text:
You're reading an old version of this documentation. For up-to-date information, please have a look at v0.7.

# aspecd.report module¶

General facilities for generating reports.

To do scientific research in terms of reproducibility and traceability it’s highly necessary to report all the steps done on a given dataset and never separate the dataset from its metadata. Similarly, a recipe in context of recipe-driven data analysis stores a lot of relevant information on what tasks have been performed on a series of datasets (and even more so the recipe history). However, having a dataset (or recipe history) containing all these metadata is only useful if there are easy ways to retrieve and present the information stored. This is the task of reports. This module provides functionality to create reports based on templates provided either by the user or by the package as such.

## “Batteries included”: Templates contained in the package¶

The “batteries included” approach of Python itself is probably responsible to a great deal for the success of Python as a language. ASpecD, similarly, tries to provide you with a sensible set of tools you need for your routine data analysis in spectroscopy. Reports are no exception to that rule.

Hence, ASpecD comes bundled with a (growing) series of templates allowing you to create reports of datasets and alike. Thus, getting access to all information stored in a single dataset is as simple as calling a single reporter, and in context of recipe-driven data analysis, it is even simpler:

- kind: report
type: LaTeXReporter
properties:
template: dataset.tex
filename: report.tex
compile: true

This would create a report of a dataset that is then stored in the file report.tex, using the template dataset.tex bundled with the ASpecD package. As you even set compile to true, it would even compile the LaTeX report, including all figures generated during cooking the recipe and referenced from within the report. Hence, you end up in your current directory with both, a LaTeX file report.tex and a PDF file report.pdf.

But what if you don’t like the way the bundled templates look like? Don’t worry, we’ve got you covered: Simply provide a relative or absolute path to your own template, even with the same name. Hence, in the above example, if you place a file dataset.tex in the directory you serve the recipe from, it will be used instead of the bundled one. Developers of other packages can do the same and provide templates with the same name as those provided by the ASpecD package. Theirs will take precedence. For details, see below.

## Output formats¶

Template engines provide means of a “separation of concerns” (a term coined by Edsger W. Dijkstra and highly important not only in software development): The data source is entirely independent of the formatting of the report, and one and the same template engine can be used to create reports in a multitude of output formats.

Currently, the ASpecD framework supports two output formats (more may be added in the future): plain text (txt) and LaTeX. The respective reporter classes are:

The bundled templates used with the respective reporters are stored within the templates/report/ directory of the ASpecD package, and here within subdirectories (txt, latex) for each of the formats. This makes it easy to add additional formats and to shorten the template paths when using the reporters.

## Choosing the language of a report¶

While ASpecD is currently designed to predominantly use English as its native language, reports support different languages, provided the templates for the respective language are contained in the package.

To select a language (different than the default, English), set the Reporter.language attribute accordingly. By default, two-letter ISO codes are used, such as en (English) and de (German). Please note that this setting will only have an effect if templates for the requested language are provided either by the ASpecD package or by the package based on ASpecD.

In context of recipe-driven data analysis, getting a report on a dataset in German rather than in English, use the following:

- kind: report
type: LaTeXReporter
properties:
language: de
template: datensatz.tex
filename: bericht.tex
compile: true

Note that the name of the template changes as well, as it would not make too much sense to have English filenames for templates written in other languages.

Important

As the ASpecD framework natively uses the English language for descriptions, parameter names and alike, providing templates in different languages will likely result in a mixture of languages within the final report.

## Package developers: Organisation of templates¶

As mentioned above, the bundled templates used with the respective reporters are stored within the templates/report/ directory of the ASpecD package. Each output format has its own subdirectory, currently existing directories are txt and latex. Furthermore, to allow for different languages, each of these directories can (and should) contain another interlayer in terms of directories representing the languages. By default, two-letter ISO codes are used, such as en (English) and de (German).

Currently, the ASpecD template organisation looks similar to the following:

templates/
report/
latex/
de/
basis.tex
datensatz.tex
...
en/
base.tex
dataset.tex
...
txt/
en/
dataset.txt
...

