Archive for February, 2010

I.2 The Language and Grammer of Mathematics, 2. Four Basic Concepts and Pages 9 – 12

Thursday, February 18th, 2010

The word “is” and its uses: 2 is the square root of four, 2 is less than 5 and 2 is an even prime number.   These simple sentences are used to elucidate the concepts of sets, functions, relations and binary operations.

We define a collection of objects to be a set.   Thereby we can discuss these collections whether finite or infinite.  Sets are used to define mathematical objects and in proving statements about mathematical reasoning.  “Is an element of” is the mathematical symbol “Γ.

Functions are defined as the process of transforming objects.  f(x) = y means the function “f” transforms the object “x” into the object “y”.  Functions have properties and there exist sets of functions, for example, Hilbert Spaces, Function Spaces and Vector Spaces.  Some functions can be inverted.  Functions have a domain and a range.  They transform objects from the domain into objects in the range.  “f: A ® B means “f” is a function with domain “A” and range “B” with “:” read as “such that”.  The image of the function “f” is the set of values actually taken by f(x) for xÎA.  The word “image” is also used as the one element f(x).

One way of writing f(x) = y is f: x # y.  Notice this “bar arrow” is for elements of sets, whereas above, the arrow without the bar is used on the sets.

An “injection” means that f(x) and f(y) are different whenever x and y are different.  

A “surjection” means that every element y of the image B is equal to f(x) for some element x of the domain A.  So, there is a function”g” where g(y) = x.

A function that is both an injection and a surjection is called a “bijection”.  A function that is a bijection has an inverse (and visa versa). 

“Is less than” is denoted by the symbol “<“.  Three examples of relations are: =, < and Î.  Strictly speaking the relation “<” in the sentence “a < b” is different if “a” and “b” are members of the positive integers than if “a” and “b” are members of the reals.  We ususally don’t even see this difference.

Sometimes relationships are between two sets.  If A is the set of positive integers and B the set of all sets of positive integers then the relationship “Δ is between the two sets, that is, A Î B. 

“Equivalence relations” regard objects as “essentially the same”.  Examples are “is similar to” and “is conguent modulo m to”.  “Similar” refers to geometric objects that can be transformed into one another by a combination of reflections, rotations, translations, and enlargements.  Two numbers that differ by a multiple of m are said to be conguent modulo m.

Relations take a set and divide it into equivalence classes where each part consists of objects that are essentially the same.

Define a relation on a set A as “~”.  An equivalence relation has the following three properties: reflexive, symmetric and transitive.  Reflexive means x ~ x for all x  Î A.  Symmetric means if x ΠA and y Î A and x ~ y then it must also be that y ~ x.  Transitive means that x ΠA, y ΠA and z Î A such that x ~ y and y ~ z then it must be true that x ~ z. 

The relations “is similar to” and “is conguent to modulo m” both have the properties of being reflexive, symmetric and transitive.  The relation “is less than” is transitive but not reflexive or symmetric.

Equivalence relations are used to make precise the notion of a quotient.

Examples of “binary operations” are: “plus”, “minus”, “times”, “divided by” and “raised to a power”.   A binary operation is a function from the set of pairs (x,y), where x Î A and y Î A, of domain A, onto the range A.  However, interestingly, we write x + y rather than +(x,y).

A binary operation may have the properties: commutative, associative, identity and inverse.  Let “*” represent a binary operation.  Commutative means x * y = y * x.  Associative means (x * y)  * z = x * (y * z).  By convention binary operations need to be defined for all x Î A.

An element e Î A is called an identity if e * x = x * e = x for all x Î A.   Examples are 0 and 1 for plus and times.

If * has an identity e and x Î A then an inverse for x is an element y Î A such that x * y = y * x = e.   Examples for plus and times are: -x and 1/x.

The basic properties of binary operations are fundamental to the structures of abstract algebra.

                      

I.2 The Language and Grammer of Mathematics and Page 8

Wednesday, February 17th, 2010

Discussion of the clarity and simplicity needed in grammer and language for use in mathematical thinking.  Discussed the word “and” and its multiple meaning in English and mathematics.  One of the most obvious meaning is “plus”.  Another use is to join two sentences, and on this it is commented: “When mathematicians are at their most formal, they simply outlaw the noun-linking use of ‘and’ …”.

