Package {superstats}

Super Statistics.

Description

Superstatistics" generally applies to analyzing complex systems where traditional statistical methods are insufficient. This approach is particularly valuable in areas like physics, finance, and environmental science. It steps in when there are underlying variables or parameters in a system that fluctuate significantly over space or time, which traditional statistics might overlook.In practical terms, superstatistics could be doing the following:

1- Analyzing Turbulent Fluid Flow: In physics, especially in the study of turbulence in fluids, superstatistics helps in understanding the statistical properties of the flow under varying conditions like temperature or pressure.

2- Financial Markets Analysis: In finance, market conditions are extremely volatile and influenced by numerous factors. Superstatistics can provide insights into risk assessment and price movements by accounting for these underlying fluctuations.

3- Environmental and Climatic Studies: Climate systems are inherently complex and dynamic. Superstatistics can be used to better understand and predict climatic changes by integrating the variability of factors like temperature, pressure, and humidity over time.

4- Biological Systems Analysis: In biology, processes like the spread of diseases or ecosystem dynamics often involve layers of complexity. Superstatistics allows for more nuanced modeling of these processes.

The key feature of superstatistics is its ability to integrate the effects of fluctuating parameters into a more comprehensive statistical framework, offering deeper insights into complex systems where traditional statistical methods may fall short.

Details

Maintainers

Michail Tsagris mtsagris@uoc.gr

Author(s)

Michail Tsagris mtsagris@uoc.gr, Manos Papadakis papadakm95@gmail.com and Omar Alzeley oazeley@uqu.edu.sa.

References

Alzeley O. (2020). On a Statistical Learning Algorithm of Superstatistics. Computational and Mathematical Methods, 2(3), e1085.

Grid of all combinations of parameter values

Description

Creates a matrix with all possible combinations of the candidate values of several parameters.

Usage

createalfa(alfax)

Arguments

Details

The function creates all possible combinations (the Cartesian product, or full grid) of the values stored in the rows of alfax, by picking exactly one value from each row. It works recursively: if alfax has a single row, the transposed row is returned; otherwise, each value of the first row is repeated and bound (cbind) with the combinations obtained from the remaining rows.

If alfax has p rows and m columns, the output has m^p rows and p columns. The first column varies slowest and the last column varies fastest. Note that the number of rows grows exponentially with the number of rows of alfax, so large inputs can require a lot of memory.

Value

A matrix with ncol(alfax)^nrow(alfax) rows and nrow(alfax) columns. Each row of the matrix is one combination of values, with the j-th element taken from the j-th row of alfax.

Author(s)

Original Matlab code by Omar Alzeley.

R implementation and documentation: Manos Papadakis and Michail Tsagris.

References

Alzeley O. (2020). On a Statistical Learning Algorithm of Superstatistics. Computational and Mathematical Methods, 2(3), e1085.

See Also

main, find_cf

Examples

## two parameters with three candidate values each
x <- rbind( c(1, 2, 3), c(10, 20, 30) )
createalfa(x)

## three parameters with two candidate values each
y <- rbind( c(0.1, 0.5), c(1, 2), c(5, 6) )
dim( createalfa(y) )  ## 8 rows, 3 columns

## a single row is simply transposed
createalfa( rbind(1:3) )

Characteristic function estimation with a learning algorithm

Description

Estimates the characteristic function of multivariate data over a grid of points using a learning (neural network type) algorithm.

Usage

find_cf(x, outrange, N, nu, sg, g)

Arguments

Details

This function finds the characteristic function of the given data x. The characteristic function is evaluated on a regular grid of N^d points, with N equally spaced values between outrange[1] and outrange[2] in each of the d dimensions. The grid is created with createalfa. The parameters nu, sg and g are the learning parameters of the algorithm.

Two networks are used, one for the real part and one for the imaginary part of the characteristic function. Their values are initialised randomly and then updated once for each observation (row of x), using the cosine and sine of the inner product between the grid points and the observation. A Gaussian kernel with standard deviation sg, evaluated between all pairs of grid points, is used in the updating of the weights.