If you do not plan to support multiple languages, you can skip the language directories. In this case, the templates are supposed to be in English language.

To change the contents of a (sub)template for your package, e.g. the “colophon.tex” template containing important information on how a report has been generated, simply provide a template with this name in the corresponding template directory within your package. As long as you are using recipe-driven data analysis and provide a default package in your recipe, the reporters will be notified and look for templates in your package before defaulting to the templates provided with the ASpecD framework.

## Background: Jinja2¶

The report functionality relies heavily on the Jinja2 template engine . Therefore, it is very useful to be generally familiar with the concepts of Jinja2. Basically, this template engine allows users to specify rather complicated replacements and logic within the template. However, logic within templates should be used sparingly, and the template should always be rendered correctly even without processing it by the template engine. Only then templates are easy to develop and adapt.

Two concepts used by Jinja2 the user of the report facilities of the ASpecD framework should be familiar with are “environment” and “context”. The former is a list of settings determining the type of delimiters used within a certain template for the control structures that are understood by Jinja2. As Jinja2 is developed with web applications (and hence HTML) in mind, those delimiters may not be feasible for other types of languages a template may be written in, such as LaTeX.

Currently, the aspecd.report module of the ASpecD framework provides a generic environment as well as dedicated Txt and LaTeX environments, implemented as respective classes:

These environments get automatically loaded by the respective reporter classes:

While the TxtEnvironment and TxtReporter classes are basically identical to the GenericEnvironment and Reporter classes, respectively, the LaTeXEnvironment and LaTeXReporter provide some heavy adaptations to rendering and even compiling LaTeX templates.

The second important concept of Jinja2 is that of the “context”: Think of it as a dictionary containing all the key–value pairs you can use to replace placeholders within a template with their actual values. In the simplest of all cases within the context of the ASpecD framework, this could be the metadata of an aspecd.dataset.Dataset.

## Module documentation¶

class aspecd.report.Reporter(template='', filename='')

Base class for reports.

To generate a report from a template, you will need a template in the first place. The path to the template (including its filename) is set in the template attribute that is either provided as parameter at initialisation or assigned later on. Similarly, you should specify an output filename, stored in the filename attribute. The variables known to the template that should be replaced by appropriate content, termed “context” in Jinja2 language, are stored in the context attribute. This is, by default, an ordered dict to preserve the order of the keys it contains. This is sometimes very useful.

Next, you want to render the template, and most often, save it to a file afterwards. If you would like to separate those steps, you can call the render() and save() methods separately. For convenience, simply call create() to render the report and save it to the file whose name you have provided.

The whole procedure may look as follows:

template = "/path/to/my/template.tex"
filename = "/path/to/my/final/report.tex"
report_ = aspecd.report.Reporter(template=template, filename=filename)
report_.create()

template

Template file used to generate report.

Type

str

filename

Name of the resulting template file.

Type

str

context

Variables of a template that are replaced with the given content.

It contains a key “sysinfo” containing system-related information, i.e. the information contained in the aspecd.system.SystemInfo class.

Furthermore, the key “template_dir” contains the (relative or absolute) path to the template provided in template. This is particularly useful for including subtemplates.

A key “timestamp” contains the current timestamp when starting to render the report.

Type

collections.OrderedDict

environment

Jinja2 environment used for rendering the template.

Defaults to a aspecd.report.GenericEnvironment object with settings for rendering generic templates.

Type

aspecd.report.GenericEnvironment

report

Actual report, i.e. rendered template

Type

str

package

Additionally to adding a template loader, the package name and its version number as well as all its dependencies get added to the sysinfo key of the context dictionary.

Type

str

package_path

Path to the templates within the package defined by package

Type

str

language

(Human) language of the templates

Usually a two-letter code. Only if the corresponding template directory exists within the package, the language will be set.

Type

str

comment

User-supplied comment describing intent, purpose, reason, …

Type

str

Parameters
• template (str) – Path to template file used to generate report.

• filename (str) – Path to the output file the report should be rendered to.

