class: center, middle, inverse, title-slide # Statistical Distributions and Simulation --- --- class: inverse, center, middle # Review: Statistical distributions --- ## Statistical distributions Recall from your previous statistics classes... - A **random variable** is a value we don't know until we take a **sample**. + Coin flip: Could be heads (0) or tails (1) + Person's height: Could be anything from 0 feet to 20 feet. + Annual income of a US worker: $0 to $1.6 billion -- - The **distribution** of the random variable tells us the **possible values** and **how likely they are** + Coin flip has 50% chance of heads, 50% chance of tails + Likelihood of heights follow a bell curve shape around 5 foot 7. + Most American workers make under $100,000 ---  ---  --- ## Statistical distributions Some distributions have names. For this class, you need to know a few of them. -- + Uniform + Normal + t-distribution + Binomial + Chi-Square --- # Uniform When you know the **range**, but not much else. All values in the range are **equally likely** to occur.  --- # Normal When you expect values to fall **near the center**. Frequency of values follows a **bell curve** shape.  --- # t-Distribution Slightly **wider bell curve** shape. Basically, the same context as the **Normal distribution**, but used more often in real data. (When the *standard deviation* is unknown.)  --- # Chi-Square Distribution Somewhat **skewed**, and only allows values **above zero**. Used in testing **count data**.  --- # Binomial Appears when you have **two possible outcomes**, and you are **counting** how many times each outcome occurred. (Common example: voting data)  --- # Statistical Distributions Things you might want to use a Statistical distribution for: -- * Find the **probability** of an **event** + If I flip 10 coins, what are the chances I get all heads? -- * Estimate a **percentage** of a **population**: + About what percent of people are above 6 feet tall? -- * Quantify the **evidence** in your **data**: + In my survey of 100 people, 67 said they were voting for Measure A. How confident am I that Measure A will pass? --- class: inverse, center, middle # Statistical distributions in R ## Simulate observations ## Make the computer do the math --- ## Distributions in R - `r` stands for for **random sampling**: generate a random value from the distribution -- - We typically use this to **simulate** date; that is, to generate pretend observations to see what happens. --- ## Random Sampling ```r runif(3) ``` ``` ## [1] 0.3929180 0.2072864 0.9098014 ``` ```r runif(3, min = 10, max = 20) ``` ``` ## [1] 16.34877 19.64085 15.12729 ``` ```r rnorm(3) ``` ``` ## [1] -0.1977894 -0.7088441 0.3207394 ``` ```r rnorm(3, mean = -100, sd = 50) ``` ``` ## [1] -37.98323 -113.03041 -159.13502 ``` --- ## Random Sampling ```r rt(3, df = 11) ``` ``` ## [1] 0.9224215 0.8096584 0.1684738 ``` ```r rbinom(3, size = 10, prob = 0.7) ``` ``` ## [1] 5 7 7 ``` ```r rchisq(3, df = 11) ``` ``` ## [1] 10.600462 9.457107 12.776857 ``` --- ## Distributions in R - `p` for **probability**: i.e. compute the chances of an observation less than `x` -- - We usually use this for calculating **p-values** --- # Probabilities ```r pnorm(1.5) ``` ``` ## [1] 0.9331928 ``` ```r pnorm(70, mean = 67, sd = 3) ``` ``` ## [1] 0.8413447 ``` ```r 1 - pnorm(70, mean = 67, sd = 3) ``` ``` ## [1] 0.1586553 ``` --- ## Try it! - What is the probability that a Chi-Square distribution with 35 degrees of freedom gives a value of 25 or less? - What is the probability that 10 coin tosses give 6 or more heads? --- ## Try it! - What is the probability that a Chi-Square distribution with 35 degrees of freedom gives a value of 25 or less? ```r pchisq(25, df = 35) ``` ``` ## [1] 0.1054672 ``` - What is the probability that 10 coin tosses give 6 or more heads? ```r 1 - pbinom(5, size = 10, prob = 0.5) ``` ``` ## [1] 0.3769531 ``` --- ## Distributions in R - `q` stands for **quantile**: given a probability p, compute the x-value such that P(X < x) = p. - the `q` functions are the "backwards" version of the `p` functions -- - We don't use this very often --- # Quantiles ```r qnorm(.95) ``` ``` ## [1] 1.644854 ``` ```r qnorm(.95, mean = 67, sd = 3) ``` ``` ## [1] 71.93456 ``` --- ## Try it! - If I randomly sample a number between 16 and 22, I can be 80% sure my number will be less than what? - What is the critical value for a 1-sided t-test at the 0.05 significance level, with a sample size of 50? --- ## Try it! - If I uniformly sample a number between 16 and 22, I can be 80% sure my number will be less than what? ```r qunif(.80, min = 16, max = 22) ``` ``` ## [1] 20.8 ``` - What is the critical value for a 1-sided t-test at the 0.05 significance level, with a sample size of 50? ```r qt(.05, df = 50) ``` ``` ## [1] -1.675905 ``` --- ## Distributions in R `d` stands for **density**: compute the height of distribution curve -- * **Discrete** distributions: probability of getting that **exact value** * **Continuous** distributions: usually meaningless --- ## Try it What is the probability of getting exactly 12 heads in 20 coin tosses, for a coin with a 70% chance of tails? --- ## Try it What is the probability of getting exactly 12 heads in 20 coin tosses, for a coin with a 70% chance of tails? ```r dbinom(12, size = 20, prob = .3) ``` ``` ## [1] 0.003859282 ``` --- class: inverse, middle, center # Making fake data --- # Simulating a dataset fake_data <- tibble(
 names = charlatan::ch_name(100),
 height = rnorm(100, mean = 67, sd = 3),
 age = runif(100, min = 15, max = 75),
 measure = rbinom(100, size = 1, prob = 0.6)
) %>%
 mutate(
 supports_measure_A = ifelse(measure == 1, "yes", "no")
 )

