text:
# Gauge invariance and it's relation to gauge bosons

1. Mar 18, 2008

### PsiPhi

Hello,
I'm currently doing a project that is concerned with the hopeful discovery of the Higgs Boson at LHC. I'll be running some code that my supervisor has produced, but before that he wanted me to understand more of the physics that is behind the Higgs mechanism.

He has proposed a question to me, "Why gauge invariance forbids mass terms for gauge bosons?"

I've been reading quite a few textbooks and i'm not too sure if this could be the answer to the problem. The answer had an explanation from the Bose-Einstein condensate (BEC) context, where it talks about definite phase of the complex wavefunction describing the BEC and the existence of such a phase breaks global gauge invariance - a symmetry associated with particle conservation.
Could this particle conservation be the reason for no mass terms in gauge invariance? Another possible answer could be a chapter I read on about the Goldstone boson.

I have a feeling that the two possible explanations that i've provided for the question are way off the mark.

Could anyone recommend a textbook, particular topics about the question or websites that may shed some light on my question?

Cheers.

2. Mar 19, 2008

### Barmecides

fermion mass terms in the lagrangian $$m\bar{\psi}\psi$$ are scalar. Maybe you should try to rewrite this term as a function of $$\psi_L$$ and $$\psi_R$$ ?

3. Mar 19, 2008

### Lester

The reason why a by-hand mass term is forbidden in a gauge theory is that this spoils gauge invariance and the theory loses renormalizability. These are two fundamental ingredients for the proper working of the theory. This difficulty can be evaded through the Higgs mechanism. The proof that a gauge theory with the Higgs mechanism is renormalizable is due to Veltman and 't Hooft as I think you know.

Cheers

Jon

4. Mar 19, 2008

### QuantumDevil

Read chapers: 54, 58 and 69. of this book -> http://www.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf
In chapters 58 and 69 it is explicitly stated that theory with mass term in Lagrangian don't forbid gauge invariance!

5. Mar 19, 2008

### superpoincare

Where do you see it ?

6. Mar 19, 2008

### QuantumDevil

Page 346 below equation 58.7

7. Mar 19, 2008

### QuantumDevil

okey, you're right it's my mistake. His question was about mass term for gauge bosons not matter fields.

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## Creating Voxel Volumes Quicky in Python

Posted in Development, How To, Programming with tags , , , on 2011/06/09 by Adam Griffiths

Creating large chunks of memory is quite tricky in Python because of a number of overheads. I’ve been attempting this myself and have found a fast method to do this which combines a few handy optimisations we would need later on for a Voxel Engine anyway.

Please be aware that the code evolves along the way, so don’t blindly copy and paste the first examples =).

Also, the WordPress theme is cutting of some text. If you copy and paste it, it is still there.

This post is continued in Position Aware RLE, RLE and Part 2.

## Pinning Eclipse to the Windows 7 “start bar”

Posted in How To, Programming with tags , , , , on 2011/05/26 by Adam Griffiths

Eclipse and Windows 7 have a bit of a feud. There is no option to pin Eclipse to the start bar. If you create a shortcut you can pin the shortcut, but when the app is run it creates a second item.

Be aware that you’ll need to edit the file as Admin if you installed Eclipse into a system location. Best way is to go Start > “WordPad” > Right Click > Run as Administrator, and then open the file from inside the application.

## Building PyBullet on Windows

Posted in How To, Programming with tags , , , on 2011/05/24 by Adam Griffiths

PyBullet is a wrapper around the popular Bullet C++ physics library.

Once you’ve build the C++ Bullet library you can then build PyBullet.

The PyBullet documentation states to use the –include-dirs directive to add the Bullet include path, but it also requires the numpy include directory which didn’t resolve automatically for me. Unfortunately you can’t just append it to the –include-dirs flag.

I resolved it with the following:

cd path\to\PyBullet

set INCLUDE=%INCLUDE%;c:\path\to\python\site-packages\numpy\core\include

python setup.py build_ext –include-dirs c:\path\to\bullet\src –library-dirs c:\path\to\bullet\msvc\2008\lib\Release

Obviously you replace the \msvc\2008 directory with the appropriate one if you didn’t build the vanilla target.

After this, copy the file PyBullet\build\lib.win32-2.6\bullet\bullet.pyd to PyBullet\bullet.

Include the PyBullet\bullet directory in your Python project and import.

You should now be able to do the following:

import bullet.bullet as bullet

print help(bullet)

## Make sucks

Posted in Programming, Rant, Uncategorized with tags , , , , , on 2011/04/03 by Adam Griffiths

I used Make extensively at my last job. You know what? Make sucks!

I’ve talked about this on Reddit and the only response I ever got was fan boi’s who have memorised the manual saying “You just need to read the manual! It’s fine!”.

## Python Multiprocessing with ZeroMQ

Posted in Programming with tags , on 2011/02/22 by Adam Griffiths

Good example of ZMQ with Python and multiple processes.

Also found an issue with the example code which I have fixed below (and included a comment on the original blog post).

format: plain


text:
# What is the apparent diameter of the sun as seen from earth?

I know this may not be physically accurate.

For my purposes, basically 3D renders, I am assuming the sun rays are parallel rays from an infinitely far lamp. If the sky is clear, what size would that lamp be?

Or in other words, how big will the sun appear if we assume it is a lamp? 1cm? 5cm?

Is there a formula for converting astronomical arc minutes / arc seconds to linear dimensions?

• "Is there a formula for converting astronomical arc minutes / arc seconds to linear dimensions?" Sure: for small angles (a few degrees or less, depends on you precision needs) convert to radians and multiple by the distance. For larger angles or for maximum precision you do a little trigonometry: $r \tan \theta$ where $r$ is the distance. – dmckee Jan 10 '13 at 13:51
• In addition to Rody Oldenhuis's answer (the first sentence of which is the most telling), I suggest you ask around in the rendering/graphics community for some succinct guide to angles and angular sizes - one must certainly exist. Often infinite distances and parallel rays are misused in this way. For an analogy, you are basically asking, "How do I convert between current and voltage if the resistance of the electrical circuit is infinite?" – user10851 Jan 11 '13 at 23:27
• as an Electrical Engineer myself, I can now relate to your analogy :-) – Ayman Jan 13 '13 at 5:31

$\ \ \ \ \frac{\alpha}{2} = \arctan{ \left( \frac{695,500}{150,000,000} \right) } \approx 0.004633 \mathrm{\ radians}$
$\ \ \ \ \ \mathrm{new\ radius} = D \cdot \tan \left( \frac{\alpha}{2} \right) = D \cdot \frac{695 \frac12}{150,000}$
where $D$ is your desired distance in km.


text:
# Player

Today 按播放鍵聽點音樂

## 歌名This Time Around歌手名 Michael Jackson

### 歌詞

This time around I'll never get bit Though you really wanna fix me This time around you're making me sick Though you really wanna get me Somebody's out Somebody's out to get me They really wanna fix me, hit me But this time around I'm taking no shit Though you really wanna get me You really wanna get me He really thought he really had Had a hold on me He really thought he really had They thought they really had control of me He really thought he really had Control of me He really thought he really had They thought they really could control me This time around I'll never get bit Though you really wanna get me This time around I'm taking no s... Though you really wanna fix me Somebody's out Somebody's out to use me They really want to use me And then falsely accuse me This time around They'll take it like spit 'Cause you really can't control me You know you can't control me He really thought he really had Had a hold on me He really thought he really had They thought they really had control of me He really thought he really had A hold on me He really thought he really had They thought they really had control of me He really thought he really had Had a hold on me He really thought he really had They thought they really had control of me He really thought he really had A hold on me He really thought he really had They thought they really had control of me Listen, I've got problems of my own Flashin' cameras, taps on my phone Even in my home I ain't safe as I should be Things always missin' Maybe it could be my friends They ain't friends if they robbin' me Stoppin' me from makin' a profit, see Apology shallow like the ocean I guess I'll resort to gun totin' If I was dead broke and smokin' I'd probably be by my lonesome I'm a killer nigga I ain't jokin' Endo smoke got me choked and I'm hopin' The fool comes slippin' So I could blow'em open This time around I changed up my flow Got rid of the rocks Got Pitts by the door I've raised other peoples to watch my back Stay away from strangers So I won't slack And I know my nigga Mike like that Baby This time around yeah... He really thought he really had Control of me He really thought he really had They thought they really had control of me He really thought he really had A hold on me He really thought he really had They thought they really had control of me He really thought he really had Had a hold on me He really thought he really had They thought they really had control of me He really thought he really had Control of me He really thought he really had They thought they really had control of me He really thought he really had A hold on me He really thought he really had They thought they really had control of me This time around yeah He really thought


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# Fairness module in dalex¶

We live in a world that is getting more divided each day. In some parts of the world, the differences and inequalities between races, ethnicities, and sometimes sexes are aggravating. The data we use for modeling is in the major part a reflection of the world it derives from. And the world can be biased, so data and therefore model will likely reflect that. The introduction to this topic is well presented in Fairness and machine learning.

