cocktail: Fits the regularization paths for the elastic net penalized...
In fastcox: Lasso and Elastic-Net Penalized Cox's Regression in High Dimensions Models using the Cocktail Algorithm
Description
Usage
Arguments
Details
Value
Author(s)
References
See Also
Examples
Description
Fits a regularization path for the elastic net penalized Cox's model at a sequence of regularization parameters lambda.
Usage
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Arguments
x
matrix of predictors, of dimension N*p; each row is an observation vector.
y
a survival time for Cox models. Currently tied failure times are not supported.
d
censor status with 1 if died and 0 if right censored.
nlambda
the number of lambda values - default is 100.
lambda.min
given as a fraction of lambda.max - the smallest value of lambda for which all coefficients are zero. The default depends on the relationship between N (the number of rows in the matrix of predictors) and p (the number of predictors). If N > p, the default is 0.0001,
close to zero. If N<p, the default is 0.01.
A very small value of lambda.min will lead to a saturated fit. It takes no effect if there is user-defined lambda sequence.
lambda
a user supplied lambda sequence. Typically, by leaving this option unspecified users can have
the program compute its own lambda sequence based on
nlambda and lambda.min. Supplying a value of
lambda overrides this. It is better to supply
a decreasing sequence of lambda values than a single (small) value, if not, the program will sort user-defined lambda sequence in decreasing order automatically.
alpha
The elasticnet mixing parameter, with 0 < α <= 1. See details.
pf
separate penalty weights can be applied to each coefficient of beta to allow
differential shrinkage. Can be 0 for some variables, which implies
no shrinkage, and results in that variable always being included in the
model. Default is 1 for all variables (and implicitly infinity for
variables listed in exclude). See details.
exclude
indices of variables to be excluded from the
model. Default is none. Equivalent to an infinite penalty factor.
dfmax
limit the maximum number of variables in the
model. Useful for very large p, if a partial path is desired. Default is p+1.
pmax
limit the maximum number of variables ever to be nonzero. For example once β enters the model, no matter how many times it exits or re-enters model through the path, it will be counted only once. Default is min(dfmax*1.2,p).
standardize
logical flag for variable standardization, prior to
fitting the model sequence. If TRUE, x matrix is normalized such that sum squares of each column <Xj,Xj>/N=1. Note that x is always centered (i.e. sum(Xj)=0) no matter standardize is TRUE or FALSE. The coefficients are always returned on
the original scale. Default is is TRUE.
eps
convergence threshold for coordinate majorization descent. Each inner
coordinate majorization descent loop continues until the relative change in any
coefficient (i.e. max(j)|beta_new[j]-beta_old[j]|^2) is less than eps. Defaults value is 1e-6.
maxit
maximum number of outer-loop iterations allowed at fixed lambda value. Default is 1e4. If models do not converge, consider increasing maxit.
Details
The algorithm estimates β based on observed data, through elastic net penalized log partial likelihood of Cox's model.
\arg\min(-loglik(Data,β)+λ*P(β))
It can compute estimates at a fine grid of values of lambdas in order to pick up a data-driven optimal lambda for fitting a 'best' final model. The penalty is a combination of l1 and l2 penalty:
P(β)=(1-α)/2||β||_2^2+α||β||_1.
alpha=1 is the lasso penalty.
For computing speed reason, if models are not converging or running slow, consider increasing eps, decreasing
nlambda, or increasing lambda.min before increasing
maxit.
FAQ:
Question: “I am not sure how are we optimizing alpha. I can get optimal lambda for each value of alpha. But how do I select optimum alpha?”
Answer: cv.cocktail only finds the optimal lambda given alpha fixed. So to
chose a good alpha you need to fit CV on a grid of alpha, say (0.1,0.3, 0.6, 0.9, 1) and let cv.cocktail choose the optimal lambda for each alpha, then you choose the (alpha, lambda) pair that corresponds to the lowest predicted deviance.
Question: “I understand your are referring to minimizing the quantity cv.cocktail\$cvm, the mean 'cross-validated error' to optimize alpha and lambda as you did in your implementation. However, I don't know what the equation of this error is and this error is not referred to in your paper either. Do you mind explaining what this is?
”
Answer: We first define the log partial-likelihood for the Cox model. Assume
\hat{β}^{[k]} is the estimate fitted on k-th fold, define the log partial likelihood function as
L(Data,\hat{β}[k])=∑_{s=1}^{S} x_{i_{s}}^{T}\hat{β}[k]-\log(∑_{i\in R_{s}}\exp(x_{i}^{T}\hat{β}[k])).
Then the log partial-likelihood deviance of the k-th fold is defined
as
D[Data,k]=-2(L(Data,\hat{β}[k])).
We now define the measurement we actually use for cross validation:
it is the difference between the log partial-likelihood deviance evaluated
on the full dataset and that evaluated on the on the dataset with
k-th fold excluded. The cross-validated error is defined as
CV-ERR[k]=D(Data[full],k)-D(Data[k^{th}\,\,fold\,\,excluded],k).