Raises
• aspecd.report.FileNotFoundError – Raised if the template file provided does not exist.

• aspecd.report.MissingFilenameError – Raised if no output file for the report is provided.

Changed in version 0.6.3: New attributes package, package_path, language

Changed in version 0.6.4: New attribute comment

render()

Render the template.

The actual rendering of the template should be implemented within the non-public method _render().

Before calling the non-public method _render(), the renderer checks the existence of the file provided as attribute template as well as whether an output filename has been provided. If this is not the case, a corresponding exception will be raised.

Raises

FileNotFoundError – Raised if the template file provided does not exist.

save()

Save report to file.

Raises

aspecd.report.MissingFilenameError – Raised if no output file for the report is provided.

create()

Create report, thus rendering template and saving result to file.

Convenience method to render the template and save the generated report to a file, thus simply calling render() and save() subsequently.

class aspecd.report.TxtReporter(template='', filename='')

Plain text reporter.

The most basic format for a report is plain text. Its probably biggest advantage is that it is intrinsically platform-independent (except of the encoding, but UTF-8 should not pose any problems any more in 2021 and after).

The perhaps biggest disadvantage is the lack of standard formatting and the overall limited formatting options, not to speak of including figures.

As such, the plain text reporter can be used for a first overview and for maximum portability. For well-formatted reports, have a look at the LaTeXReporter.

environment

Jinja2 environment used for rendering the template.

Similar to the aspecd.report.GenericEnvironment, but with the package path for template lookup set to the path for txt templates within the ASpecD package.

Type

aspecd.report.TxtEnvironment

New in version 0.6.

class aspecd.report.LaTeXReporter(template='', filename='')

LaTeX Reporter.

Often, templates for reports are written in LaTeX, and the results typeset as PDF file upon a (pdf)LaTeX run. For convenience, this class offers the necessary facilities to compile the template once written.

The whole procedure may look as follows:

template = "template.tex"
filename = "report.tex"
report_ = aspecd.report.LaTeXReporter(template=template,
filename=filename)
report_.create()
report_.compile()

This will result with a file “report.pdf” in the current directory. If you specify a relative or absolute path for the filename of the report, the resulting PDF file will be copied to that path accordingly.

Note that for compiling a temporary directory is used, such as not to clutter the current working directory with all the auxiliary files usually created during a (pdf)LaTeX run. Furthermore, currently, only a single (pdf)LaTeX run is performed with option “-interaction=nonstopmode” passed in order to not block further execution.

Important

For enhanced security, the temporary directory used for compiling the template will be removed after successful compilation. Therefore, no traces of your report should remain outside the current directory controlled by the user.

Note

Due to problems with LaTeX rendering text containing underscores, the keys in the context dict are recursively parsed and each key containing underscores converted to camel case (but preserving the case of the first character: “foo_bar” => “fooBar”). Thus, the template can be compiled using LaTeX without having to replace the placeholder variables beforehand.

environment

Jinja2 environment used for rendering the template.

Defaults to a aspecd.report.LaTeXEnvironment object with settings for rendering LaTeX templates.

Type

aspecd.report.LaTeXEnvironment

includes

List of files that need to be present for compiling the template.

These files will be copied into the temporary directory used for compiling the template.

Type

list

latex_executable

Name of/path to the LaTeX executable.

Defaults to “pdflatex”

Type

str

Parameters
• template (str) – Path to template file used to generate report.

• filename (str) – Path to the output file the report should be rendered to.

Raises

aspecd.report.LaTeXExecutableNotFoundError – Raised if the LaTeX executable could not be found

compile()

Compile LaTeX template.

The template is copied to a temporary directory and the LaTeX executable specified in latex_executable called on the report. Afterwards, the result is copied back to the original directory.

Additionally, all files necessary to compile the report are copied to the temporary directory as well.

Raises

aspecd.report.LaTeXExecutableNotFoundError – Raised if the LaTeX executable could not be found

class aspecd.report.GenericEnvironment(env=None, path='templates/report/', lang=None)

Jinja2 environment for rendering generic templates.

