Presentation given for the Boston Predictive Analytics group by Daniel Gerlanc of Enplus Advisors Inc on July 25, 2012 at the CIC. Visit us at enplusadvisors.com
Intermediate Regression Topics including variable transformations and simulation for constructing confidence intervals.
An Introduction into Anomaly Detection Using CUSUMDominik Dahlem
A gentle introduction into anomaly detection using the cumulative sum (CUSUM) algorithm. Extensive visuals are used to exemplify the inner workings of the algorithm. CUSUM relies on stationarity assumptions of the underlying process.
Crushing the Head of the Snake by Robert Brewer PyData SV 2014PyData
Big Data brings with it particular challenges in any language, mostly in performance. This talk will explain how to get immediate speedups in your Python code by exploiting both timeless programming techniques and fixes specific to Python. We will cover: I. Amongst Our Weaponry 1. How to Time and Profile Python 2. Extracting Loop invariants: constants, lookup tables, even methods! 3. Caching: memoization and heavier things II Gunfight at the O.K. Corral in Morse Code 1. Python functions vs C functions 2. Vector operations: NumPy 3. Reducing calls: loops, generators, recursion III. The Semaphore Version of Wuthering Heights 1. Using select instead of Queue 2. Serialization overhead 3. Parallelizing work
An Introduction into Anomaly Detection Using CUSUMDominik Dahlem
A gentle introduction into anomaly detection using the cumulative sum (CUSUM) algorithm. Extensive visuals are used to exemplify the inner workings of the algorithm. CUSUM relies on stationarity assumptions of the underlying process.
Crushing the Head of the Snake by Robert Brewer PyData SV 2014PyData
Big Data brings with it particular challenges in any language, mostly in performance. This talk will explain how to get immediate speedups in your Python code by exploiting both timeless programming techniques and fixes specific to Python. We will cover: I. Amongst Our Weaponry 1. How to Time and Profile Python 2. Extracting Loop invariants: constants, lookup tables, even methods! 3. Caching: memoization and heavier things II Gunfight at the O.K. Corral in Morse Code 1. Python functions vs C functions 2. Vector operations: NumPy 3. Reducing calls: loops, generators, recursion III. The Semaphore Version of Wuthering Heights 1. Using select instead of Queue 2. Serialization overhead 3. Parallelizing work
Clustering and Visualisation using R programmingNixon Mendez
Clustering Analysis is a collection of patterns into clusters based on similarity.
Here we will discuss on the following :
Microarray Data of Yeast Cell Cycle
Clustering Analysis :-
Principal Component Analysis (PCA)
Multidimensional Scaling (MDS)
K-Means
Self-Organizing Maps (SOM)
Hierarchical Clustering
Kazushi Okamoto: Families of Triangular Norm Based Kernel Function and Its Application to Kernel k-means, Joint 8th International Conference on Soft Computing and Intelligent Systems and 17th International Symposium on Advanced Intelligent Systems (SCIS-ISIS2016), 2016.08.25
In this session Max Kleiner shows four groups of the ML: Regression, Dimension Reduction, Clustering and Classification. ML recognizes patterns and laws in the learning data. Most ML projects allegedly fail due to lack of data consolidation and due to lack of hypothesis. On the basis of the well-known IRIS dataset the 4 groups with 4 algorithms each are gone through and this lack is avoided.
Detecting and Auditing for Fraud in Financial Statements Using Data AnalysisFraudBusters
Webinar series from FraudResourceNet LLC on Preventing and Detecting Fraud Using Data Analytics. Recordings of these Webinars are available for purchase from our Website fraudresourcenet.com
This Webinar focused on fraud detection using data analytic software (Excel, ACL, IDEA)
FraudResourceNet (FRN) is the only searchable portal of practical, expert fraud prevention, detection and audit information on the Web.
FRN combines the high quality, authoritative anti-fraud and audit content from the leading providers, AuditNet ® LLC and White-Collar Crime 101 LLC/FraudAware.
The two entities designed FRN as the “go-to”, easy-to-use source of “how-to” fraud prevention, detection, audit and investigation templates, guidelines, policies, training programs (recorded no CPE and live with CPE) and articles from leading subject matter experts.
FRN is a continuously expanding and improving resource, offering auditors, fraud examiners, controllers, investigators and accountants a content-rich source of cutting-edge anti-fraud tools and techniques they will want to refer to again and again.