Value
An object with S3 class cocktail.
call
the call that produced this object
beta
a p*length(lambda) matrix of coefficients, stored as a sparse matrix (dgCMatrix class, the standard class for sparse numeric matrices in the Matrix package.). To convert it into normal type matrix use as.matrix().
lambda
the actual sequence of lambda values used
df
the number of nonzero coefficients for each value of
lambda.
dim
dimension of coefficient matrix (ices)
npasses
total number of iterations (the most inner loop) summed over all lambda values
jerr
error flag, for warnings and errors, 0 if no error.
Author(s)
Yi Yang and Hui Zou
Maintainer: Yi Yang <yi.yang6@mcgill.ca>
References
Yang, Y. and Zou, H. (2013),
"A Cocktail Algorithm for Solving The Elastic Net Penalized Cox's Regression in High Dimensions", Statistics and Its Interface, 6:2, 167-173.
https://github.com/emeryyi/fastcox
See Also
Examples
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Example output
Loading required package: Matrix
$s0
NULL
$s1
[1] 1 45
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[1] 1 7 45 55
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[1] 1 7 45 55
$s6
[1] 1 7 12 45 55
$s7
[1] 1 7 12 45 55
$s8
[1] 1 7 12 45 55
$s9
[1] 1 7 12 45 55
$s10
[1] 1 7 12 17 42 45 55
$s11
[1] 1 7 12 14 17 42 45 55 59
$s12
[1] 1 7 12 14 17 42 45 55 59 74 91
$s13
[1] 1 7 12 14 17 42 45 55 59 74 91 100
$s14
[1] 1 7 10 12 14 17 42 45 55 59 74 91 100
$s15
[1] 1 2 7 10 12 14 17 28 42 45 55 59 74 91 100
$s16
[1] 1 2 7 10 12 14 17 28 42 45 55 59 74 91 100
$s17
[1] 1 2 7 10 12 14 17 28 42 45 55 59 71 74 91 100
$s18
[1] 1 2 7 10 12 14 17 28 42 45 55 59 71 74 91 98 100
$s19
[1] 1 2 7 10 12 14 17 28 42 45 55 59 71 74 91 96 98 100
$s20
[1] 1 2 7 10 12 14 17 26 28 42 45 55 56 59 71 74 91 96 98
[20] 100
$s21
[1] 1 2 7 10 12 14 17 26 28 42 45 55 56 59 71 74 91 96 98
[20] 100
$s22
[1] 1 2 5 7 10 12 14 17 26 28 42 45 55 56 59 71 74 91 96
[20] 98 100
$s23
[1] 1 2 5 7 10 12 14 17 26 28 30 42 44 45 55 56 59 71 74
[20] 84 91 96 98 100
$s24
[1] 1 2 5 7 10 12 14 17 26 28 30 31 42 44 45 55 56 59 71
[20] 74 77 84 91 96 98 100
$s25
[1] 1 2 5 7 10 12 14 17 26 28 30 31 42 44 45 55 56 59 71
[20] 74 77 84 91 96 98 100
$s26
[1] 1 2 5 7 10 12 14 17 23 26 28 30 31 42 44 45 55 56 59
[20] 71 74 77 84 91 96 98 100
$s27
[1] 1 2 5 7 10 12 14 17 23 26 28 30 31 39 42 44 45 55 56
[20] 59 71 74 77 84 88 91 96 98 100
$s28
[1] 1 2 5 7 10 12 14 17 23 26 28 30 31 39 42 44 45 55 56
[20] 59 71 74 77 84 88 91 96 98 100
$s29
[1] 1 2 5 7 10 12 14 17 23 26 28 30 31 39 42 44 45 55 56
[20] 59 71 74 75 77 84 88 91 96 98 100
$s30
[1] 1 2 5 7 10 12 14 17 23 26 28 30 31 39 42 44 45 55 56
[20] 59 71 74 75 77 84 88 91 96 98 100
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[1] 1 2 5 7 10 12 14 17 23 26 28 30 31 39 42 44 45 55 56
[20] 59 63 71 74 75 77 84 88 91 96 98 100
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[1] 1 2 5 7 10 12 14 23 26 28 30 31 39 42 44 45 55 56 59
[20] 63 71 74 75 77 84 88 91 96 98 100
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[1] 1 2 5 7 10 11 12 14 23 26 28 30 31 39 42 44 45 55 56
[20] 59 63 71 74 75 77 84 88 91 96 98 100
$s34
[1] 1 2 3 5 7 10 11 12 14 23 26 28 30 31 39 40 42 44 45
[20] 48 55 56 59 63 71 74 75 77 84 88 91 96 98 100
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[1] 1 2 3 5 7 10 11 12 14 23 26 28 30 31 39 40 42 44 45
[20] 48 55 56 59 63 71 74 75 77 84 88 91 96 98 100
$s36
[1] 1 2 3 5 7 10 11 12 14 23 26 28 30 31 39 40 42 44 45
[20] 48 55 56 59 63 71 74 75 77 84 88 91 96 98 100
$s37
[1] 1 2 3 5 7 10 11 12 14 23 26 28 30 31 39 40 42 44 45
[20] 48 55 56 59 63 71 74 75 77 84 88 91 96 98 100
$s38
[1] 1 2 3 5 7 8 10 11 12 14 16 23 26 28 30 31 39 40 42
[20] 44 45 48 49 55 56 59 62 63 65 71 74 75 77 84 88 91 96 98
[39] 100
$s39
[1] 1 2 3 5 7 8 10 11 12 14 16 23 26 28 30 31 39 40 42
[20] 44 45 48 49 55 56 59 62 63 64 65 71 74 75 77 84 88 91 96
[39] 98 100
$s40
[1] 1 2 3 5 7 8 10 11 12 14 16 23 26 28 31 39 40 42 44
[20] 45 48 49 55 56 59 62 63 64 65 71 74 75 77 84 88 91 96 98
[39] 100
$s41
[1] 1 2 3 5 7 8 10 11 12 14 16 17 23 26 28 31 32 39 40
[20] 42 44 45 48 49 55 56 59 62 63 64 65 71 74 75 77 84 88 96
[39] 98 100
$s42
[1] 1 2 3 5 7 8 10 11 12 14 16 17 23 26 28 31 32 39 40
[20] 42 44 45 48 49 55 56 59 62 63 64 65 71 74 75 77 84 88 96
[39] 98 100
$s43
[1] 1 2 3 5 7 8 10 11 12 14 16 17 23 26 28 31 32 37 39
[20] 40 42 44 45 46 48 49 55 56 58 59 62 63 64 65 71 72 74 75
