simulation¶

Minimize a function using the downhill simplex algorithm.

This algorithm only uses function values, not derivatives or second derivatives.

func : callable func(x,*args)

The objective function to be minimized.

x0 : ndarray

Initial guess.

args : tuple, optional

Extra arguments passed to func, i.e. f(x,*args).

callback : callable, optional

Called after each iteration, as callback(xk), where xk is the current parameter vector.

xtol : float, optional

Relative error in xopt acceptable for convergence.

ftol : number, optional

Relative error in func(xopt) acceptable for convergence.

maxiter : int, optional

Maximum number of iterations to perform.

maxfun : number, optional

Maximum number of function evaluations to make.

full_output : bool, optional

Set to True if fopt and warnflag outputs are desired.

disp : bool, optional

Set to True to print convergence messages.

retall : bool, optional

Set to True to return list of solutions at each iteration.

xopt : ndarray

Parameter that minimizes function.

fopt : float

Value of function at minimum: fopt = func(xopt).

iter : int

Number of iterations performed.

funcalls : int

Number of function calls made.

warnflag : int

1 : Maximum number of function evaluations made. 2 : Maximum number of iterations reached.

allvecs : list

Solution at each iteration.

minimize: Interface to minimization algorithms for multivariate

functions. See the ‘Nelder-Mead’ method in particular.

Uses a Nelder-Mead simplex algorithm to find the minimum of function of one or more variables.

This algorithm has a long history of successful use in applications. But it will usually be slower than an algorithm that uses first or second derivative information. In practice it can have poor performance in high-dimensional problems and is not robust to minimizing complicated functions. Additionally, there currently is no complete theory describing when the algorithm will successfully converge to the minimum, or how fast it will if it does.

Minimize a function using modified Powell’s method. This method only uses function values, not derivatives.

func : callable f(x,*args)

Objective function to be minimized.

x0 : ndarray

Initial guess.

args : tuple, optional

Extra arguments passed to func.

callback : callable, optional

An optional user-supplied function, called after each iteration. Called as callback(xk), where xk is the current parameter vector.

direc : ndarray, optional

Initial direction set.

xtol : float, optional

Line-search error tolerance.

ftol : float, optional

Relative error in func(xopt) acceptable for convergence.

maxiter : int, optional

Maximum number of iterations to perform.

maxfun : int, optional

Maximum number of function evaluations to make.

full_output : bool, optional

If True, fopt, xi, direc, iter, funcalls, and warnflag are returned.

disp : bool, optional

If True, print convergence messages.

retall : bool, optional

If True, return a list of the solution at each iteration.

xopt : ndarray

Parameter which minimizes func.

fopt : number

Value of function at minimum: fopt = func(xopt).

direc : ndarray

Current direction set.

iter : int

Number of iterations.

funcalls : int

Number of function calls made.

warnflag : int

Integer warning flag:

1 : Maximum number of function evaluations. 2 : Maximum number of iterations.

allvecs : list

List of solutions at each iteration.

minimize: Interface to unconstrained minimization algorithms for

multivariate functions. See the ‘Powell’ method in particular.

Uses a modification of Powell’s method to find the minimum of a function of N variables. Powell’s method is a conjugate direction method.

The algorithm has two loops. The outer loop merely iterates over the inner loop. The inner loop minimizes over each current direction in the direction set. At the end of the inner loop, if certain conditions are met, the direction that gave the largest decrease is dropped and replaced with the difference between the current estiamted x and the estimated x from the beginning of the inner-loop.

The technical conditions for replacing the direction of greatest increase amount to checking that

1. No further gain can be made along the direction of greatest increase from that iteration.
2. The direction of greatest increase accounted for a large sufficient fraction of the decrease in the function value from that iteration of the inner loop.

Powell M.J.D. (1964) An efficient method for finding the minimum of a function of several variables without calculating derivatives, Computer Journal, 7 (2):155-162.

Press W., Teukolsky S.A., Vetterling W.T., and Flannery B.P.: Numerical Recipes (any edition), Cambridge University Press

Generate a Gaussian.

x : float

The x coordinate of the center of the Gaussian (degrees)

y : float

The y coordinate of the center of the Gaussian (degrees)

s : float

The dispersion of the Gaussian (degrees)

beta : float

The amplitude of the Gaussian

deg_x : 2D array

The coordinate matrix along the horizontal dimension of the display (degrees)

deg_y : 2D array

The coordinate matrix along the vertical dimension of the display (degrees)

Returns

stim : ndarray

The 1D array containing the stimulus energies given the Gaussian coordinates

Resamples the visual stimulus

The function takes an ndarray stim_arr and resamples it by the user specified scale_factor. The stimulus array is assumed to be a three dimensional ndarray representing the stimulus, in screen pixel coordinates, over time. The first two dimensions of stim_arr together represent the exent of the visual display (pixels) and the last dimensions represents time (TRs).

stim_arr : ndarray

Array_like means all those objects – lists, nested lists, etc. – that can be converted to an array.

scale_factor : float

The scale factor by which the stimulus is resampled. The scale factor must be a float, and must be greater than 0.

mode : str, optional

Points outside the boundaries of the input are filled according to the given mode (‘constant’, ‘nearest’, ‘reflect’ or ‘wrap’). Default is ‘nearest’.

resampled_arr : ndarray

An array that is resampled according to the user-specified scale factor.

x, random=random.random -> shuffle list x in place; return None.

Optional arg random is a 0-argument function returning a random float in [0.0, 1.0); by default, the standard random.random.

A utility function for creating a sweeping bar stimulus in memory.

This function creates a standard retinotopic mapping stimulus, the sweeping bar. The user specifies some of the display and stimulus parameters. This is particularly useful for writing tests and simulating responses of visually driven voxels.

pixels_across : int

The number of pixels along the horizontal dimension of the display.

pixels_down : int

The number of pixels along the vertical dimension of the display.

viewing_distance : float

The distance between the participant and the display (cm).

screen_width : float

The width of the display (cm). This is used to compute the visual angle for determining the pixels per degree of visual angle.

thetas : array-like

An array containing the orientations of the bars that will sweep across the display. For example thetas = np.arange(0,360,360/8).

num_steps : int

The number of steps the bar makes on each sweep through the visual field.

ecc : float

The distance from fixation each bar sweep begins and ends (degees).

blanks : bool

A flag determining whether there are blank periods inserted at the beginning and the end of the stimulus run.

clip : float

The bar stimulus is created by cliping a very oblong two-dimensional Gaussian oriented orthogonally to the direction of the sweep.

bar : ndarray

An array containing the bar stimulus. The array is three-dimensional, with the first two dimensions representing the size of the display and the third dimension representing time.