Using Data Analytics to Conduct a Forensic AuditFraudBusters
Webinar series from FraudResourceNet LLC on Preventing and Detecting Fraud Using Data Analytics. Recordings of these Webinars are available for purchase from our Website fraudresourcenet.com
This Webinar focused on fraud detection using data analytic software (Excel, ACL, IDEA)
FraudResourceNet (FRN) is the only searchable portal of practical, expert fraud prevention, detection and audit information on the Web.
FRN combines the high quality, authoritative anti-fraud and audit content from the leading providers, AuditNet ® LLC and White-Collar Crime 101 LLC/FraudAware.
The two entities designed FRN as the “go-to”, easy-to-use source of “how-to” fraud prevention, detection, audit and investigation templates, guidelines, policies, training programs (recorded no CPE and live with CPE) and articles from leading subject matter experts.
FRN is a continuously expanding and improving resource, offering auditors, fraud examiners, controllers, investigators and accountants a content-rich source of cutting-edge anti-fraud tools and techniques they will want to refer to again and again.
Go Predictive Analytics, LLC is a premier data mining and predictive analytics consulting company. We remove the barriers that loom large with creating and deploying data mining solutions for high ROI.
FRN combines the high quality, authoritative anti-fraud and audit content from the leading providers, AuditNet ® LLC and White-Collar Crime 101 LLC/FraudAware.
The two entities designed FRN as the “go-to”, easy-to-use source of “how-to” fraud prevention, detection, audit and investigation templates, guidelines, policies, training programs (recorded no CPE and live with CPE) and articles from leading subject matter experts.
FRN is a continuously expanding and improving resource, offering auditors, fraud examiners, controllers, investigators and accountants a content-rich source of cutting-edge anti-fraud tools and techniques they will want to refer to again and again.
White-Collar Crime Fighter Newsletter Subscribe Now at No Cost!
FraudResourceNet has made the premier Anti-Fraud newsletter, White-Collar Crime Fighter freely available to all. All this is required is to complete the registration form with your work email address!
The widely read newsletter, White-Collar Crime Fighter brings you expert strategies and actionable advice from the most prominent experts in the fraud-fighting business. Every two months you'll learn about the latest frauds, scams and schemes... and the newest and most effective fraud-fighting tools, techniques and technologies to put to work immediately to protect your organization.
When it comes to fraud, knowledge of the countless schemes, how they work and red flags to look for will help keep you, your organization and your clients safe.
At FraudResourceNet we understand this and take great pride in providing our FREE White Collar Crime Fighter newsletter -- filled with exclusive articles and tips to provide the knowledge you need.
Make sure you stay informed. Sign up for White Collar Crime Fighter newsletter and we’ll keep you up-to-date on special promos, training opportunities, and other news and offers from FraudResourceNet!
Signing up is easy and FREE. If you have not already subscribed to our newsletter, please sign up to get started!
Sign up for the White Collar Crime Fighter Newsletter (a $99 value ... now completely FREE)
Think Like a Fraudster to Catch a FraudsterFraudBusters
FRN combines the high quality, authoritative anti-fraud and audit content from the leading providers, AuditNet ® LLC and White-Collar Crime 101 LLC/FraudAware.
The two entities designed FRN as the “go-to”, easy-to-use source of “how-to” fraud prevention, detection, audit and investigation templates, guidelines, policies, training programs (recorded no CPE and live with CPE) and articles from leading subject matter experts.
FRN is a continuously expanding and improving resource, offering auditors, fraud examiners, controllers, investigators and accountants a content-rich source of cutting-edge anti-fraud tools and techniques they will want to refer to again and again.
White-Collar Crime Fighter Newsletter Subscribe Now at No Cost!
FraudResourceNet has made the premier Anti-Fraud newsletter, White-Collar Crime Fighter freely available to all. All this is required is to complete the registration form with your work email address!
The widely read newsletter, White-Collar Crime Fighter brings you expert strategies and actionable advice from the most prominent experts in the fraud-fighting business. Every two months you'll learn about the latest frauds, scams and schemes... and the newest and most effective fraud-fighting tools, techniques and technologies to put to work immediately to protect your organization.
When it comes to fraud, knowledge of the countless schemes, how they work and red flags to look for will help keep you, your organization and your clients safe.
At FraudResourceNet we understand this and take great pride in providing our FREE White Collar Crime Fighter newsletter -- filled with exclusive articles and tips to provide the knowledge you need.
