text: For example, say we had included all personality items in a CFA with five factors, we could do the above calculations separately for each factor and obtain their composite reliabilities. Analyze Descriptive Statistics Descriptives. SPSS Statistics Test Procedure in SPSS Statistics. Applied Psychological Measurement, 21(2), 173-184. Reliability indices are traditionally rounded to two decimal places, so the reliability would be reported as .95 in this example. The basic characteristics of the goods to be embroidered and proper knowledge of the appropriate underlay stitches to be used must be well understood by an embroiderer, before accepting an order. Garson sure has a lot to offer, although [side comment] I believe I have found a chink in his armor when it comes to centering in regression (he advocates it unnecessarily). Thank you. Remember also that I said we should conduct reliability analysis on any subscales individually. The results are 0.50, 0.47 and 0.50. The following is shamelessly extracted from the following link. I came across his website because some of my colleagues use SPSS for psychometrics, esp. Click the Options button, and check “Variance” under . I'm not sure what you mean by "composite reliability," but try looking under: If you want to calculate Cronbach's alpha, here is an SPSS-based tutorial: How to calculate the Average Variance Extracted (AVE) by SPSS in SEM? It is very shortly discussed on Garson's SPSS website. Below are step-by-step commands for calculating Cronbach's alpha in SPSS. The average variance extracted (AVE) calculated as follows: Now, you should have reverse scored item 3 (see above). Centering remains useful when one wants to avoid spurious collinearity effects or to interpret the effect of a predictor as deviation from its means (e.g., age). I have some issues with an exploratory factor analysis. SPSS Reliability Example. If not with SPSS, Stata might help too. If we assume that the time1 and time2 scores are to be summed to produce a total score, we essentially have a two-item test, and we can compute a measure of internal consistency. The composite reliability formula *assumes* that fit is good (so that parameter estimates are interpretable), while fit assessment actually evaluates fit. Estimation of composite reliability for congeneric measures. any suggestions please? This calculator estimates composite reliability as: Source: Raykov, T. (1997). I'm thinking that by "composite What was the shortest-duration EVA ever? Interrater reliability (Kappa) Discriminant Validity through Variance Extracted (Factor Analysis)? Al). Did human computers use floating-point arithmetics? Can you please suggest a freeware that computes automatically the AVE, Composite Reliability of construct as well as crossloadings, outerloadings ? CR is a less biased estimate of reliability than Chonbachs Alpha, the acceptable value of CR is 0.7 and above. (See Chapter 3, Creating New Variables Using Compute in the online SPSS book.) How to write graph coordinates in German? Cronbach's alpha can be carried out in SPSS Statistics using the Reliability Analysis... procedure. What drives the test reliability of a composite based on three components? Dispersion. Thanks for contributing an answer to Cross Validated! Yes, it is. coefficient, is obtained by combining all of the true score variances and covariances in the composite of indicator variables related to constructs, and by dividing this sum by the total variance in the composite. rev 2021.1.5.38258, The best answers are voted up and rise to the top, Cross Validated works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. Very different results of principal component analysis in SPSS and Stata after rotation. Definition of Composite Reliability: Composite reliability (CR), ? You can also view the video on YouTube through this link. So the only way to include cases with more than 50% observed data would be to impute them in a separate step before you run the reliability analysis. Finally, the "average variance extracted" measures the amount of variance that is captured by the construct in relation to the amount of variance due to measurement error and can be calculated using the following formula: (summation of squared factor loadings)/(summation of squared factor loadings) (summation of error variances) (Fornell & Larcker). model used. Drag over the desired variables . You can use the below online calculator by inputting just your factor loadings from the CFA. AVE- average variance extracted (AVE) should not be less than .05, this is to show that more than half of the variances is observed (Janssens, et. Is it better to use a smaller, more accurate measuring cylinder several times or a larger, less accurate one for the same volume? I have computed Average Variance Extracted (AVE) by first squaring the factor loadings of each item, adding these scores for each variable (3 variables in total) and then divide it by the number of items each variable had (8, 5, and 3). Enter the data into SPSS as shown above in Table 3. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Muhammad Abdullahi Wase Specialist Hospital Kano Nigeria. Composite reliability assesses the strength of relations, while fit assesses the *pattern* of relations. What I would do from here is examine the syntax SPSS gives you when you perform an FA, and use the variables named to compute the average variance explained. Kind regards and good luck with your research. Menus . (1997). Join ResearchGate to find the people and research you need to help your work. Is such a high difference possible and logical between the 2 coefficients? Can anyone tell me how to calculate average variance extracted (AVE) and composite reliability (CR) of a single latent variable with 5 indicators? I need to calculate a composite reliability using three different point scales. I seem to remember that Cronbach's alpha will only equal composite score reliability for essentially tau-equivalent tests. should there be necessary revisions with the questions or better yet should the specific subpart be remove from the questionaire? How to calculate composite score with a 2-factor scale. I haven't used SPSS in some time, and I don't remember seeing an option to perform these calculations, but you can certainly do it using the syntax. Let’s have SPSS compute Cronbach’s Alpha for the three questions about those who are opposed to religion. What was the "5 minute EVA"? What is the main difference between composite reliability in Smart PLS and Cronbach Alpha in SPSS to measure the reliability? BSS scores ranged from 2.00 to 5.82, with a mean of 4.30 and standard deviation of 0.85. , which has been also referred to as McDonald’s ? and “Kurtosis and Skewness” under . Reliability Analysis on SPSS Let’s test the reliability of the SAQ using the data in SAQ.sav. If the average variance extracted is less than .50, then the variance due to measurement error is greater than the variance due to the construct. And by exclude, they mean listwise deletion. Open the data set in SPSS and click on “Analyze” in the menu. for factor analysis or reliability studies. $\begingroup$ hi.. thankyou for ur response .. yes, i mean the cronbach's alpha.. the result i get is 0.634 and N of the items is 25..i have used all my question that using likert's scale to test the reliability which is my section C question.. my friend suggest me to use negatively worded to change the value of some negative question. Secondly which correlation should i use for discriminant analysis, - Component CORRELATION Matrix VALUES WITHIN THE RESULTS OF FACTOR ANALYSIS (Oblimin Rotation). For example, say we have a survey consisting of 15 questions to measure satisfaction. SPSS: Data Entry and Reliability Commands. and component/factor loadings? Use MathJax to format equations. Let’s start with reliability. With these data now consider how one calculates in SPSS Cronbach's alpha for both scales, Task Value and Anxiety. You may obtain good composite reliability numbers when individual R2's are low. High composite reliability is a very good indication that all your items constantly measure the same construct. If we use the results from our orthogonal rotation (look back at In order to get square multiple correlation of each item, you need to find square of each item Standardized Regression Weight / Estimate. Sedangkan eror (e) berasal dari measurement eror dari masing-masing indikator : (1- Loading 2). In de psychologie maken we vooral gebruik van de "interne consistentie betrouwbaarheid" om erachter te komen of de items (bijvoorbeeld vragen op een vragenlijst) daadwerkelijk datgene meten wat er zou moeten worden gemet… In a series of SEM Journal articles, Raykov demonstrated that the composite reliability for a set of continuous items is explainable as: var(true) / [var(true) + var(residual)] within a CFA model where var(true) is equal to the sum across items of the variance explainable for each item and var(residual) is the sum of the residual error variances. What is the acceptable range for factor loading in SEM? One paper (Peterson & Kim 2012) said that although CR is a better estimate, there isn't much a difference between the values. If the one-parameter logistic model (1PL) or the Rasch model are used, the result is equivalent to adding raw … Given 0.95 is often used as a high stakes cutoff, using one reliability metric I have a decidedly low stakes assessment, and another it passes muster, I'm a little confused here. Those are T1_COLATH, T7_LIBATH, T13_SPKATH. AVE, Composite Reliability of construct. How can you make a scratched metal procedurally? Composite Reliability (CR). Estimation of composite reliability for congeneric measures. Suppose you wish to give a survey that measures job motivation by asking five questions. Dimana nilai standar loading berasal dari nilai standardized loading untuk tiap-tiap indikator (hasil output AMOS, Lisrel). 