The BetaML.Utils Module

Utils module

Provide shared utility functions for various machine learning algorithms. You don't usually need to import from this module, as each other module (Nn, Perceptron, Clusters,...) reexport it.

error(ŷ,y) - Categorical error (Int vs Int)

error(ŷ,y) - Categorical error with probabilistic prediction of a single datapoint (PMF vs Int).

error(ŷ,y) - Categorical error with probabilistic predictions of a dataset (PMF vs Int).

reshape(myNumber, dims..) - Reshape a number as a n dimensional Array

accuracy(ŷ,y) - Categorical accuracy (Int vs Int)

accuracy(ŷ,y;tol)

Categorical accuracy with probabilistic predictions of a dataset given in terms of a dictionary of probabilities (Dict{T,Float64} vs T).

Parameters:

• ŷ: A narray where each item is the estimated probability mass function in terms of a Dictionary(Item1 => Prob1, Item2 => Prob2, ...)
• y: The N array with the correct category for each point $n$.
• tol: The tollerance to the prediction, i.e. if considering "correct" only a prediction where the value with highest probability is the true value (tol = 1), or consider instead the set of tol maximum values [def: 1].

accuracy(ŷ,y;tol)

Categorical accuracy with probabilistic prediction of a single datapoint (PMF vs Int).

Use the parameter tol [def: 1] to determine the tollerance of the prediction, i.e. if considering "correct" only a prediction where the value with highest probability is the true value (tol = 1), or consider instead the set of tol maximum values.

accuracy(ŷ,y;tol,ignoreLabels)

Categorical accuracy with probabilistic predictions of a dataset (PMF vs Int).

Parameters:

• ŷ: An (N,K) matrix of probabilities that each $\hat y_n$ record with $n \in 1,....,N$ being of category $k$ with $k \in 1,...,K$.
• y: The N array with the correct category for each point $n$.
• tol: The tollerance to the prediction, i.e. if considering "correct" only a prediction where the value with highest probability is the true value (tol = 1), or consider instead the set of tol maximum values [def: 1].
• ignoreLabels: Wheter to ignore the specific label order in y. Useful for unsupervised learning algorithms where the specific label order don't make sense [def: false]

accuracy(ŷ,y;tol)

Categorical accuracy with probabilistic prediction of a single datapoint given in terms of a dictionary of probabilities (Dict{T,Float64} vs T).

Parameters:

• ŷ: The returned probability mass function in terms of a Dictionary(Item1 => Prob1, Item2 => Prob2, ...)
• tol: The tollerance to the prediction, i.e. if considering "correct" only a prediction where the value with highest probability is the true value (tol = 1), or consider instead the set of tol maximum values [def: 1].

aic(lL,k) - Akaike information criterion (lower is better)

autoJacobian(f,x;nY)

Evaluate the Jacobian using AD in the form of a (nY,nX) madrix of first derivatives

Parameters:

• f: The function to compute the Jacobian
• x: The input to the function where the jacobian has to be computed
• nY: The number of outputs of the function f [def: length(f(x))]

Return values:

• An Array{Float64,2} of the locally evaluated Jacobian

Notes:

• The nY parameter is optional. If provided it avoids having to compute f(x)

batch(n,bSize;sequential=false)

Return a vector of bSize vectors of indeces from 1 to n. Randomly unless the optional parameter sequential is used.

Example:

julia julia> Utils.batch(6,2,sequential=true) 3-element Array{Array{Int64,1},1}: [1, 2] [3, 4] [5, 6]

bic(lL,k,n) - Bayesian information criterion (lower is better)

celu(x; α=1)

https://arxiv.org/pdf/1704.07483.pdf

classCounts(x)

Return a dictionary that counts the number of each unique item (rows) in a dataset.

Cosine distance

crossEntropy(ŷ, y; weight)

Compute the (weighted) cross-entropy between the predicted and the sampled probability distributions.

To be used in classification problems.

dcelu(x; α=1)

https://arxiv.org/pdf/1704.07483.pdf

delu(x; α=1) with α > 0

https://arxiv.org/pdf/1511.07289.pdf

dmish(x)

https://arxiv.org/pdf/1908.08681v1.pdf

dplu(x;α=0.1,c=1)

Piecewise Linear Unit derivative

https://arxiv.org/pdf/1809.09534.pdf

drelu(x)

Rectified Linear Unit

https://www.cs.toronto.edu/~hinton/absps/reluICML.pdf

dsigmoid(x)

dsoftmax(x; β=1)

Derivative of the softmax function

https://eli.thegreenplace.net/2016/the-softmax-function-and-its-derivative/

dsoftplus(x)

https://en.wikipedia.org/wiki/Rectifier(neuralnetworks)#Softplus

dtanh(x)

elu(x; α=1) with α > 0

https://arxiv.org/pdf/1511.07289.pdf

entropy(x)

Calculate the entropy for a list of items (or rows).

See: https://en.wikipedia.org/wiki/Decisiontreelearning#Gini_impurity

getScaleFactors(x;skip)

Return the scale factors (for each dimensions) in order to scale a matrix X (n,d) such that each dimension has mean 0 and variance 1.

Parameters

• x: the (n × d) dimension matrix to scale on each dimension d
• skip: an array of dimension index to skip the scaling [def: []]

Return

• A touple whose first elmement is the shift and the second the multiplicative

term to make the scale.

giniImpurity(x)

Calculate the Gini Impurity for a list of items (or rows).