fake_data ``` ## # A tibble: 100 x 5 ## names height age measure supports_measure_A ## <chr> <dbl> <dbl> <int> <chr> ## 1 Dexter Erdman 70.3 16.5 0 no ## 2 Jaxson Grant 65.4 40.5 1 yes ## 3 Jaliyah Abshire 62.8 60.9 1 yes ## 4 Landyn Hand-Daniel 63.3 63.9 1 yes ## 5 Shepherd Little 71.4 64.3 0 no ## 6 Hilmer Shanahan 64.5 46.5 1 yes ## 7 Dr. Alivia Wehner 70.0 48.9 1 yes ## 8 Chaim Corkery 62.1 23.2 0 no ## 9 Elnora Beahan-Rice 71.6 44.8 1 yes ## 10 Ms. Devyn Brekke 67.1 63.9 1 yes ## # … with 90 more rows ``` --- ```r fake_data %>% ggplot(aes(x = supports_measure_A, y = age)) + geom_boxplot() ```  --- # Plotting distributions  --- # Plotting Distributions fake_data %>%
 ggplot(aes(x = height)) +
 geom_histogram(bins = 10, fill = "grey", col = "white") +
 stat_function(fun = dnorm,
 col = "cornflowerblue", lwd = 2)  --- # Plotting Distributions fake_data %>%
 ggplot(aes(x = height)) +
 geom_histogram(bins = 10, fill = "grey", col = "white") +
 stat_function(fun = ~dnorm(.x, mean = 67, sd = 3),
 col = "cornflowerblue", lwd = 2)  --- # Plotting Distributions fake_data %>%
 ggplot(aes(x = height)) +
 geom_histogram(aes(y = ..density..),
 bins = 10, fill = "grey", col = "white") +
 stat_function(fun = ~dnorm(.x, mean = 67, sd = 3),
 col = "cornflowerblue", lwd = 2)  --- # Plotting Distributions ```r fake_data %>% ggplot(aes(x = height)) + geom_histogram(aes(y = ..density..), bins = 10, fill = "grey", col = "white") + stat_function(fun = ~dnorm(.x, mean = 67, sd = 3), col = "cornflowerblue", lwd = 2) + stat_function(fun = ~dnorm(.x, mean = 67, sd = 2), col = "deeppink", lwd = 2) ```  --- # Empirical Distribution  --- # Theoretical Distributions


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Mobile Enabled
Smart Governance will adopt a mobile first approach, with robust mobile applications enabling citizens to easily access government services through their phones. Government officials and field workers are also empowered with mobile applications that enable them to deliver 24×7 governance with ease.
Real time Dashboards
to drive decisions
Real time dashboards provide governments with actionable insights on demand. Administrators and department heads are empowered with verified data that enables them to manage their work and budgets better. Field level employee reports enable effective performance management and ensures efficient usage of government resources.
Decision
Support Systems
Decision-Support Systems leverage the power of data and analytics to empower city leaders to solve fast-changing challenges, prioritize action and manage teams of execution. They can be used to improve mobility, disaster response, access to basic services, urban planning and other issues.
Ready to use
Smart governance platform will be configurable to local needs and can be quickly adopted across many cities at scale.
Open Standards and Open APIs
Smart Governance will be governed by the guiding principle of Open Standards and Open APIs which will enable inter-operability between multiple systems.
Smart Governance empowers
multiple stakeholders
Citizens
- Access governments easily using your mobile phone
- Your requests automatically reach the appropriate person
- Stay up to date with inbuilt notifications
State Governments
- Enable all cities to become smart cities
- Drive best practice sharing across cities
- Improved urban budgeting and forecasting
Local Governments
- Realize the benefits of data driven governance
- Reduced paperwork for employees
- Efficient and prioritized use of resources
System Integrators
- Smart Governance reduces total project costs
- Ready to use & configurable governance platform
- Can be integrated with multiple applications