Since the starting values are random, the results depend on the random number generator, so use set.seed for reproducibility. The kernel matrix has N^d \times N^d elements, hence the computational cost and memory requirements grow very fast with N and with the number of variables.

Value

A list including:

Author(s)

Original Matlab code by Omar Alzeley.

R implementation and documentation: Manos Papadakis and Michail Tsagris.

References

Alzeley O. (2020). On a Statistical Learning Algorithm of Superstatistics. Computational and Mathematical Methods, 2(3), e1085.

See Also

main, createalfa

Examples

set.seed(1)
## univariate data
x <- matrix( rnorm(100), ncol = 1 )
a <- find_cf(x, outrange = c(-2, 2), N = 11, nu = 0.1, sg = 0.5, g = 0.1)
cbind(a$Alfa, a$Teta, a$TetaI)

## bivariate data
y <- matrix( rnorm(100 * 2), ncol = 2 )
b <- find_cf(y, outrange = c(-1, 1), N = 5, nu = 0.1, sg = 0.5, g = 0.1)
dim(b$Alfa)  ## 25 rows, 2 columns

Characteristic function estimation with a learning algorithm: simulation example

Description

Generates data from a Gaussian or a uniform distribution, estimates their characteristic function with a learning algorithm and compares it with the theoretical characteristic function.

Usage

main(N = 20, d = 2, n = 500, nu = 1.8, sg = 0.25, g = 0.00036, dist = 1)

Arguments

Details

If dist = 0, the data are generated from a d-dimensional Gaussian distribution with mean vector (1,\ldots,1) and covariance matrix 2 I_d, and the characteristic function is evaluated over the range [-5, 5] in each dimension. Otherwise, the data are generated from the uniform distribution on [-2, 3] in each dimension, and the characteristic function is evaluated over the range [-3, 4]. The characteristic function is then estimated with find_cf and compared with the theoretical characteristic function of the distribution that generated the data. The theoretical characteristic function is used to compute the error of the estimated one.

Three graphics devices are opened (via dev.new). They contain the generated data, the real and the imaginary part of the estimated characteristic function, and the real and imaginary part of the theoretical characteristic function together with the errors (differences between the theoretical and the estimated values).

The following is the theory behind the use of uniform distributions in superstatistics. Let U[0,N] be a uniform variable on the interval [0,N]. Observe that

\mathrm{E}\, U[0,N]=\frac{1}{N}\int_0^N x \, dx = \frac{N}{2}.

Write

\begin{array}{rcl} \mathrm{E}\, e^{-hU[0,N]} & = & \displaystyle \frac{1}{N}\int_0^N e^{-hx} dx = e^{-hN/2}\frac{1}{N}\int_0^N e^{-h(x-N/2)} dx \\ & = & \displaystyle e^{-hN/2}\int_0^N e^{-hN(x/N-1/2)} \frac{dx}{N}. \end{array}

Now, by considering a new variable of integration t=\frac{x-N/2}{N},

\mathrm{E}\, e^{-hU[0,N]}=e^{-hN/2}\int_{-1/2}^{1/2}e^{-hN t} dt.

Assume that c=hN is small. Then, by Taylor expansion, e^{-c t}=1-c t+ c^2t^2/2-\cdots and so

\mathrm{E}\, e^{-hU[0,N]}=e^{-hN/2}\int_{-1/2}^{1/2} \left(1-c t+ \frac{c^2t^2}{2}-\cdots\right)dt = e^{-hN/2}\left(1 +\frac{c^2}{24} +\cdots\right).

Consider now the uniform distribution case, U[a,a+b].

Let the random variable \beta be distributed uniformly in [a, a + b] for some a \geq 0. Then, its probability density function reads as

f(\beta)= \left\lbrace \begin{array}{ll} \displaystyle \frac{1}{b}, & \mbox{if } a < \beta < a+b, \\ 0, & \mbox{otherwise.} \end{array} \right.