We propose a way in which ML engineers can easily check if their model is biased.

Fairness module is still work-in-progres and new features will be added over time.

In [1]:
import dalex as dx
import numpy as np

In [2]:
dx.__version__

Out[2]:
'1.4.0'

## Case study - german credit data¶

To showcase the abilities of the module, we will be using the German Credit Data dataset) to assign risk for each credit-seeker.

This simple task may require using an interpretable decision tree classifier.

In [3]:
from sklearn.compose import ColumnTransformer
from sklearn.pipeline import Pipeline
from sklearn.preprocessing import OneHotEncoder
from sklearn.tree import DecisionTreeClassifier

# credit data

# risk is the target
X = data.drop(columns='risk')
y = data.risk

categorical_features = ['sex', 'job', 'housing', 'saving_accounts', "checking_account", 'purpose']
numerical_features = ['age', 'duration', 'credit_amount']
categorical_transformer = Pipeline(steps=[
('onehot', OneHotEncoder(handle_unknown='ignore'))
])

preprocessor = ColumnTransformer(transformers=[
('cat', categorical_transformer, categorical_features),
('num', 'passthrough', numerical_features)
])

clf = Pipeline(steps=[
('preprocessor', preprocessor),
('classifier', DecisionTreeClassifier(max_depth=7, random_state=123))
])

clf.fit(X, y)

Out[3]:
Pipeline(steps=[('preprocessor',
ColumnTransformer(transformers=[('cat',
Pipeline(steps=[('onehot',
OneHotEncoder(handle_unknown='ignore'))]),
['sex', 'job', 'housing',
'saving_accounts',
'checking_account',
'purpose']),
('num', 'passthrough',
['age', 'duration',
'credit_amount'])])),
('classifier',
DecisionTreeClassifier(max_depth=7, random_state=123))])

We create an Explainer object to proceed with dalex functionalities.

In [4]:
exp = dx.Explainer(clf, X, y)

Preparation of a new explainer is initiated

-> data              : 1000 rows 9 cols
-> target variable   : Parameter 'y' was a pandas.Series. Converted to a numpy.ndarray.
-> target variable   : 1000 values
-> model_class       : sklearn.tree._classes.DecisionTreeClassifier (default)
-> label             : Not specified, model's class short name will be used. (default)
-> predict function  : <function yhat_proba_default at 0x00000292C4D43700> will be used (default)
-> predict function  : Accepts only pandas.DataFrame, numpy.ndarray causes problems.
-> predicted values  : min = 0.0, mean = 0.7, max = 1.0
-> model type        : classification will be used (default)
-> residual function : difference between y and yhat (default)
-> residuals         : min = -0.944, mean = 0.0, max = 0.9
-> model_info        : package sklearn

A new explainer has been created!

In [5]:
exp.model_performance().result

Out[5]:
recall precision f1 accuracy auc
DecisionTreeClassifier 0.96 0.84 0.896 0.844 0.914793

Let's say that performance is satisfying. To check if the model is biased, we will use the fairness module from dalex. Checking if the model is fair should be straightforward. Apart from the dx.Explainer, we will need 2 parameters:

• protected - array-like with subgroup values that denote a sensitive attribute (protected variable) like sex, nationality etc. The fairness metrics will be calculated for each of those subgroups and compared.
• privileged - a string representing one of the subgroups. It should be the one suspected of the most privilege.
In [6]:
# array with values like male_old, female_young, etc.
protected = data.sex + '_' + np.where(data.age < 25, 'young', 'old')

privileged = 'male_old'

## Now it is time to check fairness!¶

We use a unified dalex interface to create a fairness explanation object. Use the model_fairness() method:

In [7]:
fobject = exp.model_fairness(protected = protected, privileged=privileged)

The idea here is that ratios between scores of privileged and unprivileged metrics should be close to 1. The closer the more fair the model is. But to relax this criterion a little bit, it can be written more thoughtfully:

Where the epsilon is a value between 0 and 1, it should be a minimum acceptable value of the ratio. On default, it is 0.8, which adheres to four-fifths rule (80% rule) often looked at in hiring, for example.

In [8]:
fobject.fairness_check(epsilon = 0.8) # default epsilon

No bias was detected!

Conclusion: your model is fair in terms of checked fairness criteria.

Ratios of metrics, based on 'male_old'. Parameter 'epsilon' was set to 0.8 and therefore metrics should be within (0.8, 1.25)
TPR       ACC       PPV       FPR       STP
female_old    1.018828  1.000000  0.971963  0.921525  0.964677
female_young  0.981172  0.938824  0.891355  0.872197  0.855055
male_young    1.019874  0.977647  0.929907  0.896861  0.918392

This model cannot be called fair! Generally, each metric should be between (epsilon, 1/epsilon). Metrics are calculated for each subgroup, and then their scores are divided by the score of the privileged subgroup. That is why we omit male_old in this method. When at least 2 metrics have scores ratio outside of the epsilon range, the model may be declared unfair. In our case it cannot be decided automatically but the bias is visible and FPR (False Positive Rate) is preety important in case of risk assigning, so let's call our model unfair.

## Useful attributes¶

The result attribute is metric_scores where each row is divided by row indexed with privileged (in this case male_old).

In [9]:
# to see all scaled metric values you can run
fobject.result

Out[9]:
TPR TNR PPV NPV FNR FPR FDR FOR ACC STP
female_old 1.018828 1.063177 0.971963 1.114495 0.590909 0.921525 1.166667 0.474860 1.000000 0.964677
female_young 0.981172 1.102888 0.891355 1.071864 1.409091 0.872197 1.645833 0.670391 0.938824 0.855055
male_old 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000
male_young 1.019874 1.083032 0.929907 1.142509 0.568182 0.896861 1.416667 0.346369 0.977647 0.918392
In [10]:
# or unscaled ones via
fobject.metric_scores

Out[10]:
TPR TNR PPV NPV FNR FPR FDR FOR ACC STP
female_old 0.974 0.589 0.832 0.915 0.026 0.411 0.168 0.085 0.850 0.792
female_young 0.938 0.611 0.763 0.880 0.062 0.389 0.237 0.120 0.798 0.702
male_old 0.956 0.554 0.856 0.821 0.044 0.446 0.144 0.179 0.850 0.821
male_young 0.975 0.600 0.796 0.938 0.025 0.400 0.204 0.062 0.831 0.754

## Bias Detection Plots¶

The fairness explanation object includes plots that allow bias visualization from different perspectives:

1. fairness_check plot

2. metric_scores plot

### Fairness Check plot¶

This is a visualization of the fairness_check result.

In [11]:
fobject.plot()

Found NaN's or 0's for models: {'DecisionTreeClassifier'}
It is advisable to check 'metric_ratios'


text:
# Relation between functional measures and states in AQFT

Let $$(M,g)$$ be a globally hyperbolic spacetime and $$\phi$$ a KG field. In AQFT we consider the algebra of observables $$\mathfrak{A}$$ generated by $$\phi(f)$$ where $$f\in C^\infty_0(M)$$ is a test function.

A state is then a linear functional $$\omega : \mathfrak{A}\to\mathbb{C}$$ which satisfies $$\omega(1)=1$$ and $$\omega(a^\ast a)\in [0,+\infty)$$ for all $$a\in \mathfrak{A}$$.

Now, a state allows for the computation of the $$n$$-point functions $$W_n(f_1,\dots,f_n)=\langle \phi(f_1)\cdots \phi(f_n)\rangle_\omega=\omega(\phi(f_1)\cdots \phi(f_n)).\tag{1}$$

Now in usual treatments, one writes via functional integration

$$\langle \operatorname{T}[\phi(x_1)\cdots \phi(x_n)]\rangle=\int \varphi(x_1)\cdots \varphi(x_n)\dfrac{e^{iS[\varphi]}\mathfrak{D}\varphi}{Z}\tag{2}$$

Now if we make sense of $$Z^{-1}e^{-S[\phi]}\mathfrak{D}\phi$$ as a functional measure after passing to Euclidean time (which is possible at least for the free theory), there seems to be a one-to-one relation between time-ordered correlators and functional measures. In other words, there seems to be a one-to-one relation between the restriction of $$\omega$$ to time-ordered products and functional measures.

