README

Gonum stat GoDoc

Package stat is a statistics package for the Go language.

Documentation

Overview

    Package stat provides generalized statistical functions.

    Index

    Examples

    Constants

    This section is empty.

    Variables

    This section is empty.

    Functions

    func Bhattacharyya

    func Bhattacharyya(p, q []float64) float64

      Bhattacharyya computes the distance between the probability distributions p and q given by:

      -\ln ( \sum_i \sqrt{p_i q_i} )
      

      The lengths of p and q must be equal. It is assumed that p and q sum to 1.

      func BivariateMoment

      func BivariateMoment(r, s float64, x, y, weights []float64) float64

        BivariateMoment computes the weighted mixed moment between the samples x and y.

        E[(x - μ_x)^r*(y - μ_y)^s]
        

        No degrees of freedom correction is done. The lengths of x and y must be equal. If weights is nil then all of the weights are 1. If weights is not nil, then len(x) must equal len(weights).

        func CDF

        func CDF(q float64, c CumulantKind, x, weights []float64) float64

          CDF returns the empirical cumulative distribution function value of x, that is the fraction of the samples less than or equal to q. The exact behavior is determined by the CumulantKind. CDF is theoretically the inverse of the Quantile function, though it may not be the actual inverse for all values q and CumulantKinds.

          The x data must be sorted in increasing order. If weights is nil then all of the weights are 1. If weights is not nil, then len(x) must equal len(weights).

          CumulantKind behaviors:

          - Empirical: Returns the lowest fraction for which q is greater than or equal
          to that fraction of samples
          

          func ChiSquare

          func ChiSquare(obs, exp []float64) float64

            ChiSquare computes the chi-square distance between the observed frequencies 'obs' and expected frequencies 'exp' given by:

            \sum_i (obs_i-exp_i)^2 / exp_i
            

            The lengths of obs and exp must be equal.

            func CircularMean

            func CircularMean(x, weights []float64) float64

              CircularMean returns the circular mean of the dataset.

              atan2(\sum_i w_i * sin(alpha_i), \sum_i w_i * cos(alpha_i))
              

              If weights is nil then all of the weights are 1. If weights is not nil, then len(x) must equal len(weights).

              Example
              Output:
              
              The circular mean is 1.37037.
              

              func Correlation

              func Correlation(x, y, weights []float64) float64

                Correlation returns the weighted correlation between the samples of x and y with the given means.

                sum_i {w_i (x_i - meanX) * (y_i - meanY)} / (stdX * stdY)
                

                The lengths of x and y must be equal. If weights is nil then all of the weights are 1. If weights is not nil, then len(x) must equal len(weights).

                Example
                Output:
                
                Correlation computes the degree to which two datasets move together
                about their mean. For example, x and y above move similarly.
                Correlation is 0.59915
                

                func CorrelationMatrix

                func CorrelationMatrix(dst *mat.SymDense, x mat.Matrix, weights []float64)

                  CorrelationMatrix returns the correlation matrix calculated from a matrix of data, x, using a two-pass algorithm. The result is stored in dst.

                  If weights is not nil the weighted correlation of x is calculated. weights must have length equal to the number of rows in input data matrix and must not contain negative elements. The dst matrix must either be empty or have the same number of columns as the input data matrix.

                  func Covariance

                  func Covariance(x, y, weights []float64) float64

                    Covariance returns the weighted covariance between the samples of x and y.

                    sum_i {w_i (x_i - meanX) * (y_i - meanY)} / (sum_j {w_j} - 1)
                    

                    The lengths of x and y must be equal. If weights is nil then all of the weights are 1. If weights is not nil, then len(x) must equal len(weights).

                    Example
                    Output:
                    
                    Covariance computes the degree to which datasets move together
                    about their mean.
                    Cov = 13.8000
                    If datasets move perfectly together, the variance equals the covariance
                    Cov2 is 37.7000, VarX is 37.7000
                    

                    func CovarianceMatrix

                    func CovarianceMatrix(dst *mat.SymDense, x mat.Matrix, weights []float64)

                      CovarianceMatrix calculates the covariance matrix (also known as the variance-covariance matrix) calculated from a matrix of data, x, using a two-pass algorithm. The result is stored in dst.

                      If weights is not nil the weighted covariance of x is calculated. weights must have length equal to the number of rows in input data matrix and must not contain negative elements. The dst matrix must either be empty or have the same number of columns as the input data matrix.

                      func CrossEntropy

                      func CrossEntropy(p, q []float64) float64

                        CrossEntropy computes the cross-entropy between the two distributions specified in p and q.