2. - Averaging the items and then take correlation. Where does the phrase, "Costs an arm and a leg" come from? A big difference between Cronbach Alpha and Composite Reliability: Why? I would anticipate that alpha would be a lower bound, given the model is congeneric; however, I did not expect composite reliability to be so much higher than alpha. In many quantitative studies, composite scoring and assessing reliability are key steps in data management and analysis process. SCALE defines a scale for analysis, providing a label for the scale and specifying its component variables. I found some scholars that mentioned only the ones which are smaller than 0.2 should be considered for deletion. If SCALE is omitted, all variables named on VARIABLES are used, and the label for the scale is ALL.. Second, we create new variables to increase the reliability and validity of our measures over that for single questions. Then point your mouse at “Scale” and click on “Reliability Analysis”. How can I calculate Composite Reliability. What is Composite Reliability? I want to know if that can be used in SPSS for calculation of AVE? @chl: nice citation. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. These SPSS statistics tutorials briefly explain the use and interpretation of standard statistical analysis techniques for Medical, Pharmaceutical, Clinical Trials, Marketing or Scientific Research. Two composite reliability measures, coefficient alpha and coefficient omega with unit weights (otherwise known as construct reliability), are commonly used in structural equations modeling. Cronbach’s alpha is a convenient test used to estimate the reliability, or internal consistency, of a composite score. Sorry for the lack of a definitive answer, but I felt that some response was definitely better than none. im currently having an undergraduate thesis. Where to keep savings for home loan deposit? It is desirable that for the normal distribution of data the values of skewness should be near to 0. statmodel.com/discussion/messages/9/258.html?1237657036, faculty.chass.ncsu.edu/garson/PA765/reliab.htm, Interpreting discrepancies between R and SPSS with exploratory factor analysis. Now, what on Earth does that mean? right-clicking your SPSS factor Are there 300,000 items in the average American household, and 10,000 items in the average European household? My Alpha returned a value of 0.64 (low but I guess I can proceed since I've seen such a practice before and since authors such as Hair and Kline accept a threshold between 0.6 and 0.7). However, there are various ideas in this regard. Turn on the SPSS program and select the Variable View, furthermore, in the Name write Item_1 to Item_10. Anyway and since my factor is homogeneous but has different loadings for all the 4 items involved, I think CR would be a better alternative. Coach to clarify the contents of the Variance total of the squared multiple correlations plus the total sum of each variable, then divides it by the number of factors in that variable. Asking for help, clarification, or responding to other answers. consistency reliability. site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. High scores on the BSS indicate that a person believes that men should protect women, that men need women, or that women have positive qualities that men lack. Ways to do live polling (aka ConcepTests) during class. In analyzing the data, you want to ensure that these questions (q1 through q5) all reliably measure the same latent variable (i.e., job motivation).To test the internal consistency, you can run the Cronbach's alpha test using the reliability command in SPSS, as follows: © 2008-2021 ResearchGate GmbH. Assuming that CR is indeed correct, can I proceed any further and do a multiple regression analysis based on the reliability provided by CR and not Cronbach? How to add gradient map to Blender area light? reliability" you mean internal RELIABILITY does not add any new variables to the active dataset. CR is often advocated as an alternative option due to the usual violation of the tau-equivalency assumption by Cronbach's Alpha. But I am confused should I take the above AVE Values calculated and compare it with the correlation OR I have to square root these values (√0.50 = 0.7071; √0.47 = 0.6856; √0.50 = 0.7071) and then compare the results with the correlation. Betrouwbaarheids analyse is een breed concept. (a) Step 1. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. Applied Psychological Measurement, 21(2), 173-184. Distribution. Is there any pre- requirement for this and which statistical package I should use for this (AMOS)? http://www.thestatisticalmind.com/calculators/comprel/composite_reliability.htm, https://www.youtube.com/watch?v=MHwFCX9iyFc, https://statistics.laerd.com/spss-tutorials/cronbachs-alpha-using-spss-statistics.php, Membrane filtration as a tool in biotechnical down-stream processing, Monitoring and Control of Pharmaceutical Protein Production with Sequential Integrated Down Stream Processes. Is solder mask a valid electrical insulator? To learn more, see our tips on writing great answers. Surprisingly, my CR returned a value of 0.787 using a calculator based on the formula provided by Raykov (1997). Construct the composite variable using COMPUTE in SPSS to create it. Surveys using IBM-SPSS Notes -New Variables, Reliability, Validity, Factor Analysis Page 2 providers, to obtain a measure of the total number of visits to all providers we need to sum these individual responses. Tutorial for Reliability Analysis. MathJax reference. Some said that the items which their factor loading are below 0.3 or even below 0.4 are not valuable and should be deleted. Cc by-sa three components i felt that some response was definitely better than none agree to our terms of,. In order to stabilize the fabric or raise the design consistency reliability ( Cronbach 's alpha for both scales Task! Mcdonald ’ s ranged from 2.00 to 5.82, with a 2-factor scale there are various ideas in this.... Which has been also referred to as McDonald ’ s test the reliability on... The video on composite reliability spss through this link you can use the nearest points mouse “! The specific subpart be remove from the results below, we could see the! Tau-Equivalent tests as well as crossloadings, outerloadings ranged from 2.00 to,! Standard one and i do not want to know if that can be used in SPSS and after! That represent a variable to create a score, or responding to other answers omega with weights! Mentioned only the ones which are smaller than 0.3 in SAQ.sav on Let. E ) berasal dari measurement eror dari masing-masing indikator: ( 1- loading )... Cronbach alpha and composite reliability is a standard one and i do not want to know if that be... From the CFA psychometrics, esp this case, the convergent validity our. Scored item 3 ( see above ) average American household, and the label specified. Reliability does not add any new variables to the active dataset the main between. Extracted ( factor analysis ( 2 ), below, we could composite reliability spss all the items which their loading! Subscribe to this RSS feed, copy and paste this URL into your reader. Decimal places, so the reliability, or data point, for that variable the extraversion factor is.90 furthermore! Spss program and select the variable View, furthermore, in the Name write to. 0 2 big difference between Cronbach alpha and composite reliability for the lack of a composite with... Theoretically appropriate a convenient test used to estimate the reliability analysis... procedure measures that. Using three different point scales the BSS was 0.74,  Costs an arm and a leg come... As McDonald ’ s test the reliability would be reported as.95 in this case, composite! Given a system missing value the design step, click the Options button, and check “ ”... There any pre- requirement for this ( AMOS ) the factor loading of two items are smaller than 0.2 be! Internal consistency reliability ( Cronbach 's alpha the average American household, and the label is in... Skewness and kurtosis faculty.chass.ncsu.edu/garson/PA765/reliab.htm, Interpreting discrepancies between R and SPSS with exploratory factor analysis ) commands for calculating 's. 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# How do I fit a curve into non linear data?

