Statistical methods in bioinformatics: an introduction
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2001
|
| Series: | Statictics for biology and health
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 449 - 461 |
| Physical Description: | XIX, 476 S. graph. Darst. : 25 cm |
| ISBN: | 0387952292 |
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| 100 | 1 | |a Ewens, W. J. |d 1937- |e Verfasser |0 (DE-588)172070325 |4 aut | |
| 245 | 1 | 0 | |a Statistical methods in bioinformatics |b an introduction |c Warren J. Ewens ; Gregory R. Grant |
| 264 | 1 | |a New York [u.a.] |b Springer |c 2001 | |
| 300 | |a XIX, 476 S. |b graph. Darst. : 25 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Statictics for biology and health | |
| 500 | |a Literaturverz. S. 449 - 461 | ||
| 650 | 7 | |a Bio-informatica |2 gtt | |
| 650 | 4 | |a Bio-informatique - Méthodes statistiques | |
| 650 | 7 | |a DNA |2 gtt | |
| 650 | 7 | |a Eiwitten |2 gtt | |
| 650 | 4 | |a Estadística matemática | |
| 650 | 7 | |a Genetica |2 gtt | |
| 650 | 7 | |a Markov-processen |2 gtt | |
| 650 | 4 | |a Métodos estadísticos | |
| 650 | 7 | |a Statistiek |2 gtt | |
| 650 | 7 | |a Structuur-activiteit-relatie |2 gtt | |
| 650 | 4 | |a Statistik | |
| 650 | 4 | |a Bioinformatics |x Statistical methods | |
| 650 | 4 | |a Computational Biology | |
| 650 | 4 | |a Statistics as Topic | |
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| 700 | 1 | |a Grant, Gregory R. |d 1965- |e Verfasser |0 (DE-588)17347389X |4 aut | |
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| adam_text |
CONTENTS
PREFACE
VII
1
PROBABILITY
THEORY
(I):
ONE
RANDOM
VARIABLE
1
1.1
INTRODUCTION
.
1
1.2
DISCRETE
RANDOM
VARIABLES,
DEFINITIONS
.
3
1.2.1
PROBABILITY
DISTRIBUTIONS
AND
PARAMETERS
.
3
1.2.2
INDEPENDENCE
.
6
1.3
SIX
IMPORTANT
DISCRETE
PROBABILITY
DISTRIBUTIONS
.
7
1.3.1
ONE
BERNOULLI
TRIAL
.
7
1.3.2
THE
BINOMIAL
DISTRIBUTION
.
7
1.3.3
THE
UNIFORM
DISTRIBUTION
.
9
1.3.4
THE
GEOMETRIC
DISTRIBUTION
.
10
1.3.5
THE
NEGATIVE
BINOMIAL
AND
THE
GENERALIZED
GEOMETRIC
DISTRIBUTIONS
.
12
1.3.6
THE
POISSON
DISTRIBUTION
.
13
1.4
THE
MEAN
OF
A
DISCRETE
RANDOM
VARIABLE
.
13
1.5
THE
VARIANCE
OF
A
DISCRETE
RANDOM
VARIABLE
.
16
1.6
GENERAL
MOMENTS
OF
A
PROBABILITY
DISTRIBUTION
.
18
1.7
THE
PROBABILITY-GENERATING
FUNCTION
.
19
1.8
CONTINUOUS
RANDOM
VARIABLES
.
21
1.9
THE
MEAN,
VARIANCE,
AND
MEDIAN
OF
A
CONTINUOUS
RANDOM
VARIABLE
.
22
1.9.1
DEFINITIONS
.
22
1.9.2
CHEBYSHEV
'
S
INEQUALITY
.
23
XII
CONTENTS
1.10
FIVE
IMPORTANT
CONTINUOUS
DISTRIBUTIONS
.
24
1.10.1
THE
UNIFORM
DISTRIBUTION
.
24
1.10.2
THE
NORMAL
DISTRIBUTION
.