See: https://en.wikipedia.org/wiki/Decisiontreelearning#Information_gain

L1 norm distance (aka Manhattan Distance)

Euclidean (L2) distance

Squared Euclidean (L2) distance

LogSumExp for efficiently computing log(sum(exp.(x)))

Transform an Array{T,1} in an Array{T,2} and leave unchanged Array{T,2}.

meanDicts(dicts)

Compute the mean of the values of an array of dictionaries.

Given dicts an array of dictionaries, meanDicts first compute the union of the keys and then average the values. If the original valueas are probabilities (non-negative items summing to 1), the result is also a probability distribution.

meanRelError(ŷ,y;normDim=true,normRec=true,p=1)

Compute the mean relative error (l-1 based by default) between ŷ and y.

There are many ways to compute a mean relative error. In particular, if normRec (normDim) is set to true, the records (dimensions) are normalised, in the sense that it doesn't matter if a record (dimension) is bigger or smaller than the others, the relative error is first computed for each record (dimension) and then it is averaged. With both normDim and normRec set to false the function returns the relative mean error; with both set to true (default) it returns the mean relative error (i.e. with p=1 the "mean absolute percentage error (MAPE)") The parameter p [def: 1] controls the p-norm used to define the error.

mish(x)

https://arxiv.org/pdf/1908.08681v1.pdf

oneHotEncoder(y,d;count)

Encode arrays (or arrays of arrays) of integer data as 0/1 matrices

Parameters:

• y: The data to convert (integer, array or array of arrays of integers)
• d: The number of dimensions in the output matrik. [def: maximum(maximum.(Y))]
• count: Wether to count multiple instances on the same dimension/record or indicate just presence. [def: false]

pca(X;K,error)

Perform Principal Component Analysis returning the matrix reprojected among the dimensions of maximum variance.

Parameters:

• X : The (N,D) data to reproject
• K : The number of dimensions to maintain (with K<=D) [def: nothing]
• error: The maximum approximation error that we are willing to accept [def: 0.05]

Return:

• A named tuple with:
  • X: The reprojected (NxK) matrix with the column dimensions organized in descending order of of the proportion of explained variance
  • K: The number of dimensions retieved
  • error: The actual proportion of variance not explained in the reprojected dimensions
  • P: The (D,K) matrix of the eigenvectors associated to the K-largest eigenvalues used to reproject the data matrix
  • explVarByDim: An array of dimensions D with the share of the cumulative variance explained by dimensions (the last element being always 1.0)

Notes:

• If K is provided, the parameter error has no effect.
• If one doesn't know a priori the error that she/he is willling to accept, nor the wished number of dimensions, he/she can run this pca function with out = pca(X,K=size(X,2)) (i.e. with K=D), analise the proportions of explained cumulative variance by dimensions in out.explVarByDim, choose the number of dimensions K according to his/her needs and finally pick from the reprojected matrix only the number of dimensions needed, i.e. out.X[:,1:K].

Example:

julia> X = [1 10 100; 1.1 15 120; 0.95 23 90; 0.99 17 120; 1.05 8 90; 1.1 12 95]
6×3 Array{Float64,2}:
 1.0   10.0  100.0
 1.1   15.0  120.0
 0.95  23.0   90.0
 0.99  17.0  120.0
 1.05   8.0   90.0
 1.1   12.0   95.0
 julia> X = pca(X,error=0.05).X
6×2 Array{Float64,2}:
  3.1783   100.449
  6.80764  120.743
 16.8275    91.3551
  8.80372  120.878
  1.86179   90.3363
  5.51254   95.5965

plu(x;α=0.1,c=1)

Piecewise Linear Unit

https://arxiv.org/pdf/1809.09534.pdf

Polynomial kernel parametrised with c=0 and d=2 (i.e. a quadratic kernel). For other cᵢ and dᵢ use K = (x,y) -> polynomialKernel(x,y,c=cᵢ,d=dᵢ) as kernel function in the supporting algorithms

Radial Kernel (aka RBF kernel) parametrised with γ=1/2. For other gammas γᵢ use K = (x,y) -> radialKernel(x,y,γ=γᵢ) as kernel function in the supporting algorithms

relu(x)

Rectified Linear Unit

https://www.cs.toronto.edu/~hinton/absps/reluICML.pdf

scale(x,scaleFactors;rev)

Perform a linear scaling of x using scaling factors scaleFactors.

Parameters

• x: The (n × d) dimension matrix to scale on each dimension d
• scalingFactors: A tuple of the constant and multiplicative scaling factor

respectively [def: the scaling factors needed to scale x to mean 0 and variance 1]

• rev: Wheter to invert the scaling [def: false]

Return

• The scaled matrix

Notes:

• Also available scale!(x,scaleFactors) for in-place scaling.
• Retrieve the scale factors with the getScaleFactors() function

sigmoid(x)

softmax (x; β=1)

The input x is a vector. Return a PMF

softplus(x)

https://en.wikipedia.org/wiki/Rectifier(neuralnetworks)#Softplus

squaredCost(ŷ,y)

Compute the squared costs between a vector of prediction and one of observations as (1/2)*norm(y - ŷ)^2.

Aside the 1/2 term correspond to the squared l-2 norm distance and when it is averaged on multiple datapoints corresponds to the Mean Squared Error. It is mostly used for regression problems.

Sterling number: number of partitions of a set of n elements in k sets