Subject: Re: Bias in bootstrapings
From: Rodney Sparapani
Date: Wed, 06 Nov 1996 09:16:48 -0500
Reno Lindberg wrote:
>
> I have studied the population of palm trees in Ecuador. Through a
> projection matrix i found the population growth rate (Lambda, Dominant
> eigenvector) to be 0.9629. I made a bootsrap analysis on the
> population(5000 resamplings of the population with all parameters), and
> found the average growthrate to be 0.9484 (with the confidence intervals
> of 0.89685 and 0.98974). There is a bias of 0.0145 between the calculated
> growthrate (or observed (Lambda(o))) and the growthrate found through the
> bootstraping. Can I say that my population is declining because my upper
> limit is under 1.0 or can't I tell because if the difference between the
> upper limit and Lambda(b) added to Lamda(o) gives me an upper limit of
> 1.004. How to implement this bias into my results? And what do the bias
> tell me about the structure of my sample and the population?
> It is big quetions, but what I ask for is just som ideas!
>
> reno@pop.bio.aau.dk
Take a look at:
Better Bootstrap Confidence Intervals, by Bradley Efron,
JASA March '87, Vol. 82, No. 397, pp. 171-185
Rodney Sparapani
Subject: Mantel again
From: Rossi Jean Pierre
Date: Wed, 06 Nov 1996 16:51:01 +0100
Hello everybody
I am trying to learn about the Mantel statistic (Mantel,1967) in the
field of soil ecology. If the method is useful in the study of linear
structures, its ability to investigate relationships between matrices in
the case of strong non-linearity seems to be doubtful (Legendre and
Fortin,1989). Unfortunately, in soil biology, most of the spatial
structures encountered or not linear gradient!
Can you help?
*Mantel, N. (1967). The detection of disease clustering and a
generalized regression approach. Cancer Research, 27, 209-220.
*Legendre, P. and Fortin, M.J. (1989). Spatial pattern and ecological
analysis. Vegetatio, 80, 107-138.
Jean Pierre ROSSI Tel: (33) 1 48 02 55 01
Laboratoire d'Ecologie Fax: (33) 1 48 47 36 78
des Sols Tropicaux Email: rossijp@bondy.orstom.fr
ORSTOM Centre de Bondy
32 Av. VARAGNAT
93143 Bondy Cedex FRANCE
Subject: Re: Help with vector spaces, please.
From: ehren@macalester.edu (David L. O. Ehren)
Date: 6 Nov 1996 19:29:59 GMT
In article , ebohlman@netcom.com (Eric
Bohlman) wrote:
> Robert Gelb (rgelb@engr.csulb.edu) wrote:
> : Help with homework (vector spaces):
>
> : The problem is to find out whether a given set is a vector space.
>
> : The problem is as follows:
> : W' is the set of all ordered pairs(x,y) of real numbers that satisfy the
> : equation 2x+3y=1.
>
> : The answer at the end of the book says, that this set is not a vector
> : space.
> : Can someone explain to me why? I would consult the book but it provides
> : theoretical examples, however no practical ones.
>
> For one thing, W' isn't closed under scalar multiplication; if 2x+3y=1,
> then 2ax+3ay=a, so unless a=1, a(x,y) isn't in W'.
The graph of the equation you have is a line, so it LOOKS like a
one-dimensional vector space. As was stated, it is not closed under
scalar multiplication, but it also fails to have an identity (0,0) or even
be closed under addition: if you have members (x,y) and (z,w), then 2(x+z)
+ 3(y+w) = 1+1 = 2.
Subject: Re: Random number generation
From: wpilib+@pitt.edu (Richard F Ulrich)
Date: 6 Nov 1996 20:24:17 GMT
I posted without too much thought when I wrote,
: What you seem to be requesting is somebody's "really rotten
: pseudo random number generator"; with even the worst of those
: (among the practical ones), you can only make the future prediction
: as a statistical one, for example, "the 10th number PROBABLY will be
: SMALL..."
-- but I have not totally changed my mind despite several
comments by other people. This message, below, contains extractions
from three other notes.