24
1.10.3
THE
NORMAL
APPROXIMATION
TO
A
DISCRETE
DISTRIBUTION
27
1.10.4
THE
EXPONENTIAL
DISTRIBUTION
.
28
1.10.5
THE
GAMMA
DISTRIBUTION
.
31
1.10.6
THE
BETA
DISTRIBUTION
.
32
1.11
THE
MOMENT-GENERATING
FUNCTION
.
32
1.12
EVENTS
.
36
1.12.1
WHAT
ARE
EVENTS?
.
36
1.12.2
COMPLEMENTS,
UNIONS,
AND
INTERSECTIONS
.
36
1.12.3
PROBABILITIES
OF
EVENTS
.
37
1.12.4
CONDITIONAL
PROBABILITIES
.
39
1.12.5
INDEPENDENCE
OF
EVENTS
.
41
1.13
THE
MEMORYLESS
PROPERTY
OF
THE
GEOMETRIC
AND
THE
EXPONENTIAL
DISTRIBUTIONS
.
42
1.14
ENTROPY
AND
RELATED
CONCEPTS
.
43
1.14.1
ENTROPY
.
43
1.14.2
RELATIVE
ENTROPY
.
44
1.14.3
SCORES
AND
SUPPORT
.
45
1.15
TRANSFORMATIONS
.
46
1.16
EMPIRICAL
METHODS
.
47
2
PROBABILITY
THEORY
(II):
MANY
RANDOM
VARIABLES
55
2.1
MANY
DISCRETE
RANDOM
VARIABLES
.
55
2.1.1
THE
INDEPENDENT
CASE
.
56
2.1.2
THE
DEPENDENT
CASE
AND
THE
MULTINOMIAL
DISTRIBUTION
57
2.1.3
PROPERTIES
OF
RANDOM
VARIABLES
FROM
PGFS
.
57
2.2
MANY
CONTINUOUS
RANDOM
VARIABLES
.
59
2.2.1
THE
INDEPENDENT
CASE
.
59
2.2.2
THE
DEPENDENT
CASE
.
60
2.3
MOMENT-GENERATING
FUNCTIONS
.
60
2.4
INDICATOR
RANDOM
VARIABLES
.
62
2.4.1
DEFINITIONS
.
62
2.4.2
EXAMPLE:
SEQUENCING
EST
LIBRARIES
.
63
2.5
MARGINAL
DISTRIBUTIONS
.
66
2.6
CONDITIONAL
DISTRIBUTIONS
.
68
,
2.7
EXPECTED
VALUES
.
71
2.8
COVARIANCE
AND
CORRELATION
.
73
2.9
ASYMPTOTIC
DISTRIBUTIONS
.
75
2.10
DERIVED
RANDOM
VARIABLES
(I):
SUMS,
AVERAGES,,AND
MINIMA
75
2.10.1
SUMS
AND
AVERAGES
.
76
2.10.2
THE
MINIMUM
OF
N
RANDOM
VARIABLES
.
79
2.11
DERIVED
RANDOM
VARIABLES
(II):
THE
MAXIMUM
OF
N
RANDOM
VARIABLES
.
81
CONTENTS
XIII
2.11.1
DISTRIBUTIONAL
PROPERTIES:
CONTINUOUS
RANDOM
VARIABLES
.
81
2.11.2
DISTRIBUTIONAL
PROPERTIES:
DISCRETE
RANDOM
VARIABLES
84
2.11.3
AN
ASYMPTOTIC
FORMULA
FOR
THE
DISTRIBUTION
OF
X
MAX
87
2.11.4
BOUNDS
FOR
THE
DISTRIBUTIONS
OF
THE
MAXIMA
OF
GEOMETRIC
AND
GEOMETRIC-LIKE
RANDOM
VARIABLES
.
89
2.12
INTRODUCTION
TO
P-VALUES
FOR
Y
MAX
.
91
2.12.1
INTRODUCTION
.
91
2.12.2
P-VALUES
FOR
GEOMETRIC-LIKE
RANDOM
VARIABLES
.
.