                        func Entropy

                        func Entropy(p []float64) float64

                          Entropy computes the Shannon entropy of a distribution or the distance between two distributions. The natural logarithm is used.

                          - sum_i (p_i * log_e(p_i))
                          
                          Example
                          Output:
                          
                          Entropy is a measure of the amount of uncertainty in a distribution
                          The second bin of p is very likely to occur. It's entropy is 0.6247
                          The distribution of q is more spread out. It's entropy is 1.3195
                          Adding buckets with zero probability does not change the entropy.
                          The entropy of r is: 1.2206
                          A distribution with no uncertainty has entropy 0.0000
                          

                          func ExKurtosis

                          func ExKurtosis(x, weights []float64) float64

                            ExKurtosis returns the population excess kurtosis of the sample. The kurtosis is defined by the 4th moment of the mean divided by the squared variance. The excess kurtosis subtracts 3.0 so that the excess kurtosis of the normal distribution is zero. If weights is nil then all of the weights are 1. If weights is not nil, then len(x) must equal len(weights).

                            Example
                            Output:
                            
                            Kurtosis is a measure of the 'peakedness' of a distribution, and the
                            excess kurtosis is the kurtosis above or below that of the standard normal
                            distribution
                            ExKurtosis = -5.41200
                            Weighted ExKurtosis is -0.6779
                            

                            func GeometricMean

                            func GeometricMean(x, weights []float64) float64

                              GeometricMean returns the weighted geometric mean of the dataset

                              \prod_i {x_i ^ w_i}
                              

                              This only applies with positive x and positive weights. If weights is nil then all of the weights are 1. If weights is not nil, then len(x) must equal len(weights).

                              Example
                              Output:
                              
                              The arithmetic mean is 10.1667, but the geometric mean is 8.7637.
                              The exponential of the mean of the logs is 8.7637
                              

                              func HarmonicMean

                              func HarmonicMean(x, weights []float64) float64

                                HarmonicMean returns the weighted harmonic mean of the dataset

                                \sum_i {w_i} / ( sum_i {w_i / x_i} )
                                

                                This only applies with positive x and positive weights. If weights is nil then all of the weights are 1. If weights is not nil, then len(x) must equal len(weights).

                                Example
                                Output:
                                
                                The arithmetic mean is 10.16667, but the harmonic mean is 6.8354.
                                

                                func Hellinger

                                func Hellinger(p, q []float64) float64

                                  Hellinger computes the distance between the probability distributions p and q given by:

                                  \sqrt{ 1 - \sum_i \sqrt{p_i q_i} }
                                  

                                  The lengths of p and q must be equal. It is assumed that p and q sum to 1.

                                  func Histogram

                                  func Histogram(count, dividers, x, weights []float64) []float64

                                    Histogram sums up the weighted number of data points in each bin. The weight of data point x[i] will be placed into count[j] if dividers[j] <= x < dividers[j+1]. The "span" function in the floats package can assist with bin creation.

                                    The following conditions on the inputs apply:

                                    - The count variable must either be nil or have length of one less than dividers.
                                    - The values in dividers must be sorted (use the sort package).
                                    - The x values must be sorted.
                                    - If weights is nil then all of the weights are 1.
                                    - If weights is not nil, then len(x) must equal len(weights).
                                    
                                    Example
                                    Output:
                                    
                                    Histogram counts the amount of data in the bins specified by
                                    the dividers. In this data set, there are 7 data points less than 7 (between dividers[0]
                                    and dividers[1]), 12 data points between 7 and 20 (dividers[1] and dividers[2]),
                                    and 0 data points above 1000. Since dividers has length 5, there will be 4 bins.
                                    Hist = [7 12 72 10]
                                    
                                    For ease, the floats Span function can be used to set the dividers
                                    Hist = [11 10 10 10 10 10 10 10 10 10]
                                    
                                    Histogram also works with weighted data, and allows reusing of
                                    the count field in order to avoid extra garbage
                                    Weighted Hist = [66 165 265 365 465 565 665 765 865 965]
                                    

                                    func JensenShannon

                                    func JensenShannon(p, q []float64) float64

                                      JensenShannon computes the JensenShannon divergence between the distributions p and q. The Jensen-Shannon divergence is defined as

                                      m = 0.5 * (p + q)
                                      JS(p, q) = 0.5 ( KL(p, m) + KL(q, m) )
                                      

                                      Unlike Kullback-Liebler, the Jensen-Shannon distance is symmetric. The value is between 0 and ln(2).