==========================from latest message> (from Barry Hembree),
> | : I'm looking for a pseudo random number algorithm which enables me to
> | : choose the n:th number in the sequence without needing to generate the
> | : first n-1 numbers (where n can be quite big). Does such a thing exist?
:Myself and Bruce Collings wrote a paper on how to do what you ask for
:arbitrary length sequence RNGs. It was published in the Oct 1984 issue
:of JACM.
===========================
Assuming JACM was the Journal of the Association for Computing
Machinery, I looked, and failed to find any the article in Oct 1984,
or Oct 1983 or Oct 1985. Nor did a search of some indexes, which
included JACM, show any Barry Hembree in 1984 or 1985.
Hey, I'm making an effort to learn something here, but to what avail?
(I would have just asked for the reference and gone to check again,
but that is going to take me a day or two more, and I thought that it
was time that I wrote some update to my original response...)
Before that, there was the post by Robert E Sawyer (soen@pacbell.net):
"Given the necessary caveats about limitations of PRNGs, the request seems
a reasonable one in context. Since (almost?) every PRNG has a (large) finite
cycle length L, it makes sense to wonder whether, in theory, a generator might
somehow be designed to allow the user to select among the L rotations of the
cycle."
I agree that the math-theory I once studied about cycles suggests that
such a thing is conceivable. But the little that I know about random
number generators suggests that skipping through the cycle should be
tough. The main step, typically, is to multiply the previous Seed
by a large amount, and to extract some set of bits. On the one hand,
the "large amount" and set of bits are chosen (I think) for how
*little* information that they should carry pertaining to the previous
or the next step. On the other hand, the FAULT of linear congruential
generators is that they DO have a tendency to provide lumpiness at
certain lag-times -- if one is to discuss just THAT type of PRNG.
But in detail, it seems to me that the regenerating the finite cycle
by a different rule has probably NOT been done. (And, from my
original suspicion: if it could be done for a sequence, then I think
there is probably OTHER kinds of evidence that the sequence on hand
is not a very good one....)
It seems a whole lot tougher to imagine skipping through the cycle
if the selection/generation includes a pseudo-random list that
is generated, then selected from at (pseudo) random; which is
a refinement that I first read about in one of these Usenet groups.
I can understand the idea of somehow skipping through a cycle, if you
are given the rules for generating it. But I have to admit that I
really have little idea of what Herman Rubin was getting at, when
he posted.
"For the various pseudo-random algorithms in my ken, typically the
n-th element can be generated in time O(log(n)), and except for n
VERY large, it is not likely that better will be possible."
-- How in heaven's name does one achieve an efficiency on the
order of log(n)? That would imply that the 100th-to-come item
is not much harder to pre-define than the 1000th. Is this a
statement of theoretical principle (and can we have a hint,
or a reference), or has someone actually achieved this for some
interesting PRNGs?
Rich Ulrich, biostatistician wpilib+@pitt.edu
Western Psychiatric Inst. and Clinic Univ. of Pittsburgh
Subject: Re: Mantel again
From: wpilib+@pitt.edu (Richard F Ulrich)
Date: 6 Nov 1996 20:40:29 GMT
Rossi Jean Pierre (rossijp@bondy.orstom.fr) wrote:
: I am trying to learn about the Mantel statistic (Mantel,1967) in the
: field of soil ecology. If the method is useful in the study of linear
: structures, its ability to investigate relationships between matrices in
: the case of strong non-linearity seems to be doubtful (Legendre and
: Fortin,1989). Unfortunately, in soil biology, most of the spatial
: structures encountered or not linear gradient!
: Can you help?
-- I just looked a the Mantel article, and its whole PURPOSE is to
show you how to take into account 'strong non-linearity'. The
main, particular example is about looking at clusters in time, versus
clusters in space. He suggests using the RECIPROCALs of distance between
cases, and of how far apart they occurred in time. He shows an example
where the reciprocals work ENORMOUSLY better than using the ordinary
scales of distance==miles and time==days. Does your other reference
mis-use Mantel to suggest that he is doing something that is just
linear?