91
2.12.3
P-VALUES
FOR
THE
GENERALIZED
GEOMETRIC
DISTRIBUTION
92
2.13
LARGE
OBSERVATIONS
AND
RARE
EVENTS
.
94
2.14
ORDER
STATISTICS
.
94
2.14.1
DEFINITION
.
94
2.14.2
EXAMPLE:
THE
UNIFORM
DISTRIBUTION
.
96
.
2.14.3
THE
SAMPLE
MEDIAN
.
97
2.15
TRANSFORMATIONS
.
97
2.15.1
THEORY
.
97
2.15.2
EXAMPLE:
THE
UNIFORM
DISTRIBUTION
(CONTINUED)
.
.
98
3
STATISTICS
(I):
AN
INTRODUCTION
TO
STATISTICAL
INFERENCE
105
3.1
INTRODUCTION
.
105
3.2
CLASSICAL
AND
BAYESIAN
METHODS
.
105
3.3
CLASSICAL
ESTIMATION
METHODS
.
107
3.3.1
UNBIASED
ESTIMATION
.
107
3.3.2
BIASED
ESTIMATORS
.
109
3.4
CLASSICAL
HYPOTHESIS
TESTING
.
110
3.4.1
GENERAL
PRINCIPLES
.
110
3.4.2
P-VALUES
.
113
3.4.3
HYPOTHESIS
TESTING
EXAMPLES
.
113
3.4.4
LIKELIHOOD
RATIOS,
INFORMATION,
AND
SUPPORT
.
117
3.4.5
HYPOTHESIS
TESTING
USING
X
MAX
AS
TEST
STATISTIC
.
.
118
3.5
NONPARAMETRIC
ALTERNATIVES
TO
THE
TWO-SAMPLE
T-TEST
.
.
.
119
3.6
THE
BAYESIAN
APPROACH
TO
HYPOTHESIS
TESTING
.
122
3.7
THE
BAYESIAN
APPROACH
TO
ESTIMATION
.
124
3.8
MULTIPLE
TESTING
.
125
4
STOCHASTIC
PROCESSES
(I):
POISSON
PROCESSES
AND
MARKOV
CHAINS
129
4.1
THE
HOMOGENEOUS
POISSON
PROCESS
AND
THE
POISSON
DISTRIBUTION
.
129
4.2
THE
POISSON
AND
THE
BINOMIAL
DISTRIBUTIONS
.
132
4.3
THE
POISSON
AND
THE
GAMMA
DISTRIBUTIONS
.
133
4.4
INTRODUCTION
TO
FINITE
MARKOV
CHAINS
.
133
4.5
TRANSITION
PROBABILITIES
AND
THE
TRANSITION
PROBABILITY
MATRIX
135
4.6
MARKOV
CHAINS
WITH
ABSORBING
STATES
.
137
XIV
CONTENTS
4.7
MARKOV
CHAINS
WITH
NO
ABSORBING
STATES
.
137
4.7.1
STATIONARY
DISTRIBUTIONS
.
138
4.7.2
EXAMPLE
.
139
4.8
THE
GRAPHICAL
REPRESENTATION
OF
A
MARKOV
CHAIN
.
140
4.9
MODELING
.
141
5
THE
ANALYSIS
OF
ONE
DNA
SEQUENCE
147
5.1
SHOTGUN
SEQUENCING
.
147
5.2
MODELING
DNA
.
155
5.3
MODELING
SIGNALS
IN
DNA
.
156
5.3.1
INTRODUCTION
.
156
5.3.2
WEIGHT
MATRICES:
INDEPENDENCE
.
157
5.3.3
MARKOV
DEPENDENCIES
.
158
5.3.4
MAXIMAL
DEPENDENCE
DECOMPOSITION
.
158
5.4
LONG
REPEATS
.
160
5.5
R-SCANS
.
162
5.6
THE
ANALYSIS
OF
PATTERNS
.
165
5.7
OVERLAPS
COUNTED
.
166
5.7.1
GENERAL
COMMENTS
.
166
5.7.2
NUMBER
OF
OCCURRENCES
.