[39] 77 84 88 96 98 100
$s44
[1] 1 2 3 5 7 8 10 11 12 14 16 17 23 26 28 31 32 37 39
[20] 40 42 44 45 46 48 49 55 56 58 59 62 63 64 65 71 72 74 75
[39] 77 84 88 96 98 100
$s45
[1] 1 2 3 5 7 8 10 11 12 14 16 17 23 26 28 31 32 36 37
[20] 39 40 42 44 45 46 48 49 55 56 58 59 62 63 64 65 70 71 72
[39] 74 75 77 84 86 88 96 98 100
$s46
[1] 1 2 3 5 7 8 10 11 12 14 16 17 23 26 28 31 32 34 36
[20] 37 39 40 41 42 44 45 46 48 49 55 56 58 59 62 63 64 65 70
[39] 71 72 74 75 77 84 86 88 96 98 100
$s47
[1] 1 2 3 5 7 8 10 11 12 14 16 17 23 26 28 31 32 34 36
[20] 37 39 40 41 42 44 45 46 48 49 55 56 58 59 62 63 64 65 70
[39] 71 72 73 74 75 77 84 86 88 96 98 100
$s48
[1] 1 2 3 5 7 8 10 11 12 14 16 17 23 26 28 31 32 34 36
[20] 37 39 40 41 42 44 45 46 47 48 49 55 56 58 59 62 63 64 65
[39] 70 71 72 73 74 75 77 83 84 86 88 96 98 100
$s49
[1] 1 2 3 5 7 8 10 11 12 14 16 17 18 23 26 28 31 32 34
[20] 36 37 39 40 41 42 44 45 46 47 48 49 55 56 58 59 62 63 64
[39] 65 70 71 72 73 74 75 77 83 84 86 88 96 98 100
$s50
[1] 1 2 3 5 7 8 10 11 12 14 16 17 18 23 26 28 31 32 34
[20] 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58 59 62 63
[39] 64 65 70 71 72 73 74 75 77 83 84 86 88 96 98 100
$s51
[1] 1 2 3 5 7 8 10 11 12 14 16 17 18 23 26 28 31 32 34
[20] 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58 59 62 63
[39] 64 65 70 71 72 73 74 75 77 83 84 86 88 96 98 100
$s52
[1] 1 2 3 7 8 10 11 12 14 16 17 18 23 26 28 31 32 34 36
[20] 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58 59 62 63 64
[39] 65 66 70 71 72 73 74 75 77 83 84 86 88 96 98 100
$s53
[1] 1 2 3 7 8 10 11 12 14 16 17 18 23 26 28 31 32 34 36
[20] 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58 59 62 63 64
[39] 65 66 70 71 72 73 74 75 77 83 84 86 88 96 98 100
$s54
[1] 1 2 3 5 7 8 10 11 12 14 16 17 18 23 26 28 31 32 34
[20] 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58 59 62 63
[39] 64 65 66 70 71 72 73 74 75 77 83 84 86 88 96 98 100
$s55
[1] 1 2 3 5 7 8 10 11 12 14 16 17 18 23 26 28 31 32 34
[20] 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58 59 62 63
[39] 64 65 66 70 71 72 73 74 75 77 83 84 86 88 95 96 98 100
$s56
[1] 1 2 3 5 7 8 10 11 12 14 15 16 17 18 23 26 27 28 31
[20] 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58 59
[39] 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 88 95 96 98
[58] 100
$s57
[1] 1 2 3 5 7 8 10 11 12 14 15 16 17 18 23 26 27 28 31
[20] 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58 59
[39] 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 88 93 95 96
[58] 98 100
$s58
[1] 1 2 3 5 7 8 10 11 12 14 15 16 17 18 23 26 27 28 31
[20] 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58 59
[39] 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87 88 93 95
[58] 96 98 100
$s59
[1] 1 2 3 5 7 8 9 10 11 12 14 15 16 17 18 23 26 27 28
[20] 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58
[39] 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87 88 93
[58] 95 96 98 100
$s60
[1] 1 2 3 5 7 8 9 10 11 12 14 15 16 17 18 23 26 27 28
[20] 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58
[39] 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87 88 93
[58] 95 96 98 100
$s61
[1] 1 2 3 5 7 8 9 10 11 12 14 15 16 17 18 23 26 27 28
[20] 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58
[39] 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87 88 93
[58] 95 96 98 100
$s62
[1] 1 2 3 5 7 8 9 10 11 12 14 15 16 17 18 23 26 27 28
[20] 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58
[39] 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87 88 93
[58] 95 96 98 100
$s63
[1] 1 2 3 5 7 8 9 10 11 12 14 15 16 17 18 23 26 27 28
[20] 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58
[39] 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87 88 93
[58] 95 96 98 100
$s64
[1] 1 2 3 5 7 8 9 10 11 12 14 15 16 17 18 23 26 27 28
[20] 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58
[39] 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87 88 93
[58] 95 96 98 100
$s65
[1] 1 2 3 5 7 8 9 10 11 12 14 15 16 17 18 23 26 27 28
[20] 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58
[39] 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87 88 93
[58] 95 96 98 100
$s66
[1] 1 2 3 5 7 8 9 10 11 12 14 15 16 17 18 23 26 27 28
[20] 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58
[39] 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87 88 90
[58] 93 95 96 98 100
$s67