I did an experiment in my Uni and I collected data $$(ω,υ(ω))$$ modeled by the equation:

$$v(ω)=\frac{C}{\sqrt{(ω^2-ω_0^2 )^2 +γ^2 ω^2}}$$

where $$ω_0$$ is known. Do you know how can I fit a curve to my data $$(ω,υ(ω))$$ ? and how to find the parameter $$γ$$ through this process ?

• Can you tell us if $C$ is known? Apr 1 '19 at 6:10
• Let's assume that is not known and we have to extract $C$, $ω_0$ and $γ$. Apr 1 '19 at 18:25

I used mycurvefit.com for your problem. After creating an account (or maybe without if number of parameters is 2 or less) it lets you fit your function with at most 20 data points, which was enough. Here is an example

That correctly founds the parameter (g) close to 6.

Here are 20 data points that I have generated for $$C=10$$, $$\omega_0=10$$, and $$\gamma=6$$:

w       v(w)

5.4881  0.1294
7.1519  0.1538
6.0276  0.1366
5.4488  0.1290
4.2365  0.1164
6.4589  0.1429
4.3759  0.1176
8.9177  0.1746
9.6366  0.1716
3.8344  0.1132
7.9173  0.1655
5.2889  0.1271
5.6804  0.1319
9.2560  0.1744
0.7104  0.1004
0.8713  0.1006
0.2022  0.1000
8.3262  0.1706
7.7816  0.1636
8.7001  0.1737

Copy and paste them into the data sheet at the bottom.

P.S.: an analytical answer cannot be derived since the derivative equation (derivative = 0) of mean squared error with respect to parameter $$\gamma$$ is intractable, therefore gradient descent must be used with the help of computer (similar to what this site does).

EDIT:

I've forgot to add noise to $$v(\omega)$$, here is a noisy ($$\tilde{v}(\omega) = v(\omega)+\mathcal{N}(\mu=0, \sigma=0.01)$$) version with the same parameters:

w       v(w)
7.7132  0.1512
0.2075  0.1014
6.3365  0.1559
7.488   0.1483
4.9851  0.1039
2.248   0.0868
1.9806  0.106
7.6053  0.1848
1.6911  0.1136
0.8834  0.1174
6.8536  0.15
9.5339  0.1866
0.0395  0.0973
5.1219  0.1313
8.1262  0.1656
6.1253  0.1325
7.2176  0.1562
2.9188  0.1026
9.1777  0.1877
7.1458  0.1556

which gives $$g=5.7$$, meaning 20 data points are not enough for this level of noise or higher.

If you are more interested you can learn a framework like tensorflow to build the function and fit it to arbitrarily large number of data.

Here is a simple solution that you can compute yourself. It is trivial to solve your equation for $$γ$$ as a function of the other values. We get $$γ = \sqrt{(\biggr( \frac{C}{v(ω)} \biggl) ^2 - {(ω^2-ω_0^2)^2}) / {ω^2}}$$

It is almost true that you can substitute in your values and get several estimates for $$γ$$, which you can simply average to get a good estimate. I say almost true because as $$ω$$ approaches zero, both the numerator and denominator go to zero, so the calculation becomes unstable. As long as you stay away from $$ω=0$$, this should give good results. @Esmailian provided some nice data in his answer. Using his data I get the following estimates of $$γ$$.

6.012568  5.999664  5.994524  5.991173  6.009153  5.996748  6.019433
5.998072  6.001754  5.993656  6.002409  5.985938  5.996226  5.998363
6.474674  6.525602 14.140690  5.999124  6.000368  5.998427

Three of his $$ω$$'s are less than 1. These are the ones that give the estimates of $$γ$$

6.474674  6.525602 14.140690

The mean of the remaining estimates is 5.999859. As long as you have enough $$ω$$'s that are not near zero, this should provide a good estimate of $$γ$$.

format: plain


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# Compositing

## Facets

It is easy to make multiple plots that all share a common dataset and axis.

using Gadfly, RDatasets
iris = dataset("datasets", "iris")
plot(iris, xgroup="Species", x="SepalLength", y="SepalWidth",
Geom.subplot_grid(Geom.point))

Geom.subplot_grid can similarly arrange plots vertically, or even in a 2D grid if there are two shared axes.

Subplots have some inner and outer elements, including Guides and Scales. For example, place the guide inside Geom.subplot_grid(...) to change the subplot labels, or outside to change the outer plot labels.

haireye = dataset("datasets", "HairEyeColor")
palette = ["brown", "blue", "tan", "green"]

plot(haireye, y=:Sex, x=:Freq, color=:Eye, ygroup=:Hair,
Geom.subplot_grid(Geom.bar(position=:stack, orientation=:horizontal),
Guide.ylabel(orientation=:vertical) ),
Scale.color_discrete_manual(palette...),
Guide.colorkey(title="Eye\ncolor"),
Guide.ylabel("Hair color"), Guide.xlabel("Frequency") )

More examples can be found in the plot gallery at Geom.subplot_grid and Scale.{x,y}group.

## Stacks

To composite plots derived from different datasets, or the same data but different axes, a declarative interface is used. The Tutorial showed how such disparate plots can be horizontally arranged with hstack. Here we illustrate how to vertically stack them with vstack or arrange them in a grid with gridstack. These commands allow more customization in regards to tick marks, axis labeling, and other plot details than is available with Geom.subplot_grid.

theme1 = Theme(key_position=:none)
fig1a = plot(iris, x=:SepalLength, y=:SepalWidth, color=:Species, theme1,
alpha=[0.6], size=:PetalLength, Scale.size_area(maxvalue=7))
fig1b = plot(iris, x=:SepalLength, color=:Species, Geom.density,
Guide.ylabel("density"), Coord.cartesian(xmin=4, xmax=8), theme1)
vstack(fig1a,fig1b)


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, 18.12.2019 12:31 karolmolina49

# Find the volume of the cone. use 3.14 for π. round to the nearest tenth. a. 18,324.8 cm3 b. 1,271.7 cm3 c. 1,527.1 cm3 d. 4,581.2 cm3

### Another question on Mathematics

Mathematics, 21.06.2019 19:20
The fraction of defective integrated circuits produced in a photolithography process is being studied. a random sample of 300 circuits is tested, revealing 17 defectives. (a) calculate a 95% two-sided confidence interval on the fraction of defective circuits produced by this particular tool. round the answers to 4 decimal places. less-than-or-equal-to p less-than-or-equal-to (b) calculate a 95% upper confidence bound on the fraction of defective circuits. round the answer to 4 decimal places. p less-than-or-equal-to
Mathematics, 21.06.2019 20:30
Asmall business produces and sells balls. the fixed costs are $20 and each ball costs$4.32 to produce. each ball sells for \$8.32. write the equations for the total cost, c, and the revenue, r, then use the graphing method to determine how many balls must be sold to break even.


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# complex Difference-in-difference formula

I have 5-year sales information from a grocery store in Canada. I want to check whether an event that happened in 2017, affected the effect of the price of a product on its sales.

For example, imagine that before treatment price elasticity (i.e. the degree to which changes in price impact the unit sales of a product) for diary goods was -0.05, I am eager to know whether my treatment changed it or not. Actually, I already know that price influences sales, but I suspect that the imposition of the treatment might alter the consumers' price sensitivity.

The standard DiD model only investigates the impact of treatment on the outcome variable. While I need to see whether the treatment has affected the coefficient of the regression model (Sales ~ Price) or not. In other words, I need to check whether the treatment affected the relationship between my independent variable (Price) and dependent variable (Sales). Would you please guide me on what formula should I use as the standard difference-in-difference model does not seem to be working here?

$$log(Outcome_{it})= \beta_1 + \beta_2Treat_{i} + \beta_3 Post_{t} + \beta_4 (Treat \times Post)_{it} + \epsilon_{it}$$

can I use the below model instead?