"We have developed an amazing machine learning model, for predicting if our customers will be able to pay back their loan" – Bank* Representative
*: You properly shouldn't trust anything said.
"One small issue, our customer-facing advisors don't understand or trust the model and sometimes goes against it." – Bank Spokesperson
"Oh, we just did some visual discretizations and told them to trust it." – Bank Representative
Needless to say, they missed the point.
§2: "... the controller shall, provide the data subject with the information necessary to ensure fair and transparent processing".
§2.f: such as "the existence of automated decision-making, and at least in those cases, meaningful information about the logic involved ...".
You can't say something about memorization from a single hidden unit.
But then how?
"Intuition is developed from a
feedback loop."
"Interactive visualization proivides a feedback loop."
"Confirmation bias is when we interpret or favor information that confirms one's personal beliefs."
Transform your intuition into a quantifiable hypothesis!
tl;dr: most methods are not better than a random baseline.
Why where so many papers published, all claiming improved methods?
Confirmation bias!
Use these tools to explain models,
but be hyper-aware of confirmation bias.
Always transform your intuition into
a quantifiable hypothesis.


Artists With Songs
Name
Contact
Nº of Songs
Origin
Registered
Artists Without Songs
Name
Contact
Origin
Registered


Black to play 42
Money play
pips Black=112 White=135
(CorrectAnswer= 0/0, ErrorScore= 0)
At some point during a blitz, the "local ammunition" starts to run low and we face a tricky decision. Do we keep rushing the nearest available ammunition forwards to try to continue attacking, at the risk of overextending ourselves? Or do we start consolidating by moving our rear checkers into a better position so that we retain a strong advantage even if the blitz fails?
Here it is tempting to keep trying to close out the checker by playing 13/9, but according to the rollout, anything but 20/16 18/16 is a whopper.
Here it is tempting to keep trying to close out the checker by playing 13/9, but according to the rollout, anything but 20/16 18/16 is a whopper.
1. 20/16 18/16 eq: +0.860 77.88% (G:42.81% B:0.28%) 22.12% (G:3.94% B:0.25%) 2. 20/16 7/5 eq: +0.761 (-0.099) 74.91% (G:41.13% B:0.31%) 25.09% (G:5.25% B:0.35%) 3. 20/18 20/16 eq: +0.736 (-0.124) 74.34% (G:39.89% B:0.20%) 25.66% (G:4.92% B:0.31%) 4. 20/16 13/11 eq: +0.725 (-0.135) 73.94% (G:40.96% B:0.27%) 26.06% (G:5.87% B:0.54%) 5. 18/16 13/9 eq: +0.708 (-0.152) 72.92% (G:40.21% B:0.21%) 27.08% (G:3.95% B:0.26%) 6. 13/9 7/5 eq: +0.610 (-0.250) 70.55% (G:36.24% B:0.24%) 29.45% (G:4.27% B:0.21%) 7. 13/7 eq: +0.588 (-0.272) 70.44% (G:34.68% B:0.22%) 29.56% (G:4.74% B:0.27%) 8. 20/18 13/9 eq: +0.581 (-0.279) 70.00% (G:35.60% B:0.18%) 30.00% (G:5.29% B:0.31%)


@()(implicit request: play.api.mvc.RequestHeader) @main("Local Storage") {
This page shows a sample of use of Local Storage via HTML 5.
The code uses Modernizr to detect if Local Storage is available in the browser.
You can find more information at http://diveintohtml5.info/storage.html.
Please check the source code to see how this sample works.
Write text in the textarea. You can force a save or a removal of data via the corresponding buttons. When you refresh the page it will still be displayed at the bottom.


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All those numbers which lie between two specific numbers.
All those points which lie between two specific points along a line.


