n

Dimensionality of random variable

Input:

alpha

Transformation parameter (optional, default 0.5)

beta

Transformation parameter (optional, default 2)

kappa

Transformation parameter (optional, default 3-size(X,1))

WM

Weight vector for mean calculation

Output: W

Weight matrix for covariance calculation

c

Scaling constant

M

Initial state mean (Nx1 column vector)

Input:

P

Initial state covariance

c

Parameter returned by UT_WEIGHTS

Output:

X

Matrix where 2N+1 sigma points are as columns

M

Random variable mean (Nx1 column vector)

Input:

P

Random variable covariance (NxN pos.def. matrix)

g

Transformation function of the form g(x,param) as matrix,

inline function, function name or function reference

param

Parameters of g (optional, default empty)

mu

Estimated mean of y

S

Estimated covariance of y

Output: C

Estimated cross-covariance of x and y

SX

Sigma points of x

W

Weights as cell array

M

Mean state estimate after prediction step

P

State covariance after prediction step

Input:

Y

Measurement vector.

h

Measurement model function as a matrix H defining linear

function h(x) = H*x, inline function, function handle or

name of function in form h(x,param)

R

Measurement covariance.

param

Parameters of h.

M

Updated state mean

P

Updated state covariance

Output:

K

Computed Kalman gain

MU

Predictive mean of Y

S

Predictive covariance Y

LH

Predictive probability (likelihood) of measurement.

M

Smoothed state mean sequence

Output: P

Smoothed state covariance sequence

D

Smoother gain sequence

f

Function f(x,param) as inline, name or reference

Input:

m

Mean of the d-dimensional Gaussian distribution

P

Covariance of the Gaussian distribution

param

Parameters for the function (optional)

I

The integral

Output:

x

Evaluation points

W

Weights

Syntax:

pc = GH_PACKED_PC(x,fmmparam)

Input:

x

Evaluation point

fmmparamArray of handles and parameters to form the functions.

Output:

pc

Output values

gh_transform

gh_transform

Syntax:

[mu,S,C,SX,W] = GH_TRANSFORM(M,P,g,p,param)

M

Random variable mean (Nx1 column vector)

P

Random variable covariance (NxN pos.def. matrix)

Input:

g

Transformation function of the form g(x,param) as matrix,

inline function, function name or function reference

p

Number of points in Gauss-Hermite integration

param

Parameters of g (optional, default empty)

mu

Estimated mean of y

S

Estimated covariance of y

Output:

C

Estimated cross-covariance of x and y

SX

Sigma points of x

W

Weights as cell array

M

Nx1 mean state estimate of previous step

P

NxN state covariance of previous step

Input:

f

Dynamic model function as a matrix A defining linear

function f(x) = A*x, inline function, function handle or

name of function in form f(x,param) (optional, default

eye())

Q

Process noise of discrete model (optional, default zero)

param

Parameters of f (optional, default empty)

p

Degree of approximation (number of quadrature points)

Output:

M

Updated state mean

P

Updated state covariance

M

Mean state estimate after prediction step

P

State covariance after prediction step

Y

Measurement vector.

Input:

h

Measurement model function as a matrix H defining linear

function h(x) = H*x, inline function, function handle or

name of function in form h(x,param)

R

Measurement covariance

param

Parameters of h

p

Degree of approximation (number of quadrature points)

M

Updated state mean

P

Updated state covariance

Output:

K

Computed Kalman gain

MU

Predictive mean of Y

S

Predictive covariance Y

LH

Predictive probability (likelihood) of measurement.

Input:

Hermite filter

K state process noise covariance matrices for each step.

param

Parameters of f(.). Parameters should be a single cell ar-

ray, vector or a matrix containing the same parameters

for each step, or if different parameters are used on each

step they must be a cell array of the format { param_1,

param_2, ...}, where param_x contains the parameters for

step x as a cell array, a vector or a matrix. (optional, de-

fault empty)

p

Degree on approximation (number of quadrature points)

same_p If set to ’1’ uses the same parameters on every time step

(optional, default 1)

M

Smoothed state mean sequence

Output: P

Smoothed state covariance sequence

D

Smoother gain sequence

Syntax:

p = hermitepolynomial(n)

Input:

n

Polynomial order

Output:

p

Polynomial coefficients (starting from greatest order)

f

Function f(x,param) as inline, name or reference

n

Polynomial order

Input:

m

Mean of the d-dimensional Gaussian distribution

P

Covariance of the Gaussian distribution

param

Optional parameters for the function

I

The integral value

Output:

x

Evaluation points

W

Weights

F

Function values