$$log(Sales_{it})= \beta_1 + \beta_2Treat_{i} \times Price_{it} + \beta_3 Post_{t} \times Price_{i} + \beta_4 DiD_{it} \times Price_{it} + \epsilon_{it}$$

where DiD is:

$$DiD = Treat \times Post$$

• Welcome. Your equation is correct, so it may be a problem with how you coded the variables. Could you describe how you coded each variable in the model? Or, maybe try posting your output? Doing so will help you get the best possible answer. Jan 7 at 22:06
• Hi, thanks, I used my DiD model as below : linModel <- lm(log(sales) ~ Treat* Var1 + Post * Var1 + DiD * Var1 , data=mydata) where DiD is (Treat * Post), and I want to check whether the coefficient of the Var1 has changed. is my model correct instead of standard DiD model? Jan 7 at 23:23
• I added it to my question as well. Jan 7 at 23:31
• What is $Var_1$? Is it a covariate? And, what do you mean when you say the model is not working? Jan 7 at 23:37
• Var1 is my primary interest. I meant that the standard DiD model only investigates the impact of treatment on the outcome variable. However, I need to check whether the treatment affected the relationship between my independent variable and dependent variable. For example, imagine that before treatment price elasticity for diary good was -0.05, I am eager to know whether my treatment changed it or not. Jan 8 at 0:21


text:
# American Institute of Mathematical Sciences

2003, 2003(Special): 967-976. doi: 10.3934/proc.2003.2003.967

## Linear conservation laws of nonholonomic systems with symmetry

 1 Department of Mathematics, North Carolina State University, Raleigh, NC 27695, United States

Received  September 2002 Revised  April 2003 Published  April 2003

In nonholonomic dynamics, symmetries do not always lead to conservation laws. In this paper we study conditions for a nonholonomic system with symmetry to have conservation laws linear in momentum.
Citation: Dmitry V. Zenkov. Linear conservation laws of nonholonomic systems with symmetry. Conference Publications, 2003, 2003 (Special) : 967-976. doi: 10.3934/proc.2003.2003.967
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Impact Factor:


text:
# Tag Info

$\newcommand{\ket}[1]{\left|{#1}\right\rangle}$ The $\tfrac{\theta}{2}$ can be seen on the picture given at the end of my answer. Maybe you are puzzled because you are tempting to interpret $$\Psi=\cos\frac{\theta}{2} \lvert 0\rangle + e^{i\phi}\sin\frac{\theta}{2} \lvert 1\rangle$$ as a linear combination of vectors in the 3D Euclidean space. Note ...


text:
# Expectation of maximal Wasserstein distance between empirical distribution and a pdf

Let $$P$$ be a continuous probability distribution on $$R^d$$, $$X$$ the random variable $$\sim P$$, and $$\hat{X}$$ be n i.i.d samples drawn according to $$P$$. We have another variable $$\mu \in S^{d-1}$$. Do we have some result about $$\textbf{E}sup_{\mu \in S^{d-1}} W_2^2(\hat{X}\mu^T,X\mu^T) ?$$

• What is the sup over? – Aryeh Kontorovich Jan 26 at 18:57
• But these are random?.. – Aryeh Kontorovich Jan 26 at 19:32
• I made a mistake about the $sup$, now I give the original problem. – Will Cai Jan 28 at 1:37


text:
### StatNLP: Essential Information Theory, Part 1

Thu 15 February 2018

I'd like to blog regularly, but I often struggle to come up with topics. So what I'm going to do is blog my way through some textbooks, to work on my writing and also to make sure that I have the concepts in the books down pat. The first textbook in the series will be Foundations of Statistical Natural Language Processing by Manning and Schütze (which I'll refer to as StatNLP from here on out). And the first topic will be Essential Information Theory (section 2.2 in the book). This first post will introduce entropy and provide a few examples.

# Information Theory Basics

Information Theory as a field was developed by Claude Shannon in the 1940s. I won't go into too many details, but basically Shannon was working on a way to quantify how much information one could transmit through a channel that would inevitably lose some of the information due to some amount of noise. He introduced some key concepts that we'll define and visualize here. Information Theory as a whole is an active field, and we'll only scratch the surface here, but at the end of the series I'll provide some pointers for anyone interested in learning more.

## Preliminaries

In this post we'll be looking at probabilities of random variables. We'll define a discrete random variable as a set of outcomes $$X$$. Each potential outcome in $$X$$ has an associated probability: $$p(X=x)$$. The expectation (or expected value) of $$X$$ is a weighted sum of the outcomes according to the probabilities:

$$\mathbb{E}[X] = \sum_{x \in X} xp(x)$$

## Entropy

One of the key pieces in Information Theory is entropy. Entropy is defined as the amount of information in a random variable. The formula for entropy is

$$H(X) = - \sum_{x \in X} p(x) \log p(x)$$

Often times the logarithm in the formula is base 2, and the output is a measure of bits. Sometimes you'll see the natural log ($$\ln$$), and the output will be in nats. You can also write this formula a few other ways. If you don't like the minus sign in the beginning there, you can move it in and use the reciprocal of the logarithm:

$$H(X) = \sum_{x \in X} p(x) \log \frac{1}{p(x)}$$

$$\log \frac{1}{p(x)}$$ is known as the Shannon information content of an outcome (or information content for short). It tells you how much information you get for a specific outcome of a random variable. The previous formula looks a lot like an expectation. That's because it is! So we can rewrite the formula for entropy again as:

$$H(X) = \mathbb{E}( \log \frac{1}{p(X)})$$

Notice in this last formula that the $$p(x)$$ is now $$p(X)$$ (capital $$X$$ instead of lowercase $$x$$). When we have a lowercase $$x$$, we're looking for the probability of a specific value $$x$$ which is in the set of possible values $$X$$. In the third equation, we're calculating the expectation of the random variable $$X$$, and so we use $$p(X)$$.

So what do all of these equivalent formulas mean? When we're calculating entropy, what we're really calculating is the amount of information contained in a particular random variable. On one extreme, if we know everything about the random variable (if $$p(x) = 1$$ for one of the potential outcomes), then there is no information to be gained from the random variable. On the other extreme, if all of the outcomes are equally likely (as in a standard six-sided die) then there is a lot of information in the random variable.

I always like to think about entropy in terms of flipping a coin. Let's say we have a completely unbiased coin, so that the probability of it landing on heads is equal to the probability of it landing on tails. Our random variable is still $$X$$, and the possible values are $$x = H$$ and $$x = T$$. The probability of each outcome is $$p(x = H) = p(x = T) = 0.5$$.

Let's plug this in to our formula for entropy (we'll use the 2nd formula above).

\begin{align} H(X) &= \sum_{x \in \{H, T\}}p(x) \log \frac{1}{p(x)} \\ &= 0.5 * \log \frac{1}{0.5} + 0.5 * \log \frac{1}{0.5} \\ &= 2*(0.5 \log 2) \\ &= 1 \end{align}

So what this tells us is that every time we flip an unbiased coin, we get 1 bit of information. what if the coin was biased? For example, let's say that the probability of heads is $$0.75$$ and the probability of tails is $$0.25$$. In that case when we calculate entropy we get

\begin{align} H(X) &= \sum_{x \in \{H, T\}}p(x) \log \frac{1}{p(x)} \\ &= 0.75 * \log \frac{1}{0.75} + 0.25 * \log \frac{1}{0.25} \\ &\approx 0.81 \end{align}

We get less information if we flip a biased coin. Why is that? Because there is less uncertainty in the outcome of the coin flip. If the coin is biased, then one side is more likely to come up than the other. In the case above, if the probability of getting heads is $$0.75$$, then before you even flip it the odds are good that it will come up heads. So you don't get as much information as when the coin is unbiased, where there is more uncertainty (in fact, the maximum amount of uncertainty).

Let's look at a plot of entropy as the bias of a coin changes.

As the plot shows, the more biased the coin is, the less information we gain when we flip it. Sometimes it helps to think about entropy in terms of being surprised. When we flip a coin biased to land on heads, we're less surprised when it does land on heads (lower entropy). But when a coin is unbiased, each flip is a surprise (high entropy).

# Conclusion

Entropy is a key building block for Information Theory, and it is also very useful in Machine Learning and NLP (more in a later post). Next time we'll look at entropy for more than one random variable, and define joint entropy, conditional entropy, and (probably) KL-Divergence.