: *Mantel, N. (1967). The detection of disease clustering and a
: generalized regression approach. Cancer Research, 27, 209-220.
: *Legendre, P. and Fortin, M.J. (1989). Spatial pattern and ecological
: analysis. Vegetatio, 80, 107-138.
Rich Ulrich, biostatistician wpilib+@pitt.edu
Western Psychiatric Inst. and Clinic Univ. of Pittsburgh
Subject: Multiple Regression
From: tschmitz@hpu.edu
Date: 6 Nov 1996 21:21:09 GMT
Could someone please explain how to understand this in English?
Particularyly the t-stats and f-stat. I know what they are, but I am confused by the output.
Multiple Linier Regrerssion of GDP based on IBTS, V, and JS Y = GDP
X1 = IBTS
X2 = V
SUMMARY OUTPUT X3 = JS
Regression Statistics
Multiple R 0.996378
R Square 0.992769
Adjusted R Square 0.992504
Standard Error 0.000122
Observations 86
ANOVA
df SS MS F Significance F
Regression 3 0.000166 0.0000554 3752.701 0.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000123
Residual 82 0.00000121 0.0000000148
Total 85 0.000167
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.000% Upper 95.000%
Intercept -588.06 79.31423 -7.4143 0.0000000000998 -745.841 -430.279 -745.841 -430.279
X Variable 1 266.7569 16.97887 15.71112 0.0000000000000000000000000188 232.9805 300.5333 232.9805 300.5333
X Variable 2 343.1546 96.82104 3.544215 0.000653 150.5466 535.7626 150.5466 535.7626
X Variable 3 -20.86 6.245629 -3.33994 0.001263 -33.2846 -8.43549 -33.2846 -8.43549
-----------------------------------------------------------------------
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Subject: Re: Random number generation
From: d91johol@isy.liu.se (John Olsson)
Date: 7 Nov 1996 11:52:51 GMT
I got a personal reply from Barry W. Brown, who said that he and some other persons
had made a program package based on (among other papers)
"Implementing a Random Number Package with Splitting Facilities"
Pierre L'Ecuyer and Serge Côté
ACM Transactions on Mathematical Software
Vol. 17, No. 1, March 1991, Pages 98-111
The algorithm described in this paper allows the user to advance the state of the
generator by 2^k values.
Part of the REAME file that comes with the source:
RANLIB.C
Library of C Routines for Random Number Generation
README
Compiled and Written by:
Barry W. Brown
James Lovato
Department of Biomathematics, Box 237
The University of Texas, M.D. Anderson Cancer Center
1515 Holcombe Boulevard
Houston, TX 77030
This work was supported by grant CA-16672 from the National Cancer Institute.
THANKS TO OUR SUPPORTERS
This work was supported in part by grant CA-16672 from the National
Cancer Institute. We are grateful to Larry and Pat McNeil of Corpus
Cristi for their generous support. Some equipment used in this effort
was provided by IBM as part of a cooperative study agreement; we thank
them.
SUMMARY OF RANLIB
The bottom level routines provide 32 virtual random number generators.
Each generator can provide 1,048,576 blocks of numbers, and each block
is of length 1,073,741,824. Any generator can be set to the beginning
or end of the current block or to its starting value. Packaging is
provided so that if these capabilities are not needed, a single
generator with period 2.3 X 10^18 is seen.
Using this base, routines are provided that return:
(1) Beta random deviates
(2) Chi-square random deviates
(3) Exponential random deviates
(4) F random deviates
(5) Gamma random deviates
(6) Multivariate normal random deviates (mean and covariance
matrix specified)
(7) Noncentral chi-square random deviates
(8) Noncentral F random deviates
(9) Univariate normal random deviates
(10) Random permutations of an integer array
(11) Real uniform random deviates between specified limits
(12) Binomial random deviates
(13) Negative Binomial random deviates
(14) Multinomial random deviates
(15) Poisson random deviates
(16) Integer uniform deviates between specified limits
(17) Seeds for the random number generator calculated from a
character string
COMMENTS ON THE C VERSION OF RANLIB
The C version was obtained by converting the original Fortran RANLIB
to C using PROMULA.FORTRAN and performing some hand crafting of the
result. Information on PROMULA.FORTRAN can be obtained from
PROMULA Development Corporation
3620 N. High Street, Suite 301
Columbus, Ohio 43214
(614) 263-5454
RANLIB.C was tested using the xlc compiler under AIX 3.1 on an IBM
RS/6000. The code was also examined with lint on the same system.