Analysis Thinking and The Branches of Mathematics and Pages 2 -7

Tuesday, February 16th, 2010

Pages 2 – 7 up to Section I.2.

Algebra thinking is about finite things and analysis thinking is about the infinite.

The Main Branches of Mathematics are: Algebra, Number Theory, Geometry, Algebraic Geometry, Analysis, Logic, Combinatorics, Theoretical Computer Science, Probability and Methematical Physics.

Algebra – concerned with number systems, polynomials, groups, fields, vector spaces and rings.

Number Theory – properties of the positive integers, equations in integers or algebraic number theory and analytic number theory which is rooted in the study of prime numbers.

 Geometry – study of manifolds, topology (exact distances don’t matter) and differential topology (distances matter).

Algebraic Geometry – manifolds described by polynomials.

Analysis – partial differential equations, infinite dimension vector spaces (Banach spaces, Hilbert spaces, C*-algebras, Von Neuman algebras) and dynamics (motion of planets).

 Logic – fundamental questions about mathematics – set theory, category theory, model theory and logic concerning “rules of deduction”.  Results include Godel’s Incompleteness Theorems and The Independence of the Continuum Hypothesis. 

Combinatorics – counting things, discrete structures (also discrete mathematics) and structures with few constraints.

Theoretical Computer Science – efficiency of computation an application is cryptography.

Probability – phenomena that are so complex they are best described by probabalistic statements.  Problems are often in fields of biology, economics, computer science and physics.  Examples include, phase transition – probabalistic models of critical phenomena and Brownian motion.

Mathematical Physics – pysicist often lead the way with non-rigourous aurguments that are then taken up and rigourously proved by mathematicians.  Examples are: vertex operator algebras, mirror symmetry, general relativity and the Einstein equations and operator algebras.

Reading the Book “The Princeton Companion to Mathematics” edited by Timothy Gowers.

Monday, February 15th, 2010

Reading the entire book starting last night.

Read Page 1 and Page 2 up until section 1.2.

Algebra is about letters and geometry is about a visual conception of a problem.

There is often a correspondance between these two ways of looking at a problem.

Forecasting Auto sales in Brazil by Make/model, city and month.

Wednesday, February 3rd, 2010

Over 200,000 individual forecasts!

My Proficiency Levels for Statistical Analysis in Business and Industry

Tuesday, February 2nd, 2010

Definition:

Use the Company process to provide Statistical Analysis services for an internal or external client. 

           

Proficiency Basic

·         Define basic descriptive statistics (mean, median, mode, range, covariance, population standard deviation) and terms, such as: Euclidean Distance, Mahalanobis Distance, Missing Values versus Zeros, Product Popularity Computations, Quantiles and Rank.

·         Explain basic descriptive statistics and terms to clients.

·         Recognize when ones own basic statistical knowledge is inadequate to resolve a problem.  Be willing and know how to bring in appropriate resources.

·         Execute, with guidance, a simple statistical analysis software program.

·         Know how to create scatter plots, bar charts and histograms.

·         Collect or assemble data for statistical analysis.

 

Proficiency Basic Plus

·         Have knowledge and/or practical experience in introductory level statistical analysis including the following concepts: Assignable Causes and Actions, Avoiding Lying with Statistics and Maps, Chernoff Faces, Coded/Dummy Variables, Coefficient of Determination, Combinatorial Analysis, Computation of Expected, Confidence Interval for a datum, Confidence Intervals, Data Basics, Discrete Distributions, Elementary Probability Theory, Elementary Statistical Sampling, Gaussian/Normal Distribution, Geometric Mean, Graphical Display/Analysis, Harmonic Mean, Histograms, Interpolation, Introduction to trend cycle forecasting, Multivariate Star Plots, Lack of Fit in Regression Analysis, Method of Least Squares, Market Share Analysis, Odds Ratio, One-sample t-test, One-way tables, Original Data Plots, Over parameterized model, Paired t-test, Percentiles, Pie Chart Replacements, Quartiles, R – Pearson correlation coefficient, Randomness, Ranks, Raw Data Plots, Residuals, Run tests, Sign Test, Spearman R, Student’s t-test, Travel Distance or Travel Time, Trimmed Means, Types of Data, Uniform distribution, Univariate Regression Analysis and Weighted Mean.

·         Define intermediate descriptive statistics (correlation, r-squared, chi-squared, geometric mean, harmonic mean, t-statistic, F-statistic, sample standard deviation and those included in the introductory concepts above).

