correlation. Communications in Statistics: Simulation and Computation,16(1), 199-208.

(3) Moore, B. A. and C. H. Reilly. 1993. Randomly Generating Synthetic Optimization Problems with Explicitly Induced Correlation. OSU/ISE Working Paper Series Number 1993-002. The Ohio State University, Columbus, Ohio.

(4) Devroye, L. 1986. Non-Uniform Random Variate Generation. Springer-

Verlag, New York.