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Correlation Coefficients

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pearsonr

Pearson correlation coefficient. 
function pearsonr(
  x: Tensor,
  y: Tensor
): [number, number]
​
x
Tensor
required
First tensor
​
y
Tensor
required
Second tensor (must have same size as x)
​
return
[number, number]
Tuple of [correlation coefficient, two-tailed p-value]
  • Correlation coefficient is in range [-1, 1]
  • P-value tests the null hypothesis that the correlation is zero
Measures linear correlation between two variables. Statistical Context:
  • r = 1: Perfect positive linear relationship
  • r = 0: No linear relationship
  • r = -1: Perfect negative linear relationship
Pearson’s r measures the strength and direction of the linear relationship. It’s computed as r = cov(X,Y) / (std(X) * std(Y)). The p-value is computed using the t-distribution with n-2 degrees of freedom under the null hypothesis that the population correlation is zero. Assumptions:
  • Linear relationship between variables
  • Both variables approximately normally distributed
  • No significant outliers
  • Homoscedasticity (constant variance)
NaN inputs propagate to NaN correlation. Requires at least 2 paired samples. 
const x = tensor([1, 2, 3, 4, 5]);
const y = tensor([2, 4, 6, 8, 10]);
const [r, p] = pearsonr(x, y);  // r = 1.0 (perfect linear)

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spearmanr

Spearman’s rank correlation coefficient. 
function spearmanr(
  x: Tensor,
  y: Tensor
): [number, number]
​
x
Tensor
required
First tensor
​
y
Tensor
required
Second tensor (must have same size as x)
​
return
[number, number]
Tuple of [correlation coefficient, p-value]
  • ρ (rho) is in range [-1, 1]
  • P-value tests the null hypothesis of no monotonic relationship
Non-parametric measure of monotonic relationship between two variables. Computed as Pearson correlation of rank values. Statistical Context:
  • ρ = 1: Perfect monotonic increasing relationship
  • ρ = 0: No monotonic relationship
  • ρ = -1: Perfect monotonic decreasing relationship
Spearman’s rank correlation measures monotonic relationships (whether linear or not). It’s computed by ranking the data and then applying Pearson correlation to the ranks. More robust to outliers than Pearson. Advantages over Pearson:
  • Detects non-linear monotonic relationships
  • Robust to outliers
  • No assumption of normality
  • Works well with ordinal data
Ties are assigned average ranks. Requires at least 2 paired samples. 
const x = tensor([1, 2, 3, 4, 5]);
const y = tensor([2, 4, 6, 8, 10]);
const [rho, p] = spearmanr(x, y);  // rho = 1.0 (perfect monotonic)

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kendalltau

Kendall’s tau correlation coefficient. 
function kendalltau(
  x: Tensor,
  y: Tensor
): [number, number]
​
x
Tensor
required
First tensor
​
y
Tensor
required
Second tensor (must have same size as x)
​
return
[number, number]
Tuple of [tau coefficient, p-value]
  • τ (tau) is in range [-1, 1]
  • P-value uses normal approximation with tie-corrected variance
Non-parametric measure of ordinal association based on concordant/discordant pairs. Statistical Context:
  • τ = 1: All pairs concordant (perfect agreement)
  • τ = 0: Equal concordant and discordant pairs
  • τ = -1: All pairs discordant (perfect disagreement)
Kendall’s tau is based on the number of concordant pairs (both increase together) minus discordant pairs (one increases while the other decreases), normalized by total pairs. This implementation uses the tau-b variant with tie correction. Advantages:
  • More robust to outliers than Spearman
  • Better for small sample sizes
  • Has a direct interpretation (probability of concordance minus probability of discordance)
  • More appropriate when many tied ranks exist
Complexity: O(n²) - suitable for moderate sample sizes (n < 10,000). For larger datasets, consider using Spearman’s rho instead. Ties are excluded from concordant/discordant counts and reduce the denominator. 
const x = tensor([1, 2, 3, 4, 5]);
const y = tensor([1, 3, 2, 4, 5]);
const [tau, p] = kendalltau(x, y);  // Mostly concordant

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Covariance and Correlation Matrices

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corrcoef

Computes the Pearson correlation coefficient matrix. 
function corrcoef(
  x: Tensor,
  y?: Tensor
): Tensor
​
x
Tensor
required
Input tensor (1D or 2D). If 2D, each column is treated as a variable.
​
y
Tensor
Optional second tensor. If provided, computes correlation between x and y.
​
return
Tensor
Correlation matrix (symmetric with 1s on diagonal)
  • For two 1D tensors: 2×2 matrix
  • For 2D tensor: n×n matrix where n is number of variables (columns)
For two variables, returns 2×2 correlation matrix. For a 2D tensor, treats each column as a variable and computes pairwise correlations. Statistical Context: The correlation matrix shows all pairwise Pearson correlations. The diagonal is always 1 (each variable perfectly correlates with itself). The matrix is symmetric (corr(X,Y) = corr(Y,X)). Use Cases:
  • Identifying multicollinearity in regression analysis
  • Feature selection in machine learning
  • Understanding relationships between multiple variables
  • Exploratory data analysis
const x = tensor([1, 2, 3, 4, 5]);
const y = tensor([2, 4, 5, 4, 5]);
corrcoef(x, y);  // [[1.0, 0.8], [0.8, 1.0]]

const data = tensor([[1, 2], [3, 4], [5, 6]]);
corrcoef(data);  // 2×2 correlation matrix for 2 variables

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cov

Computes the covariance matrix. 
function cov(
  x: Tensor,
  y?: Tensor,
  ddof?: number
): Tensor
​
x
Tensor
required
Input tensor (1D or 2D). If 2D, each column is treated as a variable.
​
y
Tensor
Optional second tensor. If provided, computes covariance between x and y.
​
ddof
number
default:"1"
Delta degrees of freedom. Use 0 for population covariance, 1 for sample covariance (default).
​
return
Tensor
Covariance matrix (symmetric)
  • For two 1D tensors: 2×2 matrix with variances on diagonal and covariances off-diagonal
  • For 2D tensor: n×n matrix where n is number of variables (columns)
Covariance measures how two variables change together. Statistical Context:
  • Positive covariance: Variables tend to increase together
  • Negative covariance: As one increases, the other tends to decrease
  • Near-zero covariance: No linear relationship
Covariance is unbounded and depends on the scale of the variables, making it difficult to interpret. The diagonal contains variances. For standardized measure, use correlation instead. Relationship to Correlation: 
corr(X,Y) = cov(X,Y) / (std(X) * std(Y))
Use Cases:
  • Portfolio theory (covariance of asset returns)
  • Principal Component Analysis (PCA)
  • Multivariate statistical analysis
  • Linear discriminant analysis
const x = tensor([1, 2, 3, 4, 5]);
const y = tensor([2, 4, 5, 4, 5]);
cov(x, y);  // 2×2 covariance matrix

const data = tensor([[1, 2], [3, 4], [5, 6]]);
cov(data);  // 2×2 covariance matrix for 2 variables
cov(data, undefined, 0);  // Population covariance

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