1 time series and stochastic processes

Random variable sequence Y t:t=0,±1,±2,±3,... is called a stochastic process and is used as a model for observing time series. 2 mean, variance and covariance

For the random process Y t:t=0,±1,±2,±3,..., the mean function is defined as follows:

Μt =e (Y t), t=0,±1,±2,...

That is, μt is the expectation of the process at t time.

The self covariance function γt,s is defined as follows:

Γt,s =cov (y t, y s), t,s=0,±1,±2,..

where Cov (y T, y s) =e[(y t−μt) (y s−μs)]

The autocorrelation function ρt,s is defined as follows:

Ρt,s =corr (y t, y s), t,s=0,±1,±2,..

Of these: Corr (y T, y s) =cov (y T, y S) var (y T) var (y s) −−−−−−−−−−−−−√1 random walks

Make e 1, E 2,... For the mean value of 0, the variance is an independent σι2 sequence of random variables, and the observed time series Y t:t=1,2,... The structure is as follows:

Y t =y t−1 +e T, initial condition is y 1 =e 1

Note: Over time, the mean is unchanged, the variance increases linearly with time, and the positive correlation of Y values is getting stronger at the adjacent point. 2 Sliding Average

Suppose construction Y T is as follows:

Y t =e t +e t−1 2

It can be proved that all t have ρt,t−k equal. And then the concept of smoothness is derived. 3 Smoothness of 1 smoothness

The basic idea of smoothness: the statistical laws that determine process characteristics do not change with time. In a sense, the process is at the equilibrium point of statistics.

If all the time delay K and T 1, t 2,..., t n all have y T 1, y T 2,..., y t N and y t 1−k, y t 2−k,..., y T n−k the union distribution is the same, then the process procedure Y t For the strict and stable.

A stochastic process Y T is called a weak (second order moment) stable condition is:

1. The mean function is constant at all times

2.γt,t−k =γt0,k, for all time t and hysteresis K 2 white Noise

The sequence e T, which is defined as a random variable with independent distribution, is strictly stable.

It is assumed that the white noise process has 0 mean value and the variance is σι2 3 random cnoidal

Define a procedure:

Y-T =cos[2π (T12 +θ)]t=0,±1,±2,...

Where the Θ (one-time) is selected from the uniform distribution of intervals 0 to 1

According to the mean value and variance, it can be proved that the process is also stable.

In summary: For a given time series, only the time series graph based on observational data is difficult to assess whether the smoothness is a reasonable assumption.