# wnorm: Calculate Weighted Norm of series In Rssa: A Collection of Methods for Singular Spectrum Analysis

## Description

Function calculates the W-norm for input objects or for objects stored in input ssa obect.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12``` ```## S3 method for class '1d.ssa' wnorm(x, ...) ## S3 method for class 'nd.ssa' wnorm(x, ...) ## S3 method for class 'toeplitz.ssa' wnorm(x, ...) ## S3 method for class 'mssa' wnorm(x, ...) ## Default S3 method: wnorm(x, L = (N + 1) %/% 2, ...) ## S3 method for class 'complex' wnorm(x, L = (N + 1) %/% 2, ...) ```

## Arguments

 `x` the input object. This might be ssa object for ssa method, or just a series. `L` window length. `...` arguments to be passed to methods.

## Details

`L`-weighted norm of series is Frobenius norm of its `L`-trajectory matrix. So, if `x` is vector (series), the result of `wnorm(x, L)` is equal to ```sqrt(sum(hankel(x, L)^2)```, but in fact is calculated much more efficiently. For 1d SSA and Toeplitz SSA `wnorm(x)` calculates weighted norm for stored original input series and stored window length.

`L`-weighted norm of 2d array is Frobenius norm of its ```L[1] * L[2]```-trajectory hankel-block-hankel matrix. For 2d SSA this method calculates weighted norm for stored original input array and stored 2d-window lengths.

## References

Golyandina, N., Nekrutkin, V. and Zhigljavsky, A. (2001): Analysis of Time Series Structure: SSA and related techniques. Chapman and Hall/CRC. ISBN 1584881941

`ssa-input`, `hankel`, `wcor`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ```wnorm(co2, 20) # Construct ssa-object for 'co2' with default parameters but don't decompose ss <- ssa(co2, force.decompose = FALSE) wnorm(ss) # Artificial image for 2D SSA mx <- outer(1:50, 1:50, function(i, j) sin(2*pi * i/17) * cos(2*pi * j/7) + exp(i/25 - j/20)) + rnorm(50^2, sd = 0.1) # Construct ssa-object for 'mx' with default parameters but don't decompose s <- ssa(mx, kind = "2d-ssa", force.decompose = FALSE) wnorm(s) ```