166
5.7.3
DISTANCE
BETWEEN
OCCURRENCES
.
169
5.8
OVERLAPS
NOT
COUNTED
.
173
5.8.1
GENERAL
COMMENTS
.
173
5.8.2
DISTANCE
BETWEEN
RECURRENCES
.
173
5.8.3
NUMBER
OF
RECURRENCES
.
174
5.9
GENERALIZATIONS
.
175
5.9.1
ARBITRARY
NUCLEOTIDE
PROBABILITIES
.
175
5.9.2
ARBITRARY
WORD
LENGTH
.
176
6
THE
ANALYSIS
OF
MULTIPLE
DNA
OR
PROTEIN
SEQUENCES
181
6.1
TWO
SEQUENCES:
FREQUENCY
COMPARISONS
.
181
6.2
ALIGNMENTS
.
183
6.3
SIMPLE
TESTS
FOR
SIGNIFICANT
SIMILARITY
IN
AN
ALIGNMENT
.
.
.
184
6.4
ALIGNMENT
ALGORITHMS
FOR
TWO
SEQUENCES
.
189
6.4.1
INTRODUCTION
.
189
6.4.2
GAPPED
GLOBAL
COMPARISONS
AND
DYNAMIC
PROGRAMMING
ALGORITHMS
.
191
6.4.3
FITTING
ONE
SEQUENCE
INTO
ANOTHER
USING
A
LINEAR
GAP
MODEL
.
194
6.4.4
LOCAL
ALIGNMENTS
WITH
A
LINEAR
GAP
MODEL
.
195
6.4.5
OTHER
GAP
MODELS
.
196
6.4.6
LIMITATIONS
OF
THE
DYNAMIC
PROGRAMMING
ALIGNMENT
ALGORITHMS
.
198
6.5
PROTEIN
SEQUENCES
AND
SUBSTITUTION
MATRICES
.
199
6.5.1
INTRODUCTION
.
199
CONTENTS
XV
6.5.2
BLOSUM
SUBSTITUTION
MATRICES
.
200
6.5.3
PAM
SUBSTITUTION
MATRICES
.
205
6.5.4
A
SIMPLE
SYMMETRIC
PAM
MATRIX
.
210
6.6
MULTIPLE
SEQUENCES
.
.
.
.
211
7
STOCHASTIC
PROCESSES
(II):
RANDOM
WALKS
219
7.1
INTRODUCTION
.
219
7.2
THE
SIMPLE
RANDOM
WALK
.
221
7.2.1
INTRODUCTION
.
221
7.3
THE
DIFFERENCE
EQUATION
APPROACH
.
222
7.3.1
ABSORPTION
PROBABILITIES
.
222
7.3.2
MEAN
NUMBER
OF
STEPS
TAKEN
UNTIL
THE
WALK
STOPS
223
7.4
THE
MOMENT-GENERATING
FUNCTION
APPROACH
.
224
7.4.1
ABSORPTION
PROBABILITIES
.
224
7.4.2
MEAN
NUMBER
OF
STEPS
UNTIL
THE
WALK
STOPS
.
226
7.4.3
AN
ASYMPTOTIC
CASE
.
227
7.5
GENERAL
WALKS
.
228
7.6
GENERAL
WALKS:
ASYMPTOTIC
THEORY
.
230
7.6.1
INTRODUCTION
.
230
7.6.2
THE
RENEWAL
THEOREM
.
231
7.6.3
UNRESTRICTED
WALKS
.
231
7.6.4
RESTRICTED
WALKS
.
234
8
STATISTICS
(II):
CLASSICAL
ESTIMATION
AND
HYPOTHESIS
TESTING
237
8.1
INTRODUCTION
.
237
8.2
CLASSICAL
ESTIMATION
THEORY
.
237
8.2.1
CRITERIA
FOR
"
GOOD
"
ESTIMATORS
.
.
237
8.2.2
MAXIMUM
LIKELIHOOD
ESTIMATION
.