Mathematical Operations
[...]
The notation “_x_ modulo _y_” (_y_ must be finite and non-zero) computes a value _k_ of the same sign as _y_ (or zero) such that abs(_k_) < abs(_y_) and _x_ - _k_ = _q_ × _y_ for some integer _q_.
The mathematical function remainder(_x_, _y_) produces the mathematical value whose sign is the sign of _x_ and whose magnitude is abs(_x_) modulo _y_.
[...]
Mathematical functions min, max, abs, remainder, and floor are not defined for Numbers and BigInts, and any usage of those methods that have non-mathematical value arguments would be an editorial error in this specification.
[...]


Library: Foundation
Package: Logging
Header: Poco/EventLogChannel.h
Description
This Windows-only channel works with the Windows NT Event Log service.
To work properly, the EventLogChannel class requires that either the PocoFoundation.dll or the PocoMsg.dll Dynamic Link Library containing the message definition resources can be found in $PATH.
Inheritance
Direct Base Classes: Channel
All Base Classes: Channel, Configurable, RefCountedObject
Member Summary
Member Functions: close, findLibrary, getCategory, getProperty, getType, log, open, setProperty, setUpRegistry
Inherited Functions: close, duplicate, getProperty, log, open, referenceCount, release, setProperty
Constructors
EventLogChannel
Creates the EventLogChannel. The name of the current application (or more correctly, the name of its executable) is taken as event source name.
EventLogChannel
EventLogChannel(
const std::string & name
);
Creates the EventLogChannel with the given event source name.
EventLogChannel
EventLogChannel(
const std::string & name,
const std::string & host
);
Creates an EventLogChannel with the given event source name that routes messages to the given host.
Destructor
~EventLogChannel
~EventLogChannel();
Member Functions
close
void close();
Closes the EventLogChannel.
See also: Poco::Channel::close()
getProperty
std::string getProperty(
const std::string & name
) const;
Returns the value of the given property.
See also: Poco::Channel::getProperty()
log
void log(
const Message & msg
);
Logs the given message to the Windows Event Log.
The message type and priority are mapped to appropriate values for Event Log type and category.
See also: Poco::Channel::log()
open
void open();
Opens the EventLogChannel. If necessary, the required registry entries to register a message resource DLL are made.
See also: Poco::Channel::open()
setProperty
void setProperty(
const std::string & name,
const std::string & value
);
Sets or changes a configuration property.
The following properties are supported:
- name: The name of the event source.
- loghost: The name of the host where the Event Log service is running. The default is "localhost".
- host: same as host.
- logfile: The name of the log file. The default is "Application".
See also: Poco::Channel::setProperty()
findLibrary
static std::string findLibrary(
const char * name
);
getCategory
static int getCategory(
const Message & msg
);
getType
static int getType(
const Message & msg
);
setUpRegistry
void setUpRegistry() const;
Variables
PROP_HOST
static const std::string PROP_HOST;
PROP_LOGFILE
static const std::string PROP_LOGFILE;
PROP_LOGHOST
static const std::string PROP_LOGHOST;
PROP_NAME
static const std::string PROP_NAME;


DVST3 is dedicated to the highest quality of calculating costs escaping abuse delivered with empathy, care and understanding
Project Lead
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React Two Developer
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Work In Progress Documentation
Implemented of the timeline platform
The implementation of the timeline platform could be divided into the collection of ‘Lines‘ and ‘TextLabels’. ‘BufferGeometry’ and ‘LineMaterial’ are used to generate lines as documented in three.js best practice, meanwhile, wrapping it around the react component and using props to make it reusable. The same implementation approach applies for ‘TextLabels’ as well to make it versatile. Then, these components were mapped based on the PoI date and pre-set platform settings.
Co-designed new event systems by reducing states
After some implementation and discussion between two of me and Jerry, we figured: 1.Making every object in 3D is easier than having some UI elements overlay 2. Just scrolling through the timeline could be felt weak in terms of interaction 3. We could make better use of our environment 4. Implementing camera movement and positioning some additional elements on the scene is less work compared to their effect So, we discussed about what is feasible and what would be more interactive, and we came up some changes that alter the interaction and design. The interaction scenario is and rough camera angle is shown in the drawing here.