The RANLIB test programs were also successfully run using the gcc
compiler (see below) on a Solbourne.
RANLIB.C can be obtained from statlib. Send mail whose message is
'send ranlib.c.shar from general' to statlib@lib.stat.cmu.edu.
RANLIB.C can also be obtained by anonymous ftp to odin.mda.uth.tmc.edu
(129.106.3.17) where is is available as
/pub/unix/ranlib.c.tar.Z
For obvious reasons, the original RANLIB (in Fortran) has
been renamed to
/pub/unix/ranlib.f.tar.Z
on the same machine.
Subject: Re: Multiple Regression
From: jdebord@MicroNet.fr (Jean Debord)
Date: Thu, 07 Nov 1996 18:41:06 GMT
tschmitz@hpu.edu wrote:
>Could someone please explain how to understand this in English?
>Particularyly the t-stats and f-stat. I know what they are, but I am confused by the output.
> [... computer output snipped ...]
t is the ratio of the parameter to its standard error. The t test
compares the value of the parameter with zero. If the associated
probability (P-value) is sufficiently low, you may admit (with a risk
equal to the probability) that the parameter differs significantly
from zero. Usually one chooses probability values of 0.05 (5%), 0.01
(1%) and 0.001 (0.1%) as the reference values to which the observed
probability is compared. If the probability is > 0.05, the parameter
does not differ significantly from zero : this means that the
influence of the corresponding variable is negligible, so that you may
wish to discard this variable from the regression.
Your results show that the influence of variables 1 and 2 is
significant at the 0.1% level, and that the influence of variable 3 is
significant at the 1% level. So, you don't need to discard any
variable.
The coefficient of determination r² (R square) is the ratio of the
"explained" sum of squares (SSe) to the total sum of squares (SSt) :
r² = SSe / SSt
SSt = Sum [y(i) - m]²
SSe = Sum [ycalc(i) - m]²
where :
m is the average of the y(i) : m = [Sum y(i)] / n
n the number of observations
ycalc(i) the estimated y value for the i-th observation.
The adjusted coefficient of determination (r²aj) is such that :
r²aj = 1 - [(n - 1) / (n - p)] (1 - r²)
where p is the number of regression parameters
This coefficient takes into account the number of regression
parameters with respect to the number of observations.
The residual standard error (s) is such that :
s² = SSr / (n - p)
s² = residual variance
SSr = Residual sum of squares = Sum [y(i) - ycalc(i)]²
F is the ratio of the explained variance to the residual variance :
F = [SSe / (p - 1)] / s²
In the case of linear regression : SSt = SSe + SSr
(p - 1) and (n - p) are the numbers of degrees of freedom (df)
associated with the explained and residual variance, respectively.
In your case : n = 86, p = 4, p - 1 = 3, n - p = 82, and the F is
significant at the 0.1% level
I hope this helps.
Best regards.
Jean Debord
Faculte de Medecine, Limoges, France
Subject: Wall Street Quant Position
From: David Rothman
Date: Thu, 07 Nov 1996 13:01:10 +0100
The trading arm of a major investment firm is seeking a quantitative
specialist for its New York based Analytical Equity Trading Group to
work with its senior professionals in the on-going development of
sophisticated statistical/econometric trading models and strategies.