                                      func Kendall

                                      func Kendall(x, y, weights []float64) float64

                                        Kendall returns the weighted Tau-a Kendall correlation between the samples of x and y. The Kendall correlation measures the quantity of concordant and discordant pairs of numbers. If weights are specified then each pair is weighted by weights[i] * weights[j] and the final sum is normalized to stay between -1 and 1. The lengths of x and y must be equal. If weights is nil then all of the weights are 1. If weights is not nil, then len(x) must equal len(weights).

                                        Example
                                        Output:
                                        
                                        Kendall correlation computes the number of ordered pairs
                                        between two datasets.
                                        Kendall correlation is 0.25000
                                        

                                        func KolmogorovSmirnov

                                        func KolmogorovSmirnov(x, xWeights, y, yWeights []float64) float64

                                          KolmogorovSmirnov computes the largest distance between two empirical CDFs. Each dataset x and y consists of sample locations and counts, xWeights and yWeights, respectively.

                                          x and y may have different lengths, though len(x) must equal len(xWeights), and len(y) must equal len(yWeights). Both x and y must be sorted.

                                          Special cases are:

                                          = 0 if len(x) == len(y) == 0
                                          = 1 if len(x) == 0, len(y) != 0 or len(x) != 0 and len(y) == 0
                                          

                                          func KullbackLeibler

                                          func KullbackLeibler(p, q []float64) float64

                                            KullbackLeibler computes the Kullback-Leibler distance between the distributions p and q. The natural logarithm is used.

                                            sum_i(p_i * log(p_i / q_i))
                                            

                                            Note that the Kullback-Leibler distance is not symmetric; KullbackLeibler(p,q) != KullbackLeibler(q,p)

                                            Example
                                            Output:
                                            
                                            

                                            func LinearRegression

                                            func LinearRegression(x, y, weights []float64, origin bool) (alpha, beta float64)

                                              LinearRegression computes the best-fit line

                                              y = alpha + beta*x
                                              

                                              to the data in x and y with the given weights. If origin is true, the regression is forced to pass through the origin.

                                              Specifically, LinearRegression computes the values of alpha and beta such that the total residual

                                              \sum_i w[i]*(y[i] - alpha - beta*x[i])^2
                                              

                                              is minimized. If origin is true, then alpha is forced to be zero.

                                              The lengths of x and y must be equal. If weights is nil then all of the weights are 1. If weights is not nil, then len(x) must equal len(weights).

                                              Example
                                              Output:
                                              
                                              Estimated slope is:  0.988572424633503
                                              Estimated offset is: 3.0001541344029676
                                              R^2: 0.9999991095061128
                                              

                                              func Mahalanobis

                                              func Mahalanobis(x, y mat.Vector, chol *mat.Cholesky) float64

                                                Mahalanobis computes the Mahalanobis distance

                                                D = sqrt((x-y)ᵀ * Σ^-1 * (x-y))
                                                

                                                between the column vectors x and y given the cholesky decomposition of Σ. Mahalanobis returns NaN if the linear solve fails.

                                                See https://en.wikipedia.org/wiki/Mahalanobis_distance for more information.

                                                func Mean

                                                func Mean(x, weights []float64) float64

                                                  Mean computes the weighted mean of the data set.

                                                  sum_i {w_i * x_i} / sum_i {w_i}
                                                  

                                                  If weights is nil then all of the weights are 1. If weights is not nil, then len(x) must equal len(weights).

                                                  Example
                                                  Output:
                                                  
                                                  The mean of the samples is 3.5500
                                                  The weighted mean of the samples is 1.9000
                                                  The mean of x2 is 1.9000
                                                  The weights act as if there were more samples of that number
                                                  

                                                  func MeanStdDev

                                                  func MeanStdDev(x, weights []float64) (mean, std float64)

                                                    MeanStdDev returns the sample mean and unbiased standard deviation When weights sum to 1 or less, a biased variance estimator should be used.