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GREAT FOOD. GREAT VIEW. GREAT SAVINGS.
Skip the beaches and see the planes from a boat this summer. Get
ready for some high-flying fun featuring the Blue Angels during the
Air and Water Show.(Weather permitting)
Use promo code AUGUST and save 20% on select Air & Water Show
Cruises:*
Afternoon Air & Water Preview Lunch
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– 8/15**
Afternoon Air & Water Show Lunch
(http://www.mysticbluecruises.com/chicago/search?timeOfDay=Any&metro=Chicago&startDate=08%2F17%2F2014&paxSize=1-19&brands=Mystic)
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show events on this day.
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Devices in China Case Study
Devices by Gender
Devices by Brand
Devices by Age
Devices by Date
Total Number of Devices
Average of Devices
Built by
Datafresh


August ??, 1999
by Terry L. Howe
Well, it was a night run, so no pictures of us going to the lake at
night, but Sean and I headed back up to the lake in the morning...
Eric Resko broke down before the first obstacle, but his friend
made it up in his TJ (not to the lake). Keith broke in his TJ
on this trail as well, it had less than a 1000 miles on it. He
made it up to the water fall, but when he saw it, he had second
thoughts.
For me, the highlight of the trip was catching big air going up the
waterfall at night. It was raining out and all I could see was the
rain and clouds in my head lights.
|
|
|
Our camping area near tree line.
|
|
|
|
Sean climbing the water fall.
|
|
|
|
Sean catches the big air line coming down the water fall.
|
|
|
|
Sean goes down one of the last obstacles, no big air this year.
|


{% include "header.html" %}
Session Name
Session Details
Session Trainee
Trainee Description
Session File
{% include "footer.html" %}


STI surveillance Health sexually transmitted HIV {border-style global px;} #ffffff;} World Available [cited at: infections STIs Global collab: health national June data prevalence Organization infection countries populations syphilis AIDS Department including men px; support {font-family HIV, font-size incidence key prevention women Control Disease double; information programme reporting resistance systems Centers Int advocacy antimicrobial border-color border-width burden case cases congenital control estimates investment limited monitoring risk services solid; strengthening surveys transmission {font-style {font-weight ;():–.


HTTPMethod
public enum HTTPMethod : String
HTTP method definitions.
-
HTTP method options
Declaration
Swift
case options = "OPTIONS"
-
HTTP method get
Declaration
Swift
case get = "GET"
-
HTTP method head
Declaration
Swift
case head = "HEAD"
-
HTTP method post
Declaration
Swift
case post = "POST"
-
HTTP method put
Declaration
Swift
case put = "PUT"
-
HTTP method patch
Declaration
Swift
case patch = "PATCH"
-
HTTP method delete
Declaration
Swift
case delete = "DELETE"
-
HTTP method trace
Declaration
Swift
case trace = "TRACE"
-
HTTP method connect
Declaration
Swift
case connect = "CONNECT"


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Warehouse Keeper Holvas:
Hmmm, what would Barbado like to have?
Well, he is a musician and has expressed an interest in learning to play a new instrument... Yes, I believe he mentioned that he wanted to learn to play the flute!