[1] 1 2 3 5 7 8 9 10 11 12 14 15 16 17 18 23 26 27 28
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[39] 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87 88 90
[58] 93 95 96 98 100
$s68
[1] 1 2 3 5 7 8 9 10 11 12 14 15 16 17 18 23 26 27 28
[20] 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58
[39] 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87 88 90
[58] 93 95 96 98 100
$s69
[1] 1 2 3 5 7 8 9 10 11 12 14 15 16 17 18 23 26 27 28
[20] 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58
[39] 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87 88 90
[58] 93 95 96 98 100
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[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
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[39] 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87
[58] 88 90 93 95 96 98 100
$s71
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55
[39] 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87
[58] 88 90 93 95 96 98 100
$s72
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55
[39] 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87
[58] 88 90 93 95 96 98 100
$s73
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55
[39] 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87
[58] 88 90 93 95 96 98 100
$s74
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55
[39] 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87
[58] 88 90 93 94 95 96 98 100
$s75
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 98 100
$s76
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 98 100
$s77
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 98 100
$s78
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 98 100
$s79
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 98 100
$s80
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 98 100
$s81
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 98 100
$s82
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 98 100
$s83
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 98 100
$s84
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 97 98 100
$s85
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 97 98 100
$s86
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 97 98 100
$s87
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 97 98 100
$s88
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 97 98 100
$s89
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 97 98 100
$s90
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 97 98 100
$s91
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 97 98 100
$s92
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 97 98 100
$s93
[1] 1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26
[20] 27 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52
[39] 54 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84
[58] 86 87 88 90 93 94 95 96 97 98 100
$s94
[1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 23
[20] 26 27 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49
[39] 52 54 55 56 58 59 62 63 64 65 66 67 70 71 72 73 74 75 77
[58] 83 84 86 87 88 90 93 94 95 96 97 98 100
$s95
[1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 23
[20] 26 27 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49
[39] 52 54 55 56 58 59 60 62 63 64 65 66 67 70 71 72 73 74 75
[58] 77 83 84 86 87 88 90 93 94 95 96 97 98 100
$s96
[1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 23
[20] 26 27 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49
[39] 52 54 55 56 58 59 60 62 63 64 65 66 67 70 71 72 73 74 75
[58] 77 83 84 86 87 88 90 92 93 94 95 96 97 98 100
$s97
[1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 23
[20] 26 27 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49
[39] 52 54 55 56 58 59 60 62 63 64 65 66 67 70 71 72 73 74 75
[58] 77 83 84 86 87 88 90 92 93 94 95 96 97 98 100
$s98
[1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 23
[20] 26 27 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49
[39] 52 54 55 56 58 59 60 62 63 64 65 66 67 70 71 72 73 74 75
[58] 77 83 84 86 87 88 90 92 93 94 95 96 97 98 100
$s99
[1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 23
[20] 26 27 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49
[39] 52 54 55 56 58 59 60 62 63 64 65 66 67 70 71 72 73 74 75
[58] 77 83 84 86 87 88 90 92 93 94 95 96 97 98 100
fastcox documentation built on May 2, 2019, 10:25 a.m.
