This is documentation for Mathematica 9, which was
based on an earlier version of the Wolfram Language.
MATHEMATICA GUIDE

# New in 9.0: Mathematics & Algorithms

Mathematica 9 adds major extensions and integration across the system. Probability and statistics adds survival and reliability analysis as well as a unified treatment for all random processes. Graphs and networks adds network flows, social network analysis, and major performance enhancements. Control systems adds PID autotuning as well as full support for descriptor and delay systems. Numerical differential equations now includes hybrid systems and parametric systems, as well as advanced support for differential algebraic equations.

## ReferenceReference

### Descriptive Statistics - Dependency Measures »

SpearmanRho Spearman's rank correlation for data and distributions

KendallTau Kendall's rank correlation for data and distributions

### Hypothesis Tests - Independence Tests »

IndependenceTest test whether two vectors are independent

CorrelationTest test whether two vectors are uncorrelated

### New Parametric Distributions

HypoexponentialDistribution hypoexponential distribution

### Derived Distributions »

SplicedDistribution splicing several distributions such as different body and tail

CompoundPoissonDistribution Poisson distribution with given jump size distribution

OrderDistribution generalized to different and multivariate distributions

### New Data Models

EventData censored (left, right, interval) and truncated data

WeightedData weighted data

Mean descriptive statistics works for censored and weighted data

EstimatedDistribution parametric distribution estimation from censored data

EmpiricalDistribution nonparametric distributions from censored data

### Survival Analysis »

SurvivalModelFit survival functions with confidence bands from censored data

CoxModelFit Cox proportional hazards model from censored data

LogRankTest hypothesis test for equal hazard rates

### Reliability Modeling and Analysis »

ReliabilityDistribution reliability block diagram model for a system

FailureDistribution fault tree model for a system

StandbyDistribution standby reliability distribution (cold, hot, ...)

BirnbaumImportance measure to pinpoint subsystem to improve for reliability

### Random Processes »

RandomFunction simulate a random process

TemporalData represent one or several time series data

EstimatedProcess process estimated from time series data

SliceDistribution distribution for a time slice of a random process

### Differential Equations with Events »

WhenEvent discrete actions in differential equations when an event occurs

NDSolve now supports differential equations with events

### Differential Equations with Parameters

ParametricNDSolveValue solve differential equations depending on parameters

### Differential Algebraic Equations

NDSolve can now solve high-index DAEs

### Graphs & Networks »

FindPostmanTour find a tour that traverses each edge at least once

### Social Network Analysis »

FindGraphCommunities find communities within a graph

GlobalClusteringCoefficient compute the global clustering coefficient

### Network Flows and Cuts »

FindMaximumFlow find the maximum flow from a source vertex to a target vertex

FindMinimumCostFlow find the minimum cost flows

FindMinimumCut find the minimum cut for a graph

### Random Graphs »

GraphPropertyDistribution automatically compute properties of random graphs

SpatialGraphDistribution higher-performance random graph generation

### Control Systems

PIDTune automatic design of PID controllers

### Models with Time Delays »

SystemsModelDelay represent delays in state-space and transfer-function models

### Models with Algebraic Constraints »

DescriptorStateSpace whether to use descriptor state-space representation

### Fast Symbolic Linear and Polynomial Algebra

LinearSolve fast methods for sparse symbolic and exact systems

Expand fast methods for polynomials with numeric coefficients

### New Matrix Functions

MatrixLog matrix log function

MatrixFunction matrix function for a general function

### New Algebraic Numbers

CubeRoot real cube root

Surd real root

### Coordinate Systems

CoordinateChartData properties of coordinate systems

TransformedField transform a field between coordinate systems

### Symbolic Tensor Algebra »

Arrays domain of arrays with given properties

### Symmetries and Symmetrized Arrays

SymmetrizedArray compact representation of arrays with symmetry