object Combinatorics extends Error
The Combinatorics
object provides several common combinatorics functions,
such as factorial permutations, combinations, gamma and beta functions.
- Alphabetic
- By Inheritance
- Combinatorics
- Error
- AnyRef
- Any
- Hide All
- Show All
- Public
- All
Value Members
-
final
def
!=(arg0: Any): Boolean
- Definition Classes
- AnyRef → Any
-
final
def
##(): Int
- Definition Classes
- AnyRef → Any
-
final
def
==(arg0: Any): Boolean
- Definition Classes
- AnyRef → Any
-
final
def
asInstanceOf[T0]: T0
- Definition Classes
- Any
-
def
betaF(a: Double, b: Double): Double
Compute the beta function 'B(a, b)' for the following two cases: (1) when 'a' or 'b' are integers and (2) when 'a' or 'b' are integers + 1/2.
Compute the beta function 'B(a, b)' for the following two cases: (1) when 'a' or 'b' are integers and (2) when 'a' or 'b' are integers + 1/2.
- a
the first parameter, a real number satisfying (1) or (2)
- b
the second parameter, a real number satisfying (1) or (2)
- See also
http://mathworld.wolfram.com/BetaFunction.html
-
def
choose(n: Int, k: Int, l: Int): Long
Compute trinomial coefficients: 'n' choose '(k, l'), combinations of 'n' things, '(k, l)' at a time, using Pascal's Tetrahedron.
Compute trinomial coefficients: 'n' choose '(k, l'), combinations of 'n' things, '(k, l)' at a time, using Pascal's Tetrahedron. Ex: Given 'n' balls, counts ways in which 'k' are chosen for group 1 and 'l' are chosen for group 2.
- n
the total number of items
- k
the of items to choose
- l
the of items to choose (requires 0 <= k + l <= n)
- See also
http://people.sju.edu/~pklingsb/bintrin.pdf
-
def
choose(n: Int, k: Int): Long
Compute binomial coefficients: 'n' choose 'k', combinations of 'n' things, 'k' at a time, using Pascal's Triangle.
Compute binomial coefficients: 'n' choose 'k', combinations of 'n' things, 'k' at a time, using Pascal's Triangle.
- n
the total number of items
- k
the of items to choose (requires k <= n)
- See also
http://www.mathsisfun.com/pascals-triangle.html
-
def
chose(n: Int, k: Int): Long
Compute 'n' choose 'k' (combinations of 'n' things, 'k' at a time).
Compute 'n' choose 'k' (combinations of 'n' things, 'k' at a time). A more efficient implementation is given below.
- n
the total number of items
- k
the of items to choose (requires k <= n)
-
def
clone(): AnyRef
- Attributes
- protected[java.lang]
- Definition Classes
- AnyRef
- Annotations
- @throws( ... )
-
final
def
eq(arg0: AnyRef): Boolean
- Definition Classes
- AnyRef
-
def
equals(arg0: Any): Boolean
- Definition Classes
- AnyRef → Any
-
def
fac(k: Int): Double
Compute 'k' factorial 'k!' using three techniques (requires 'k <= 170').
Compute 'k' factorial 'k!' using three techniques (requires 'k <= 170').
- k
the nonnegative integer-valued argument to the factorial function
-
def
finalize(): Unit
- Attributes
- protected[java.lang]
- Definition Classes
- AnyRef
- Annotations
- @throws( classOf[java.lang.Throwable] )
-
final
def
flaw(method: String, message: String): Unit
Show the flaw by printing the error message.
Show the flaw by printing the error message.
- method
the method where the error occurred
- message
the error message
- Definition Classes
- Error
-
def
gammaF(a: Double): Double
Compute the gamma function 'gamma (a)' using the Lanczos Approximation.
Compute the gamma function 'gamma (a)' using the Lanczos Approximation.
- a
the parameter, a real number
- See also
http://en.wikipedia.org/wiki/Lanczos_approximation
-
final
def
getClass(): Class[_]
- Definition Classes
- AnyRef → Any
-
def
hashCode(): Int
- Definition Classes
- AnyRef → Any
-
def
hyp2f1(a: Double, b: Double, c: Double, z: Double): Double
Compute the Gauss Hypergeometric function '2F1(a, b, c; z)' using a power series expansion.
Compute the Gauss Hypergeometric function '2F1(a, b, c; z)' using a power series expansion.
- a
the first parameter, a real/complex number
- b
the second parameter, a real/complex number
- c
the third parameter, a real/complex number, may not be a negative integer
- z
the variable, a real/complex number s.t. |z| < 1
- See also
people.maths.ox.ac.uk/porterm/research/pearson_final.pdf
en.wikipedia.org/wiki/Hypergeometric_function For faster or more robust algorithms,
http://dx.doi.org/10.1016/j.cpc.2007.11.007
-
def
iBetaF(z: Double, a: Double, b: Double): Double
Compute the incomplete beta function 'B(z; a, b)', a generalization of the beta function 'z = 1'.
Compute the incomplete beta function 'B(z; a, b)', a generalization of the beta function 'z = 1'.
- z