As a last note, I wrote this post in RMarkdown, which was surprisingly easy to integrate with this site, which runs on Pelican. All I needed was the RMD Reader Pelican Plugin.


text:
# Bioshock remastered – A shocking turn of events

One of the extra content things that came with Bioshock remastered, I really enjoyed this, this is also my second attempt, which is why it didn’t take me so long, although it should take 4mins apparently. May be a bit stuttery in parts, but until I get my new PC we’ll have to put up with such things for a while.


text:
# My Weblog

## SPOJ 6560. N DIV PHI_N (Hard)

SPOJ 6560. N DIV PHI_N (Hard) also accepted using the same idea accept using a bit memoization. Precalculation of product of primes in prime table $<=$ 10^25000 and linear search is enough to get accepted.Its first 6000 primes whose products are sufficient . Accepted Haskell code. I also tried a slightly improved Rabin Miller primality testing with idea give topcoder forum for faster modular exponentiation for large numbers and code with isProbable function got accepted with 3.95 and time limit exceed other time.

```import Data.List
import Data.Maybe
import Data.Bits
import qualified Data.ByteString.Char8 as BS

powM::Integer->Integer->Integer->Integer
powM a d n
| d == 0 = 1
| d == 1 = mod a n
| otherwise = mod q n  where
p = powM  ( mod ( a^2 ) n ) ( shiftR d 1 ) n
q = if (.&.) d 1 == 1 then mod ( a * p ) n else p

powL :: Integer -> Integer -> Integer -> Integer
powL a d n
| d <= 10^10 = powM a d n
| otherwise =  mod ( powM ( powL a ( div d 1000000 ) n ) 1000000 n   *  powM a ( mod d 1000000 ) n ) n

isProbable::Integer-> Bool
isProbable n |n<=1 = False
|n==2 = True
|even n = False
|otherwise	= rabinMiller [2,3] n s d where
s = until ( \x -> testBit ( n - 1) ( fromIntegral x ) )  ( +1 ) 0
d = div ( n - 1 ) (  shiftL 1 ( fromIntegral s )  )
--d=until (\x->mod x 2/=0) (`div` 2) (n-1) --(n-1=2^s*d)
--s=until (\x->d*2^x==pred n) (+1) 0  --counts the power

rabinMiller::[Integer]->Integer->Integer->Integer-> Bool
rabinMiller [] _ _ _ = True
rabinMiller (a:xs) n s d =
if a == n then True
else  case x==1 of
True -> rabinMiller xs n s d
_ -> if any ( == pred n) . take ( fromIntegral s ) . iterate ( \e -> mod (e^2) n ) \$  x then rabinMiller xs n s d
else  False
where x=powL a d n

prime :: [Integer]
prime = 2:3:filter isPrime [2*k+1 | k<-[2..]]

isPrime :: Integer -> Bool
isPrime n = all ((/=0). mod n  ).takeWhile ( <= ( truncate.sqrt.fromIntegral \$ n) ) \$ prime

anstable ::[Integer]
anstable =  scanl1 (*)   prime

helpSolve :: [Integer] -> Integer -> Integer -> Integer
helpSolve (x:xs) n ret
| x > n = ret
| otherwise = helpSolve xs n x

solve :: Integer -> BS.ByteString
solve 1 = BS.pack "1"
solve n = BS.pack.show \$ a  where
a = helpSolve ( tail anstable ) n ( head anstable )

readInt :: BS.ByteString -> Integer

main = BS.interact \$ BS.unlines.map ( solve .readInt ) .  tail . BS.lines
```


text:
# How to make a parabola and a line in picture environment?

I'd like to make this picture using the picture environment of LaTeX and I have no idea how.

The picture environment might not be the typical tool to draw graphs of parabolas, but because you ask for it:

\documentclass{article}

\usepackage{xcolor}

\begin{document}

\setlength{\unitlength}{1.5cm}
\begin{picture}(4,3)

\put(0,0){\vector(1,0){4}}
\put(0,0){\vector(0,1){3}}
\put(0,0){\line(2,1){3.5}}

{%
\thicklines
\color{blue}%
\qbezier(0,0)(1.5,3)(3,1.5)
}

\put(-0.25,2.9){y}
\put(3.9,-0.25){x}
\put(3.1,1.3){(x,y)}
\put(1.5,1){R}

{%
\color{orange}%
\multiput(3,0)(0,0.2){8}{\line(0,1){0.1}}
}

\qbezier(0.8,0)(0.8,0.2)(0.7,0.34)
\put(0.9,0.1){$\Phi$}

\qbezier(0.71,0)(0.7,0.35)(0.35,0.6)
\put(0.3,0.3){$\Theta$}

{%
\color{orange}%
\multiput(0,0)(0.25,0.5){6}{\line(1,2){0.12}}
}
\end{picture}

\end{document}


text:
# Draw user defined print margin

How can one draw a box on each page (KOMA class srcreprt) to show a user defined print margin. E.g. the box should be 25 mm from each page margin (which might be different from the actual layout setting which can be shown with \usepackage{showframe})

\documentclass{scrreprt}
\usepackage{blindtext}
\usepackage{showframe}

\begin{document}

\Blinddocument

\end{document}

-
This is related to tex.stackexchange.com/questions/171459/… and the crop package –  Christian Hupfer Apr 15 at 9:00

Try this code:

\documentclass{scrreprt}
\usepackage{blindtext}
\usepackage{tikzpagenodes}
\usetikzlibrary{calc}

\usepackage{eso-pic}

\begin{tikzpicture}[remember picture, overlay]
\draw[line width=1pt]
($(current page.north west) +(25mm,-25mm)$)
rectangle
($(current page.south east) +(-25mm,25mm)$);
\end{tikzpicture}%
}
\begin{document}

\Blinddocument

\end{document}

Output:

-

With background:

\documentclass{scrreprt}
\usepackage{blindtext}
\usepackage{background}
\backgroundsetup{
angle=0,
scale=1,
opacity=1,
color=black,
contents={%
\begin{tikzpicture}[remember picture, overlay]
\draw ([xshift=25mm,yshift=25mm]current page.south west) rectangle ([xshift=-25mm,yshift=-25mm)]current page.north east);
\end{tikzpicture}%
}
}

\begin{document}
\Blinddocument

\end{document}

-

After a few modification of the karlkoeller's code I came up with the solution I was looking for:

\documentclass{scrreprt}
\usepackage{blindtext}

\usepackage{tikzpagenodes}
\usetikzlibrary{calc}

\usepackage{eso-pic}

\begin{tikzpicture}[remember picture, overlay]
\draw[line width=1pt]
($(current page.north west) +(25mm,-25mm)$)
rectangle
($(current page.south east) +(-25mm,25mm)$);
\end{tikzpicture}%
}

\begin{document}

\Blinddocument

\end{document}

-
Sorry, I didn't read "from page margin". I've modified my answer. –  karlkoeller Apr 15 at 9:40
@karlkoeller Eh!. I too followed suit. Damn. :) –  Harish Kumar Apr 15 at 9:59


text:
### Super Science Friends: how to save Newton

There is a new animated series in the city(1): Super Science Friends. Created by Brett Jubinville of Tinman Creative Studios, it was funded the first short through Kickstarter, while the rest of the episodes can be supported on Patreon.
The series is evidently inspired by Super Friends, produced by Hanna-Barbera for DC Comics, with the place of the heroes of JLA taken by some iconic scientists: Marie Curie, with a radioactive ring similar to Green Lantern’s one; Charles Darwin, able to transform into any animal, like Beast Boy from the Teen Titans; Nikola Tesla, with electromagnetic powers, like Magneto from Marvel universe, historical X-Men enemy; Sigmund Freud, father of psychoanalysis with oversight of lust, to be seen as an alternative to Aquaman with his oversight of aquatic creatures.

### Star Trek Network

Using data from tv series and movie, Star Trek Viz relizes some network plots with the connections between the characters of one of the most famous sci-fi serie.