                                                    func MeanVariance

                                                    func MeanVariance(x, weights []float64) (mean, variance float64)

                                                      MeanVariance computes the sample mean and unbiased variance, where the mean and variance are

                                                      \sum_i w_i * x_i / (sum_i w_i)
                                                      \sum_i w_i (x_i - mean)^2 / (sum_i w_i - 1)
                                                      

                                                      respectively. If weights is nil then all of the weights are 1. If weights is not nil, then len(x) must equal len(weights). When weights sum to 1 or less, a biased variance estimator should be used.

                                                      func Mode

                                                      func Mode(x, weights []float64) (val float64, count float64)

                                                        Mode returns the most common value in the dataset specified by x and the given weights. Strict float64 equality is used when comparing values, so users should take caution. If several values are the mode, any of them may be returned.

                                                        func Moment

                                                        func Moment(moment float64, x, weights []float64) float64

                                                          Moment computes the weighted n^th moment of the samples,

                                                          E[(x - μ)^N]
                                                          

                                                          No degrees of freedom correction is done. If weights is nil then all of the weights are 1. If weights is not nil, then len(x) must equal len(weights).

                                                          func MomentAbout

                                                          func MomentAbout(moment float64, x []float64, mean float64, weights []float64) float64

                                                            MomentAbout computes the weighted n^th weighted moment of the samples about the given mean \mu,

                                                            E[(x - μ)^N]
                                                            

                                                            No degrees of freedom correction is done. If weights is nil then all of the weights are 1. If weights is not nil, then len(x) must equal len(weights).

                                                            func Quantile

                                                            func Quantile(p float64, c CumulantKind, x, weights []float64) float64

                                                              Quantile returns the sample of x such that x is greater than or equal to the fraction p of samples. The exact behavior is determined by the CumulantKind, and p should be a number between 0 and 1. Quantile is theoretically the inverse of the CDF function, though it may not be the actual inverse for all values p and CumulantKinds.

                                                              The x data must be sorted in increasing order. If weights is nil then all of the weights are 1. If weights is not nil, then len(x) must equal len(weights).

                                                              CumulantKind behaviors:

                                                              - Empirical: Returns the lowest value q for which q is greater than or equal
                                                              to the fraction p of samples
                                                              - LinInterp: Returns the linearly interpolated value
                                                              

                                                              func RNoughtSquared

                                                              func RNoughtSquared(x, y, weights []float64, beta float64) float64

                                                                RNoughtSquared returns the coefficient of determination defined as

                                                                R₀^2 = \sum_i w[i]*(beta*x[i])^2 / \sum_i w[i]*y[i]^2
                                                                

                                                                for the line

                                                                y = beta*x
                                                                

                                                                and the data in x and y with the given weights. RNoughtSquared should only be used for best-fit lines regressed through the origin.

                                                                The lengths of x and y must be equal. If weights is nil then all of the weights are 1. If weights is not nil, then len(x) must equal len(weights).

                                                                func ROC

                                                                func ROC(cutoffs, y []float64, classes []bool, weights []float64) (tpr, fpr, thresh []float64)

                                                                  ROC returns paired false positive rate (FPR) and true positive rate (TPR) values corresponding to cutoff points on the receiver operator characteristic (ROC) curve obtained when y is treated as a binary classifier for classes with weights. The cutoff thresholds used to calculate the ROC are returned in thresh such that tpr[i] and fpr[i] are the true and false positive rates for y >= thresh[i].

                                                                  The input y and cutoffs must be sorted, and values in y must correspond to values in classes and weights. SortWeightedLabeled can be used to sort y together with classes and weights.

                                                                  For a given cutoff value, observations corresponding to entries in y greater than the cutoff value are classified as false, while those less than or equal to the cutoff value are classified as true. These assigned class labels are compared with the true values in the classes slice and used to calculate the FPR and TPR.

                                                                  If weights is nil, all weights are treated as 1.

                                                                  If cutoffs is nil or empty, all possible cutoffs are calculated, resulting in fpr and tpr having length one greater than the number of unique values in y. Otherwise fpr and tpr will be returned with the same length as cutoffs. floats.Span can be used to generate equally spaced cutoffs.

