MULTIVARIATE_T

Overview

The MULTIVARIATE_T function computes the probability density function (PDF), cumulative distribution function (CDF), or draws random samples from the multivariate t-distribution. This distribution generalizes the Student’s t-distribution to multiple dimensions and is commonly used in statistics for modeling data with heavier tails than the multivariate normal distribution. The PDF for the multivariate t-distribution is:

f(x) = \frac{\Gamma\left(\frac{\nu + d}{2}\right)}{\Gamma\left(\frac{\nu}{2}\right) \nu^{d/2} \pi^{d/2} |\Sigma|^{1/2}} \left[1 + \frac{1}{\nu}(x - \mu)^T \Sigma^{-1} (x - \mu)\right]^{-\frac{\nu + d}{2}}

where $x$ is a $d$ -dimensional vector, $\mu$ is the location vector, $\Sigma$ is the shape (covariance) matrix, and $\nu$ is the degrees of freedom. For more details, see the scipy.stats.multivariate_t documentation .

This wrapper exposes only the most commonly used parameters: location, shape, degrees of freedom, and method. Advanced options such as singularity checks, random seed, and integration controls are omitted for simplicity. This example function is provided as-is without any representation of accuracy.

Usage

To use the function in Excel:


=MULTIVARIATE_T(x, [loc], [shape], [df], [method], [size])

x (2D list, required): Table of points at which to evaluate the function or fit the distribution. Each row is a point, each column is a dimension.
loc (2D column vector, optional, default=zero vector): Location of the distribution. Must have the same number of rows as columns in x.
shape (2D list, optional, default=identity matrix): Positive semidefinite shape (covariance) matrix. Must be square with size equal to the number of columns in x.
df (float, optional, default=1): Degrees of freedom. Must be greater than zero.
method (string, optional, default=“pdf”): Which method to compute: pdf, cdf, or rvs.
size (integer, optional): Number of samples to draw if method is rvs.

The function returns a 2D list of results for each input point, or an error message (string) if the input is invalid. For rvs, the output is a 2D list of random samples. For pdf and cdf, the output is a column vector of results for each input point.

Examples

Example 1: PDF at Origin

Inputs:

x		loc		shape			df	method
0.0	0.0	0.0	0.0	1.0	0.0		2	pdf
				0.0	1.0

Excel formula:


=MULTIVARIATE_T({0,0}, {0;0}, {1,0;0,1}, 2, "pdf")

Expected output:

Result
0.159

Example 2: CDF at Origin

Inputs:

x		loc		shape			df	method
0.0	0.0	0.0	0.0	1.0	0.0		2	cdf
				0.0	1.0

Excel formula:


=MULTIVARIATE_T({0,0}, {0;0}, {1,0;0,1}, 2, "cdf")

Expected output:

Result
0.250

Example 3: Random Samples

Inputs:

x		loc		shape			df	method	size
0.0	0.0	0.0	0.0	1.0	0.0		2	rvs	2
				0.0	1.0

Excel formula:


=MULTIVARIATE_T({0,0}, {0;0}, {1,0;0,1}, 2, "rvs", 2)

Expected output:

Sample 1 (dim 1)	Sample 1 (dim 2)
…	…
…	…

(Two rows, each with two columns; values will vary.)

Example 4: PDF with Default Arguments

Inputs:

x		method
1.0	1.0	pdf

Excel formula:


=MULTIVARIATE_T({1,1}, "pdf")

Expected output:

Result
0.031

Python Code


from scipy.stats import multivariate_t as scipy_multivariate_t
from typing import List, Optional, Union
 
def multivariate_t(
    x: List[List[float]],
    loc: Optional[List[List[float]]] = None,
    shape: Optional[List[List[float]]] = None,
    df: Optional[float] = 1,
    method: str = 'pdf',
    size: Optional[int] = None
) -> Union[List[List[Optional[float]]], str]:
    """
    Computes the PDF, CDF, or draws random samples from a multivariate t-distribution.
 
    Args:
        x: 2D list of float values. Points at which to evaluate the function or fit the distribution.
        loc: 2D list of float values (column vector). Location of the distribution. Default is zero vector.
        shape: 2D list of float values. Positive semidefinite shape matrix of the distribution. Default is identity matrix.
        df: Degrees of freedom of the distribution. Must be greater than zero. Default is 1.
        method: Which method to compute (str): 'pdf', 'cdf', 'rvs'. Default is 'pdf'.
        size: Number of samples to draw if method is 'rvs'. Optional.
 
    Returns:
        2D list of results for each input point, or an error message (str) if input is invalid.
 
    This example function is provided as-is without any representation of accuracy.
    """
    # Validate x
    if not isinstance(x, list) or not all(isinstance(row, list) and all(isinstance(val, (int, float)) for val in row) for row in x):
        return "Invalid input: x must be a 2D list of floats."
    if len(x) == 0 or len(x[0]) == 0:
        return "Invalid input: x must be a non-empty 2D list."
    dim = len(x[0])
    # Validate loc
    if loc is not None:
        if not (isinstance(loc, list) and all(isinstance(row, list) and len(row) == 1 and isinstance(row[0], (int, float)) for row in loc)):
            return "Invalid input: loc must be a 2D column vector (list of lists with one float each)."
        if len(loc) != dim:
            return "Invalid input: loc must have the same number of rows as columns in x."
        loc_vec = [row[0] for row in loc]
    else:
        loc_vec = [0.0] * dim
    # Validate shape
    if shape is not None:
        if not (isinstance(shape, list) and all(isinstance(row, list) and len(row) == dim and all(isinstance(val, (int, float)) for val in row) for row in shape)):
            return "Invalid input: shape must be a 2D list of floats with shape (dim, dim)."
        if len(shape) != dim:
            return "Invalid input: shape must have the same number of rows as columns in x."
        shape_mat = shape
    else:
        shape_mat = [[float(i == j) for j in range(dim)] for i in range(dim)]
    # Validate df
    try:
        df_val = float(df) if df is not None else 1.0
    except Exception:
        return "Invalid input: df must be a float."
    if df_val <= 0:
        return "Invalid input: df must be greater than zero."
    # Validate method
    if method not in ('pdf', 'cdf', 'rvs'):
        return "Invalid input: method must be 'pdf', 'cdf', or 'rvs'."
    # Validate size
    if method == 'rvs':
        if size is None:
            return "Invalid input: size must be specified for 'rvs' method."
        try:
            size_val = int(size)
        except Exception:
            return "Invalid input: size must be an integer."
        if size_val <= 0:
            return "Invalid input: size must be a positive integer."
    # Create distribution
    try:
        dist = scipy_multivariate_t(loc=loc_vec, shape=shape_mat, df=df_val)
    except Exception as e:
        return f"scipy.stats.multivariate_t error: {e}"
    # Compute result
    try:
        if method == 'pdf':
            result = [[float(dist.pdf(row))] for row in x]
        elif method == 'cdf':
            result = [[float(dist.cdf(row))] for row in x]
        elif method == 'rvs':
            samples = dist.rvs(size=size_val)
            # If samples is 1D, wrap as 2D list
            if dim == 1:
                result = [[float(val)] for val in samples]
            else:
                result = [[float(val) for val in row] for row in samples]
        else:
            return "Invalid method."
    except Exception as e:
        return f"scipy.stats.multivariate_t {method} error: {e}"
    # Check for invalid output values
    import math
    for row in result:
        for val in row:
            if isinstance(val, float) and (math.isnan(val) or math.isinf(val)):
                return "Invalid output: result contains NaN or infinite values."
    return result

Example Workbook

Link to Workbook