Linear predictive mathematical model has applications in many fields, including time series analysis, digital signal processing, optimization methods and the like.
Consider a sequence $ x [n] $, the simplest approach is to assume a linear prediction value for each value of the sequence number and the previous sequence number have linear relationship between the coefficient set $ a $. I.e.
Defined error function $ e [n] equiv x [n] - hat {x} [n] $, using minimum mean square error reaches a minimum value control error criterion:
Max Typically the maximum value of $ A $ our derivation, so to obtain the derivative function is zero extreme points. Makes it
Because the $ e [n] = x [n] + ax [n-1] $
and so,
Substituting this back into the equation to get
Since the definition of the discrete correlation coefficient (see below) with
Order was brought,
Therefore
The expression back to the minimum formula, you can get the formula for solving the minimum value is:
Such a first order linear prediction filter as shown below:
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alt="A first order linear prediction filter "/>
P-order situation
For the first p elements where linear prediction is performed,
There is also a residual definition:
We make $ a_0 = 1 $, the residuals formula for summation notation
In order to obtain a matrix representation, we remember
The matrix representation can be obtained linear prediction model:
Residual sum of squares
Parameter $ boldsymbol {a} $ partial derivative,
get
AR linear prediction model
AR model system, each output $ x (n) $ is a linear combination of the past p outputs.
Where $ p $ order of the AR model, $ w (n) $ 0 for the mean, variance $ sigma_w ^ 2 $ Gaussian white noise.
Given by a time series of parameters of the AR model for solving a_i $ $ $ p $ and order are two key issues for linear prediction.
We give an estimate:
And error:
begging:
make:
Wherein, $ lin {1,2, cdots, p} $
Therefore
Derived in accordance with this condition is actually extreme point boundary points:
Some basic concepts used in combing
Autocorrelation and cross correlation
Cross-correlation is a measure of similarity between two time series of values of two different times. Or call correlation of two random sequences. Even provided between different time gap is $ tau $. $ Int f ^ * (t) g (t + tau) dt $ denotes cross-correlation.
f (t) and g (t) corresponds correlatively f * (- t) and g (t) convolve.
Represents the degree of similarity between the autocorrelation of the sequence itself at different times is defined as follows:
In the case of discrete,