                                                                  More details about ROC curves are available at https://en.wikipedia.org/wiki/Receiver_operating_characteristic

                                                                  Example (AUC)
                                                                  Output:
                                                                  
                                                                  true  positive rate: [0 0 0.5 0.5 1]
                                                                  false positive rate: [0 0.5 0.5 1 1]
                                                                  auc: 0.25
                                                                  
                                                                  Example (EquallySpacedCutoffs)
                                                                  Output:
                                                                  
                                                                  true  positive rate: [0 0.333 0.333 0.583 0.583 0.583 0.667 0.667 1]
                                                                  false positive rate: [0 0 0 0 1 1 1 1 1]
                                                                  
                                                                  Example (KnownCutoffs)
                                                                  Output:
                                                                  
                                                                  true  positive rate: [0.875 0.875 1]
                                                                  false positive rate: [0.6 0.6 1]
                                                                  
                                                                  Example (Threshold)
                                                                  Output:
                                                                  
                                                                  true  positive rate: [0 0.5 0.5 1 1]
                                                                  false positive rate: [0 0 0.5 0.5 1]
                                                                  cutoff thresholds: [+Inf 0.8 0.4 0.35 0.1]
                                                                  
                                                                  Example (Unsorted)
                                                                  Output:
                                                                  
                                                                  true  positive rate: [0 0.25 0.5 0.875 0.875 1 1]
                                                                  false positive rate: [0 0 0 0 0.6 0.6 1]
                                                                  
                                                                  Example (Unweighted)
                                                                  Output:
                                                                  
                                                                  true  positive rate: [0 0.25 0.5 0.75 0.75 1 1]
                                                                  false positive rate: [0 0 0 0 0.5 0.5 1]
                                                                  
                                                                  Example (Weighted)
                                                                  Output:
                                                                  
                                                                  true  positive rate: [0 0.25 0.5 0.875 0.875 1 1]
                                                                  false positive rate: [0 0 0 0 0.6 0.6 1]
                                                                  

                                                                  func RSquared

                                                                  func RSquared(x, y, weights []float64, alpha, beta float64) float64

                                                                    RSquared returns the coefficient of determination defined as

                                                                    R^2 = 1 - \sum_i w[i]*(y[i] - alpha - beta*x[i])^2 / \sum_i w[i]*(y[i] - mean(y))^2
                                                                    

                                                                    for the line

                                                                    y = alpha + beta*x
                                                                    

                                                                    and the data in x and y with the given weights.

                                                                    The lengths of x and y must be equal. If weights is nil then all of the weights are 1. If weights is not nil, then len(x) must equal len(weights).

                                                                    func RSquaredFrom

                                                                    func RSquaredFrom(estimates, values, weights []float64) float64

                                                                      RSquaredFrom returns the coefficient of determination defined as

                                                                      R^2 = 1 - \sum_i w[i]*(estimate[i] - value[i])^2 / \sum_i w[i]*(value[i] - mean(values))^2
                                                                      

                                                                      and the data in estimates and values with the given weights.

                                                                      The lengths of estimates and values must be equal. If weights is nil then all of the weights are 1. If weights is not nil, then len(values) must equal len(weights).

                                                                      func Skew

                                                                      func Skew(x, weights []float64) float64

                                                                        Skew computes the skewness of the sample data. If weights is nil then all of the weights are 1. If weights is not nil, then len(x) must equal len(weights). When weights sum to 1 or less, a biased variance estimator should be used.

                                                                        func SortWeighted

                                                                        func SortWeighted(x, weights []float64)

                                                                          SortWeighted rearranges the data in x along with their corresponding weights so that the x data are sorted. The data is sorted in place. Weights may be nil, but if weights is non-nil then it must have the same length as x.

                                                                          func SortWeightedLabeled

                                                                          func SortWeightedLabeled(x []float64, labels []bool, weights []float64)

                                                                            SortWeightedLabeled rearranges the data in x along with their corresponding weights and boolean labels so that the x data are sorted. The data is sorted in place. Weights and labels may be nil, if either is non-nil it must have the same length as x.

                                                                            func StdDev

                                                                            func StdDev(x, weights []float64) float64

                                                                              StdDev returns the sample standard deviation.

                                                                              Example
                                                                              Output:
                                                                              
                                                                              The standard deviation of the samples is 8.8034
                                                                              The weighted standard deviation of the samples is 10.5733
                                                                              

                                                                              func StdErr

                                                                              func StdErr(std, sampleSize float64) float64

                                                                                StdErr returns the standard error in the mean with the given values.

                                                                                Example
                                                                                Output:
                                                                                
                                                                                The standard deviation is 10.5733 and there are 18 samples, so the mean
                                                                                is likely 4.1667 ± 2.4921.
                                                                                

                                                                                func StdScore

                                                                                func StdScore(x, mean, std float64) float64

                                                                                  StdScore returns the standard score (a.k.a. z-score, z-value) for the value x with the given mean and standard deviation, i.e.