The mean and the variance of this distribution are therefore

\beta_0 = a + \frac{b}{2}, \qquad \sigma^2 = \frac{b^2}{12}. \qquad (1)

The superstatistics for this more general case can be found by the same argument. First, one should note that the generalised Boltzmann factor is defined by Beck and Cohen (2003) as

B = \int_0^\infty f(\beta) \exp(-\beta E) d\beta,

direct integration of which gives

\int_0^\infty f(\beta) \exp(-\beta E) d\beta = \int_a^{a + b} \frac{1}{b} \exp(-\beta E) d\beta = \frac{\exp(-a E)}{b E} \left[1 - \exp(-b E)\right].

Substituting a from (1) into the final expression of the integral, we have

B = \exp(-E \beta_0) \frac{\exp\left(E \frac{b}{2}\right) - \exp\left(-E \frac{b}{2}\right)}{b E}.

Taking into account the following series expansions for small bE:

\frac{\exp\left(\pm E \frac{b}{2}\right)}{b E} \approx \frac{1}{bE} \pm \frac{1}{2} + \frac{bE}{8} \pm \frac{(bE)^2}{48} + \cdots

we have

\frac{\exp\left(E \frac{b}{2}\right) - \exp\left(- E \frac{b}{2}\right)}{b E} \approx 1 + \frac{(bE)^2}{24} + \frac{(bE)^4}{1920} + \cdots

Substituting this into the expression for B, we arrive at the expansion

B \approx \exp(-\beta_0 E) \left[1 + \frac{(b E)^2}{24} + \frac{(b E)^4}{1920} + \cdots\right].

Let x be a discrete random variable with a discrete uniform distribution with parameter N, i.e. x \in \lbrace a + 1, a + 2, \ldots, a + N \rbrace. Then, the probability mass function is given by f(x) = \frac{1}{N} and the mean is given by

\begin{array}{rcl} x_0 & = & \displaystyle \sum x f(x) = \frac{1}{N}\sum_{x = a + 1}^{a + N} x = \frac{1}{N}\left( (a + 1) N + \sum_{j = 0}^{N - 1} j\right) \\ & = & \displaystyle \frac{1}{N}\left( (a + 1)N + \frac{N (N - 1)}{2}\right) = a + 1 + \frac{N - 1}{2} = a + \frac{N + 1}{2}. \end{array}

Let I_{z, N} = \frac{1}{N}\sum_{j = a + 1}^{a + N} z^{j}. Then, taking into account that a = x_0 - \frac{N + 1}{2}, we have

\begin{array}{rcl} I_{z, N} & = & \displaystyle \frac{z^{a + 1} + z^{a + 2} + \cdots + z^{a + N}}{N} \\ & = & \displaystyle \frac{z^{x_0 - \frac{N + 1}{2} + 1} + z^{x_0 - \frac{N + 1}{2} + 2} + \cdots + z^{x_0 - \frac{N + 1}{2} + N}}{N} \\ & = & \displaystyle z^{x_0} \frac{z^{- \frac{N + 1}{2}} \left(z + z^2 + \cdots + z^N\right)}{N} \\ & = & \displaystyle z^{x_0} \frac{z^{- \frac{N + 1}{2}} z \left(z^N - 1\right)}{N \left(z - 1\right)} \\ & = & \displaystyle \frac{z^{x_0}}{z - 1} \frac{z^{\frac{1 + N}{2}} - z^{\frac{1 - N}{2}}}{N}. \end{array}

Note that we can also use the exponential form z^{\frac{N}{2}} = \exp\left[\frac{N}{2} \ln z\right] to write

I_{z, N} = \frac{z^{x_0 + \frac{1}{2}}}{z - 1} \frac{\exp\left[\frac{N}{2} \ln z\right] - \exp\left[-\frac{N}{2} \ln z\right]}{N}.