### Four all new exotic particles

LHCb has recently observed four new exotic-like particles in the decay of the $B^+$:
he properties of these structures are consistent with their interpretation as four-quark particles, which are considered as "exotic", (hence the "exotic-like" name in the title), although the details of the four quark $c{\bar c}s{\bar s}$ binding mechanism is still under discussion.
Read also paper 1 and paper 2

### Juno: Journey into the unknown

The Juno Mission is travelling in space and it'll arrive near Jupiter today. Waiting this new space event, watch the trailer of the mission:

### The Newton Medal is (bit) late

I think there is a subtle black humor in the 2016 Isaac Newton Medal, that was awarded by Tom Kibble:
The award is recognition of his contributions to mankind through his insight into the origins of mass and also through establishing astroparticle physics as a new branch of physics.
Kibble died on the 2nd June 2016 and he cannot retire the prize, but in every case we can remember his most important contribution, Global Conservation Laws and Massless Particles with Gerald Guralnik and Carl Richard Hagen about the Brout-Englert-Higgs mechanism:
An intersting reading about the story behind the paper is The History of the Theory of the Spontaneous Breaking by Guralnik:
Shortly thereafter, as we were literally placing the manuscript in the envelope to be sent to PRL, Kibble came into the office bearing two papers by Higgs and the one by Englert and Brout. These had just arrived in the then very slow and unreliable (because of strikes and the peculiarities of Imperial College) mail. We were very surprised and even amazed. We had no idea that there was any competing interest in the problem, particularly outside of the United States. Hagen and I quickly glanced at these papers and thought that, while they aimed at the same point, they did not form a serious chall enge to our work.

G. S. Guralnik, C. R. Hagen, T. W. B. Kibble, 1964, 'Global Conservation Laws and Massless Particles', Physical Review Letters, vol. 13, no. 20, pp. 585-587 http://dx.doi.org/10.1103/physrevlett.13.585 (sci-hib)
Gerald S. Guralnik, 2009, The History of the Guralnik, Hagen and Kibble development of the Theory of Spontaneous Symmetry Breaking and Gauge Particles, International Journal of Modern Physics A, vol. 24, no. 14, pp. 2601-2627 http://dx.doi.org/10.1142/s0217751x09045431 (arXiv)

### Another bit of gravity

After the first detection of gravitational waves from merged black holes, LIGO detected a new signal:
The two LIGO gravitational wave detectors in Hanford Washington and Livingston Louisiana have caught a second robust signal from two black holes in their final orbits and then their coalescence into a single black hole. This event, dubbed GW151226, was seen on December 26th at 03:38:53 (in Universal Coordinated Time, also known as Greenwich Mean Time), near the end of LIGO's first observing period ("O1"), and was immediately nicknamed "the Boxing Day event".
The inferred component masses are consistent with values dynamically measured in x-ray binaries, but are obtained through the independent measurement process of gravitational-wave detection. Although it is challenging to constrain the spins of the initial black holes, we can conclude that at least one black hole had spin greater than 0.2. These recent detections in Advanced LIGO's first observing period have revealed a population of binary black holes that heralds the opening of the field of gravitational-wave astronomy.
About the first observation, GW150914, you can read Binary Black Hole Mergers in the first Advanced LIGO Observing Run and Dynamical formation of the GW150914 binary black hole (sci-hub) (or Black hole pairs spat out of mosh pits make gravitational waves).

### How to produce stationary wave fields

A cylindrical surface of stationary light from
Zamboni-Rached, M., Recami, E., & Hernández-Figueroa, H. (2005). Theory of "frozen waves": modeling the shape of stationary wave fields Journal of the Optical Society of America A, 22 (11) DOI: 10.1364/JOSAA.22.002465 (arXiv)
On the 15th december, 2005, was published an interesting patent (I found it via Research Gate), Method and apparatus for producing stationary intense wave fields of arbitrary shape by Erasmo Recami, Michel Rached Zamboni, Hugo Enrique Ernandez Figueros, Valerio Abate, Cesar Augusto Dartora, Kleber Suza Nobrega, Marco Mattiuzzi:
Method for producing a stationary wave field of arbitrary shape comprising the steps of defining at least one volume being limited in the direction of the axis of propagation of a beam, of the type $0 \leq z \leq L$; defining an intensity pattern within the said region $0 \leq z \leq L$ by a function $F(z)$, describing the said localized and stationary intensity pattern, which is approximated by means of a Fourier expansion or by a similar expansion in terms of (trigonometric) orthogonal functions; providing a generic superposition of Bessel or other beams highly transversally confined; calculating the maximum number of superimposed Bessel beams the amplitudes, the phase velocities and the relative phases of each Bessel beam of the superposition, and the transverse and longitudinal wavenumbers of each Bessel beam of the superposition.
The invention is based on the following theoretical papers:

### Alice underground: the door, the quaternion and the relativity

Alice underground is the first version of Aline in the wonderland by Lewis Carroll. The original manuscript, illustrated by Carroll himself, was given to the little Alice Liddell for Christmas in 1864 and picked up the story that he had told to Alice and her sisters Lorina and Edith during a summer's afternoon, precisely on July the 4th, 1862. This first version of the carrollian fantasy novel is, ultimately, a restricted version of Alice, where various characters and episodes completely absent in Underground are added, such as the Duchess or the team composed by the Mad Hatter, the March Hare and the Dormouse.
The intial, interesting considerations about underground is about the importance of the trees and the doors: following the suggestion by Adele Cammarata(3), we can assume that the tree and the door that Alice cross to enter the garden of the Queen of Hearts, completely absent in Wonderland, is linked with the Celtic tradition. Indeed the oak is one of the sacred trees of the druids, symbolizing a link between heaven and earth(1). In this way the oak, which in Celtic was called duir, is a real door that connects people with the gods, but also ourselves with our inner part. So, from an etymological point of view, a carved door in a tree trunk is a Celtic symbol used to identify the Alice's passage towards a more stable phase after the size's changes of the previous scenes.
These changes in size, alluding both to the transition to adulthood, in perfect connection with the Druidic symbolism, and with the more classic homothetic transformations, i.e. the transformations which, without changing the proportions of a geometric figure, change its size. All these changes remain unchanged in the transition to the second version, including the meeting with the Caterpillar, who continues to ask Alice:
Who are you?

### Doubt like a salmon

An happy #towelday to all readers!
If you has ready and appreciate also one book by Douglas Adams, I think that you could have a little idea of what the people who knew him proved after his death. So, Peter Guzzardi composed The Salmond of Doubt with the idea to make an hearthful tribute to the writer of the Hitchhiker's Guide using unpublished articles, interviews and the third unfinished novel of the series of Dirk Gently, which also gives its name to collection.

from the italian cover of the book, by Franco Brambilla

### Transit of Mercury

We speak about astronomical transit or simply a transit when an astronomical body move between the observer and the celestial body that he observes. In particular the transit was used by Planck to discover exolpanets, but we can use also to observe some events that occur in our solar system. And between 9th and 10th May it occurs the transit of Mercury between Sun and Earth.
In the figure below (via EclipseWise) you can see the visibility of the transit:
More details on: NASA | NakedEyePlanets.com

### How to become a superhero

from Amazing Fantasy by Stan Lee and Steve Ditko
We analyze a collaboration network based on the Marvel Universe comic books. First, we consider the system as a binary network, where two characters are connected if they appear in the same publication. The analysis of degree correlations reveals that, in contrast to most real social networks, the Marvel Universe presents a disassortative mixing on the degree. Then, we use a weight measure to study the system as a weighted network. This allows us to find and characterize well defined communities. Through the analysis of the community structure and the clustering as a function of the degree we show that the network presents a hierarchical structure. Finally, we comment on possible mechanisms responsible for the particular motifs observed.
Gleiser, P. (2007). How to become a superhero Journal of Statistical Mechanics: Theory and Experiment, 2007 (09) DOI: 10.1088/1742-5468/2007/09/P09020 (arXiv)

### A mathematical theory of communication

A mathematical theory of communication is a paper by Claude Shannon published in two part in July and October 1948. The paper posed the basis for the modern information theory and the basi elements of communication:
• An information source that produces a message
• A transmitter that operates on the message to create a signal which can be sent through a channel
• A channel, which is the medium over which the signal, carrying the information that composes the message, is sent
• A receiver, which transforms the signal back into the message intended for delivery
• A destination, which can be a person or a machine, for whom or which the message is intended

Shannon, C. (1948). A Mathematical Theory of Communication Bell System Technical Journal, 27 (3), 379-423 DOI: 10.1002/j.1538-7305.1948.tb01338.x (pdf)
Shannon, C. (1948). A Mathematical Theory of Communication Bell System Technical Journal, 27 (4), 623-656 DOI: 10.1002/j.1538-7305.1948.tb00917.x (pdf)

