WSP11333

• File Number: 8273.100
• Description: Colorado River Basin Salinity Control - Federal Agency Reports - BOR
• Basin: Colorado Mainstem
• Water Division: 5
• Date: 5/1/1975
• Title: Application of Stochastic Hydrology to Simulate Streamflow and Salinity in the Colorado River
• Water Supply Pro - Doc Type: Report/Study

Yi - r!t Yi-l +~ l-R~ . zi

l\)
,....
'"OJ
l\)

or K-th order Markov model,

(6)

k
Yi - L ajYi-j + ~ l-R2 . zi

j-l

(7)

In the equations, rl is the first serial correlation coefficient and
terms are autoregressive coefficients, all estimated from tpe sample
The term;zi is the unexplained portion of the standardized series
standard~zed to have a variance of one, R2 is the coefficient of
determination and is the decimal percent explained variance by this
correlation. Calculating the zi series we obtain a standardized,
serially 'independent series as follows:

the aj
data.

k
zi - [Yi - L ajYi-J;(t l_R2
j-l

(8)

The resutting series exhibits all the properties of a random process
similar do white noise if the first-order model was adequate. In
other wonds it has no serial dependence, no trends, and no periodic
properti_s. However, if two or more series are involved, there may
still re~in some correlation between the zi series resulting from
the orig~nal series, This aspect is discussed in the following section,

2.1.4 Correlation

When the ,random components of two time series are correlated, they
may showisome degree of dependence. This may be quantitized by a
least-squares regression analysis of the series. The degree of
correlat~on is measured by r, the correlation coefficient. For
standardized data (with mean - 0 and variance - 1.0) the intercept
is zero ~nd the regression coefficient (slope) equals the correlation
coefficient, r. Figure 3 indicates this relationship.

y=f (X)

0.0

'1':.'. :, ;
I, .,
1
I

0.0

x

Figure 3. Plot of two linearly
correlated variables.

9