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Description Usage Arguments Details Value Author(s) References See Also Examples
Description
Fits a regularization path for the elastic net penalized Cox's model at a sequence of regularization parameters lambda.
Usage
1 2 3 4 5 6 7 8 9 10 11 12 |
Arguments
x |
matrix of predictors, of dimension N*p; each row is an observation vector. |
y |
a survival time for Cox models. Currently tied failure times are not supported. |
d |
censor status with 1 if died and 0 if right censored. |
nlambda |
the number of |
lambda.min |
given as a fraction of |
lambda |
a user supplied |
alpha |
The elasticnet mixing parameter, with 0 < α <= 1. See details. |
pf |
separate penalty weights can be applied to each coefficient of beta to allow
differential shrinkage. Can be 0 for some variables, which implies
no shrinkage, and results in that variable always being included in the
model. Default is 1 for all variables (and implicitly infinity for
variables listed in |
exclude |
indices of variables to be excluded from the model. Default is none. Equivalent to an infinite penalty factor. |
dfmax |
limit the maximum number of variables in the model. Useful for very large p, if a partial path is desired. Default is p+1. |
pmax |
limit the maximum number of variables ever to be nonzero. For example once β enters the model, no matter how many times it exits or re-enters model through the path, it will be counted only once. Default is |
standardize |
logical flag for variable standardization, prior to
fitting the model sequence. If |
eps |
convergence threshold for coordinate majorization descent. Each inner
coordinate majorization descent loop continues until the relative change in any
coefficient (i.e. max(j)|beta_new[j]-beta_old[j]|^2) is less than |
maxit |
maximum number of outer-loop iterations allowed at fixed lambda value. Default is 1e4. If models do not converge, consider increasing |
Details
The algorithm estimates β based on observed data, through elastic net penalized log partial likelihood of Cox's model.
\arg\min(-loglik(Data,β)+λ*P(β))
It can compute estimates at a fine grid of values of lambdas in order to pick up a data-driven optimal lambda for fitting a 'best' final model. The penalty is a combination of l1 and l2 penalty:
P(β)=(1-α)/2||β||_2^2+α||β||_1.
alpha=1 is the lasso penalty.
For computing speed reason, if models are not converging or running slow, consider increasing eps, decreasing
nlambda, or increasing lambda.min before increasing
maxit.
FAQ:
Question: “I am not sure how are we optimizing alpha. I can get optimal lambda for each value of alpha. But how do I select optimum alpha?”
Answer: cv.cocktail only finds the optimal lambda given alpha fixed. So to
chose a good alpha you need to fit CV on a grid of alpha, say (0.1,0.3, 0.6, 0.9, 1) and let cv.cocktail choose the optimal lambda for each alpha, then you choose the (alpha, lambda) pair that corresponds to the lowest predicted deviance.
Question: “I understand your are referring to minimizing the quantity cv.cocktail\$cvm, the mean 'cross-validated error' to optimize alpha and lambda as you did in your implementation. However, I don't know what the equation of this error is and this error is not referred to in your paper either. Do you mind explaining what this is?
”
Answer: We first define the log partial-likelihood for the Cox model. Assume \hat{β}^{[k]} is the estimate fitted on k-th fold, define the log partial likelihood function as
L(Data,\hat{β}[k])=∑_{s=1}^{S} x_{i_{s}}^{T}\hat{β}[k]-\log(∑_{i\in R_{s}}\exp(x_{i}^{T}\hat{β}[k])).
Then the log partial-likelihood deviance of the k-th fold is defined as
D[Data,k]=-2(L(Data,\hat{β}[k])).
We now define the measurement we actually use for cross validation: it is the difference between the log partial-likelihood deviance evaluated on the full dataset and that evaluated on the on the dataset with k-th fold excluded. The cross-validated error is defined as
CV-ERR[k]=D(Data[full],k)-D(Data[k^{th}\,\,fold\,\,excluded],k).