### The Marvel Universe as a real social network

Cover of the first number of Marvel Team-Up, comic book dedicated to the collaborations between Marvel heroes
We investigate the structure of the Marvel Universe collaboration network, where two Marvel characters are considered linked if they jointly appear in the same Marvel comic book. We show that this network is clearly not a random network, and that it has most, but not all, characteristics of "real-life" collaboration networks, such as movie actors or scientific collaboration networks. The study of this artificial universe that tries to look like a real one, helps to understand that there are underlying principles that make real-life networks have definite characteristics.
R. Alberich, J. Miro-Julia, & F. Rossello (2002). Marvel Universe looks almost like a real social network arXiv arXiv: cond-mat/0202174v1

### Postcards from Pluto

The New Horizons mission has provided a lot of observations about Pluto, the farthermost known celestial object in our Solar System. Thanks to these observations, astronomers can now describe some interesting Pluto's characteristics.
The Pluto system is surprisingly complex, comprising six objects that orbit their common center of mass in approximately a single plane and in nearly circular orbits. (...)
All four of Pluto's small moons are highly elongated objects with surprisingly high surface reflectances (albedos) suggestive of a water-ice surface composition. Kerberos appears to have a double-lobed shape, possibly formed by the merger of two smaller bodies. Crater counts for Nix and Hydra imply surface ages of at least 4 billion years. Nix and Hydra have mostly neutral (i.e., gray) colors, but an apparent crater on Nix's surface is redder than the rest of the surface; this finding suggests either that the impacting body had a different composition or that material with a different composition was excavated from below Nix's surface. All four small moons have rotational periods much shorter than their orbital periods, and their rotational poles are clustered nearly orthogonal to the direction of the common rotational poles of Pluto and Charon.

### A prize for a theorem

Pierre de Fermat was born in 1601 in Beaumont-de-Lomagne in southwestern France. He was not even a mathematician; he was a civil servant who devoted himself to mathematics as a hobby. He was regarded as one of the most gifted self-taught mathematicians who ever lived.
I think that this quotation from The girl who played with fire by Stieg Larson, the second novel from the series Millennium, was the perfect introduction to a post about the Fermat's Last Theorem and Andrew Wiles, who proofed it in 1993. And he awarded the Abel Prize just some days ago.
Simply a joke?

Pierre de Fermat by Bernarda Bryson
Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos ejusdem nominis fas est dividere: cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.(1)
This is what Pierre de Fermat wrote in the margin of Arithmetica by Diofanto and it establish the impossibility to find solutions in the field of natural numbers for the diophantine equation(2) that generalizes the Pythagorean theorem: $a^n + b^n = c^n$ Following the Pythagorean theorem, for $n = 2$ the Fermat's equation has solutions in the field of natural numbers, called Pythagorean triples. For $n$ greater than 2 the story, instead, becomes more complicated, so that the same Fermat, while writing what follows, never provided a full demonstration, but only for the special case of $n = 4$:
I have discovered a truly marvelous proof of this theorem, which can't be contained in the too narrow page margin.

### Ludolph van Ceulen: in searching of pi

One of the most important mathematicians for $\pi$, was Ludolph van Ceulen, German mathematician born 28 January 1540 in Hildesheim. His father, Johannes Van Ceulen, was a small trader who could not afford advanced education for a son who showed some interest in mathematics. The main difficulty for Ludolph studies was the Latin, the language in which they were written the basic texts as well as the more recent ones in mathematics and science at the time. And Latin was a subject to be advanced studies.
Another fundamental challenge was the particular historical period in which Ludolph lived. At that time, in fact, life for Protestants was rather complicated: the Spanish Inquisition was, in fact, powerful enough to extend his long hands even in Germany. The Van Ceulen's, as Protestants, were forced, like many in the same conditions, to migrate to the most welcoming Netherlands of Prince William of Orange.
On the other hand Ludolph himself was a traveler: immediately after his father's death became a little travel first in the region of Livonia (Latvia and Estonia in our age), then to Antwerp to visit his brother Gert and then to Delft in the Netherlands where he settled for a time, since there was born one of his five daughter on May 4th, 1578.
His wife, Mariken Jansen, died in 1590, but Ludolph remarriage on June 17th of that year with Adriana Simondochter, widow of Bartholomew Cloot, accounting and math teacher, with whom he had generated eight children, for a total of 13 mouths to feed. The two families, Cloot and Van Ceulen, were in a close relations of friendship, so it's pretty obvious to imagine that the marriage between Ludolph and Adriana was the best solution to avoid losing a strong relationship.

### The tabulating machine

Herman Hollerith, an American inventor, was born on the 29th February 1860. His most famous invention was the electromechanical tabulation of data:
At the urging of John Shaw Billings, Hollerith developed a mechanism using electrical connections to trigger a counter, recording information. A key idea was that data could be encoded by the locations of holes in a card. Hollerith determined that data punched in specified locations on a card, in the now-familiar rows and columns, could be counted or sorted mechanically. A description of this system, An Electric Tabulating System (1889), was submitted by Hollerith to Columbia University as his doctoral thesis, and is reprinted in Randell's book.
In the patent of his invention, we can read:
The herein-described method of compiling statistics, which consists in recording separate statistical items pertaining to the individual by holes or combinations of holes punched in sheets of electrically non-conducting material, and bearing a specific relation to each other and to a standard, and then counting or tallying such statistical items separately or in combination by means of mechanical counters operated by electro-magnets the circuits through which are controlled by the perforated sheets, substantially as and for the purpose set forth.
In 1896 Hollerith founded The Tabulating Machine Company, that in 1911, with four others company, became the Computing-Tabulating-Recording Company, renamed International Business Machines (IBM) in 1924.

### Save research in Italy

About three weeks ago, Nature published a letter by Giorgio Parisi and signed by 69 italian scientists about the state of the research in Italy. This is the text:
We call for the European Union to push governments into keeping their research funding above subsistence level. This will ensure that scientists from across Europe can compete for Horizon 2020 research funding, not just those from the United Kingdom, Germany and Scandinavia.
Europe's research money is divided between the European Commission and national governments. The commission funds large, transnational collaborative networks in mostly applied areas of research, and the governments support small-scale, bottom-up science and their own strategic research programmes.
Some member states are not keeping their part of the bargain. Italy, for example, seriously neglects its research base. The Italian National Research Council has not overseen basic research for decades, being itself starved of resources. University funding has dwindled to a bare minimum. The ministerial initiative known as PRIN (Research Projects of National Interest) has been defunct since 2012, apart from a few limited programmes for young researchers.
This year's PRIN allocation of a euro92-million (US$100-million) funding call to cover all research areas is too little, too late. Compare this with the annual French National Research Agency's allocation of up to euro1 billion, or with Italy's euro900-million annual contribution to the EU Seventh Framework Programme that ran in 2007–13. That resulted in a net annual loss of euro300 million for Italian science. To prevent distorted development in research among EU countries, national policies must be coherent and guarantee a balanced use of resources. You can sign the petition on change.org. ### "We are hearing the universe" When we describe a geometric space, we need to define a metric, or in other words a way to measure distances: in particular in general relativity we use the tensor metric,$g_{\mu \nu}\$. Supposing the existence of gravitational waves, it is possible to calculate their effect on the radiation coming from some cosmic objects like a binary or a couple of merging black holes.
On September 14, 2015, within the first two days of Advanced LIGO's operation, the researchers detected a signal so strong that it could be seen by eye. The most intense portion of the signal lasted for about 0.2 s and was observed in both detectors, with a combined signal-to-noise ratio of 24. Fittingly, this first gravitational wave signal, dubbed GW150914, arrived less than two months before the 100-year anniversary of the publication of Einstein's general relativity theory.(1)

### A bit of math behind Ant-Man

I write also for Lo Spazio Bianco (The White Space), a web-zine about comics. Thanks to Carlo Coratelli, that contacted Sarofsky Corp., we can realize an interview with Erin Sarofsky, President, Owner and Executive Creative Director of the company. They realized some titles for Marvel Studios movies, in oparticular Ant-Man, and I proposed two questions about this movie. The questions, with Erin's answers, are the following, but you can read the complete interview here:
We loved the opening animation in Ant-Man. It seems it's inspired by the Power[s] of Ten of the Eames. Is it really like that?
Ab-So-Lutely! We were very inspired by the Eames film. Luckily though, our universe is the Marvel Universe... so being 100% accurate was not necessary.
About Ant-Man, we also loved the work with the mathematical modeling you used to produce the animation and we think it would be quite interesting to elaborate about the actual realization method. How does it work?
We actually worked in reverse. We planned our moves and then did the math after the fact. We used the surface of the grass as our zero; anything above that is positive and anything below is negative.
Andy Zazzera, CG Director on the job, and our wicked smart math guy did all the estimations. (Again, nothing is exact because our world is fake... but the math is legit).
For a while I was tempted to quest something of more detailed about the math developing, but I thought that it could not so interesting for our regular readers. However, I'm convinced that it is not for Field of Science's readers!