Make sure you stay informed. Sign up for White Collar Crime Fighter newsletter and we’ll keep you up-to-date on special promos, training opportunities, and other news and offers from FraudResourceNet!
Signing up is easy and FREE. If you have not already subscribed to our newsletter, please sign up to get started!
Sign up for the White Collar Crime Fighter Newsletter (a $99 value ... now completely FREE)
Clustering and Visualisation using R programmingNixon Mendez
Clustering Analysis is a collection of patterns into clusters based on similarity.
Here we will discuss on the following :
Microarray Data of Yeast Cell Cycle
Clustering Analysis :-
Principal Component Analysis (PCA)
Multidimensional Scaling (MDS)
K-Means
Self-Organizing Maps (SOM)
Hierarchical Clustering
Kazushi Okamoto: Families of Triangular Norm Based Kernel Function and Its Application to Kernel k-means, Joint 8th International Conference on Soft Computing and Intelligent Systems and 17th International Symposium on Advanced Intelligent Systems (SCIS-ISIS2016), 2016.08.25
In this session Max Kleiner shows four groups of the ML: Regression, Dimension Reduction, Clustering and Classification. ML recognizes patterns and laws in the learning data. Most ML projects allegedly fail due to lack of data consolidation and due to lack of hypothesis. On the basis of the well-known IRIS dataset the 4 groups with 4 algorithms each are gone through and this lack is avoided.
Detecting and Auditing for Fraud in Financial Statements Using Data AnalysisFraudBusters
Webinar series from FraudResourceNet LLC on Preventing and Detecting Fraud Using Data Analytics. Recordings of these Webinars are available for purchase from our Website fraudresourcenet.com
This Webinar focused on fraud detection using data analytic software (Excel, ACL, IDEA)
FraudResourceNet (FRN) is the only searchable portal of practical, expert fraud prevention, detection and audit information on the Web.
FRN combines the high quality, authoritative anti-fraud and audit content from the leading providers, AuditNet ® LLC and White-Collar Crime 101 LLC/FraudAware.
The two entities designed FRN as the “go-to”, easy-to-use source of “how-to” fraud prevention, detection, audit and investigation templates, guidelines, policies, training programs (recorded no CPE and live with CPE) and articles from leading subject matter experts.
FRN is a continuously expanding and improving resource, offering auditors, fraud examiners, controllers, investigators and accountants a content-rich source of cutting-edge anti-fraud tools and techniques they will want to refer to again and again.
Using Data Analytics to Conduct a Forensic AuditFraudBusters
Webinar series from FraudResourceNet LLC on Preventing and Detecting Fraud Using Data Analytics. Recordings of these Webinars are available for purchase from our Website fraudresourcenet.com
This Webinar focused on fraud detection using data analytic software (Excel, ACL, IDEA)
FraudResourceNet (FRN) is the only searchable portal of practical, expert fraud prevention, detection and audit information on the Web.
FRN combines the high quality, authoritative anti-fraud and audit content from the leading providers, AuditNet ® LLC and White-Collar Crime 101 LLC/FraudAware.
The two entities designed FRN as the “go-to”, easy-to-use source of “how-to” fraud prevention, detection, audit and investigation templates, guidelines, policies, training programs (recorded no CPE and live with CPE) and articles from leading subject matter experts.
FRN is a continuously expanding and improving resource, offering auditors, fraud examiners, controllers, investigators and accountants a content-rich source of cutting-edge anti-fraud tools and techniques they will want to refer to again and again.
Go Predictive Analytics, LLC is a premier data mining and predictive analytics consulting company. We remove the barriers that loom large with creating and deploying data mining solutions for high ROI.
FRN combines the high quality, authoritative anti-fraud and audit content from the leading providers, AuditNet ® LLC and White-Collar Crime 101 LLC/FraudAware.
The two entities designed FRN as the “go-to”, easy-to-use source of “how-to” fraud prevention, detection, audit and investigation templates, guidelines, policies, training programs (recorded no CPE and live with CPE) and articles from leading subject matter experts.
FRN is a continuously expanding and improving resource, offering auditors, fraud examiners, controllers, investigators and accountants a content-rich source of cutting-edge anti-fraud tools and techniques they will want to refer to again and again.
White-Collar Crime Fighter Newsletter Subscribe Now at No Cost!