This makes me wonder, can this be extended to the state $$\omega$$ itself? Is there some sort of correspondence between states and functional measures so that, for instance, a gaussian state would be in correspondence to a gaussian measure?

Or is this relation between time-ordered products and measures not even one-to-one or even well-defined?

What is the relation between algebraic states and functional measures, if there is any relation?

• Very interesting question. I am however wondering if this can be made sufficiently rigorous even for the vacuum state: passing from Euclidean time to Lorentzian is not completely trivial, since the Osterwalder-Schroder theorem contains some rather non-trivial condition on Euclidean correlators that is needed to ensure Wightman functions are distributions (I have in mind the O-S II paper). I don't know anything about whether this condition can be interpreted naturally in terms of functional measures. – Peter Kravchuk Jul 6 at 19:24
• Identity (2) is wrong because the left-hand side is not symmetric if interchanging $f_k$ and $f_k$ (a causal propagator takes place), whereas the right-hand side is symmetric by construction. Passing to the Euclidean time is a problem in a generic curved globally hyperbolic spacetime... Maybe in static spacetimes the question could make some sense, but it is not completely clear how the correlation functions are related with the algebraic state. For Gaussian states it seems to make sense if the states are static. – Valter Moretti Jul 7 at 12:33
• @ValterMoretti, I think the question is interesting even in flat space. I also do not understand your comment about identity (2) because there are no $f_k$ in it. – Peter Kravchuk Jul 7 at 22:51
• Yes, you are right sorry, just replace $f_k$ for $x_k$ in my commnent. Indeed, I include Minkowki spacetime in the class of static spacetimes. – Valter Moretti Jul 8 at 5:15
• What I wanted to remark is that, in the Lorenzian case, the left-hand side of (2) needs a T product. – Valter Moretti Jul 8 at 5:26


text:
# Area of ring section closed within a rectangle

I wish to find out the area of a section of a ring which can be acted on my a rectangle 100mm wide by 60mm height. I know the inner diameter,ID and outer diameter,OD of the ring and the width of the ring $\frac{OD-ID}{2}$ will always be below 60mm so the ring will always be within the rectangle.

I thought it would be a simple take one section area away from the other... but it isn't.

http://bayimg.com/EagDkAAeE

-
I worked this out in the end by myself.. easier than I thought. Work out the segments areas of both the ID and OD circle. Work out the rectangle area of the remaining central area of the circle. Then minus the larger area from the smaller area. –  Dan Nov 23 '12 at 15:56


text:
The first session in our statistical learning with Python series will briefly touch on some of the core components of Python’s scientific computing stack that we will use extensively later in the course. We will not only introduce two important libraries for data wrangling, numpy and pandas, but also show how to create plots using matplotlib. Please note that this is not a thorough introduction to these libraries; instead, we would like to point out what basic functionality they provide and how they differ from their counterparts in R.

But before we get into the details we will briefly describe how to setup a Python environment and what packages you need to install in order to run the code examples in this notebook.

Requirements

Python environment

We strongly recommend that you use a bundled Python distribution such as Anaconda. We have assembled a quick installation guide for Mac, Linux, and Windows in a previous blog post.

To run the R examples in this code you also need:

rpy2>=2.3.1

You can find instructions how to install rpy2 here . If you have an working R environment on your machine the following command should install rpy2:

$pip install -U rpy2 To test if rpy2 was installed correctly run: $ python -m 'rpy2.tests'

If you run on Anaconda and it complains that it misses libreadline.so please install the following conda package:

$conda install python=2.7.5=2 IPython IPython is an interactive computing environment for Python. It is a great tool for interactive data analysis and programming in general. Amongst other things it features a web-based notebook server that supports code, documentation, inline plots, and much more. In fact, all blog posts in this series will be written using IPython notebooks with the advantage that you can simply download it from here and either run it locally or view it on nbviewer. Manipulating Data The goal of this session is to get familiar with the basics of how to work with data in Python. The basic data containers that are used to manipulate data in Python are n-dimensional arrays that act either as vectors, matrices, or tensors. In contrast to statistical computing environments like R, the fundamental data structures for data analysis in Python are not built into the computing environment but are available via dedicated 3rd party libraries. These libraries are called numpy and pandas. Numpy Numpy is the lingua-franca in the Python scientific computing ecosystem. It basically provides an n-dimensional array object that holds elements of a specific dtype (eg. numpy.float64 or numpy.int32). Most packages that we will discuss in this series will directly operate on arrays. Numpy also provides common operations on arrays such as element-wise arithmetic, indexing/slicing, and basic linear algebra (dot product, matrix decompositions, …). Below we show some basic working with numpy arrays: In [36]: from __future__ import division # always use floating point division import numpy as np # convention, use alias np # a one dimensional array x = np.array([2, 7, 5]) print 'x:', x # print x # a sequence starting from 4 to 12 with a step size of 3 y = np.arange(4, 12, 3) print 'y:', y # element-wise operations on arrays print 'x + y:', x + y print 'x / y:', x / y print 'x ^ y:', x ** y # python uses ** for exponentiation x: [2 7 5] y: [ 4 7 10] x + y: [ 6 14 15] x / y: [ 0.5 1. 0.5] x ^ y: [ 16 823543 9765625] If you need any help on operations such as np.arange you can access its documentation by either typing help(np.arange) or — if you use IPython — write a '?' after the command: np.arange?. You can index and slice an array using square brackets []. To slice an array, numpy uses Python’s slicing syntax x[start:end:step] where step is the step size which is optional. If you omit start or end it will use the beginning or end, respectively. Python uses exclusive semantics meaning that the element with position end is not included in the result. Indexing can be done either by position or by using a boolean mask: In [37]: print x[1] # second element of x print x[1:3] # slice of x that includes second and third elements print print x[-2] # indexing using negative indices - starts from -1 print x[-np.array([1, 2])] # fancy indexing using index array print print x[np.array([False, True, True])] # indexing using boolean mask 7 [7 5] 7 [5 7] [7 5] For two or more dimensional arrays we just add slicing/indexing arguments, to select the whole dimension you can simply put a colon (:). In [38]: # reshape sequence to 2d array (=matrix) where rows hold contiguous sequences # then transpose so that columns hold contiguous sub sequences z_temp = np.arange(1, 13).reshape((3,4)) print "z_temp" print z_temp print # transpose z = z_temp.T print "z = z_temp.T (transpose of z_temp)" print z print # slicing along two dimensions a = z[2:4, 1:3] print "a = z[2:4, 1:3]" print a print # slicing along 2nd dimension b = z[:, 1:3] print "b = z[:, 1:3]" print b print # first column, returns 1d array c = z[:, 0] print "c = z[:, 0]" print c # one dimensional print # first column but return 2d array (remember: exclusive semantics) cc = z[:, 0:1] print "cc = z[:, 0:1]" print cc # two dimensional; column vector z_temp [[ 1 2 3 4] [ 5 6 7 8] [ 9 10 11 12]] z = z_temp.T (transpose of z_temp) [[ 1 5 9] [ 2 6 10] [ 3 7 11] [ 4 8 12]] a = z[2:4, 1:3] [[ 7 11] [ 8 12]] b = z[:, 1:3] [[ 5 9] [ 6 10] [ 7 11] [ 8 12]] c = z[:, 0] [1 2 3 4] cc = z[:, 0:1] [[1] [2] [3] [4]] To get information on the dimensionality and shape of an array you will find the following methods useful: In [39]: print z.shape # number of elements along each axis (=dimension) print z.ndim # number of dimensions print z[:, 0].ndim # return first column as 1d array (4, 3) 2 1 In numpy, slicing will return a new array that is basically a view on the original array, thus, it doesn’t require copying any memory. Indexing (in numpy often called fancy indexing), on the other hand, always copies the underlying memory. Differences between R and Python R differentiates between vectors and matrices whereas in numpy both are unified by the n-dimensional numpy.ndarray class. There are a number of crucial differences in how indexing and slicing are handled in Python vs. R. Note that the examples below require the Python package rpy2 to be installed. In [66]: # allows execution of R code in IPython try: %load_ext rmagic except ImportError: print "Please install rpy2 to run the R/Python comparision code examples" The rmagic extension is already loaded. To reload it, use: %reload_ext rmagic Python uses 0-based indexing whereas indices in R start from 1: In [41]: x = np.arange(5) # seq has excl semantics x[0] Out[41]: 0 In [65]: %%R # tells IPython that the following lines will be R code x <- seq(0, 4) # seq has incl semantics print(x[1]) [1] 0 Python uses exclusive semantics for slicing whereas R uses inclusive semantics: In [43]: x[0:2] # doesnt include index 2 Out[43]: array([0, 1]) In [44]: %%R x <- seq(0, 4) # seq has incl semantics print(x[1:2]) # includes index 2 [1] 0 1 Negative indices have different semantics: in Python they are used to index from the end on an array whereas in R they are used to drop positions: In [45]: x[-2] # second element from the end Out[45]: 3 In [46]: %%R x <- seq(0, 4) # seq has incl semantics print(x[-2]) # drop 2nd position, ie 1 [1] 0 2 3 4 If you index on a specific position of a matrix both R and Python will return a vector (ie. array with one less dimension). If you want to retain the dimensionality, R supports a drop=FALSE argument whereas in Python you have to use slicing instead: In [47]: X = np.arange(4).reshape((2, 2)).T # 2d array X[0:1, :] # still 2d array - slice selects one element Out[47]: array([[0, 2]]) In [48]: %%R X = matrix(seq(0, 3), 2, 2) print(X[1, , drop=FALSE]) # use drop=FALSE  [,1] [,2] [1,] 0 2 Pandas As numpy, pandas provides a key data structure: the pandas.DataFrame; as can be inferred from the name it behaves very much like an R data frame. Pandas data frames address three deficiencies of arrays: 1. they hold heterogenous data; each column can have its own numpy.dtype, 2. the axes of a DataFrame are labeled with column names and row indices, 3. and, they account for missing values which this is not directly supported by arrays. Data frames are extremely useful for data munging. They provide a large range of operations such as filter, join, and group-by aggregation. Below we briefly show some of the core functionality of pandas data frames using some sample data from the website of the book “Introduction to Statistical Learning”: In [49]: import pandas as pd # convention, alias pd # Load car dataset auto = pd.read_csv("http://www-bcf.usc.edu/~gareth/ISL/Auto.csv") auto.head() # print the first lines Out[49]: mpg cylinders displacement horsepower weight acceleration year origin name 0 18 8 307 130 3504 12.0 70 1 chevrolet chevelle malibu 1 15 8 350 165 3693 11.5 70 1 buick skylark 320 2 18 8 318 150 3436 11.0 70 1 plymouth satellite 3 16 8 304 150 3433 12.0 70 1 amc rebel sst 4 17 8 302 140 3449 10.5 70 1 ford torino One of the first things you should do when you work with a new dataset is to look at some summary statistics such as mean, min, max, the number of missing values and quantiles. For this, pandas provides the convenience method pd.DataFrame.describe: In [50]: auto.describe() Out[50]: mpg cylinders displacement weight acceleration year origin count 397.000000 397.000000 397.000000 397.000000 397.000000 397.000000 397.000000 mean 23.515869 5.458438 193.532746 2970.261965 15.555668 75.994962 1.574307 std 7.825804 1.701577 104.379583 847.904119 2.749995 3.690005 0.802549 min 9.000000 3.000000 68.000000 1613.000000 8.000000 70.000000 1.000000 25% 17.500000 4.000000 104.000000 2223.000000 13.800000 73.000000 1.000000 50% 23.000000 4.000000 146.000000 2800.000000 15.500000 76.000000 1.000000 75% 29.000000 8.000000 262.000000 3609.000000 17.100000 79.000000 2.000000 max 46.600000 8.000000 455.000000 5140.000000 24.800000 82.000000 3.000000 You can use the dot . or bracket [] notation to access columns of the dataset. To add new columns you have to use the bracket [] notation: In [51]: mpg = auto.mpg # get mpg column weight = auto['weight'] # get weight column auto['mpg_per_weight'] = mpg / weight print auto[['mpg', 'weight', 'mpg_per_weight']].head()  mpg weight mpg_per_weight 0 18 3504 0.005137 1 15 3693 0.004062 2 18 3436 0.005239 3 16 3433 0.004661 4 17 3449 0.004929 Columns and rows of a data frame are labeled, to access/manipulate the labels either use pd.DataFrame.columns or pd.DataFrame.index: In [52]: print(auto.columns) print(auto.index[:10]) Index([u'mpg', u'cylinders', u'displacement', u'horsepower', u'weight', u'acceleration', u'year', u'origin', u'name', u'mpg_per_weight'], dtype=object) Int64Index([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], dtype=int64) Indexing and slicing work similar as for numpy arrays just that you can also use column and row labels instead of positions: In [53]: auto.ix[0:5, ['weight', 'mpg']] # select the first 5 rows and two columns weight and mpg Out[53]: weight mpg 0 3504 18 1 3693 15 2 3436 18 3 3433 16 4 3449 17 5 4341 15 For more information on pandas please consult the excellent online documentation or the references at the end of this post. Differences between R and Python The major difference between the data frame in R and pandas from a user’s point of view is that pandas uses an object-oriented interface (ie methods) whereas R uses a functional interface: In [54]: auto.head(2) Out[54]: mpg cylinders displacement horsepower weight acceleration year origin name mpg_per_weight 0 18 8 307 130 3504 12.0 70 1 chevrolet chevelle malibu 0.005137 1 15 8 350 165 3693 11.5 70 1 buick skylark 320 0.004062 In [55]: # this command pushes the pandas.DataFrame auto to R-land %Rpush auto In [56]: %%R auto = data.frame(auto) print(head(auto, 2))  mpg cylinders displacement horsepower weight acceleration year origin 0 18 8 307 130 3504 12.0 70 1 1 15 8 350 165 3693 11.5 70 1 name mpg_per_weight 0 chevrolet chevelle malibu 0.005136986 1 buick skylark 320 0.004061738 Below is a table that shows some of methods that pandas DataFrame provides and the corresponding functions in R: Tables Pandas R first rows auto.head() head(auto) summary statistics auto.describe() summary(auto) column access auto['mpg'] auto$mpg
row access auto.iloc[0] auto[1,]
column deletion del auto['mpg'] auto$mpg = NULL column dtypes auto.dtypes str(auto) unique elements auto.year.unique() unique(auto$year)
value counts auto.year.value_counts() table(auto\$year)

Plotting using Matplotlib

Like R there are several different options for creating statistical graphics in Python, including Chaco and Bokeh, but the most common plotting libary is Matplotlib. Here is a quick introduction on how to create graphics in Python similar to those created using the base R functions.

In [57]:
%pylab inline

import pandas as pd
import matplotlib.pyplot as plt

data = np.random.randn(500)  # array of 500 random numbers
Populating the interactive namespace from numpy and matplotlib

To make a histogram you can use the hist() command

In [58]:
plt.hist(data)
plt.ylabel("Counts")
plt.title("The Gaussian Distribution")
Out[58]:
<matplotlib.text.Text at 0x5e46d50>

Like R, you can specify various options to change the plotting behavior. For example, to make a histogram of frequency rather than of raw counts you pass the argument normed=True

You can also easily make a scatter plot

In [59]:
x = np.random.randn(50)
y = np.random.randn(50)

plt.plot(x, y, 'bo')  # b for blue, o for circles
plt.xlabel("x")
plt.ylabel("y")
plt.title("A scatterplot")
Out[59]:
<matplotlib.text.Text at 0x5e5a890>

Matplotlib supports Matlab-style plotting commands, where you can quickly specify color (b for blue, r for red, k for black etc.) and a symbol for the plotting character ('-' for solid lines, '--' for dashed lines, '*' for stars, …)

In [60]:
s = np.arange(11)
plt.plot(s, s ** 2, 'r--')
Out[60]:
[<matplotlib.lines.Line2D at 0x687bd50>]

There is also a scatter command that also creates scatterplots

In [61]:
plt.scatter(x, y)
Out[61]:
<matplotlib.collections.PathCollection at 0x667a390>

Boxplots are very useful to compare two distributions

In [62]:
plt.boxplot([x, y])  # Pass a list of two arrays to plot them side-by-side
plt.title("Two box plots, side-by-side")
Out[62]:
<matplotlib.text.Text at 0x6155410>

Matplotlib and Pandas

Pandas provides a convenience interface to matplotlib, you can create plots by using the pd.DataFrame.plot() method

In [63]:
# create a scatterplot of weight vs "miles per galone"
auto.plot(x='weight', y='mpg', style='bo')
plt.title("Scatterplot of weight and mpg")

# create a histogram of "miles per galone"
plt.figure()
auto.hist('mpg')
plt.title("Histogram of mpg (miles per galone)")
Out[63]:
<matplotlib.text.Text at 0x64e8190>
<matplotlib.figure.Figure at 0x6890c10>

Finally, pandas has built in support for creating scatterplot matrices and much more.