The environment does not change any of the jinja settings except of the loaders. Here, a list of loaders using jinja2.ChoiceLoader is implemented. Using this loader makes it possible to search subsequently in different places for a template. Here, the first hit is used, therefore, the sequence of loaders is crucial.

Currently, there are two loaders implemented, in exactly this sequence:

1. jinja2.FileSystemLoader

Looking for templates in the current directory and using an absolute path

2. jinja2.PackageLoader

Looking for templates in the aspecd package in the package path “templates/report/”, i.e. the base directory for all report templates of the ASpecD package.

Additional package loaders can be inserted before the package loader for the ASpecD package using the method add_package_loader(). This allows packages based on the ASpecD framework to define templates with the same name as those in ASpecD and thus to load the templates provided by the derived package rather than those from ASpecD.

path

Path to the templates within the package

Default: “templates/report”

Type

str

Parameters

Changed in version 0.6.3: New attribute package_path, new parameters env, path, lang

set_language()

Only if a language is set in the class and the corresponding template directory exists, the loaders will be updated. As it seems, there is currently no way to adjust the package_path of a jinja2.PackageLoader, hence the loaders are replaced with new loaders with adjusted package_path.

add_package_loader(package_name='', package_path='')

The package loader will be inserted before the loader for the ASpecD package. This allows packages based on the ASpecD framework to define templates with the same name as those in ASpecD and thus to load the templates provided by the derived package rather than those from ASpecD.

Only in case of the given package path to exist the package loader will be added.

Parameters
• package_name (str) – Name of the package to add the loader for

• package_path (str) –

Path to the templates within the package

Defaults to package_path

New in version 0.6.3.

class aspecd.report.TxtEnvironment(lang=None)

Jinja2 environment for rendering generic text templates.

The environment does not change any of the jinja settings except of the loaders. Here, a list of loaders using jinja2.ChoiceLoader is implemented. Using this loader makes it possible to search subsequently in different places for a template. Here, the first hit is used, therefore, the sequence of loaders is crucial.

Currently, there are two loaders implemented, in exactly this sequence:

1. jinja2.FileSystemLoader

Looking for templates in the current directory and using an absolute path

2. jinja2.PackageLoader

Looking for templates in the aspecd package in the package path “templates/report/txt/”, i.e. the directory for all bare text report templates of the ASpecD package.

Changed in version 0.6.3: Now based on GenericEnvironment

class aspecd.report.LaTeXEnvironment(lang=None)

Jinja2 environment for rendering LaTeX-based templates.

This environment is designed for using templates written in LaTeX that can be rendered by LaTeX without having their control code replaced by Jinja2. While variables are usually output in LaTeX, control structures are prefixed by a LaTeX comment character (%). For convenience, the following table lists all the variables currently set within this environment.

Variable

Value

block_start_string

%{

block_end_string

}%

variable_start_string

{@

variable_end_string

}

comment_start_string

%#{

comment_end_string

}

line_statement_prefix

%%

line_comment_prefix

%#

trim_blocks

True

autoescape

False

While every measure is taken to keep the above information as accurate as possible, for authoritative information the reader is referred to the actual source code.

Besides extensively modifying the control codes used within the template, the environment implements a list of loaders using jinja2.ChoiceLoader. Using this loader makes it possible to search subsequently in different places for a template. Here, the first hit is used, therefore, the sequence of loaders is crucial.

Currently, there are two loaders implemented, in exactly this sequence:

1. jinja2.FileSystemLoader

Looking for templates in the current directory and using an absolute path

2. jinja2.PackageLoader

Looking for templates in the aspecd package in the package path “templates/report/latex/”, i.e. the directory for all bare text report templates of the ASpecD package.

Changed in version 0.6.3: Now based on GenericEnvironment


text:
# Is this the correct way to hedge two securities against each other?