the variable, a real/complex number s.t. 0 <= |z| <= 1
- a
the first parameter, a real/complex number > 0
- b
the second parameter, a real/complex number > 0
- See also
http://mathworld.wolfram.com/IncompleteBetaFunction.html
-
final
def
isInstanceOf[T0]: Boolean
- Definition Classes
- Any
-
val
lfac: Array[Long]
Table of all factorial numbers that can be represented as a long (64-bit) integer
-
def
logfac(k: Int): Double
Compute the natural log factorial 'ln (k!)' so 'k! = exp (logfac (k))'.
Compute the natural log factorial 'ln (k!)' so 'k! = exp (logfac (k))'. The formula is a log transformation of Ramanujan's Factorial Approximation.
- k
the value to take the log factorial of
-
def
mfac(k: Int): Long
For small 'k', compute 'k' factorial by iterative multiplication.
For small 'k', compute 'k' factorial by iterative multiplication.
k! = k * (k-1) * ... * 2 * 1- k
the nonnegative integer-valued argument to the factorial function
-
def
mortici(k: Int): Double
Compute 'k!' using Mortici's Factorial Approximation (more accurate than Stirling's 2nd Order Factorial Approximation).
Compute 'k!' using Mortici's Factorial Approximation (more accurate than Stirling's 2nd Order Factorial Approximation).
- k
the nonnegative integer-valued argument to the factorial function
- See also
http://but.unitbv.ro/BU2010/Series%20III/BULETIN%20III%20PDF/Mathematics/Mortici.pdf
-
final
def
ne(arg0: AnyRef): Boolean
- Definition Classes
- AnyRef
-
final
def
notify(): Unit
- Definition Classes
- AnyRef
-
final
def
notifyAll(): Unit
- Definition Classes
- AnyRef
-
val
pascalTet: Array[Array[Array[Int]]]
Initial part of Pascal's Tetrahedron, precomputed to speed calculations (Trinomial Coefficients)
Initial part of Pascal's Tetrahedron, precomputed to speed calculations (Trinomial Coefficients)
- See also
https://sites.google.com/site/pascalloids/pascal-s-pyramid-3-var
-
val
pascalTri: Array[Array[Int]]
Initial part of Pascal's Triangle, precomputed to speed calculations (Binomial Coefficients)
-
def
perm(n: Int, k: Int): Long
Compute permutations of 'k' items selected from 'n' total items.
Compute permutations of 'k' items selected from 'n' total items.
- n
the total number of items
- k
the of items selected
-
def
rBetaC(z: Double, a: Double, b: Double): Double
Compute the complement of the regularized (incomplete) beta function '1.0 - I(z; a, b) = I(1.0 - z; b, a)'.
Compute the complement of the regularized (incomplete) beta function '1.0 - I(z; a, b) = I(1.0 - z; b, a)'.
- z
the variable, a real/complex number s.t. 0 <= |z| <= 1
- a
the first parameter, a real/complex number > 0
- b
the second parameter, a real/complex number > 0
-
def
rBetaF(z: Double, a: Double, b: Double): Double
Compute the regularized (incomplete) beta function 'I(z; a, b)'.
Compute the regularized (incomplete) beta function 'I(z; a, b)'.
- z
the variable, a real/complex number s.t. 0 <= |z| <= 1
- a
the first parameter, a real/complex number > 0
- b
the second parameter, a real/complex number > 0
- See also
http://mathworld.wolfram.com/RegularizedBetaFunction.html
-
def
ramanujan(k: Int): Double
Compute 'k!' using Ramanujan's Factorial Approximation (more accurate than Mortici's Factorial Approximation).
Compute 'k!' using Ramanujan's Factorial Approximation (more accurate than Mortici's Factorial Approximation).
- k
the nonnegative integer-valued argument to the factorial function
- See also
http://files.ele-math.com/articles/jmi-05-53.pdf
-
def
rfac(k: Int, x: Double = 1.0): Double
Compute the 'k'th degree rising factorial of 'x'.
Compute the 'k'th degree rising factorial of 'x'. When 'x = 1', this is the regular factorial function 'k!'. Also known as Pochhammer's symbol. Caveat: only works when 'k' is a nonnegative integer
- k
the number of factors in the product
- x
the base number to start the product
-
def
stirling(k: Int): Double
Compute 'k!' using Stirling's 2-nd Order Factorial Approximation.
Compute 'k!' using Stirling's 2-nd Order Factorial Approximation.
- k
the nonnegative integer-valued argument to the factorial function
- See also
http://en.wikipedia.org/wiki/Stirling%27s_approximation
-
final
def
synchronized[T0](arg0: ⇒ T0): T0
- Definition Classes
- AnyRef
-
def
toString(): String
- Definition Classes
- AnyRef → Any
-
final
def
wait(): Unit
- Definition Classes
- AnyRef
- Annotations
- @throws( ... )
-
final
def
wait(arg0: Long, arg1: Int): Unit
- Definition Classes
- AnyRef
- Annotations
- @throws( ... )
-
final
def
wait(arg0: Long): Unit
- Definition Classes
- AnyRef
- Annotations
- @throws( ... )