Value
An object with S3 class cocktail.
call |
the call that produced this object |
beta |
a p*length(lambda) matrix of coefficients, stored as a sparse matrix ( |
lambda |
the actual sequence of |
df |
the number of nonzero coefficients for each value of
|
dim |
dimension of coefficient matrix (ices) |
npasses |
total number of iterations (the most inner loop) summed over all lambda values |
jerr |
error flag, for warnings and errors, 0 if no error. |
Author(s)
Yi Yang and Hui Zou
Maintainer: Yi Yang <yi.yang6@mcgill.ca>
References
Yang, Y. and Zou, H. (2013),
"A Cocktail Algorithm for Solving The Elastic Net Penalized Cox's Regression in High Dimensions", Statistics and Its Interface, 6:2, 167-173.
https://github.com/emeryyi/fastcox
See Also
Examples
1 2 3 4 |
Example output
Loading required package: Matrix
$s0
NULL
$s1
[1] 1 45
$s2
[1] 1 45
$s3
[1] 1 7 45
$s4
[1] 1 7 45 55
$s5
[1] 1 7 45 55
$s6
[1] 1 7 12 45 55
$s7
[1] 1 7 12 45 55
$s8
[1] 1 7 12 45 55
$s9
[1] 1 7 12 45 55
$s10
[1] 1 7 12 17 42 45 55
$s11
[1] 1 7 12 14 17 42 45 55 59
$s12
[1] 1 7 12 14 17 42 45 55 59 74 91
$s13
[1] 1 7 12 14 17 42 45 55 59 74 91 100
$s14
[1] 1 7 10 12 14 17 42 45 55 59 74 91 100
$s15
[1] 1 2 7 10 12 14 17 28 42 45 55 59 74 91 100
$s16
[1] 1 2 7 10 12 14 17 28 42 45 55 59 74 91 100
$s17
[1] 1 2 7 10 12 14 17 28 42 45 55 59 71 74 91 100
$s18
[1] 1 2 7 10 12 14 17 28 42 45 55 59 71 74 91 98 100
$s19
[1] 1 2 7 10 12 14 17 28 42 45 55 59 71 74 91 96 98 100
$s20
[1] 1 2 7 10 12 14 17 26 28 42 45 55 56 59 71 74 91 96 98
[20] 100
$s21
[1] 1 2 7 10 12 14 17 26 28 42 45 55 56 59 71 74 91 96 98
[20] 100
$s22
[1] 1 2 5 7 10 12 14 17 26 28 42 45 55 56 59 71 74 91 96
[20] 98 100
$s23
[1] 1 2 5 7 10 12 14 17 26 28 30 42 44 45 55 56 59 71 74
[20] 84 91 96 98 100
$s24
[1] 1 2 5 7 10 12 14 17 26 28 30 31 42 44 45 55 56 59 71
[20] 74 77 84 91 96 98 100
$s25
[1] 1 2 5 7 10 12 14 17 26 28 30 31 42 44 45 55 56 59 71
[20] 74 77 84 91 96 98 100
$s26
[1] 1 2 5 7 10 12 14 17 23 26 28 30 31 42 44 45 55 56 59
[20] 71 74 77 84 91 96 98 100
$s27
[1] 1 2 5 7 10 12 14 17 23 26 28 30 31 39 42 44 45 55 56
[20] 59 71 74 77 84 88 91 96 98 100
$s28
[1] 1 2 5 7 10 12 14 17 23 26 28 30 31 39 42 44 45 55 56
[20] 59 71 74 77 84 88 91 96 98 100
$s29
[1] 1 2 5 7 10 12 14 17 23 26 28 30 31 39 42 44 45 55 56
[20] 59 71 74 75 77 84 88 91 96 98 100
$s30
[1] 1 2 5 7 10 12 14 17 23 26 28 30 31 39 42 44 45 55 56
[20] 59 71 74 75 77 84 88 91 96 98 100
$s31
[1] 1 2 5 7 10 12 14 17 23 26 28 30 31 39 42 44 45 55 56
[20] 59 63 71 74 75 77 84 88 91 96 98 100
$s32
[1] 1 2 5 7 10 12 14 23 26 28 30 31 39 42 44 45 55 56 59
[20] 63 71 74 75 77 84 88 91 96 98 100
$s33
[1] 1 2 5 7 10 11 12 14 23 26 28 30 31 39 42 44 45 55 56
[20] 59 63 71 74 75 77 84 88 91 96 98 100
$s34
[1] 1 2 3 5 7 10 11 12 14 23 26 28 30 31 39 40 42 44 45
[20] 48 55 56 59 63 71 74 75 77 84 88 91 96 98 100
$s35
[1] 1 2 3 5 7 10 11 12 14 23 26 28 30 31 39 40 42 44 45
[20] 48 55 56 59 63 71 74 75 77 84 88 91 96 98 100
$s36
[1] 1 2 3 5 7 10 11 12 14 23 26 28 30 31 39 40 42 44 45
[20] 48 55 56 59 63 71 74 75 77 84 88 91 96 98 100
$s37
[1] 1 2 3 5 7 10 11 12 14 23 26 28 30 31 39 40 42 44 45
[20] 48 55 56 59 63 71 74 75 77 84 88 91 96 98 100
$s38
[1] 1 2 3 5 7 8 10 11 12 14 16 23 26 28 30 31 39 40 42
[20] 44 45 48 49 55 56 59 62 63 65 71 74 75 77 84 88 91 96 98
[39] 100
$s39
[1] 1 2 3 5 7 8 10 11 12 14 16 23 26 28 30 31 39 40 42
[20] 44 45 48 49 55 56 59 62 63 64 65 71 74 75 77 84 88 91 96
[39] 98 100
$s40
[1] 1 2 3 5 7 8 10 11 12 14 16 23 26 28 31 39 40 42 44
[20] 45 48 49 55 56 59 62 63 64 65 71 74 75 77 84 88 91 96 98
[39] 100
$s41
[1] 1 2 3 5 7 8 10 11 12 14 16 17 23 26 28 31 32 39 40
[20] 42 44 45 48 49 55 56 59 62 63 64 65 71 74 75 77 84 88 96
[39] 98 100
$s42
[1] 1 2 3 5 7 8 10 11 12 14 16 17 23 26 28 31 32 39 40
[20] 42 44 45 48 49 55 56 59 62 63 64 65 71 74 75 77 84 88 96
[39] 98 100
$s43
[1] 1 2 3 5 7 8 10 11 12 14 16 17 23 26 28 31 32 37 39
[20] 40 42 44 45 46 48 49 55 56 58 59 62 63 64 65 71 72 74 75
[39] 77 84 88 96 98 100
$s44
[1] 1 2 3 5 7 8 10 11 12 14 16 17 23 26 28 31 32 37 39
[20] 40 42 44 45 46 48 49 55 56 58 59 62 63 64 65 71 72 74 75
[39] 77 84 88 96 98 100
$s45
[1] 1 2 3 5 7 8 10 11 12 14 16 17 23 26 28 31 32 36 37
[20] 39 40 42 44 45 46 48 49 55 56 58 59 62 63 64 65 70 71 72
[39] 74 75 77 84 86 88 96 98 100