239
8.3
CLASSICAL
HYPOTHESIS
TESTING:
SIMPLE
FIXED
SAMPLE
SIZE
TESTS
246
8.3.1
THE
LIKELIHOOD
RATIO
.
246
8.4
CLASSICAL
HYPOTHESIS
TESTING:
COMPOSITE
FIXED
SAMPLE
SIZE
TESTS
.
249
8.4.1
INTRODUCTION
.
249
8.4.2
THE
-
2
LOG
A
APPROXIMATION
.
254
8.5
SEQUENTIAL
ANALYSIS
.
258
8.5.1
THE
SEQUENTIAL
PROBABILITY
RATIO
TEST
.
258
8.5.2
THE
POWER
CURVE
FOR
A
SEQUENTIAL
TEST
.
261
8.5.3
THE
MEAN
SAMPLE
SIZE
.
262
8.5.4
THE
EFFECT
OF
BOUNDARY
OVERLAP
.
264
9
BLAST
269
9.1
INTRODUCTION
.
269
9.2
THE
COMPARISON
OF
TWO
ALIGNED
SEQUENCES
.
270
9.2.1
INTRODUCTION
.
270
XVI
CONTENTS
9.2.2
LADDER
POINTS
AND
EXCURSIONS
.
270
9.2.3
PARAMETER
CALCULATIONS
.
271
9.2.4
THE
CHOICE
OF
A
SCORE
.
273
9.2.5
BOUNDS
FOR
THE
P
VALUE
.
275
9.2.6
THE
NORMALIZED
AND
THE
BIT
SCORES
.
276
9.2.7
THE
NUMBER
OF
HIGH-SCORING
EXCURSIONS
.
278
9.2.8
THE
KARLIN-ALTSCHUL
SUM
STATISTIC
.
279
9.3
THE
COMPARISON
OF
TWO
UNALIGNED
SEQUENCES
.
281
9.3.1
INTRODUCTION
.
281
9.3.2
THEORETICAL
AND
EMPIRICAL
BACKGROUND
.
281
9.3.3
EDGE
EFFECTS
.
283
9.3.4
MULTIPLE
TESTING
.
285
9.4
THE
COMPARISON
OF
A
QUERY
SEQUENCE
AGAINST
A
DATABASE
.
286
9.5
PRINTOUTS
.
288
9.5.1
EXAMPLE
.
288
9.5.2
A
MORE
COMPLICATED
EXAMPLE
.
291
9.6
MINIMUM
SIGNIFICANCE
LENGTHS
.
294
9.6.1
A
CORRECT
CHOICE
OF
N
.
294
9.6.2
AN
INCORRECT
CHOICE
OF
N
.
296
9.7
BLAST:
A
PARAMETRIC
OR
A
NONPARAMETRIC
TEST?
.
298
9.8
RELATION
TO
SEQUENTIAL
ANALYSIS
.
298
9.9
FURTHER
DEVELOPMENTS
OF
BLAST
AND
FURTHER
THEORY
.
.
.
300
10
STOCHASTIC
PROCESSES
(III):
MARKOV
CHAINS
303
10.1
INTRODUCTION
.
303
10.2
MARKOV
CHAINS
WITH
NO
ABSORBING
STATES
.
304
10.2.1
INTRODUCTION
.
304
10.2.2
CONVERGENCE
TO
THE
STATIONARY
DISTRIBUTION
.
304
10.2.3
STATIONARY
DISTRIBUTIONS:
A
NUMERICAL
EXAMPLE
.
.
.
305
10.2.4
REVERSIBILITY
AND
DETAILED
BALANCE
.
306
10.3
HIGHER-ORDER
MARKOV
DEPENDENCE
.
308
10.3.1
TESTING
FOR
HIGHER-ORDER
MARKOV
DEPENDENCE
.
308
10.3.2
TESTING
FOR
A
UNIFORM
STATIONARY
DISTRIBUTION
.
308
10.4
PATTERNS
IN
SEQUENCES
WITH
FIRST-ORDER
MARKOV
DEPENDENCE
309
10.5
MARKOV
CHAIN
MONTE
CARLO
.