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The easiest method of combining code is copying and pasting the code from the operating systems clipboard. This method only works however when the code is written in the same language and version.
When copying and pasting code from a source where the code is written in a different language or version, changes need to be made to the code, to ensure that the code will work. These adjustments could consist of changing terminology or making grammatical changes, to ensure the code complies with the lexicon.
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In all cases, it is unethical to use code created by others without ensuring all intellectual property rights are upheld. This can be done through a a copyright notice, that acknowledges that the code has been created by someone else, and the new application complies with an other licensing agreements.


Arduino Autonomous Car
September 2014 - June 2015
Description
Programmed algorithms in Arduino Language (C/C++ based) to detect and maneuver around obstacles
Built and configured basic electric circuits with embedded Arduino UNO, servo motors, and ultrasonic sensors
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vignettes/regioncode-vignette.Rmd
regioncode-vignette.Rmd
Inspired by Vincent Arel-Bundock’s well-known countrycode
, we created regioncode
to achieve similar functions specifically for China studies. regioncode
enables seamlessly converting regions’ formal names, common-used names, and administrative division codes between each other.
regioncode
?The Chinese government gives unique geocodes for each county, city (prefecture), and provincial-level administrative unit. The so-called “administrative division codes” were consistently adjusted to matched national and regional plans of development. Geocode adjustments disturb researchers when they merge data with different versions of geocodes or region names. Especially, when researchers render statistical data on Chinese map, different geocodes between map data and statistical data may cause mess-up data output or visualization.
The package is developed to conquer such difficulties to match regional data across years more conveniently and correctly. In the current version, regioncode
enables seamlessly converting formal names, common-used names, and division codes of Chinese prefecture regions(named ‘地级市’ in Chinese) between each other and across thirty-four years from 1986 to 2019.
To install:
install.packages("regioncode")
.remotes::install_github("sammo3182/regioncode")
.The toy data was created based on a chunk from Yuhua Wang’s China’s Corruption Investigations Dataset
. The data includes information on almost 20,000 officials who were investigated during Xi Jinping’s anti-corruption campaign. We randomly drew an eighteen-line sample. The division codes in the original data were based on the 2019 version. We kept the variables of prefectural names and division codes. We added a column of to short names of the prefectures to further illustrate how the software works.
In regioncode
package, we named administrative division codes as code
, regions’ formal names as name
, and their commonly used short names as sname
. The current version enable to convert any pair of them mutually:
regioncoce
function accept numeric and character vectors as the input division codes and region names respectively. To achieve an accurate conversion, users have to specify the year of the source data correctly in the argument year_from
. Then they can set the year they want the output is. That’s it. See the following example to convert the 2019-version codes to the 1999 version:
# original geocodes. It's 2019 version
corruption$prefecture_id
# after conversion. It's 1999 version
regioncode(data_input = corruption$prefecture_id,
year_from = 2019,
year_to = 1999)
In some cases, the original data may only have division codes or region names, but users needs the other form or both formats of data. In such cases, regioncode
offers a function to convert division codes from any year to region names in any year. Users only need to alter the converting method, for example, to “code2name” in order to convert division codes to region names.
regioncode(data_input = corruption$prefecture_id,
year_from = 2019,
year_to = 1999,
method = "code2name")
Similarly, one can get the code from names, or in a less-often case get the names in a different year from the names from a given year. Users need to change the method
argument to “name2code” or “name2name” to achieve these conversions.
corruption$prefecture
regioncode(data_input = corruption$prefecture,
year_from = 2019,
year_to = 1999,
method = "name2code")
regioncode(data_input = corruption$prefecture,
year_from = 2019,
year_to = 1999,
method = "name2name")
regioncode
provides two advanced functions to achieve more complicated conversions. One of the occasions occurs when the data source includes only common-used short names of the cities instead of the full, official ones. regioncode
can still accomplish the conversion in this case when the users specify the incompleteName
to “from”. (regioncode
can also produce short names from inputs of full or short names and division code. See the Details of the help file for more information.)
# Full, official names
corruption$prefecture
# Incomplete names
corruption$prefecture_sname
# Converting
regioncode(data_input = corruption$prefecture_sname,
year_from = 2019,
year_to = 1999,
method = "name2code",
incompleteName = "from")
Another advanced application involves in the case when the municipalities directly under the central government (“zhixiashi” in Chinese Pinyin). This is common for national survey data. regioncode
can fit this case with no problem as long as the user sets the argument zhixiashi
as TRUE.
# In the sample data, the division code of municipalities were coded as NA. Filling the codes of municipalities with their provinces' codes.
code_zhixiashi <- c("110000", "120000", "310000", "400000")
corruption <- corruption %>%
mutate(prefecture_id = ifelse(province_id %in% code_zhixiashi, province_id, prefecture_id))
# Converting
regioncode(data_input = corruption$prefecture_id,
year_from = 2019,
year_to = 1999,
zhixiashi = TRUE)
regioncode
rovides a convenient way to convert Chinese administrative division codes, official names, and common-used short names between each other. This vignette offers a quick view of package features and a short tutorial for users.
The development of the package is ongoing. Future versions aim to add more administrative level choice, from province level to county level. Data are also enriching. Please contact us with any questions, bug reports, and comments.
Dr. Yue Hu
Department of Political Science,
Tsinghua University,
Email<EMAIL_ADDRESS>Website: https://sammo3182.github.io
Yufei Sun
Department of Political Science,
Tsinghua University,
Email<EMAIL_ADDRESS>Wenquan Wu
Department of Political Science,
Tsinghua University,
Email: wuwq20@mails.tsinghua.edu.cn


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