In [64]:
from pandas.tools.plotting import scatter_matrix
_ = scatter_matrix(auto[['mpg', 'cylinders', 'displacement']], figsize=(14, 10))

Matplotlib has a rich set of features to manipulate and style statistical graphics. Over the next few weeks we will cover many of them to help you make charts that you find visually appealing, but for now this should be enough to get you up and running in Python.

Python Scientific Lecture Notes

For a more in-depth discussion of the Python scientific computing ecosystem we strongly recommend the Python Scientific Lecture Notes. The lecture notes contain lots of code examples from applied science such as signal processing, image processing, and machine learning.

Python for Data Analysis

Wes Mckinney, the original author of pandas, wrote a great book on using Python for data analysis. It is not only the primary reference to pandas but also features a concise yet profound introduction to Python, numpy and matplotlib.

This post was written by Peter Prettenhofer and Mark Steadman.  Please post any feedback, comments, or questions below or send us an email at <firstname>@datarobot.com.

6 Responses to “Introduction to Python for Statistical Learning”

1. Ian Atkinson

Hi Peter,

In the text after In[37], you give
x ^ y: [ 16 823543 9765625]
It should be
x ** y: [ 16 823543 9765625]
The ^ operator gives a bitwise XOR.

Sorry to be such a quibbler!

– Ian.

Hello Ian,

Thanks for your feedback! Notation is always tricky and the fact that different programming languages use different notation does not help with the confusion. When python 3 is universal with built in Unicode support we can also easily print a superscript y on the command line, which would make clear to the user what is intended. I personally think that the caret notation is more common than the double asterisk notation to indicate exponentiation, and it allows us to emphasize that python uses ** rather than ^ for exponentiation, unlike, for example, R, which supports both. We could have written the comment as part of the print statement, which would have avoided the confusion between the intent of the statement and the line as a verbatim copy of the python code that was run.

Mark


text:
The San Antonio Family Diabetes/Gallbladder Study was initiated to identify susceptibility genes for type 2 diabetes. Evidence was previously reported of linkage to diabetes on 10q with suggestive evidence on 3p and 9p in a genome-wide scan of 440 individuals from 27 pedigrees ascertained through a single diabetic proband. Subsequently, the study was expanded to include 906 individuals from 39 extended Mexican-American pedigrees, two additional examination cycles ∼5.3 and 7.6 years after baseline, and genotypes for a new set of genome-wide markers. Therefore, we completed a second genome-wide linkage scan. Using information from a participant’s most recent exam, the prevalence of diabetes in nonprobands was 21.8%. We performed genome-wide variance components-based genetic analysis on the discrete trait diabetes using a liability model and on the quantitative Martingale residual obtained from modeling age of diabetes diagnosis using Cox proportional hazard models. Controlling for age and age2, our strongest evidence for linkage to the trait diabetes and the quantitative Martingale residual was on chromosome 3p at marker D3S2406 with multipoint empirical logarithm of odds scores of 1.87 and 3.76, respectively. In summary, we report evidence for linkage to diabetes on chromosome 3p in a region previously identified in at least three independent populations.

Type 2 diabetes, a common multifactorial metabolic disease, is caused by both environmental and genetic factors. The incidence of type 2 diabetes continues to rise and increasingly affects individuals of all ages across all ethnic groups (1, 2). Individuals from certain ethnic groups including Mexican Americans have an increased propensity toward developing type 2 diabetes. Furthermore, not only is the risk of diabetes two- to threefold higher in Mexican Americans than in non-Hispanic whites, but diabetes is more severe and more deadly in Mexican Americans than in non-Hispanic whites (3, 4).

Genetic studies consistently indicate that diabetes is familial in nature. However, despite >20 genome-wide linkage analyses of type 2 diabetes, the genes influencing susceptibility to the common forms of type 2 diabetes remain largely unknown (521). Moreover, the increased risk and severity of type 2 diabetes in Mexican Americans may indicate an increased genetic susceptibility (22, 23). Previously, in the San Antonio Family Diabetes Study, an extended pedigree study composed of Mexican-American families ascertained through a single proband with diabetes, Duggirala et al. (24) reported evidence of linkage to diabetes on 10q and suggestive evidence of linkage to diabetes on 3p and 9p. This earlier study was based on a single clinical examination of 440 individuals from 27 extended pedigrees with genotypic information from 379 microsatellite markers across the genome (24). Subsequently, the study has been expanded to include two additional examination cycles during which previous participants were invited to return, existing pedigrees were expanded, and new pedigrees were added. In addition, the Center for Inherited Disease Research (CIDR) has recently completed genotyping of 382 highly polymorphic markers distributed throughout the autosomes at ∼10-cM intervals on over 96% of the 906 individuals who have participated in at least one clinical examination. Therefore, our objective was to perform a genome-wide linkage analysis of type 2 diabetes in the updated San Antonio Family Diabetes/Gallbladder Study (SAFDGS).

The SAFDGS design and population.

The baseline SAFDGS population consisted of 579 individuals distributed across 31 extended Mexican-American families recruited and first examined between 1991 and 1994. Probands in the SAFDGS were low-income Mexican-American men and women identified as having diabetes in a prior population-based epidemiological study, the San Antonio Heart Study (25). In addition to having diabetes, the proband was required to have at least six first-degree relatives, excluding parents, who were at least 18 years old and living in the San Antonio area. First-, second-, and third-degree relatives of the proband were invited to participate.

Of the original 579 participants, 500 individuals returned for the first follow-up examination of the SAFDGS population that occurred between 1996 and 1999. In addition, 67 family members who were unable to participate in the baseline SAFDGS were recruited and participated in the first recall of the SAFDGS. Finally, between 1998 and 2001, a second follow-up examination occurred (known as the San Antonio Family Gallbladder Study), during which 265 new participants were recruited including both additional family members of the 31 original families as well as 8 additional families. In summary, there are 911 SAFDGS participants from 39 families who have participated in at least one of three clinic examinations. Of these 911 participants, 5 people have been excluded either because of chromosomal abnormalities or because of irresolvable pedigree uncertainties; hence, there are a total of 906 SAFDGS participants with phenotypic information available. The institutional review board of the University of Texas Health Science Center at San Antonio approved the study, and all subjects gave informed consent.

SAFDGS baseline and follow-up examinations.

Each examination consisted of a standardized medical examination that included interviews, anthropometry, a fasting venipuncture, and an oral glucose tolerance test. Trained interviewers obtained information on medical history and medication use. Examinations occurred in the morning, after participants had fasted the 12 h before their examination. Measurement of BMI, waist circumference, fasting plasma glucose, and plasma glucose 2 h after a standardized oral glucose load have been previously described in detail (26).

Two definitions of diabetes were used. The first, referred to throughout this study as diabetes (fasting), required a fasting plasma glucose ≥7.0 mmol/l (126 mg/dl) (27), whereas the second, referred to as diabetes (World Health Organization [WHO]), required either a fasting plasma glucose ≥7.0 mmol/l (126 mg/dl) or a 2-h glucose after an oral glucose tolerance test ≥11.1 mmol/l (200 mg/dl) (28). For each definition, participants who did not meet these criteria but who self-reported physician-diagnosed diabetes and who reported current therapy with either oral antidiabetic agents or insulin were also considered to have diabetes in this investigation. For each definition, information from a participant’s most recent examination was used to define his or her diabetes status with the caveat that individuals diagnosed with diabetes who when later examined no longer had diabetes were considered missing. Finally, age of diabetes diagnosis was modeled as a proxy for age of diabetes onset: for previously diagnosed diabetic participants, self-reported age of diagnosis was used as the age of onset; for diabetic participants initially diagnosed at a SAFDGS examination, the participant’s reported age at that examination was used as the age of onset.

Genotyping and calculation of multipoint identities by descent.

A complete genomic scan using 382 highly polymorphic markers distributed throughout the autosomes at ∼10-cM intervals has been completed by the CIDR on 872 of 906 participants. In 39 families, genotype data were available across five generations in 5 families, four generations in 19 families, three generations in 11 families, and two generations in 4 families. Genotype data were cleaned for both Mendelian and spurious double-recombinant errors using SimWalk2 (29). SimWalk2 provides a probability of error for each individual for each genotype based on the genotypes of other family members, the allele frequencies, and the marker map. For Mendelian error cleaning, genotypes were blanked at descending error probability thresholds, in increments of 0.1, until no more inconsistencies existed, resulting in the elimination of 1,247 of 364,934 genotypes, a blanking rate of 0.34%. For double recombinant cleaning, all genotypes with an error probability of ≥0.25 were blanked, resulting in the elimination of an additional 436 genotypes, a blanking rate of 0.12%. MultiMap/CRI-MAP (30, 31) was used to construct sex-averaged marker maps using the cleaned genotype data. Allele frequencies were estimated by maximum likelihood methods implemented in SOLAR (sequential oligogenic linkage analysis routines) (32), and matrices of multipoint identity-by-descent sharing probabilities were estimated using Markov chain Monte Carlo methods implemented by LOKI (33).