                                                                                  (x - mean) / std
                                                                                  

                                                                                  func Variance

                                                                                  func Variance(x, weights []float64) float64

                                                                                    Variance computes the unbiased weighted sample variance:

                                                                                    \sum_i w_i (x_i - mean)^2 / (sum_i w_i - 1)
                                                                                    

                                                                                    If weights is nil then all of the weights are 1. If weights is not nil, then len(x) must equal len(weights). When weights sum to 1 or less, a biased variance estimator should be used.

                                                                                    Example
                                                                                    Output:
                                                                                    
                                                                                    The variance of the samples is 77.5000
                                                                                    The weighted variance of the samples is 111.7941
                                                                                    

                                                                                    Types

                                                                                    type CC

                                                                                    type CC struct {
                                                                                    	// contains filtered or unexported fields
                                                                                    }

                                                                                      CC is a type for computing the canonical correlations of a pair of matrices. The results of the canonical correlation analysis are only valid if the call to CanonicalCorrelations was successful.

                                                                                      Example
                                                                                      Output:
                                                                                      
                                                                                      corRaw = ⎡-0.2192   0.3527   0.5828  -0.3883⎤
                                                                                               ⎢-0.3917   0.6448   0.7208  -0.4837⎥
                                                                                               ⎢-0.3022   0.7315   0.6680  -0.4273⎥
                                                                                               ⎢ 0.2052  -0.7479  -0.5344   0.2499⎥
                                                                                               ⎢-0.2098   0.4560   0.9102  -0.3816⎥
                                                                                               ⎢-0.3555   0.2615   0.4609  -0.5078⎥
                                                                                               ⎣ 0.1281  -0.2735  -0.4418   0.3335⎦
                                                                                      
                                                                                      corSph = ⎡ 0.0118   0.0525   0.2300  -0.1363⎤
                                                                                               ⎢-0.1810   0.3213   0.3814  -0.1412⎥
                                                                                               ⎢ 0.0166   0.2241   0.0104  -0.2235⎥
                                                                                               ⎢ 0.0346  -0.5481  -0.0034  -0.1994⎥
                                                                                               ⎢ 0.0303  -0.0956   0.7152   0.2039⎥
                                                                                               ⎢-0.0298  -0.0022   0.0739  -0.3703⎥
                                                                                               ⎣-0.1226  -0.0746  -0.3899   0.1541⎦
                                                                                      
                                                                                      ccors = [0.9451 0.6787 0.5714 0.2010]
                                                                                      
                                                                                      pVecs = ⎡-0.2574   0.0158   0.2122  -0.0946⎤
                                                                                              ⎢-0.4837   0.3837   0.1474   0.6597⎥
                                                                                              ⎢-0.0801   0.3494   0.3287  -0.2862⎥
                                                                                              ⎢ 0.1278  -0.7337   0.4851   0.2248⎥
                                                                                              ⎢-0.6969  -0.4342  -0.3603   0.0291⎥
                                                                                              ⎢-0.0991   0.0503   0.6384   0.1022⎥
                                                                                              ⎣ 0.4260   0.0323  -0.2290   0.6419⎦
                                                                                      
                                                                                      qVecs = ⎡ 0.0182  -0.1583  -0.0067  -0.9872⎤
                                                                                              ⎢-0.2348   0.9483  -0.1462  -0.1554⎥
                                                                                              ⎢-0.9701  -0.2406  -0.0252   0.0209⎥
                                                                                              ⎣ 0.0593  -0.1330  -0.9889   0.0291⎦
                                                                                      
                                                                                      phiVs = ⎡-0.0027   0.0093   0.0490  -0.0155⎤
                                                                                              ⎢-0.0429  -0.0242   0.0361   0.1839⎥
                                                                                              ⎢-1.2248   5.6031   5.8094  -4.7927⎥
                                                                                              ⎢-0.0044  -0.3424   0.4470   0.1150⎥
                                                                                              ⎢-0.0742  -0.1193  -0.1116   0.0022⎥
                                                                                              ⎢-0.0233   0.1046   0.3853  -0.0161⎥
                                                                                              ⎣ 0.0001   0.0005  -0.0030   0.0082⎦
                                                                                      
                                                                                      psiVs = ⎡ 0.0302  -0.3002   0.0878  -1.9583⎤
                                                                                              ⎢-0.0065   0.0392  -0.0118  -0.0061⎥
                                                                                              ⎢-0.0052  -0.0046  -0.0023   0.0008⎥
                                                                                              ⎣ 0.0020   0.0037  -0.1293   0.1038⎦
                                                                                      

                                                                                      func (*CC) CanonicalCorrelations

                                                                                      func (c *CC) CanonicalCorrelations(x, y mat.Matrix, weights []float64) error

                                                                                        CanonicalCorrelations performs a canonical correlation analysis of the input data x and y, columns of which should be interpretable as two sets of measurements on the same observations (rows). These observations are optionally weighted by weights. The result of the analysis is stored in the receiver if the analysis is successful.