FraudResourceNet has made the premier Anti-Fraud newsletter, White-Collar Crime Fighter freely available to all. All this is required is to complete the registration form with your work email address!
The widely read newsletter, White-Collar Crime Fighter brings you expert strategies and actionable advice from the most prominent experts in the fraud-fighting business. Every two months you'll learn about the latest frauds, scams and schemes... and the newest and most effective fraud-fighting tools, techniques and technologies to put to work immediately to protect your organization.
When it comes to fraud, knowledge of the countless schemes, how they work and red flags to look for will help keep you, your organization and your clients safe.
At FraudResourceNet we understand this and take great pride in providing our FREE White Collar Crime Fighter newsletter -- filled with exclusive articles and tips to provide the knowledge you need.
Make sure you stay informed. Sign up for White Collar Crime Fighter newsletter and we’ll keep you up-to-date on special promos, training opportunities, and other news and offers from FraudResourceNet!
Signing up is easy and FREE. If you have not already subscribed to our newsletter, please sign up to get started!
Sign up for the White Collar Crime Fighter Newsletter (a $99 value ... now completely FREE)
Think Like a Fraudster to Catch a FraudsterFraudBusters
FRN combines the high quality, authoritative anti-fraud and audit content from the leading providers, AuditNet ® LLC and White-Collar Crime 101 LLC/FraudAware.
The two entities designed FRN as the “go-to”, easy-to-use source of “how-to” fraud prevention, detection, audit and investigation templates, guidelines, policies, training programs (recorded no CPE and live with CPE) and articles from leading subject matter experts.
FRN is a continuously expanding and improving resource, offering auditors, fraud examiners, controllers, investigators and accountants a content-rich source of cutting-edge anti-fraud tools and techniques they will want to refer to again and again.
White-Collar Crime Fighter Newsletter Subscribe Now at No Cost!
FraudResourceNet has made the premier Anti-Fraud newsletter, White-Collar Crime Fighter freely available to all. All this is required is to complete the registration form with your work email address!
The widely read newsletter, White-Collar Crime Fighter brings you expert strategies and actionable advice from the most prominent experts in the fraud-fighting business. Every two months you'll learn about the latest frauds, scams and schemes... and the newest and most effective fraud-fighting tools, techniques and technologies to put to work immediately to protect your organization.
When it comes to fraud, knowledge of the countless schemes, how they work and red flags to look for will help keep you, your organization and your clients safe.
At FraudResourceNet we understand this and take great pride in providing our FREE White Collar Crime Fighter newsletter -- filled with exclusive articles and tips to provide the knowledge you need.
Make sure you stay informed. Sign up for White Collar Crime Fighter newsletter and we’ll keep you up-to-date on special promos, training opportunities, and other news and offers from FraudResourceNet!
Signing up is easy and FREE. If you have not already subscribed to our newsletter, please sign up to get started!
Sign up for the White Collar Crime Fighter Newsletter (a $99 value ... now completely FREE)
Using Data Analytics to Find and Deter Procure to Pay FraudFraudBusters
FRN combines the high quality, authoritative anti-fraud and audit content from the leading providers, AuditNet ® LLC and White-Collar Crime 101 LLC/FraudAware.
The two entities designed FRN as the “go-to”, easy-to-use source of “how-to” fraud prevention, detection, audit and investigation templates, guidelines, policies, training programs (recorded no CPE and live with CPE) and articles from leading subject matter experts.
FRN is a continuously expanding and improving resource, offering auditors, fraud examiners, controllers, investigators and accountants a content-rich source of cutting-edge anti-fraud tools and techniques they will want to refer to again and again.
White-Collar Crime Fighter Newsletter Subscribe Now at No Cost!
FraudResourceNet has made the premier Anti-Fraud newsletter, White-Collar Crime Fighter freely available to all. All this is required is to complete the registration form with your work email address!
The widely read newsletter, White-Collar Crime Fighter brings you expert strategies and actionable advice from the most prominent experts in the fraud-fighting business. Every two months you'll learn about the latest frauds, scams and schemes... and the newest and most effective fraud-fighting tools, techniques and technologies to put to work immediately to protect your organization.
When it comes to fraud, knowledge of the countless schemes, how they work and red flags to look for will help keep you, your organization and your clients safe.
At FraudResourceNet we understand this and take great pride in providing our FREE White Collar Crime Fighter newsletter -- filled with exclusive articles and tips to provide the knowledge you need.