310
10.5.1
THE
HASTINGS-METROPOLIS
ALGORITHM
.
311
10.5.2
GIBBS
SAMPLING
.
311
10.5.3
SIMULATED
ANNEALING
.
314
10.6
MARKOV
CHAINS
WITH
ABSORBING
STATES
.
317
10.6.1
THEORY
.
317
10.6.2
EXAMPLES
.
318
10.7
CONTINUOUS-TIME
MARKOV
CHAINS
.
321
10.7.1
DEFINITIONS
.
321
10.7.2
TIME-DEPENDENT
SOLUTIONS
.
322
10.7.3
THE
STATIONARY
DISTRIBUTION
.
323
CONTENTS
XVII
10.7.4
DETAILED
BALANCE
.
.
323
10.7.5
EXPONENTIAL
HOLDING
TIMES
.
323
10.7.6
THE
EMBEDDED
CHAIN
.
U
.
.
324
11
HIDDEN
MARKOV
MODELS
327
11.1
WHAT
IS
A
HIDDEN
MARKOV
MODEL?
.
327
11.2
THREE
ALGORITHMS
.
329
11.2.1
THE
FORWARD
AND
BACKWARD
ALGORITHMS
.
329
11.2.2
THE
VITERBI
ALGORITHM
.
331
11.2.3
THE
ESTIMATION
ALGORITHMS
.
332
11.3
APPLICATIONS
.
335
11.3.1
MODELING
PROTEIN
FAMILIES
.
335
11.3.2
MULTIPLE
SEQUENCE
ALIGNMENTS
.
.
337
11.3.3
PFAM
.
339
11.3.4
GENE
FINDING
.
340
12
COMPUTATIONALLY
INTENSIVE
METHODS
349
12.1
INTRODUCTION
.
349
12.2
ESTIMATION
.
350
12.2.1
CLASSICAL
ESTIMATION
METHODS
.
350
12.2.2
BOOTSTRAP
ESTIMATION
AND
CONFIDENCE
INTERVALS
.
.
.
351
,12.3
HYPOTHESIS
TESTING:
THE
BOOTSTRAP
ALTERNATIVE
TO
THE
TWO-SAMPLE
T-TEST
.
354
12.4
MULTIPLE
TESTING
REVISITED
.
356
12.4.1
INTRODUCTION
.
356
12.4.2
STEP-DOWN
METHODS:
T-TESTS
.
356
12.4.3
STEP-DOWN
METHODS:
A
PERMUTATION
TEST
.
359
12.4.4
STEP-DOWN
METHODS
APPLIED
TO
EXPRESSION
ARRAYS
.
360
13
EVOLUTIONARY
MODELS
365
13.1
MODELS
OF
NUCLEOTIDE
SUBSTITUTION
.
.
.
365
13.2
DISCRETE-TIME
MODELS
.
366
13.2.1
THE
JUKES-CANTOR
MODEL
.
366
13.2.2
THE
KIMURA
MODELS
.
368
13.2.3
FURTHER
GENERALIZATIONS
OF
THE
KIMURA
MODELS
.
.
.
369
13.2.4
THE
FELSENSTEIN
MODEL
.
,
.
370
13.2.5
THE
HKY
MODEL
371
13.2.6
OTHER
MODELS
.
372
13.2.7
THE
REVERSIBILITY
CRITERION
.
372
13.2.8
THE
SIMPLE
SYMMETRIC
PAM
MODEL
.
.
373
13.3
CONTINUOUS-TIME
MODELS
.
374
13.3.1
THE
CONTINUOUS-TIME
JUKES-CANTOR
MODEL
.
374
13.3.2
THE
CONTINUOUS-TIME
KIMURA
MODEL
.
378
13.3.3
THE
CONTINUOUS-TIME
FELSENSTEIN
MODEL
.
381
13.3.4
THE
CONTINUOUS-TIME
HKY
MODEL
.
382
XVIII
CONTENTS
13.3.5
THE
CONTINUOUS-TIME
PAM
MODEL
.