Let's say I believe that $ts_1$ and $ts_2$ move together and I would like to pairs trade them. Am I correct in understanding that to hedge them against each other I would get their $Var_1$, $Var_2$, and $Cov_1,_2$ all in USD, then I would buy 1000 USD worth of $ts_1$ and then find how much money to put into shorting $ts_2$ by doing $\beta = Cov_1,_2$/$Var_1$, and then doing $\$\_position\_ts_2 = $\$1000* $\beta$?

• What you are looking for is called the hedge ratio. For your formula to make sense, you need to be dividing by $\beta$. – msitt Apr 3 '17 at 4:28
• Just to make sure I follow, you mean $USD\_position\_ts_2 = 1000 USD/ \beta$ where $\beta = Cov_1,_2$/$Var_1$? – user1367204 Apr 3 '17 at 4:30

Let's call one security the $security_{market}$, and another $security_{unique}$. I'm taking for granted that you have done your research and believe that you found a good pair of securities for hedging.
1. Convert each security time-series from a price to a daily percent change. Call this $pct\_change_{market}$ and $pct\_change_{unique}$.
2. Get $Var(pct\_change_{market}$), $Var(pct\_change_{unique})$ and $Cov(pct\_change_{market}, pct\_change_{unique}$).
3. $\beta$ = $Cov(pct\_change_{market}, pct\_change_{unique}$)/$Var(pct\_change_{market}$).
4. Buy \$100,000 (or whatever amount) of$security_{unique}$. Now you need to know how many dollars to sink into$security_{market}$. 5. Multiply \$100,000 by the $\beta$, this will be the amount of dollars to spend on your hedge.
6. Divide the number of dollars you came up with in step 5 by the price of $security_{market}$. Just to be clear, numerator is number of dollars, denominator is price of $security_{market}$. This will be your position for the hedge.

format: plain


text:
# direct.influence: Calculate the significance of direct influences of variant... In cape: Combined Analysis of Pleiotropy and Epistasis

## Description

This function recasts the variant-to-eigentrait effects in terms of variant-to-phenotype effects. It multiplies the β-coefficient matrices of each variant (i) and each phenotype (j) (β_{i}^{j}) by the singular value matrices (V \cdot W^{T}) obtained from the singular value decomposition performed in get.eigentraits. β_{i}^{j} = V \cdot W^{T}. It also uses the permutation data from the pairwise scan (pairscan) to calculate an empirical p value for the influence of each marker pair on each phenotype. The empirical p values are then adjusted for multiple testing using Holm's step-down procedure.

## Usage

 1 2 3 direct.influence(data.obj, pairscan.obj, transform.to.phenospace = TRUE, pval.correction = c("holm", "fdr", "lfdr"), verbose = FALSE, save.permutations = FALSE)

## Arguments

 data.obj The object in which all results are stored. See read.population. pairscan.obj The object in which the results from the pairscan are stored. See pairscan. transform.to.phenospace A logical value. If TRUE, the influence of each marker on each eigentrait is transformed to the influence of each marker on each of the original phenotypes. If FALSE, no transformation is made. If the pair scan was done on eigentraits, the influence of each marker on each eigentrait is calculated. If the pair scan was done on raw phenotypes, the influence of each marker on each phenotype is calculated. The default behavior is to transform variant influences on eigentraits to variant influences on phenotypes. pval.correction One of "holm", "fdr", or "lfdr", indicating whether the p value correction method used should be the Holm step-down procedure, false discovery rate or local false discovery rate respectively. verbose A logical value. If TRUE, the progress of the function is printed to the screen. If FALSE, the default, nothing is printed. save.permutations A logical value indicating whether the data from permutations should be saved. Saving the permutations requires more memory but can be helpful in diagnostics. If save.permutations is TRUE all permutation data are saved in an object called "permutation.data.RData".

## Value

This function adds a total of six objects to the data object. First, a flag (transform.to.phenospace) is added to the data object to indicate whether variant influences were transformed to phenotype space.

The results from the pairwise scan and the permutations of the pairwise scan are converted to phenospace if specified. These actions each add one object each to the data object (var.to.pheno.influence and var.to.pheno.influence.perm). Each element is itself a list of matrices corresponding to the original phenotypes. Each matrix contains one row per marker pair (or permutation of a marker pair) and contains the influence coefficient and standard error of the influence coefficient for each pair.