In addition, in instances in which a statistically significant linkage signal was identified, additional microsatellite markers were typed in the 705 individuals for whom we have large quantities of DNA because of the lymphoblastoid cell lines available for these individuals. The locations of the highly polymorphic microsatellite markers used were determined from the UCSC Human Genome Browser July 2003 assembly. PCR primer sequences for each marker were obtained from the genome database. Forward primers were labeled with fluorescent dyes. Multiplex PCR was performed using 240 ng pf high–molecular weight genomic DNA isolated from lymphoblastoid cell lines. The resulting PCR products were run on an ABI 3100 Avant Genetic Analyzer, and alleles were called using the GeneMapper analysis software.

Statistical analyses.

Using a variance decomposition approach implemented in SOLAR (32), we performed genetic analysis on the discrete trait diabetes, using a liability threshold model (6). This approach assumes that an individual belongs to a specific disease class if an underlying genetically determined risk or liability exceeds a certain threshold on a normally distributed liability curve. The liability is assumed to have an underlying multivariate normal distribution. The correlation in liability between individuals i and j is given by:

${\rho}_{\mathrm{ij}}\ {=}\ {\Pi}_{\mathrm{mij}}h^{2}_{\mathrm{m}}\ {+}\ 2{\Phi}\ _{\mathrm{ij}}h^{2}\ {+}\ I_{\mathrm{ij}}e^{2}$

where ρij is the correlation in liability to disease between individuals i and j; Πmij is the proportion of alleles that individuals i and j share identical by decent at a marker, m, linked to a quantitative trait locus; h2m is the heritability attributed to the quantitative trait locus near the marker locus; Φij is the kinship coefficient for individuals i and j; h2 is the heritability attributed to residual additive polygenic effects; Iij is an identity matrix (which equals 1 if i equals j and 0 if i does not equal j); and e2 is equal to 1 − (h2m + h2). Maximum likelihood techniques were used to estimate variance components and covariate effects simultaneously.

In addition to performing genetic analysis on the discrete trait diabetes, we used SAS to model age of diabetes diagnosis as a proxy for age of diabetes onset with a Cox proportional hazards model (34). In the Cox proportional hazards models, self-reported age of diagnosis was used as the time of the event for previously diagnosed diabetic participants; for diabetic participants initially diagnosed at a SAFDGS examination, the participant’s reported age at that examination was used as the time of the event; and finally, nondiabetic participants were censored at their most recent SAFDGS examination age. In addition, participants were left censored at their age 1 day before they entered the study or, in the case of an individual who entered with diabetes, 1 day before their age of reported diabetes onset. Moreover, the left censoring age (with a quadratic term) was adjusted for in the Cox proportional hazards model. Finally, using SOLAR, we performed standard multipoint variance components linkage analysis on the Martingale residual from the Cox proportional hazards model, a quantitative trait (34). In summary, Martingale residuals obtained from Cox proportional hazards models were used to model age of diabetes onset while accounting for age of entry into the study. Additional covariates of interest were incorporated into the SOLAR model.

A total of 906 individuals, aged 17–97 years, were examined. Individuals from 39 families and 32 unrelated individuals (i.e., individuals without offspring but married into the 39 families) were included. The unrelated individuals were included to help evaluate covariate effects. The median number of members per pedigree for the 39 families was 17 and ranged from 2 to 64. Of the 906 participants, 10 were excluded from the analyses for diabetes (fasting) because they were initially diagnosed but at a subsequent examination failed to meet the criteria, whereas an additional 5 were excluded because of missing information (corresponding numbers for diabetes [WHO] were 13 and 8, respectively). Diabetes (fasting) was identified in 108 of 503 nonproband women and 78 of 349 nonproband men, whereas diabetes (WHO) was identified in 125 of 499 nonproband women and 90 of 347 nonproband men. For the trait diabetes (fasting) excluding parent-offspring pairs (which are uninformative for linkage), there are 9,505 relative pairs, with varying degrees of relatedness. Of these 9,505 relative pairs, 438 are affected-affected pairs, 5,913 are unaffected-unaffected pairs, and 3,154 are affected-unaffected pairs. Table 1 includes characteristics of the study participants with and without diabetes (fasting). Overall, point estimates indicate participants with diabetes (fasting) were older and had a higher BMI and a larger waist circumference (Table 1).

The age- and age2-adjusted heritability (h2 ± SE) for diabetes (fasting) was 0.52 ± 0.121 (P < 0.0001), whereas the heritability for the Martingale residual (fasting) was 0.18 ± 0.050 (P < 0.0001). The Martingale residual for diabetes (fasting) age of onset meets the prerequisites of the variance component method used (all residuals within 4 SD; kurtosis, −0.49; residual kurtosis, −0.48; and skewness, 0.53). The results of the multipoint genome-wide linkage scans of the trait diabetes (fasting), adjusted for age and age2, and the Martingale residual (fasting) are shown in Fig. 1.

The results of the multipoint genome-wide linkage scan of the trait diabetes (fasting) adjusted for age and age2 revealed several potentially interesting chromosomal regions: the first on chromosome 3 and the second on chromosome 15. Moreover, the results of the multipoint genome-wide linkage scan of the Martingale residual (fasting) revealed a single statistically significant empirical LOD of 3.76 (P = 0.00002) on chromosome 3p near marker D3S2406, in the same region implicated in the analysis of the discrete trait diabetes (Fig. 2). Additional areas identified as potentially interesting are found in Table 2. Moreover, birth decade and waist circumference were covariates significantly associated with the trait diabetes (fasting) as well as the Martingale residual (fasting) modeling age of diabetes onset. Empirical LODs corresponding to the region on chromosome 3p identified as potentially interesting were slightly attenuated after further adjustment for birth decade (trait diabetes, empirical LOD = 1.93; Martingale residuals, empirical LOD = 3.55) and somewhat attenuated after further adjustment for both birth decade and waist circumference (trait diabetes, empirical LOD = 1.29; Martingale residuals, empirical LOD = 2.68).

The age- and age2-adjusted heritability (h2 ± SE) for diabetes (WHO) was 0.56 ± 0.112 (P < 0.0001), whereas the heritability for the Martingale residual (WHO) was 0.23 ± 0.051 (P < 0.0001). The Martingale residual for diabetes (WHO) age of onset meets the prerequisites of the variance component method used (within 4 SD; kurtosis, −0.69; residual kurtosis, −0.68; and skewness, 0.38). The results of the multipoint genome-wide linkage scans of the trait as well as the Martingale residual for diabetes (WHO) were similar to those for diabetes (fasting) and are reported in Table 2. Moreover, empirical LODs corresponding to the region on chromosome 3p previously identified were slightly attenuated after further adjustment for birth decade (trait diabetes, empirical LOD = 2.07; Martingale residuals, empirical LOD = 3.03) and somewhat attenuated after further adjustment for both birth decade and waist circumference (trait diabetes, empirical LOD = 1.40; Martingale residuals, empirical LOD = 2.15).

In addition to multipoint linkage analysis, two-point linkage analysis with each marker was completed for diabetes (fasting and WHO) adjusted only for age and age2 as well as the (fasting and WHO) Martingale residual traits (Table 3). Furthermore, to follow up the linkage signal on 3p, we typed 10 additional microsatellite markers in the 705 individuals for whom we have ample DNA through lymphoblastoid cell lines. The markers encompassed a region of 3p from 49.2 to 81.9 Mb. Two-point empirical LODs for the 10 newly typed microsatellite markers on chromosome 3 in the region of interest are found in Table 3. Moreover, multipoint analysis including these additional markers resulted in a linkage profile that was similar to the original but slightly narrowed. The peak linkage signal for the (fasting) Martingale residual trait was at 112 cM (empirical LOD of 3.55) using the new map.

In summary, our strongest evidence for linkage to diabetes occurs on chromosome 3p near marker D3S2406 (103 cM, Marshfield map). This signal is present regardless of whether the diabetes definition includes or excludes information on 2-h glucose levels and is strengthened by accounting for age of diabetes onset.