Make sure you stay informed. Sign up for White Collar Crime Fighter newsletter and we’ll keep you up-to-date on special promos, training opportunities, and other news and offers from FraudResourceNet!
Signing up is easy and FREE. If you have not already subscribed to our newsletter, please sign up to get started!
Sign up for the White Collar Crime Fighter Newsletter (a $99 value ... now completely FREE)
Increasing volumes of electronically stored information (ESI) in litigation have created the need for faster and more effective review procedures, software, and systems. Larger cases mean that 'eyes on' linear review of all documents just isn't possible sometimes. Document-intensive matters require consideration and adoption of technology-enhanced approaches to leverage attorney and staff time in an efficient and effective process. Modern technology-enhanced review tools maximize attorney review time of key evidence, accelerate production timelines, and better control discovery costs.
This webinar will discuss current best practices to streamline and speed review, including improved keyword search, multi-index search, clustering/grouped multi-doc coding, and predictive coding.
[Tutorial] building machine learning models for predictive maintenance applic...PAPIs.io
This talk introduces the landscape and challenges of predictive maintenance applications in the industry, illustrates how to formulate (data labeling and feature engineering) the problem with three machine learning models (regression, binary classification, multi-class classification) using a publicly available aircraft engine run-to-failure data set, and showcases how the models can be conveniently trained and compared with different algorithms in Azure ML.
En la sesión se mostró cómo configurar una carga de trabajo, basada en SUSE, para utilizar en Azure. Además se realizó la instalación de SUSE Manager en Azure y se mostró cómo es posible gestionar todas nuestras cargas de trabajo, incluyendo las que conservamos en centros de datos privados y las que ya han sido migradas a Azure, desde la instancia de SUSE Manager desplegada en Azure.
Varies Ranges : from 0 to 150 mm Wc up to 0 to 600 Bar
Differential (Dead Band) : Fixed (Within 10% of set value)
Temperature Range : -10 to 80 Degree C (for high temperature use of impulse tube is advisable)
Accuracy : +/- 1% to 2% FS
No. of Switches : one no. or, two nos.
Type of Switches : Micro Switches - SPDT
Sensing Element : Diaphragm (PTFE or Neoprene or SS 316) or Piston
Enclosure : Dustproof IP 40, Weatherproof to IP 54 / IP 65 / IP 66, Flameproof
Contenedores Docker en SUSE
Presentación incluida en el evento OpenExpo 2016. 2 Junio 2016
Se presenta cómo funcionan los contenedores Docker en un entorno empresarial desde el punto de vista de desarrollo y de operaciones. Además se describe porqué es necesario un orquestrador en un centro de datos basado en contenedores. En la ponencia se mostrarán las herramientas del ecosistema de SUSE para los entornos basados en contenedores Docker y cómo encajan todas las piezas respecto al centro de datos definido por software.
Javier Martínez Nohales, Responsable Técnico en SUSE Spain
Various Manifolds are necessary during the maintenance of the instruments like Pressure Gauges - Switches & Transmitters, DP Gauges - Switches and Transmitters.
We have various types of Three Valve Manifolds in SS 316, CS, Monel 400 and Hestelloy C material.
Type of Three Valve Manifolds are R type, T type and H type.
Mounting Bracket also can be provided for the manifolds.
This talk is a look into some of the surprising performance cases in Java -- with the goal of illustrating a few simple truths about the nature of compilers.
1 PROBABILITY DISTRIBUTIONS R. BEHBOUDI Triangu.docxaulasnilda
1
PROBABILITY DISTRIBUTIONS
R. BEHBOUDI
Triangular Probability Distribution
The triangular probability distribution (also called: “a lack of knowledge distribution”) is a
simplistic continuous model that is mainly used in situations when there is only limited sample
data and information about a population. It is based on the knowledge of a minimum (a lower
value), a maximum (an upper value), and a mode (peak) between those two values. For this
reason, this distribution is very popular in simulation processes related to business decision
models, project management models, financial models, and for modeling noises in digital audio
and video data.