$s46
[1] 1 2 3 5 7 8 10 11 12 14 16 17 23 26 28 31 32 34 36
[20] 37 39 40 41 42 44 45 46 48 49 55 56 58 59 62 63 64 65 70
[39] 71 72 74 75 77 84 86 88 96 98 100
$s47
[1] 1 2 3 5 7 8 10 11 12 14 16 17 23 26 28 31 32 34 36
[20] 37 39 40 41 42 44 45 46 48 49 55 56 58 59 62 63 64 65 70
[39] 71 72 73 74 75 77 84 86 88 96 98 100
$s48
[1] 1 2 3 5 7 8 10 11 12 14 16 17 23 26 28 31 32 34 36
[20] 37 39 40 41 42 44 45 46 47 48 49 55 56 58 59 62 63 64 65
[39] 70 71 72 73 74 75 77 83 84 86 88 96 98 100
$s49
[1] 1 2 3 5 7 8 10 11 12 14 16 17 18 23 26 28 31 32 34
[20] 36 37 39 40 41 42 44 45 46 47 48 49 55 56 58 59 62 63 64
[39] 65 70 71 72 73 74 75 77 83 84 86 88 96 98 100
$s50
[1] 1 2 3 5 7 8 10 11 12 14 16 17 18 23 26 28 31 32 34
[20] 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58 59 62 63
[39] 64 65 70 71 72 73 74 75 77 83 84 86 88 96 98 100
$s51
[1] 1 2 3 5 7 8 10 11 12 14 16 17 18 23 26 28 31 32 34
[20] 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58 59 62 63
[39] 64 65 70 71 72 73 74 75 77 83 84 86 88 96 98 100
$s52
[1] 1 2 3 7 8 10 11 12 14 16 17 18 23 26 28 31 32 34 36
[20] 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58 59 62 63 64
[39] 65 66 70 71 72 73 74 75 77 83 84 86 88 96 98 100
$s53
[1] 1 2 3 7 8 10 11 12 14 16 17 18 23 26 28 31 32 34 36
[20] 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58 59 62 63 64
[39] 65 66 70 71 72 73 74 75 77 83 84 86 88 96 98 100
$s54
[1] 1 2 3 5 7 8 10 11 12 14 16 17 18 23 26 28 31 32 34
[20] 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58 59 62 63
[39] 64 65 66 70 71 72 73 74 75 77 83 84 86 88 96 98 100
$s55
[1] 1 2 3 5 7 8 10 11 12 14 16 17 18 23 26 28 31 32 34
[20] 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58 59 62 63
[39] 64 65 66 70 71 72 73 74 75 77 83 84 86 88 95 96 98 100
$s56
[1] 1 2 3 5 7 8 10 11 12 14 15 16 17 18 23 26 27 28 31
[20] 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58 59
[39] 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 88 95 96 98
[58] 100
$s57
[1] 1 2 3 5 7 8 10 11 12 14 15 16 17 18 23 26 27 28 31
[20] 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58 59
[39] 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 88 93 95 96
[58] 98 100
$s58
[1] 1 2 3 5 7 8 10 11 12 14 15 16 17 18 23 26 27 28 31
[20] 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58 59
[39] 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87 88 93 95
[58] 96 98 100
$s59
[1] 1 2 3 5 7 8 9 10 11 12 14 15 16 17 18 23 26 27 28
[20] 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58
[39] 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87 88 93
[58] 95 96 98 100
$s60
[1] 1 2 3 5 7 8 9 10 11 12 14 15 16 17 18 23 26 27 28
[20] 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58
[39] 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87 88 93
[58] 95 96 98 100
$s61
[1] 1 2 3 5 7 8 9 10 11 12 14 15 16 17 18 23 26 27 28
[20] 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58
[39] 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87 88 93
[58] 95 96 98 100
$s62
[1] 1 2 3 5 7 8 9 10 11 12 14 15 16 17 18 23 26 27 28
[20] 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58
[39] 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87 88 93
[58] 95 96 98 100
$s63
[1] 1 2 3 5 7 8 9 10 11 12 14 15 16 17 18 23 26 27 28
[20] 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58
[39] 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87 88 93
[58] 95 96 98 100
$s64
[1] 1 2 3 5 7 8 9 10 11 12 14 15 16 17 18 23 26 27 28
[20] 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58
[39] 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87 88 93
[58] 95 96 98 100
$s65
[1] 1 2 3 5 7 8 9 10 11 12 14 15 16 17 18 23 26 27 28
[20] 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58
[39] 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87 88 93
[58] 95 96 98 100
$s66
[1] 1 2 3 5 7 8 9 10 11 12 14 15 16 17 18 23 26 27 28
[20] 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58
[39] 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87 88 90
[58] 93 95 96 98 100
$s67
[1] 1 2 3 5 7 8 9 10 11 12 14 15 16 17 18 23 26 27 28
[20] 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58
[39] 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87 88 90
[58] 93 95 96 98 100
$s68