·         Know the following statistical areas at an introductory level.  Areas include: computation of expected, confidence intervals, data basics, discrete distributions, elementary probability theory, elementary statistical sampling, graphical display/analysis, introduction to trend-cycle forecasting, market share analysis, one sample t-test, one-way tables, paired t-test and sign test.

·         Know and be able to explain to a client univariate regression analysis.

·         Statistical consulting on a basic level – personal communication skills.

·         Understand and be able to explain a statistical hypothesis testing.

·         With guidance, create statistical analysis output using SAS, Excel and other statistical software.

Proficiency Advanced

·         Have knowledge and/or practical experience in intermediate level statistical analysis (equivalent to a Statistics Bachelor’s degree) including the following concepts: Analysis of Variance, Bayesian Inference, Chi-square Test, Cluster Analysis, Confidence Interval for the mean, Confidence Interval Theory, Confidence Level, Continuous Distributions, Contouring, Mallows Cp statistic, Data Smoothing, Exploratory Data Analysis (Data Mining), External Model Validation, Extrapolation, Frequentist Statistics, Gravity Model Application, Hypothesis Testing, Intermediate Probability Theory, Internal Model Validation, Kendall Tau, K-Means Algorithm, K-Nearest Neighbor algorithm, Kruskall-Wallis Test, Kurtosis, Lack of Fit Tests, Level of Significance (alpha), Matrix Algebra, Maximum Likelihood Method, Metric Spaces (Generalized Distances), Minimax, Multiple or Multivariate Regression, Multi-way tables, Normality Tests, Outliers, Overfitting, Parallel Coordinate Plots, p-level (statistical significance), PRESS Statistic, Quality Control, Residual Analysis, Robust Analysis, Scheffe’s test, Shapiro-Wilks test, Shewart Control Charts, Short run control charts, Signal to Noise ratio, Site Selection, Skewness, Standard error of a proportion, Standardized residuals, Statistical Graphics, Statistical Inference, Statistical Power, Statistical Significance (p-level), Stepwise regression, Studentized residuals, Univariate Time Series Analysis, Time Series Analysis – Seasonal Factors, Type I Error, Type II Error, Weighted Least Squares, Weighted Variance and Wilcoxin Test.

·         Define advanced descriptive statistics (auto-correlation, Kendall Tau, outlier test statistics, robustness statistics and others involved in the intermediate level concepts above).

·         Explain the sources of variation and their impact upon the product or process.

·         Know the intermediate level statistical analysis areas at an intermediate level.  Areas include: analysis of variance, Bayesian inference, chi-square test, confidence interval theory, continuous distributions, exploratory data analysis (data mining), frequentist statistics, gravity model application, hypothesis testing, intermediate probability theory, lack of fit tests, matrix algebra, maximum likelihood method, multi-way tables, normality tests, residual analysis, robust analysis, site selection, statistical graphics, statistical inference, stepwise regression, univariate time series analysis, time series analysis – seasonal factors, Type I error, Type II error and weighted least squares.

·         Know and be able to explain multivariate regression analysis to a client.

·         Statistical consulting at an intermediate level – problem identification.

·         Know the stages of statistical analysis methodology (refer to Company Report #15 – Statistical Modeling Process).

     

Proficiency Advanced Plus

·         Have knowledge and/or practical experience in advanced level of statistical analysis (equivalent to a Statistics Masters’ Degree) including the following concepts: Bayesian Analysis, Categorical Data Analysis, Categorical Trees, Census Data Analysis, Classification Trees, Cook’s Distance, Cross-Validation, Data Mining, Density Estimation, Discrete Multivariate Analysis, Discriminant Function Analysis, Distributions, Duncan’s Test, Dunnett’s Test, Experimental Design, Exploratory Data Analysis, Factor Analysis, Forecasting, General Linear Models (GLIM), Gravity Modeling Theory, Hazard Function, Inductive versus Deductive Reasoning, Local Minima/Maxima, Logit Regression, Model Criticism, Multidimensional Scaling, Multinomial Logit and Probit Regression, Multivariate Time Series Analysis, Network Analysis, Newman-Kuels Test, Nonlinear Regression Analysis, Panel Data, Parallel Coordinate Analysis, Predictive Modeling, Principal component analysis, Probability Theory, Probit Regression, Quality and Productivity, Random Variables, Regression Trees, Reliability, Response surface, Sample survey, Segmentation Theory and Analysis, Spectral plot, Spatial Interaction Models, Spline fitting, Statistical Consulting Skills (technical jargon, asking good questions, listening skills, negotiating fair exchanges, talking about statistics, and resolving difficult situations), Statistical Software (S-Plus, SAS, Excel, etc.), Survival Data Analysis, Variance components (In Mixed Model ANOVA), Visual Display of Quantitative Information, Wald Statistic, Weighted Regression Analysis and Yates Corrected Chi-Square.