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Home > Standard Error > Standard Error Times 1.96

# Standard Error Times 1.96

Scenario 1. They report that, in a sample of 400 patients, the new drug lowers cholesterol by an average of 20 units (mg/dL). The following is a table of function calls that return 1.96 in some commonly used applications: Application Function call Excel NORM.S.INV(0.975) MATLAB norminv(0.975) R qnorm(0.975) scipy scipy.stats.norm.ppf(0.975) SPSS x = COMPUTE This probability is usually used expressed as a fraction of 1 rather than of 100, and written as p Standard deviations thus set limits about which probability statements can be made. Check This Out

The mean time difference for all 47 subjects is 16.362 seconds and the standard deviation is 7.470 seconds. Hyattsville, MD: U.S. As an example of the use of the relative standard error, consider two surveys of household income that both result in a sample mean of \$50,000. For example, the U.S.

If X has a standard normal distribution, i.e. These are the 95% limits. For example, suppose you work for the Department of Natural Resources and you want to estimate, with 95% confidence, the mean (average) length of all walleye fingerlings in a fish hatchery df 0.95 0.99 2 4.303 9.925 3 3.182 5.841 4 2.776 4.604 5 2.571 4.032 8 2.306 3.355 10 2.228 3.169 20 2.086 2.845 50 2.009 2.678 100 1.984 2.626 You

1. These limits were computed by adding and subtracting 1.96 standard deviations to/from the mean of 90 as follows: 90 - (1.96)(12) = 66.48 90 + (1.96)(12) = 113.52 The value
2. Archived from the original on 5 February 2008.
3. JSTOR2682923. ^ Sokal and Rohlf (1981) Biometry: Principles and Practice of Statistics in Biological Research , 2nd ed.
5. The standard deviation of the age was 9.27 years.
6. When the sample size is large, say 100 or above, the t distribution is very similar to the standard normal distribution.
8. Why not some other confidence level?
9. Table 2.

Since the samples are different, so are the confidence intervals. That is, talk about the results in terms of what the person in the problem is trying to find out -- statisticians call this interpreting the results "in the context of Exact probability test 10. Join 31 other followers Recent Posts Statistical Methods - McNemar'sTest Statistical Methods - Chi-Square and 2×2tables Statistical Methods - Standard Error and ConfidenceIntervals Epidemiology - Attributable Risk (including AR% PAR +PAR%)

X ~ N(0,1), P ( X > 1.96 ) = 0.025 , {\displaystyle \mathrm {P} (X>1.96)=0.025,\,} P ( X < 1.96 ) = 0.975 , {\displaystyle \mathrm {P} (X<1.96)=0.975,\,} and as To take another example, the mean diastolic blood pressure of printers was found to be 88 mmHg and the standard deviation 4.5 mmHg. However, it is much more efficient to use the mean 2 SD, unless the data set is quite large (say >400). With this standard error we can get 95% confidence intervals on the two percentages: 60.8 (1.96 x 4.46) = 52.1 and 69.5 39.2 (1.96 x 4.46) = 30.5 and 47.9.

Edwards Deming. If you had wanted to compute the 99% confidence interval, you would have set the shaded area to 0.99 and the result would have been 2.58. pp.748–759. As the sample size increases, the dispersion of the sample means clusters more closely around the population mean and the standard error decreases.

The distribution of the mean age in all possible samples is called the sampling distribution of the mean. Further reading Gardner, Martin J; Altman, Douglas G, eds. (1989), Statistics with confidence, BMJ Books, ISBN978-0-7279-0222-1 Retrieved from "https://en.wikipedia.org/w/index.php?title=1.96&oldid=750265657" Categories: Estimation theoryNormal distributionHidden categories: Use dmy dates from July 2013Articles with Answers chapter4 Q1.pdf 4.2 What is the 95% confidence interval for the mean of the population from which this sample count of parasites was drawn? Thus the variation between samples depends partly also on the size of the sample.

As a result, we need to use a distribution that takes into account that spread of possible σ's. his comment is here What is the sampling distribution of the mean for a sample size of 9? The system returned: (22) Invalid argument The remote host or network may be down. The shaded area represents the middle 95% of the distribution and stretches from 66.48 to 113.52.

This may sound unrealistic, and it is. This is also the standard error of the percentage of female patients with appendicitis, since the formula remains the same if p is replaced by 100-p. If X has a standard normal distribution, i.e. this contact form American Statistician.

Archived from the original on 12 February 2008. Table 2: Probabilities of multiples of standard deviation for a normal distribution Number of standard deviations (z) Probability of getting an observation at least as far from the mean (two sided As noted above, if random samples are drawn from a population, their means will vary from one to another.

## There is much confusion over the interpretation of the probability attached to confidence intervals.

Rumsey If you know the standard deviation for a population, then you can calculate a confidence interval (CI) for the mean, or average, of that population. Retrieved 2008-02-04. How many standard deviations does this represent? To calculate a CI for the population mean (average), under these conditions, do the following: Determine the confidence level and find the appropriate z*-value.

Using a sample to estimate the standard error In the examples so far, the population standard deviation σ was assumed to be known. Suppose you take a random sample of 100 fingerlings and determine that the average length is 7.5 inches; assume the population standard deviation is 2.3 inches. For the purpose of this example, the 9,732 runners who completed the 2012 run are the entire population of interest. navigate here In fact, data organizations often set reliability standards that their data must reach before publication.

Of course, T / n {\displaystyle T/n} is the sample mean x ¯ {\displaystyle {\bar {x}}} .


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RStudio addins for several common testing-related tasks during package development, such as switching between a source file and an associated test file, or running unit tests in a single test file (hotkeyable in RStudio!). testthis also provides utility function to manage tests in subdirectories of the test/testthis directory.

## Details

For details please refer to the README

## Setting the project path

Testthis uses usethis::proj_get() to detect project root of the current working directory. You can override the project root with usethis::proj_set().

## Package options

Package options can be set with options(). You can add options() to your .Rprofile to make them permanent across sessions.

testthis.sep

Default separator to use when creating test files with test_skeleton(). Defaults to _, must be either _ or -; i.e whether you want your files to be named test_foofunction.R or test-foofunction.R

testthis.integration_tests_path, testthis.acceptance_tests_path, testthis.manual_tests_path

Default paths used by the functions testthis::use_integration_tests(), testthis::test_integration(), etc...

## Testthis Tags

test_this tags are special comments that modify the behaviour of the functions supplied by this package. They are of the form #* @tag <value>. Please not that only some test_this tags really require a <value>.

Valid tags for script files in the /R/ dir (pkg/R/*.R)

• @testfile <filename>: manually specify associated test file. Should usually start with test_. This is used by test_this(), lest_this() and open_testfile().

Valid tags or test files (pkg/tests/testthat/test_*.R)

• @testing <functionname>: mark functionname as tested. Should usually go next the associated test_that() call. This is used by test_coverage().

usethis::edit_r_profile()