After the coefficients have been transformed to phenotype space, each marker is considered individually and its influence on each phenotype across all marker pair contexts is tabulated. This is done for both the pairwise scan and the permutations of the pairwise scans and adds two new objects (var.to.pheno.test.stat and var.to.pheno.test.stat.perm) to the data object. Each object is a list containing one element for each of the original phenotypes. Each element contains a table in which all instances of each marker are listed along with that marker's direct phenotypic influence, the standard error of the influence, and the t statistic (β/σ) of the influence.

Finally, because each marker can only have one influence on each phenotype, the influences from each marker pair context are filtered to report only the maximum influence of each marker on each phenotype across all marker pair contexts. This process adds an object to the data object called max.var.to.pheno.influence. This object is a list containing one element per phenotype. It tabulates the maximum influence of each marker on each phenotype, as well as the empirical and Holm's corrected p values associated with each influence.

## References

Carter, G. W., Hays, M., Sherman, A., & Galitski, T. (2012). Use of pleiotropy to model genetic interactions in a population. PLoS genetics, 8(10), e1003010. doi:10.1371/journal.pgen.1003010 Holm, S. (1979). A simple sequentially rejective multiple test procedure. Scandinavian journal of statistics, pages 65-70.

cape documentation built on May 2, 2019, 3:27 a.m.


text:
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When the true underlying distribution is known to be Gaussian, although with unknown σ, then the resulting estimated distribution follows the Student t-distribution. Review authors should look for evidence of which one, and might use a t distribution if in doubt. The sample mean x ¯ {\displaystyle {\bar {x}}} = 37.25 is greater than the true population mean μ {\displaystyle \mu } = 33.88 years. Anmelden Transkript Statistik 22.628 Aufrufe 54 Dieses Video gefällt dir?

As the sample size increases, the sampling distribution become more narrow, and the standard error decreases. Anmelden 55 7 Dieses Video gefällt dir nicht? If the population standard deviation is finite, the standard error of the mean of the sample will tend to zero with increasing sample size, because the estimate of the population mean The standard deviation of the age for the 16 runners is 10.23.

Is powered by WordPress using a bavotasan.com design. The next graph shows the sampling distribution of the mean (the distribution of the 20,000 sample means) superimposed on the distribution of ages for the 9,732 women. Wenn du bei YouTube angemeldet bist, kannst du dieses Video zu einer Playlist hinzufügen. Jobs for R usersFinance Manager @ Seattle, U.S.Data Scientist – AnalyticsTransportation Market Research Analyst @ Arlington, U.S.Data AnalystData Scientist for Madlan @ Tel Aviv, IsraelBioinformatics Specialist @ San Francisco, U.S.Postdoctoral Scholar

Statistical Notes. Flag as... doi:10.2307/2340569. One standard deviation about the central tendency covers approximately 68 percent of the data, 2 standard deviation 95 percent of the data, and 3 standard deviation 99.7 percent of the data.

share|improve this answer edited Oct 3 '12 at 12:53 answered Sep 13 '11 at 14:12 Macro 24.3k496130 add a comment| Your Answer draft saved draft discarded Sign up or log The true standard error of the mean, using σ = 9.27, is σ x ¯   = σ n = 9.27 16 = 2.32 {\displaystyle \sigma _{\bar {x}}\ ={\frac {\sigma }{\sqrt T-distributions are slightly different from Gaussian, and vary depending on the size of the sample. Schließen Ja, ich möchte sie behalten Rückgängig machen Schließen Dieses Video ist nicht verfügbar.

Ecology 76(2): 628 – 639. ^ Klein, RJ. "Healthy People 2010 criteria for data suppression" (PDF). Next, consider all possible samples of 16 runners from the population of 9,732 runners. In the case above, the mean μ is simply (12+55+74+79+90)/5 = 62. Die Bewertungsfunktion ist nach Ausleihen des Videos verfügbar.