QUALIFICATIONS:
The successful candidate will have in-depth knowledge of financial
economics, time series econometrics, stochastic processes and the
requisite skills necessary to design and implement strategies in a
sophisticated computer environment. Comfort in dealing with
Probabilistic notions such as Random Walk, Brownian Motion and
Martingale Theory, combined with Econometric ideas such as Stationarity,
Cointegration, Error-Correction Models and Arch/Garch is essential.
This position would be ideal for someone with prior experience in a
related field, and/or academic training near or at the Ph.D. level.
CONTACT:
E-mail: nyrtd@ny.ubs.com
Please reply via email with either a resume or a short informal
description of yourself. Please include a day & evening phone number.
We are an Equal Opportunity employer.
Subject: Confounding variables in regression
From: Dan Kehler <005769k@ace.acadiau.ca>
Date: Thu, 07 Nov 1996 15:29:58 -0800
We've been struggling with this question for a while:
Given the common situation of wanting to know the effect of a variable,
X2 on Y, independent of the the effect of X1 on Y, when X1 and X2 are
correlated, which of the following two methods is more appropriate?
method 1:
model 1 = Y ~ X1
model 2 = residuals(model 1) ~ X2 , test for significance of X2
Method 2:
model 1 = Y ~ X1 + X2, test for significance of X2.
This is what I think is going on:
These two methods are quite different, and the difference between them
depends on the degree of correlation betwen X1 and X2. In the first
method, the two parameters, b1 and b2 are estimated separately, such that
the entire variability in Y shared between X1 anfd X2 is allocated to the
parameter estimate for X1. In the second method the two parameter
estimates compete for the variability in Y shared between X1 and X2.
Which method is preferable and why?
We'd be very grateful for some insight or a useful reference.
Thanks,
Dan Kehler
Acadia University
Wolfville, NS, CANADA
B0P IX0
Subject: Confounding variables in regression
From: Dan Kehler <005769k@ace.acadiau.ca>
Date: Thu, 07 Nov 1996 15:29:44 -0800
We've been struggling with this question for a while:
Given the common situation of wanting to know the effect of a variable,
X2 on Y, independent of the the effect of X1 on Y, when X1 and X2 are
correlated, which of the following two methods is more appropriate?
method 1:
model 1 = Y ~ X1
model 2 = residuals(model 1) ~ X2 , test for significance of X2
Method 2:
model 1 = Y ~ X1 + X2, test for significance of X2.
This is what I think is going on:
These two methods are quite different, and the difference between them
depends on the degree of correlation betwen X1 and X2. In the first
method, the two parameters, b1 and b2 are estimated separately, such that
the entire variability in Y shared between X1 anfd X2 is allocated to the
parameter estimate for X1. In the second method the two parameter
estimates compete for the variability in Y shared between X1 and X2.
Which method is preferable and why?
We'd be very grateful for some insight or a useful reference.
Thanks,
Dan Kehler
Acadia University
Wolfville, NS, CANADA
B0P IX0
Subject: Confounding variables in regression
From: Dan Kehler <005769k@ace.acadiau.ca>
Date: Thu, 07 Nov 1996 15:29:37 -0800
We've been struggling with this question for a while:
Given the common situation of wanting to know the effect of a variable,
X2 on Y, independent of the the effect of X1 on Y, when X1 and X2 are
correlated, which of the following two methods is more appropriate?
method 1:
model 1 = Y ~ X1
model 2 = residuals(model 1) ~ X2 , test for significance of X2
Method 2:
model 1 = Y ~ X1 + X2, test for significance of X2.
This is what I think is going on:
These two methods are quite different, and the difference between them
depends on the degree of correlation betwen X1 and X2. In the first
method, the two parameters, b1 and b2 are estimated separately, such that
the entire variability in Y shared between X1 anfd X2 is allocated to the
parameter estimate for X1. In the second method the two parameter
estimates compete for the variability in Y shared between X1 and X2.
Which method is preferable and why?
We'd be very grateful for some insight or a useful reference.