Taking into account the expansions above, we obtain

\begin{array}{rcl} \exp\left[\frac{N}{2} \ln z\right] - \exp\left[-\frac{N}{2} \ln z\right] & = & \displaystyle \sum_{n = 0}^\infty \frac{1}{n!} \left[\left(\frac{N}{2}\right)^n \ln^n z - \left(-\frac{N}{2}\right)^n \ln^n z \right] \\ & = & \displaystyle \sum_{n = 0}^\infty \frac{1}{n!} \left[\left(\frac{N}{2}\right)^n - \left(-1\right)^n \left(\frac{N}{2}\right)^n \right] \ln^n z. \end{array}

Evidently,

\left(\frac{N}{2}\right)^n - \left(-1\right)^n \left(\frac{N}{2}\right)^n = \left\lbrace \begin{array}{ll} \displaystyle 2\left(\frac{N}{2}\right)^{2k + 1}, & n = 2k + 1, \\ 0, & n = 2k. \end{array} \right.

Therefore, finally, we arrive at

I_{z, N} = \frac{2z^{x_0 + \frac{1}{2}}}{N\left(z - 1\right)} \sum_{k = 0}^\infty \frac{1}{\left(2k + 1\right)!} \left(\frac{N}{2}\right)^{2k + 1} \ln^{2k + 1} z.

The Boltzmann factor for discrete uniform distributions reads as

B = \frac{1}{N} \sum_{j = a + 1}^{a + N} \exp(-j E).

Then, taking into account that a = x_0 - \frac{N + 1}{2}, we have

\begin{array}{rcl} B & = & \displaystyle \frac{\exp\left[\left(\frac{1}{2} - x_0\right) E\right]}{\exp(E) - 1} \frac{\exp\left(\frac{1}{2} N E\right) - \exp\left(-\frac{1}{2} N E\right)}{N} \\ & = & \displaystyle \frac{2}{N} \frac{\exp\left[\left(\frac{1}{2} - x_0\right) E\right]}{\exp(E) - 1} \sinh\left(\frac{1}{2} N E\right). \end{array}

Taking into account that \sinh(x) = x + \frac{x^3}{3!} + \cdots = \sum_{n = 0}^\infty \frac{x^{2 n + 1}}{(2n + 1)!}, we obtain

\begin{array}{rcl} B & = & \displaystyle \frac{\exp\left[\left(\frac{1}{2} - x_0\right) E\right]}{\exp(E) - 1} \cdot \frac{2}{N} \left[ \frac{1}{2} N E + \frac{(N E)^3}{2^3 \cdot 3!} + \frac{(N E)^5}{2^5 \cdot 5!} + \cdots \right] \\ & = & \displaystyle \frac{E \exp\left[\left(\frac{1}{2} - x_0\right) E\right]}{\exp(E) - 1} \left[ 1 + \frac{(N E)^2}{24} + \frac{(N E)^4}{1920} + \cdots \right]. \end{array}

Value

A named numeric vector with two values: the error of the real part ("Re error") and the error of the imaginary part ("Im error") of the estimated characteristic function. Each error is the sum of the squared differences between the estimated and the theoretical values over the grid points.

As a side effect, three plots (graphics devices) are produced when d is 1 or 2, see the details.

Author(s)

Original Matlab code by Omar Alzeley.

R implementation and documentation: Manos Papadakis and Michail Tsagris.

References

Alzeley O. (2020). On a Statistical Learning Algorithm of Superstatistics. Computational and Mathematical Methods, 2(3), e1085.

Beck C. and Cohen E.G.D. (2003). Superstatistics. Physica A, 322: 267–275.

See Also

find_cf, createalfa

Examples

## bivariate uniform data
main(N = 10, d = 2, n = 100)

## univariate Gaussian data
main(N = 10, d = 1, n = 100, dist = 0)