                                                                                        Canonical correlation analysis finds associations between two sets of variables on the same observations by finding linear combinations of the two sphered datasets that maximize the correlation between them.

                                                                                        Some notation: let Xc and Yc denote the centered input data matrices x and y (column means subtracted from each column), let Sx and Sy denote the sample covariance matrices within x and y respectively, and let Sxy denote the covariance matrix between x and y. The sphered data can then be expressed as Xc * Sx^{-1/2} and Yc * Sy^{-1/2} respectively, and the correlation matrix between the sphered data is called the canonical correlation matrix, Sx^{-1/2} * Sxy * Sy^{-1/2}. In cases where S^{-1/2} is ambiguous for some covariance matrix S, S^{-1/2} is taken to be E * D^{-1/2} * Eᵀ where S can be eigendecomposed as S = E * D * Eᵀ.

                                                                                        The canonical correlations are the correlations between the corresponding pairs of canonical variables and can be obtained with c.Corrs(). Canonical variables can be obtained by projecting the sphered data into the left and right eigenvectors of the canonical correlation matrix, and these eigenvectors can be obtained with c.Left(m, true) and c.Right(m, true) respectively. The canonical variables can also be obtained directly from the centered raw data by using the back-transformed eigenvectors which can be obtained with c.Left(m, false) and c.Right(m, false) respectively.

                                                                                        The first pair of left and right eigenvectors of the canonical correlation matrix can be interpreted as directions into which the respective sphered data can be projected such that the correlation between the two projections is maximized. The second pair and onwards solve the same optimization but under the constraint that they are uncorrelated (orthogonal in sphered space) to previous projections.

                                                                                        CanonicalCorrelations will panic if the inputs x and y do not have the same number of rows.

                                                                                        The slice weights is used to weight the observations. If weights is nil, each weight is considered to have a value of one, otherwise the length of weights must match the number of observations (rows of both x and y) or CanonicalCorrelations will panic.

                                                                                        More details can be found at https://en.wikipedia.org/wiki/Canonical_correlation or in Chapter 3 of Koch, Inge. Analysis of multivariate and high-dimensional data. Vol. 32. Cambridge University Press, 2013. ISBN: 9780521887939

                                                                                        func (*CC) CorrsTo

                                                                                        func (c *CC) CorrsTo(dst []float64) []float64

                                                                                          CorrsTo returns the canonical correlations, using dst if it is not nil. If dst is not nil and len(dst) does not match the number of columns in the y input matrix, Corrs will panic.

                                                                                          func (*CC) LeftTo

                                                                                          func (c *CC) LeftTo(dst *mat.Dense, spheredSpace bool)

                                                                                            LeftTo returns the left eigenvectors of the canonical correlation matrix if spheredSpace is true. If spheredSpace is false it returns these eigenvectors back-transformed to the original data space.

                                                                                            If dst is empty, LeftTo will resize dst to be xd×yd. When dst is non-empty, LeftTo will panic if dst is not xd×yd. LeftTo will also panic if the receiver does not contain a successful CC.

                                                                                            func (*CC) RightTo

                                                                                            func (c *CC) RightTo(dst *mat.Dense, spheredSpace bool)

                                                                                              RightTo returns the right eigenvectors of the canonical correlation matrix if spheredSpace is true. If spheredSpace is false it returns these eigenvectors back-transformed to the original data space.

                                                                                              If dst is empty, RightTo will resize dst to be yd×yd. When dst is non-empty, RightTo will panic if dst is not yd×yd. RightTo will also panic if the receiver does not contain a successful CC.