Thanks,
Dan Kehler
Acadia University
Wolfville, NS, CANADA
B0P IX0
Subject: Confounding variables in regression
From: Dan Kehler <005769k@ace.acadiau.ca>
Date: Thu, 07 Nov 1996 15:32:34 -0800
We've been struggling with this question for a while:
Given the common situation of wanting to know the effect of a variable,
X2 on Y, independent of the the effect of X1 on Y, when X1 and X2 are
correlated, which of the following two methods is more appropriate?
method 1:
model 1 = Y ~ X1
model 2 = residuals(model 1) ~ X2 , test for significance of X2
Method 2:
model 1 = Y ~ X1 + X2, test for significance of X2.
This is what I think is going on:
These two methods are quite different, and the difference between them
depends on the degree of correlation betwen X1 and X2. In the first
method, the two parameters, b1 and b2 are estimated separately, such that
the entire variability in Y shared between X1 anfd X2 is allocated to the
parameter estimate for X1. In the second method the two parameter
estimates compete for the variability in Y shared between X1 and X2.
Which method is preferable and why?
We'd be very grateful for some insight or a useful reference.
Thanks,
Dan Kehler
Acadia University
Wolfville, NS, CANADA
B0P IX0
Subject: Q: pmf of cos .... (need help)
From: wxu@cs.utexas.edu (Wei Xu)
Date: 7 Nov 1996 14:13:16 -0600
Hi netters,
I have the following question need your help.
Let R and D be two n-dimensional random vectors:
R = [r_1,...,r_n] and D = [d_1,...,d_n].
Their jumps or increments (i.e., r_{i+1} - r_i and d_{i+1} - d_i for all i )
are iid (independent identically distributed) random variables,
say, zero-mean Laplacian variables.
Let m1 and m2 be averaged values of R and D:
m1 = (r_1 + ...+r_n)/n,
m2 = (d_1 + ... + d_n)/n,
Define two new vectors
R1 = (r_1 - m1, ..., r_n - m1)
D1 = (d_1 - m2, ..., d_n - m2).
Now, I want to know the pmf of the following variable Y:
Y = cos(angle between R1 and D1) = /sqrt()
where means dot product of two vectors x and y.
Does anybody know how or where( which books or papers)
I can find the answer?
Please give me an email if you can help.
Thank you very much!
Wei Xu
Subject: Re: linear regression with errors
From: Jim Hunter
Date: Thu, 07 Nov 1996 14:43:39 -0500
M.A. Cremonini wrote:
>
> Hi,
> I must solve a problem like this:
>
> Ax=B
>
> A is my data matrix ans B is a column vector with results.
> B is formed by numbers which can span a certain range i.e. all
> the numbers within
> the range can be solutions of the corresponding equation:
>
> |a11 a12| |x1| |B1 +/- e1|
> |a21 a22| | | = |B2 +/- e2|
> |a31 a32| |x2| |B3 +/- e3|
> |a41 a42| | | |B4 +/- e4|
>
> For example:
>
> if A11*x1+A12*x2 = Q and Q is within the range (B1-e1 - B1+e1) the result
> is the one searched for.
>
> Is there a 'one shot' way to solve this system (I mean just using matrix
> algebra and without using a minimization procedure)?
> Thank you for your help.
> Mauro
>
In a sense there's a "one shot" method. You can restate the problem
in a standard minimax form as follows:
Let E be a diagonal matrix of the tolerances e1, e2, ...
Then the problem is:
given A and B, find x and d so that
Ax = B + Ed
with |d(i)| <= 1.
Assuming each e(i) > 0, we then need to find some x so that
|| Fx - C ||oo <= 1.
Here F = E^-1*A, C = E^-1*B, and ||*|| is the infinity or
maximum norm. So, what you what to do is to find
min || Fx - C ||oo.
x
Your original problem has a solution iff the vector that
minimizes this has a residual R = Ax' - b so that
|| R ||oo <= 1.
You should be able to find software to handle this on the
net somewhere, possibly try netlib. Look for functions concerned
with "minimax", "min-max", or Remez exchange.
Note: If you're a Matlab user, the Matlab remez function won't
be of use. That's a specialized version of the Remez algorithm
for designing digital filters.
----
Jim