·         Recognize when ones own statistical knowledge is inadequate to resolve a problem.  Be willing and know how to bring in appropriate resources.

·         Know the advanced level of statistical analysis concepts, listed above, at some level.

·         Statistical consulting at an advanced level – a level at which the statistical consulting services may be sold to the client.

·         Determine what statistical analysis packages will be useful for the project.

·         Know how to interpret output from all statistical analysis packages.

·         Lead a complex Statistical Analysis including establishing an overall strategy, providing on-going guidance, reviewing results and solving complex issues quickly.

·         Educate clients and other employees on the pros and cons of various Statistical Analysis approaches.

·         Manage the execution of a Statistical Analysis including delivery dates, quality and project costs.

·         Present Statistical Analysis results in a skilled and professional manner.

·         Influence client decisions regarding Statistical Analysis.

·         Sought out by clients and employees as an experienced, knowledgeable resource in Statistical Analysis.

·         Respond knowledgeably and professionally to requests or challenges under pressure.

·         Understand current changes in the industry and how they will impact Statistical Analysis.

·         Understand the various legal guidelines/statutes/precedences that impact Statistical Analysis.

·         Qualify and testify as an “expert” in issues of statistics and statistical analysis.

·         Produce two or more written documents, in the “Company Statistical Report” series, per year.

·         Attend at least one relevant professional meeting (such as, The Joint Statistical Meetings) per year.

     

Proficiency Expert

·         Have knowledge and/or practical experience in the expert level of statistical analysis (equivalent to a Statistics Ph.D. degree) including the following concepts: Approximation Theory, Artificial Intelligence, Bayesian Statistics, Bonferroni test, Censoring, Display of Statistical Equations and Technical Information (TeX, LaTeX, Mathtype, etc.), Econometric Models/Modeling, Entropy, Estimable Functions, Expert Systems, Fractals, Function Estimation, Function Smoothing, Functional Analysis, Genetic Algorithm, Group Theory, Graph Theory, Hilbert Space Theory, Interval Analysis, Latent Variable Models, Markov Chain Monte Carlo, Mathematical Statistics, Multivariate density Estimation, Network Flows, Network Optimization, Neural Networks, Non parametric methods, Nonlinear Estimation, Optimization Methods/Routines, Orthogonal Polynomials, Predictive Data Mining, Quality Theory, Signal detection theory, Simulations, Simplex algorithm, Spatial Data Analysis, Special Functions, Statistical Control Theory, Statistical Signal Processing, Statistics and the Law, Stochastic Processes, Structural Equation Modeling, The Scientific Method, Theoretical Statistics, Transmitted Variation and Wavelets.

·         Recognized by Company and/or clients as an “expert” in Statistical Analysis.

·         Know and be able to utilize the mathematical underpinnings of statistical methods.

·         Statistical Consulting Skills at an expert level including: problem identification, opportunity identification, application field technical jargon, asking good questions, interpretation of complex ideas into understandable examples, listening skills, negotiating fair exchanges, talking about statistics, and resolving difficult situations.

·         Respond knowledgeably and professionally to requests or challenges under significant pressure.

·         Train and coach new “experts” in advanced Statistical Analysis skills.

·         Keep abreast of research in the field of statistics at a professional level.

·         Attend and participate (present a paper, presentation, discussion or poster session) in at least two relevant professional meetings per Year (such as, The Joint Statistical Meetings).

Brazil Geographic Distribution

Tuesday, February 2nd, 2010

I just finished distributing automotive registrations from metroploitan areas to census geography for four years and two automotive vehicle classifications within all of Brazil.

Construction of PI?

Monday, February 1st, 2010

Why is is not possible to construct with straight edge, a compass, and a line of length one a line of length pi?