                                                                                              type CumulantKind

                                                                                              type CumulantKind int

                                                                                                CumulantKind specifies the behavior for calculating the empirical CDF or Quantile

                                                                                                const (
                                                                                                	// Empirical treats the distribution as the actual empirical distribution.
                                                                                                	Empirical CumulantKind = 1
                                                                                                	// LinInterp linearly interpolates the empirical distribution between sample values, with a flat extrapolation.
                                                                                                	LinInterp CumulantKind = 4
                                                                                                )

                                                                                                  List of supported CumulantKind values for the Quantile function. Constant values should match the R nomenclature. See https://en.wikipedia.org/wiki/Quantile#Estimating_the_quantiles_of_a_population

                                                                                                  type PC

                                                                                                  type PC struct {
                                                                                                  	// contains filtered or unexported fields
                                                                                                  }

                                                                                                    PC is a type for computing and extracting the principal components of a matrix. The results of the principal components analysis are only valid if the call to PrincipalComponents was successful.

                                                                                                    Example
                                                                                                    Output:
                                                                                                    
                                                                                                    variances = [0.1666 0.0207 0.0079 0.0019]
                                                                                                    
                                                                                                    proj = ⎡-6.1686   1.4659⎤
                                                                                                           ⎢-5.6767   1.6459⎥
                                                                                                           ⎢-5.6699   1.3642⎥
                                                                                                           ⎢-5.5643   1.3816⎥
                                                                                                           ⎢-6.1734   1.3309⎥
                                                                                                           ⎢-6.7278   1.4021⎥
                                                                                                           ⎢-5.7743   1.1498⎥
                                                                                                           ⎢-6.0466   1.4714⎥
                                                                                                           ⎢-5.2709   1.3570⎥
                                                                                                           ⎣-5.7533   1.6207⎦
                                                                                                    

                                                                                                    func (*PC) PrincipalComponents

                                                                                                    func (c *PC) PrincipalComponents(a mat.Matrix, weights []float64) (ok bool)

                                                                                                      PrincipalComponents performs a weighted principal components analysis on the matrix of the input data which is represented as an n×d matrix a where each row is an observation and each column is a variable.

                                                                                                      PrincipalComponents centers the variables but does not scale the variance.

                                                                                                      The weights slice is used to weight the observations. If weights is nil, each weight is considered to have a value of one, otherwise the length of weights must match the number of observations or PrincipalComponents will panic.

                                                                                                      PrincipalComponents returns whether the analysis was successful.

                                                                                                      func (*PC) VarsTo

                                                                                                      func (c *PC) VarsTo(dst []float64) []float64

                                                                                                        VarsTo returns the column variances of the principal component scores, b * vecs, where b is a matrix with centered columns. Variances are returned in descending order. If dst is not nil it is used to store the variances and returned. Vars will panic if the receiver has not successfully performed a principal components analysis or dst is not nil and the length of dst is not min(n, d).

                                                                                                        func (*PC) VectorsTo

                                                                                                        func (c *PC) VectorsTo(dst *mat.Dense)

                                                                                                          VectorsTo returns the component direction vectors of a principal components analysis. The vectors are returned in the columns of a d×min(n, d) matrix.

                                                                                                          If dst is empty, VectorsTo will resize dst to be d×min(n, d). When dst is non-empty, VectorsTo will panic if dst is not d×min(n, d). VectorsTo will also panic if the receiver does not contain a successful PC.

                                                                                                          Directories

                                                                                                          Path Synopsis
                                                                                                          Package combin implements routines involving combinatorics (permutations, combinations, etc.).
                                                                                                          Package combin implements routines involving combinatorics (permutations, combinations, etc.).
                                                                                                          Package distmat provides probability distributions over matrices.
                                                                                                          Package distmat provides probability distributions over matrices.
                                                                                                          Package distmv provides multivariate random distribution types.
                                                                                                          Package distmv provides multivariate random distribution types.
                                                                                                          Package distuv provides univariate random distribution types.
                                                                                                          Package distuv provides univariate random distribution types.
                                                                                                          Package mds provides multidimensional scaling functions.
                                                                                                          Package mds provides multidimensional scaling functions.
                                                                                                          Package samplemv implements advanced sampling routines from explicit and implicit probability distributions.
                                                                                                          Package samplemv implements advanced sampling routines from explicit and implicit probability distributions.
                                                                                                          Package sampleuv implements advanced sampling routines from explicit and implicit probability distributions.
                                                                                                          Package sampleuv implements advanced sampling routines from explicit and implicit probability distributions.
                                                                                                          Package spatial provides spatial statistical functions.
                                                                                                          Package spatial provides spatial statistical functions.