382
14
PHYLOGENETIC
TREE
ESTIMATION
385
14.1
INTRODUCTION
.
385
14.2
DISTANCES
.
387
14.3
TREE
RECONSTRUCTION:
THE
ULTRAMETRIC
CASE
.
388
14.4
TREE
RECONSTRUCTION:
NEIGHBOR-JOINING
.
393
14.5
SURROGATE
DISTANCES
.
397
14.6
TREE
RECONSTRUCTION:
PARSIMONY
.
398
14.7
TREE
RECONSTRUCTION:
MAXIMUM
LIKELIHOOD
.
400
14.8
EXAMPLE
.
404
14.9
MODELING,
ESTIMATION,
AND
HYPOTHESIS
TESTING
.
408
14.9.1
ESTIMATION
AND
HYPOTHESIS
TESTING
.
.
408
14.9.2
BOOTSTRAPPING
.
410
14.9.3
AN
EXAMPLE
.
411
14.9.4
ASSUMPTIONS
AND
PROBLEMS
.
415
14.9.5
PHYLOGENETIC
MODELS
AND
HYPOTHESIS
TESTING
.
416
A
BASIC
NOTIONS
IN
BIOLOGY
423
B
MATHEMATICAL
FORMULAE
AND
RESULTS
427
B.L
NUMBERS
AND
INTERVALS
.
427
B.2
SETS
AND
SET
NOTATION
.
428
B.3
FACTORIALS
.
428
B.4
BINOMIAL
COEFFICIENTS
.
429
B.5
THE
BINOMIAL
THEOREM
.
429
B.6
PERMUTATIONS
AND
COMBINATIONS
.
429
B.7
LIMITS
.
430
B.8
ASYMPTOTICS
.
431
B.9
STIRLING
'
S
APPROXIMATION
.
433
B.10
ENTROPY
AS
INFORMATION
.
433
B.LL
INFINITE
SERIES
.
434
B.12
TAYLOR
SERIES
.
436
B.13
UNIQUENESS
OF
TAYLOR
SERIES
.
.
.
439
B.14
LAURENT
SERIES
.
440
B.15
NUMERICAL
SOLUTIONS
OF
EQUATIONS
.
440
B.16
STATISTICAL
DIFFERENTIALS
.
441
B.17
THE
GAMMA
FUNCTION
.
441
B.18
PROOFS
BY
INDUCTION
.
443
B.19
LINEAR
ALGEBRA
AND
MATRICES
.
443
C
COMPUTATIONAL
ASPECTS
OF
THE
BINOMIAL
AND
GENERALIZED
GEOMETRIC
DISTRIBUTION
FUNCTIONS
445
D
BLAST:
SUMS
OF
NORMALIZED
SCORES
447
CONTENTS
XIX
REFERENCES
449
AUTHOR
INDEX
463
INDEX
466 |
| any_adam_object | 1 |
| author | Ewens, W. J. 1937- Grant, Gregory R. 1965- |
| author_GND | (DE-588)172070325 (DE-588)17347389X |
| author_facet | Ewens, W. J. 1937- Grant, Gregory R. 1965- |
| author_role | aut aut |
| author_sort | Ewens, W. J. 1937- |
| author_variant | w j e wj wje g r g gr grg |
| building | Verbundindex |
| bvnumber | BV013833431 |
| callnumber-first | R - Medicine |
| callnumber-label | R858 |
| callnumber-raw | R858 |
| callnumber-search | R858 |
| callnumber-sort | R 3858 |
| callnumber-subject | R - General Medicine |
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| ctrlnum | (OCoLC)248196733 (DE-599)BVBBV013833431 |
| dewey-full | 570/.1/5195 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 570 - Biology |
| dewey-raw | 570/.1/5195 |
| dewey-search | 570/.1/5195 |
| dewey-sort | 3570 11 45195 |
| dewey-tens | 570 - Biology |
| discipline | Biologie Agrar-/Forst-/Ernährungs-/Haushaltswissenschaft / Gartenbau |
| format | Book |
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| id | DE-604.BV013833431 |
| illustrated | Illustrated |
| indexdate | 2024-08-18T00:29:16Z |
| institution | BVB |
| isbn | 0387952292 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009461443 |
| oclc_num | 248196733 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-M347 DE-526 DE-634 DE-11 DE-188 DE-578 DE-B768 |
| owner_facet | DE-355 DE-BY-UBR DE-M347 DE-526 DE-634 DE-11 DE-188 DE-578 DE-B768 |
| physical | XIX, 476 S. graph. Darst. : 25 cm |
| publishDate | 2001 |
| publishDateSearch | 2001 |
| publishDateSort | 2001 |
| publisher | Springer |
| record_format | marc |
| series2 | Statictics for biology and health |