For the diabetes (fasting) age of onset model, the 1.5-LOD support interval (∼95% CI) around our peak spanned ∼20 cM from marker D3S4542 to marker D3S4529 before typing the 10 additional microsatellite markers. However, with the 10 additional microsatellite markers, the 1.5-LOD support interval was narrowed and spans an ∼10-cM region from marker D3S1296 to marker D3S3633 that harbors ∼40 identified (known and predicted) genes.

The San Antonio Family Diabetes Study has previously reported evidence for linkage to type 2 diabetes and nominal evidence for linkage to fasting-specific insulin in the 3p region implicated in this expanded SAFDGS. Specifically, Duggirala et al. (24) in 1999 reported a multipoint LOD of 2.56 for diabetes near marker GATA128C02 (Marshfield map, 112 cM) in 440 San Antonio Family Diabetes study participants who were examined at baseline and in 2001 reported a multipoint LOD ≥ 1.00 for fasting-specific insulin in 310 nondiabetic San Antonio Family Diabetes study participants at the same locus (38). Moreover, the San Antonio Family Heart Study comprised of a similar study population reports a LOD for serum insulin in 391 nondiabetic individuals of 3.07 on chromosome 3p in the region flanked by D3S1600 (86.0 cM, Marshfield map) and D3S1285 (91.0 cM, Marshfield map) (39). Similar to the SAFDGS, the San Antonio Family Heart study is an extended pedigree study of low-income Mexican Americans in San Antonio, Texas. Differences between the two studies include ascertainment without regard to health status of the proband in the San Antonio Family Heart Study as well as a younger mean age at baseline in the San Antonio Family Heart Study.

Moreover, previously in the San Antonio Family Diabetes Study, Duggirala et al. (24) not only reported suggestive evidence of linkage to diabetes on 3p but reported significant evidence of linkage on 10q as well as suggestive evidence of linkage on 9p. Contrary to the earlier findings concerning 10q that have been replicated in a number of other study populations examining diabetes and related phenotypes (16, 37, 4042), in the expanded SAFDGS, we do not find evidence of linkage to diabetes on 10q; however, we do replicate the earlier findings on both 3p and 9p. Explanations for discrepancies between previous and current findings on 10q include 1) changes in how diabetes is defined (i.e., the new fasting glucose threshold, 126 vs. 140 mg/dl), 2) changes in the study population (i.e., new pedigrees have been added and old pedigrees have been expanded) that could introduce further heterogeneity, 3) collection of phenotypic information at three versus a single time point and the subsequent aging of the study population, 4) the new set of microsatellite markers, and 5) secular trends in the prevalence of diabetes (1). Therefore, to help identify which explanation was most plausible, we completed a genome-wide linkage analysis on the 579 individuals at the San Antonio Family Diabetes study baseline examination using the new definitions of diabetes and the new genotypic information from CIDR. Results from this analysis replicate the previous 10q findings (i.e., Martingale residual [fasting], LOD = 3.00; Martingale residual [WHO], LOD = 2.57). Hence, neither the new definition of the phenotype nor the new marker set accounts for the loss of the 10q signal. Its disappearance, therefore, must relate to the expanded population, aging of our population, or secular trends in the incidence of diabetes. We attempted to control for the latter by controlling for birth decade. Different genes are likely to influence diabetes susceptibility at different ages; hence, the aging of our population over the three visits may have influenced the observed linkage signals. Moreover, because the gene pool is relatively stable, the rapid increase in the prevalence of diabetes in recent years reflects environmental changes that may call into play additional diabetes susceptibility genes through gene-environment interaction.

Other studies from outside San Antonio, Texas, have also reported evidence for linkage to type 2 diabetes or type 2 diabetes–related traits on chromosome 3p in the region implicated in this study. Our 3p signal has previously been identified with type 2 diabetes in a Chinese population (maximum reported LOD of 2.27) (21), a family study focused on identifying genes associated with maturity-onset diabetes of the young (maximum reported heterogeneity LOD of 1.81) (8), and in an ordered subset analysis of a Japanese population in which families were rank ordered based on average maximal BMI of sib-pairs and maximal BMI was adjusted for (maximum reported LOD of 2.42) (17). Additionally, our 3p region has been identified as having slight evidence for linkage to type 2 diabetes in several studies (43, 44), to fasting glucose in the Framingham Offspring study (16), and to acute insulin response in the Pima Indians (45). Finally, four additional studies have reported evidence of linkage to type 2 diabetes near our region, ∼50 cM toward the p-terminal end of chromosome 3 (12, 13, 20, 46).

Ideally, in an extended pedigree study of a discrete trait, complete information on who acquired the trait and their age of onset would be available. Therefore, when available, we used serial data to define diabetes status. This enabled us to exclude individuals likely misclassified who were identified as having diabetes at one examination but who at a subsequent examination no longer met the criteria for having diabetes (∼1% of the population for either definition). In addition, because we used information across three examinations that occurred over 10 years, we were able to identify a greater number of individuals as having diabetes (i.e., those who were initially nondiabetic but who developed diabetes at a later examination) as well as to determine more accurately their age of diabetes onset. Finally, by combining information across the three examinations, we maximized the number of individuals included in the analyses. One limitation of combining information collected over a 10-year period is that it tends to magnify the effect of secular trends. To help overcome this limitation, we adjusted for birth decade to help account for secular trends in the incidence of diabetes.

In comparison with modeling the discrete trait diabetes, which includes information only on whether or not an individual has diabetes (0.1), the Martingale residual can be viewed as a continuous measure that incorporates information based on a participant’s age at entry into the study, whether they develop diabetes during the study, age of diabetes diagnosis if they develop diabetes, and age at the end of the study if they do not develop diabetes. For example, in our study, a participant that enters the study at age 21 and develops diabetes (fasting) at age 25 has a Martingale residual of 0.96, whereas a participant who enters the study at age 64 and is 72 when the study ends and still does not have diabetes has a Martingale residual of −0.71. Therefore, use of the Martingale residual allows us to model age of diabetes diagnosis because it assigns a value to individuals based on their age of diabetes diagnosis if they have diabetes or based on their obtained age without developing diabetes if they don’t have diabetes. In summary, for each definition of diabetes (fasting and WHO), the LOD for the Martingale residual is quite a bit higher than the LOD for the trait; however, this is not unexpected because the Martingale residual is more precise, incorporating additional information.

We examined two definitions for diabetes because basing the definition on fasting only avoids the problems associated with 2-h glucose levels in the definition that increases the number of individuals with missing information on diabetes (due to missing 2-h values) and that has lower repeatability. However, it is reassuring that evidence for linkage to diabetes (fasting) is similar to evidence for linkage to diabetes (WHO). The slightly higher LODs for the Martingale residual for diabetes (fasting) relative to the Martingale residual for diabetes (WHO) (3.76 vs. 3.29 for the Martingale residual) may be explained by the lower percentage of people with missing data as well as the higher repeatability of the diabetes (fasting) definition. Moreover, although adjusting for birth decade to help account for secular trends in the incidence of diabetes only slightly attenuated the signals, additionally adjusting for waist circumference further reduced the LODs 24.5 and 29.0% for the diabetes age of onset model for fasting and WHO definitions, respectively. Interestingly, there is no evidence for linkage to waist circumference in the SAFDGS in this region on chromosome 3 (K.J.H., unpublished data).

A number of genes in the region of linkage on chromosome 3p encode proteins that appear to have biological relevance to the pathophysiology of diabetes. Their function points to a role in either insulin secretion or insulin action. Succinyl-CoA synthetase (SUCLG2, MIM 603922) is expressed in the mitochondria and in the cytosol of pancreatic islet cells as well as numerous other tissues (47). Alterations in SUCLG2 could affect key functional steps in mitochondrial metabolism leading to insulin secretion (4750). Forkhead box P1 (MIM 605515) is a novel member of a subfamily of winged helix/forkhead/HNF3 transcription factors (51). Mutations in two related members, FOXP3 and Foxo1 (mouse homolog of FOXO1), have been shown to be involved in the pathogenesis of diabetes (52, 53). Glycogen-branching enzyme (MIM 607839) plays a role in glycogen synthesis and storage, a critical factor in energy homeostasis (54). Glycogen-branching enzyme is an attractive candidate because there is substantial evidence to suggest that impaired glycogen synthesis (nonoxidative glucose disposal) could be a primary metabolic defect in pre-diabetic individuals (55).

In summary, our genome-wide multipoint linkage scan for type 2 diabetes in a Mexican-American population found evidence for linkage on chromosome 3p. Moreover, this finding replicates prior linkage findings to type 2 diabetes and fasting-specific insulin in the San Antonio Family Diabetes Study, as well as replicates three independent studies that have reported evidence for linkage to type 2 diabetes in this region and a single independent study that has reported evidence for linkage to serum insulin in this region.