The unbiased standard error plots as the ρ=0 diagonal line with log-log slope -½. What is the mean of a data at 5% standard error? The means of samples of size n, randomly drawn from a normally distributed source population, belong to a normally distributed sampling distribution whose overall mean is equal to the mean of Wird geladen...

And if so, is this formula appropriate? $$SE = \frac{SD}{\sqrt{N}}$$ standard-deviation standard-error share|improve this question edited Jul 16 '12 at 11:34 Macro 24.3k496130 asked Sep 13 '11 at 13:54 Bern 86113 Tips Calculations of the mean, standard deviation, and standard error are most useful for analysis of normally distributed data. WiedergabelisteWarteschlangeWiedergabelisteWarteschlange Alle entfernenBeenden Wird geladen... It is important to check that the confidence interval is symmetrical about the mean (the distance between the lower limit and the mean is the same as the distance between the

Correction for finite population The formula given above for the standard error assumes that the sample size is much smaller than the population size, so that the population can be considered Because these 16 runners are a sample from the population of 9,732 runners, 37.25 is the sample mean, and 10.23 is the sample standard deviation, s. Wird geladen... Über YouTube Presse Urheberrecht YouTuber Werbung Entwickler +YouTube Nutzungsbedingungen Datenschutz Richtlinien und Sicherheit Feedback senden Probier mal was Neues aus! How to build a satellite network around Kerbin which guarantees full coverage?

more than two times) by colleagues if they should plot/use the standard deviation or the standard error, here is a small post trying to clarify the meaning of these two metrics Standard error = σ/sqrt(n) So for the example above, if this were a sampling of 5 students from a class of 50 and the 50 students had a standard deviation of An easy to use online standard deviation calculator Warnings Check your math carefully. plot(seq(-3.2,3.2,length=50),dnorm(seq(-3,3,length=50),0,1),type="l",xlab="",ylab="",ylim=c(0,0.5)) segments(x0 = c(-3,3),y0 = c(-1,-1),x1 = c(-3,3),y1=c(1,1)) text(x=0,y=0.45,labels = expression("99.7% of the data within 3" ~ sigma)) arrows(x0=c(-2,2),y0=c(0.45,0.45),x1=c(-3,3),y1=c(0.45,0.45)) segments(x0 = c(-2,2),y0 = c(-1,-1),x1 = c(-2,2),y1=c(0.4,0.4)) text(x=0,y=0.3,labels = expression("95% of the

set.seed(20151204) #generate some random data x<-rnorm(10) #compute the standard deviation sd(x) 1.144105 For normally distributed data the standard deviation has some extra information, namely the 68-95-99.7 rule which tells us the Calculations for the control group are performed in a similar way. Write an Article 143 R news and tutorials contributed by (580) R bloggers Home About RSS add your blog! It is very easy to make mistakes or enter numbers incorrectly.

All the R Ladies One Way Analysis of Variance Exercises GoodReads: Machine Learning (Part 3) Danger, Caution H2O steam is very hot!! The mean age for the 16 runners in this particular sample is 37.25. The divisor, 3.92, in the formula above would be replaced by 2 × 2.0639 = 4.128. Anzeige Autoplay Wenn Autoplay aktiviert ist, wird die Wiedergabe automatisch mit einem der aktuellen Videovorschläge fortgesetzt.

Again, the following applies to confidence intervals for mean values calculated within an intervention group and not for estimates of differences between interventions (for these, see Section 7.7.3.3). A medical research team tests a new drug to lower cholesterol. It can only be calculated if the mean is a non-zero value.


text:
A quadratic equation is an expression of the form $y=ax^2 + bx + c$. Solving one of these equations means finding the value of $x$ that returns $y$ equal to zero.

x = seq(-3, 3, len = 100)
a = 1
b = -2
c = -4
y = a * x^2 + b * x + c
y[1:10]

##  [1] 11.000 10.519 10.045  9.579  9.119  8.668  8.223  7.786  7.356  6.934

plot(x, y)
abline(h = 0)