| spelling | Ewens, W. J. 1937- Verfasser (DE-588)172070325 aut Statistical methods in bioinformatics an introduction Warren J. Ewens ; Gregory R. Grant New York [u.a.] Springer 2001 XIX, 476 S. graph. Darst. : 25 cm txt rdacontent n rdamedia nc rdacarrier Statictics for biology and health Literaturverz. S. 449 - 461 Bio-informatica gtt Bio-informatique - Méthodes statistiques DNA gtt Eiwitten gtt Estadística matemática Genetica gtt Markov-processen gtt Métodos estadísticos Statistiek gtt Structuur-activiteit-relatie gtt Statistik Bioinformatics Statistical methods Computational Biology Statistics as Topic Statistik (DE-588)4056995-0 gnd rswk-swf Bioinformatik (DE-588)4611085-9 gnd rswk-swf Bioinformatik (DE-588)4611085-9 s Statistik (DE-588)4056995-0 s DE-604 Grant, Gregory R. 1965- Verfasser (DE-588)17347389X aut DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009461443&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ewens, W. J. 1937- Grant, Gregory R. 1965- Statistical methods in bioinformatics an introduction Bio-informatica gtt Bio-informatique - Méthodes statistiques DNA gtt Eiwitten gtt Estadística matemática Genetica gtt Markov-processen gtt Métodos estadísticos Statistiek gtt Structuur-activiteit-relatie gtt Statistik Bioinformatics Statistical methods Computational Biology Statistics as Topic Statistik (DE-588)4056995-0 gnd Bioinformatik (DE-588)4611085-9 gnd |
| subject_GND | (DE-588)4056995-0 (DE-588)4611085-9 |
| title | Statistical methods in bioinformatics an introduction |
| title_auth | Statistical methods in bioinformatics an introduction |
| title_exact_search | Statistical methods in bioinformatics an introduction |
| title_full | Statistical methods in bioinformatics an introduction Warren J. Ewens ; Gregory R. Grant |
| title_fullStr | Statistical methods in bioinformatics an introduction Warren J. Ewens ; Gregory R. Grant |
| title_full_unstemmed | Statistical methods in bioinformatics an introduction Warren J. Ewens ; Gregory R. Grant |
| title_short | Statistical methods in bioinformatics |
| title_sort | statistical methods in bioinformatics an introduction |
| title_sub | an introduction |
| topic | Bio-informatica gtt Bio-informatique - Méthodes statistiques DNA gtt Eiwitten gtt Estadística matemática Genetica gtt Markov-processen gtt Métodos estadísticos Statistiek gtt Structuur-activiteit-relatie gtt Statistik Bioinformatics Statistical methods Computational Biology Statistics as Topic Statistik (DE-588)4056995-0 gnd Bioinformatik (DE-588)4611085-9 gnd |
| topic_facet | Bio-informatica Bio-informatique - Méthodes statistiques DNA Eiwitten Estadística matemática Genetica Markov-processen Métodos estadísticos Statistiek Structuur-activiteit-relatie Statistik Bioinformatics Statistical methods Computational Biology Statistics as Topic Bioinformatik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009461443&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT ewenswj statisticalmethodsinbioinformaticsanintroduction AT grantgregoryr statisticalmethodsinbioinformaticsanintroduction |