The probability density function (𝒑𝒅𝒇) of the triangular random variable 𝑿 is given by:
𝒇(𝒙) = {
𝟐
(𝒃−𝒂)(𝒄−𝒂)
(𝒙 − 𝒂) 𝒊𝒇 𝒙 ≤ 𝒄
𝟐
(𝒃−𝒂)(𝒃−𝒄)
(𝒃 − 𝒙) 𝒊𝒇 𝒙 > 𝒄
(1)
The following are some of the important numerical characteristics of the triangular distribution:
2
PROBABILITY DISTRIBUTIONS
R. BEHBOUDI
𝒎𝒆𝒂𝒏 = 𝑬(𝒙) = 𝝁 =
𝒂+𝒃+𝒄
𝟑
(2)
𝒎𝒆𝒅𝒊𝒂𝒏 = 𝒎 =
{
𝒂 + √𝟎.𝟓 (𝒃 − 𝒂)(𝒄 − 𝒂) 𝒊𝒇 𝒄 <
𝒂+𝒃
𝟐
𝒄 𝒊𝒇 𝒄 =
𝒂+𝒃
𝟐
𝒃 − √𝟎.𝟓 (𝒃 − 𝒂)(𝒃 − 𝒄) 𝒊𝒇 𝒄 ≥
𝒂+𝒃
𝟐
(3)
𝒗𝒂𝒓𝒊𝒂𝒏𝒄𝒆 = 𝝈𝟐 =
𝒂𝟐+𝒃𝟐+𝒄𝟐−𝒂𝒃−𝒂𝒄−𝒃𝒄
𝟏𝟖
(4)
The 𝒄𝒅𝒇 (Cumulative density function) 𝑷(𝑿 ≤ 𝒙) of the triangular random variable is:
𝑭(𝒙) =
{
𝟏
(𝒃−𝒂)(𝒄−𝒂)
(𝒙 − 𝒂)𝟐 𝒊𝒇 𝒙 < 𝒄
𝒄−𝒂
𝒃−𝒂
𝒊𝒇 𝒙 = 𝒄
𝟏 −
𝟏
(𝒃−𝒂)(𝒃−𝒄)
(𝒃 − 𝒙)𝟐 𝒊𝒇 𝒙 > 𝒄
(5)
For example, the following is a display of the cumulative density function of a triangular random variable
with a minimum value of 2, a maximum of 8, and with a peak at 4.
3
PROBABILITY DISTRIBUTIONS
R. BEHBOUDI
Random Number Generation of Triangular Random Variables:
The CDF expression given by formula (5) can be used to generate random values according to a specific
triangular distribution. In this method, first a standard uniform random value 𝒓 is created. This value is
then used as a cumulative probability and replaces 𝑭(𝒙) in formula (5). The formula is then solved for
the random variable 𝒙. The following rule describes this random number generation:
𝒙 = {
𝒂 + √𝒓 (𝒃 − 𝒂)(𝒄 − 𝒂) 𝒊𝒇 𝒓 ≤
𝒄−𝒂
𝒃−𝒂
𝒃 − √(𝟏 − 𝒓)(𝒃 − 𝒂)(𝒃 − 𝒄) 𝒊𝒇 𝒓 >
𝒄−𝒂
𝒃−𝒂
(6)
Example:
In this example, we will simulate ten million triangular random values in R. We will then compare the
numerical characteristics of this randomly generated set with the expected values.
1. Specify the specification of the triangular distribution:
> a<-2
>
1 PROBABILITY DISTRIBUTIONS R. BEHBOUDI Triangu.docxjeremylockett77
1
PROBABILITY DISTRIBUTIONS
R. BEHBOUDI
Triangular Probability Distribution
The triangular probability distribution (also called: “a lack of knowledge distribution”) is a
simplistic continuous model that is mainly used in situations when there is only limited sample
data and information about a population. It is based on the knowledge of a minimum (a lower
value), a maximum (an upper value), and a mode (peak) between those two values. For this
reason, this distribution is very popular in simulation processes related to business decision
models, project management models, financial models, and for modeling noises in digital audio
and video data.