[1] 1 2 3 5 7 8 9 10 11 12 14 15 16 17 18 23 26 27 28
[20] 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58
[39] 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87 88 90
[58] 93 95 96 98 100
$s69
[1] 1 2 3 5 7 8 9 10 11 12 14 15 16 17 18 23 26 27 28
[20] 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55 56 58
[39] 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87 88 90
[58] 93 95 96 98 100
$s70
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55
[39] 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87
[58] 88 90 93 95 96 98 100
$s71
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55
[39] 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87
[58] 88 90 93 95 96 98 100
$s72
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55
[39] 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87
[58] 88 90 93 95 96 98 100
$s73
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55
[39] 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87
[58] 88 90 93 95 96 98 100
$s74
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 54 55
[39] 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86 87
[58] 88 90 93 94 95 96 98 100
$s75
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 98 100
$s76
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 98 100
$s77
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 98 100
$s78
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 98 100
$s79
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 98 100
$s80
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 98 100
$s81
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 98 100
$s82
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 98 100
$s83
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 98 100
$s84
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 97 98 100
$s85
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 97 98 100
$s86
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 97 98 100
$s87
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 97 98 100
$s88
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 97 98 100
$s89
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 97 98 100
$s90
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 97 98 100
$s91
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 97 98 100
$s92
[1] 1 2 3 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26 27
[20] 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52 54
[39] 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84 86
[58] 87 88 90 93 94 95 96 97 98 100
$s93
[1] 1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 17 18 23 26
[20] 27 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49 52
[39] 54 55 56 58 59 62 63 64 65 66 70 71 72 73 74 75 77 83 84
[58] 86 87 88 90 93 94 95 96 97 98 100
$s94
[1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 23
[20] 26 27 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49
[39] 52 54 55 56 58 59 62 63 64 65 66 67 70 71 72 73 74 75 77
[58] 83 84 86 87 88 90 93 94 95 96 97 98 100
$s95
[1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 23
[20] 26 27 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49
[39] 52 54 55 56 58 59 60 62 63 64 65 66 67 70 71 72 73 74 75
[58] 77 83 84 86 87 88 90 93 94 95 96 97 98 100
$s96
[1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 23
[20] 26 27 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49
[39] 52 54 55 56 58 59 60 62 63 64 65 66 67 70 71 72 73 74 75
[58] 77 83 84 86 87 88 90 92 93 94 95 96 97 98 100
$s97
[1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 23
[20] 26 27 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49
[39] 52 54 55 56 58 59 60 62 63 64 65 66 67 70 71 72 73 74 75
[58] 77 83 84 86 87 88 90 92 93 94 95 96 97 98 100
$s98
[1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 23
[20] 26 27 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49
[39] 52 54 55 56 58 59 60 62 63 64 65 66 67 70 71 72 73 74 75
[58] 77 83 84 86 87 88 90 92 93 94 95 96 97 98 100
$s99
[1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 23
[20] 26 27 28 29 31 32 34 36 37 39 40 41 42 44 45 46 47 48 49
[39] 52 54 55 56 58 59 60 62 63 64 65 66 67 70 71 72 73 74 75
[58] 77 83 84 86 87 88 90 92 93 94 95 96 97 98 100
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