FIG. 1.

Multipoint genome-wide linkage scan of the trait type 2 diabetes (fasting, blue, liability to disease) with adjustment for age and age2 and the Martingale residual modeling age of diabetes onset (orange).

FIG. 1.

Multipoint genome-wide linkage scan of the trait type 2 diabetes (fasting, blue, liability to disease) with adjustment for age and age2 and the Martingale residual modeling age of diabetes onset (orange).

Close modal
FIG. 2.

Chromosome 3 multipoint linkage of the trait type 2 diabetes (fasting, dashed line, liability to disease) with adjustment for age and age2 and the Martingale residual modeling age of diabetes onset (solid line).

FIG. 2.

Chromosome 3 multipoint linkage of the trait type 2 diabetes (fasting, dashed line, liability to disease) with adjustment for age and age2 and the Martingale residual modeling age of diabetes onset (solid line).

Close modal
TABLE 1

Characteristics of individuals with and without type 2 diabetes (fasting) in the SAFDGS

ProbandsFamily members
n 39 666 186
Diabetes status (fasting) Yes No Yes
Male (%) 38.5 40.7 41.9
Diabetes (WHO) (%) 100 3.8 100
Age (years) 63.2 40.6 55.9
Birth year 1935 1958 1942
Age of diabetes onset (years) 50.7 — 47.5
Fasting glucose (mg/dl) 171 95 184
2-h glucose (mg/dl) 367 118 325
BMI (kg/m231.3 30.1 32.5
Waist circumference (mm) 1,059 974 1,067
ProbandsFamily members
n 39 666 186
Diabetes status (fasting) Yes No Yes
Male (%) 38.5 40.7 41.9
Diabetes (WHO) (%) 100 3.8 100
Age (years) 63.2 40.6 55.9
Birth year 1935 1958 1942
Age of diabetes onset (years) 50.7 — 47.5
Fasting glucose (mg/dl) 171 95 184
2-h glucose (mg/dl) 367 118 325
BMI (kg/m231.3 30.1 32.5
Waist circumference (mm) 1,059 974 1,067

Data are means or %.

TABLE 2

Multipoint empirical LODs of ≥1.18 for the trait diabetes adjusted for age and age2 and the Martingale residual modeling age of diabetes onset

ChromosomecM (Haldane)Nearest markerEmpirical LOD
Diabetes (fasting)
3 116 D3S2406 1.87
15 47 D15S659 1.51
Martingale residual (fasting)
3 114 D3S2406 3.76
9 28 GATA187D09 1.49
11 16 D11S1999 1.57
15 45 D15S659 1.22
Diabetes (WHO)
3 117 D3S2406 1.90
4 114 D4S1647 1.43
11 61 D11S1993 2.25
15 43 D15S659 1.47
Martingale residual (WHO)
3 115 D3S2406 3.29
9 40 D9S925 1.83
11 61 D11S1993 2.25
15 45 D15S659 1.42
ChromosomecM (Haldane)Nearest markerEmpirical LOD
Diabetes (fasting)
3 116 D3S2406 1.87
15 47 D15S659 1.51
Martingale residual (fasting)
3 114 D3S2406 3.76
9 28 GATA187D09 1.49
11 16 D11S1999 1.57
15 45 D15S659 1.22
Diabetes (WHO)
3 117 D3S2406 1.90
4 114 D4S1647 1.43
11 61 D11S1993 2.25
15 43 D15S659 1.47
Martingale residual (WHO)
3 115 D3S2406 3.29
9 40 D9S925 1.83
11 61 D11S1993 2.25
15 45 D15S659 1.42

Data are P < 0.01.

TABLE 3

Two-point empirical LODs of ≥1.18, empirical LODs for markers in 3p region of interest, and empirical LODs for 10 newly typed microsatellite markers on chromosome 3 in the region of interest

MarkercM (Haldane)Fasting
WHO
Diabetes*MRDiabetes*MR
Chromosome 3p
D3S2409 78 0.27 0.28 0.07 0.10
D3S3561 78 0.17 0.43 0.33 0.58
D3S1578 78 0.65 0.99 0.48 0.61
D3S3721 82 0.33 0.15 0.43 0.19
D3S1592 84 1.07 0.83 1.27 0.91
D3S3631 89 0.27 0.29 0.10 0.09
D3S1600 93 0.11 0.39 0.01 0.24
D3S4542 97 0.32 0.37 0.12 0.15
D3S3697 103 0.40 1.00 0.28 0.87
D3S1296 106 0.29 0.92 0.10 0.39
D3S3551 107 1.42 3.30 1.02 2.14
D3S2406 111 1.54 4.11 1.79 3.81
D3S3633 117 0.38 0.83 0.95 0.98
D3S1542 117 0.96 1.33 1.60 1.39
D3S4529 120 0.74 1.62 1.10 1.58
D3S2459 128 0.13 0.28 0.01 0.00
Other locus
D1S1619 301 1.07 0.41 1.33 0.53
D4S2361 102 1.16 0.62 1.68 0.64
D4S1652 231 0.59 0.64 1.44 0.97
D5S1470 49 0.10 0.75 0.52 1.58
D8S1477 65 1.38 1.16 1.41 0.46
GATA187D09 24 0.57 1.48 0.00 0.55
D11S1999 18 0.62 1.31 0.80 1.37
D11S1344 63 0.33 0.57 2.03 2.52
D11S912 145 0.24 1.30 0.63 1.51
GATA91H06 62 0.66 0.73 1.05 1.25
D12S1064 105 0.39 1.41 0.33 1.87
MarkercM (Haldane)Fasting
WHO
Diabetes*MRDiabetes*MR
Chromosome 3p
D3S2409 78 0.27 0.28 0.07 0.10
D3S3561 78 0.17 0.43 0.33 0.58
D3S1578 78 0.65 0.99 0.48 0.61
D3S3721 82 0.33 0.15 0.43 0.19
D3S1592 84 1.07 0.83 1.27 0.91
D3S3631 89 0.27 0.29 0.10 0.09
D3S1600 93 0.11 0.39 0.01 0.24
D3S4542 97 0.32 0.37 0.12 0.15
D3S3697 103 0.40 1.00 0.28 0.87
D3S1296 106 0.29 0.92 0.10 0.39
D3S3551 107 1.42 3.30 1.02 2.14
D3S2406 111 1.54 4.11 1.79 3.81
D3S3633 117 0.38 0.83 0.95 0.98
D3S1542 117 0.96 1.33 1.60 1.39
D3S4529 120 0.74 1.62 1.10 1.58
D3S2459 128 0.13 0.28 0.01 0.00
Other locus
D1S1619 301 1.07 0.41 1.33 0.53
D4S2361 102 1.16 0.62 1.68 0.64
D4S1652 231 0.59 0.64 1.44 0.97
D5S1470 49 0.10 0.75 0.52 1.58
D8S1477 65 1.38 1.16 1.41 0.46
GATA187D09 24 0.57 1.48 0.00 0.55
D11S1999 18 0.62 1.31 0.80 1.37
D11S1344 63 0.33 0.57 2.03 2.52
D11S912 145 0.24 1.30 0.63 1.51
GATA91H06 62 0.66 0.73 1.05 1.25
D12S1064 105 0.39 1.41 0.33 1.87

Bold values are P < 0.01. MR, Martingale residual trait.

*

Newly typed microsatellite markers.

The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

This work was supported by National Institutes of Health Grants R01-DK-042273, R01-DK-047482, R01-DK-53889, and MH-59490. K.J.H. was partially supported by an American Diabetes Association Mentor-Based Award, an American Diabetes Association Research Award, and National Institute of Diabetes and Digestive and Kidney Diseases Grant K01-DK-064867. D.M.L. was supported by an American Diabetes Association Junior Faculty Award. CIDR is fully funded through a federal contract from the National Institutes of Health to The Johns Hopkins University (contract N01-HG-65403).

Genotyping services were provided by the CIDR. Yasmin Ench, Korri Weldon, and Jeanette Hamlington provided excellent technical assistance. We thank the participants of the SAFDGS families for their support and cooperation.

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text:
What is the chemical formula when zinc chloride is dissolved in water and forms tetrahedral chloro aqua complexes where the reduction in coordination number from 6 to 4 further reduces the strength of the O-H bonds in the solvated water molecules ?

Jun 13, 2018

ZnCl_2(s) + 4H_2O(l) rarr [Zn{(OH)_2}_4]"^(2+) + 2Cl^-
There might be 4 aqua ligands around zinc, there might be 6. Certainly the zinc complex would have a measurable $p {K}_{a}$.