The probability density function (𝒑𝒅𝒇) of the triangular random variable 𝑿 is given by:
𝒇(𝒙) = {
𝟐
(𝒃−𝒂)(𝒄−𝒂)
(𝒙 − 𝒂) 𝒊𝒇 𝒙 ≤ 𝒄
𝟐
(𝒃−𝒂)(𝒃−𝒄)
(𝒃 − 𝒙) 𝒊𝒇 𝒙 > 𝒄
(1)
The following are some of the important numerical characteristics of the triangular distribution:
2
PROBABILITY DISTRIBUTIONS
R. BEHBOUDI
𝒎𝒆𝒂𝒏 = 𝑬(𝒙) = 𝝁 =
𝒂+𝒃+𝒄
𝟑
(2)
𝒎𝒆𝒅𝒊𝒂𝒏 = 𝒎 =
{
𝒂 + √𝟎.𝟓 (𝒃 − 𝒂)(𝒄 − 𝒂) 𝒊𝒇 𝒄 <
𝒂+𝒃
𝟐
𝒄 𝒊𝒇 𝒄 =
𝒂+𝒃
𝟐
𝒃 − √𝟎.𝟓 (𝒃 − 𝒂)(𝒃 − 𝒄) 𝒊𝒇 𝒄 ≥
𝒂+𝒃
𝟐
(3)
𝒗𝒂𝒓𝒊𝒂𝒏𝒄𝒆 = 𝝈𝟐 =
𝒂𝟐+𝒃𝟐+𝒄𝟐−𝒂𝒃−𝒂𝒄−𝒃𝒄
𝟏𝟖
(4)
The 𝒄𝒅𝒇 (Cumulative density function) 𝑷(𝑿 ≤ 𝒙) of the triangular random variable is:
𝑭(𝒙) =
{
𝟏
(𝒃−𝒂)(𝒄−𝒂)
(𝒙 − 𝒂)𝟐 𝒊𝒇 𝒙 < 𝒄
𝒄−𝒂
𝒃−𝒂
𝒊𝒇 𝒙 = 𝒄
𝟏 −
𝟏
(𝒃−𝒂)(𝒃−𝒄)
(𝒃 − 𝒙)𝟐 𝒊𝒇 𝒙 > 𝒄
(5)
For example, the following is a display of the cumulative density function of a triangular random variable
with a minimum value of 2, a maximum of 8, and with a peak at 4.
3
PROBABILITY DISTRIBUTIONS
R. BEHBOUDI
Random Number Generation of Triangular Random Variables:
The CDF expression given by formula (5) can be used to generate random values according to a specific
triangular distribution. In this method, first a standard uniform random value 𝒓 is created. This value is
then used as a cumulative probability and replaces 𝑭(𝒙) in formula (5). The formula is then solved for
the random variable 𝒙. The following rule describes this random number generation:
𝒙 = {
𝒂 + √𝒓 (𝒃 − 𝒂)(𝒄 − 𝒂) 𝒊𝒇 𝒓 ≤
𝒄−𝒂
𝒃−𝒂
𝒃 − √(𝟏 − 𝒓)(𝒃 − 𝒂)(𝒃 − 𝒄) 𝒊𝒇 𝒓 >
𝒄−𝒂
𝒃−𝒂
(6)
Example:
In this example, we will simulate ten million triangular random values in R. We will then compare the
numerical characteristics of this randomly generated set with the expected values.
1. Specify the specification of the triangular distribution:
> a<-2
>
This tutor shows the train and test set split with histogram and a probability density function in scikit-learn on synthetic datasets. The dataset is very simple as a reference of understanding.
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6. Lattice Plots
> xyplot(jitter(rings) ~ shell.wt | sex, abalone, grid=T, pch=".",
subset=volume < 0.2,
panel=function(x, y, ...) {
panel.lmline(x, y, ...)
panel.xyplot(x, y, ...)
},
ylab="rings")
ggplot2 is a newer package that can be used to create similar plots.
9. Simple Model
> fit.1 <- lm(rings ~ sex + shell.wt, abalone)
> summary(fit.1)
Call:
lm(formula = rings ~ sex + shell.wt, data = abalone)
Residuals:
Min 1Q Median 3Q Max
-5.750 -1.592 -0.535 0.886 15.736
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.2423 0.0799 78.08 <2e-16 ***
sex 0.9142 0.0984 9.29 <2e-16 ***
shell.wt 12.8581 0.3300 38.96 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.5 on 4174 degrees of freedom
10. Centering with z-scores
Subtract the mean from each input and
divide by 1 or 2 standard deviations
Dummy/Proxy variables may be centered as
well
11. Center Values
> abalone.adj <- abalone[, c(outcome, predictors)]
for (i in predictors) {
abalone.adj[[i]] <-
(abalone.adj[[i]] - mean(abalone.adj[[i]])) / (2 * sd(abalone.adj[[i]]))
}
Also look into the ‘scale’ function