Let's zoom in a bit

x = seq(-2, -1, len = 100)
y = a * x^2 + b * x + c
plot(x, y, type = "l")
abline(h = 0, col = "blue", lwd = 3)

Let's make a function to do some of this

quadratic = function(x, a, b, c) {
return(a * x^2 + b * x + c)
}

##  [1] 4.000 3.939 3.879 3.819 3.759 3.700 3.640 3.581 3.522 3.463

Now we can do...

x = seq(-10, 10, len = 100)
plot(x, quadratic(x, a, b, c), type = "l")
abline(h = 0, col = "blue", lwd = 2)

And we can see this equation has two solutions where it crosses $y=0$. Let's compute them exactly.

The solutions of a quadratic equation is given by the well-known formula: $$x = \frac{{- b \pm \sqrt {b^2 - 4ac}}}{{2a}}$$ so we will duplicate this in R.

d = sqrt(b^2 - 4 * a * c)
s1 = (-b + d)/(2 * a)
s2 = (-b - d)/(2 * a)
s1

## [1] 3.236

s2

## [1] -1.236

plot(x, quadratic(x, a, b, c), type = "l")
abline(h = 0, col = "blue", lwd = 2)
abline(v = c(s1, s2), col = "green", lwd = 2)

Again, we should put this into a function.

quadsolve = function(a, b, c) {
d = sqrt(b^2 - 4 * a * c)
s1 = (-b + d)/(2 * a)
s2 = (-b - d)/(2 * a)
return(c(s1, s2))
}

## [1]  3.236 -1.236

quadsolve(-3, 1, 5)

## [1] -1.135  1.468

We can test our function by feeding the answer from quadsolve into quadratic and seeing if we get zero.

quadratic(quadsolve(a, b, c), a, b, c)

## [1] 0 0

quadratic(quadsolve(-3, 1, 5), -3, 1, 5)

## [1] 1.776e-15 8.882e-16

It's not exactly zero. But close enough.

So now we can solve any quadratic equation. Or can we?

quadsolve(1, 2, 3)

## Warning: NaNs produced

## [1] NaN NaN

What has happened here? Let's plot it, with some different limits on the y axis:

x = seq(-5, 5, len = 100)
plot(x, quadratic(x, 1, 2, 3), ylim = c(0, 10))
abline(h = 0, col = "blue", lwd = 2)

Now we can see that the curve never crosses the $x$ axis, and so there are no real solutions to the equation. If d is negative in the expression, R returns NaN for "not a number". We can test for this and create an error. We'll also make another improvement by vectorising the plus-or-minus calculations.

quadsolve = function(a, b, c) {
d2 = b^2 - 4 * a * c
if (d2 < 0) {
stop("No real solutions")
}
d = sqrt(d2)
s = (-b + c(-1, 1) * d)/(2 * a)
return(s)
}
quadsolve(1, 2, 3)  # expect error

## Error: No real solutions

Although there are no real solutions in this case, we can find solutions with complex numbers. In this case we should probably test and return a warning. We can persuade sqrt to not return NaN by converting its argument to a complex number.

quadsolve = function(a, b, c) {
d2 = b^2 - 4 * a * c
if (d2 < 0) {
warning("No real solutions")
d2 = as.complex(d2)
}
d = sqrt(d2)
s = (-b + c(-1, 1) * d)/(2 * a)
return(s)
}

This should now return complex numbers for quadratics that don't cross the $x$ axis.

quadsolve(1, 2, 3)

## Warning: No real solutions

## [1] -1-1.414i -1+1.414i

and we can check the solutions produce zero values of the function. In this case we get near-zero complex values.

quadratic(quadsolve(1, 2, 3), 1, 2, 3)

## Warning: No real solutions

## [1] -4.441e-16+0i -4.441e-16+0i

## Further Thoughts

• Now write a function to evaluate the cubic function, $y=ax^3+bx^2+cx+d$ and plot a few curves.
• How would you write a function to compute a polynomial of any order?
• Read the help for the curve function and plot some of the graphs in the previous examples