updates to use of "data" in nls()/nlrob()
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@ -4,12 +4,13 @@
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# Purpose: A Bioinformatics Course:
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# Purpose: A Bioinformatics Course:
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# R code accompanying the FND-STA-Probability_distribution unit.
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# R code accompanying the FND-STA-Probability_distribution unit.
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#
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#
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# Version: 1.4
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# Version: 1.6
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#
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#
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# Date: 2017-10 - 2020-11
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# Date: 2017-10 - 2020-11
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# Author: Boris Steipe (boris.steipe@utoronto.ca)
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# Author: Boris Steipe (boris.steipe@utoronto.ca)
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#
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#
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# Versions:
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# Versions:
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# 1.6 Revise use of data argument to nls()/nlrob()
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# 1.5 Extensive additions on Poisson and binomial distributions
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# 1.5 Extensive additions on Poisson and binomial distributions
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# and regression fits
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# and regression fits
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# 1.4 2020 Maintenance
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# 1.4 2020 Maintenance
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@ -35,31 +36,31 @@
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#TOC>
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#TOC>
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#TOC> Section Title Line
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#TOC> Section Title Line
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#TOC> --------------------------------------------------------------------------
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#TOC> --------------------------------------------------------------------------
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#TOC> 1 Introduction 67
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#TOC> 1 Introduction 68
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#TOC> 2 Three fundamental distributions 130
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#TOC> 2 Three fundamental distributions 131
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#TOC> 2.1 The Poisson Distribution 133
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#TOC> 2.1 The Poisson Distribution 134
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#TOC> 2.2 The hypergeometric distribution 227
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#TOC> 2.2 The hypergeometric distribution 228
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#TOC> 2.2.1 Digression: qqplot() and residuals 375
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#TOC> 2.2.1 Digression: qqplot() and residuals 376
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#TOC> 2.2.2 Fitting functions 433
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#TOC> 2.2.2 Fitting functions 436
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#TOC> 2.2.2.1 Fit a normal distribution (using nls() ) 503
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#TOC> 2.2.2.1 Fit a normal distribution (using nls() ) 506
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#TOC> 2.2.2.2 Fit a normal distribution (using nlrob()) ) 520
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#TOC> 2.2.2.2 Fit a normal distribution (using nlrob()) ) 525
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#TOC> 2.2.2.3 Extreme Value distributions: Gumbel 547
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#TOC> 2.2.2.3 Extreme Value distributions: Gumbel 551
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#TOC> 2.2.2.4 Extreme Value distributions: Weibull 570
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#TOC> 2.2.2.4 Extreme Value distributions: Weibull 576
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#TOC> 2.2.2.5 Logistic distribution 613
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#TOC> 2.2.2.5 Logistic distribution 618
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#TOC> 2.2.2.6 Log-Logistic distribution 643
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#TOC> 2.2.2.6 Log-Logistic distribution 647
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#TOC> 2.2.2.7 Fitting a negative binomial distribution 673
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#TOC> 2.2.2.7 Fitting a negative binomial distribution 676
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#TOC> 2.2.2.8 Fitting a binomial distribution 726
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#TOC> 2.2.2.8 Fitting a binomial distribution 729
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#TOC> 2.3 The uniform distribution 774
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#TOC> 2.3 The uniform distribution 841
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#TOC> 2.4 The Normal Distribution 794
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#TOC> 2.4 The Normal Distribution 861
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#TOC> 3 quantile-quantile comparison 835
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#TOC> 3 quantile-quantile comparison 902
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#TOC> 3.1 qqnorm() 845
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#TOC> 3.1 qqnorm() 912
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#TOC> 3.2 qqplot() 911
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#TOC> 3.2 qqplot() 978
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#TOC> 4 Quantifying the difference 928
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#TOC> 4 Quantifying the difference 995
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#TOC> 4.1 Chi2 test for discrete distributions 963
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#TOC> 4.1 Chi2 test for discrete distributions 1029
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#TOC> 4.2 Kullback-Leibler divergence 1054
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#TOC> 4.2 Kullback-Leibler divergence 1120
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#TOC> 4.2.1 An example from tossing dice 1065
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#TOC> 4.2.1 An example from tossing dice 1131
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#TOC> 4.2.2 An example from lognormal distributions 1188
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#TOC> 4.2.2 An example from lognormal distributions 1253
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#TOC> 4.3 Continuous distributions: Kolmogorov-Smirnov test 1231
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#TOC> 4.3 Continuous distributions: Kolmogorov-Smirnov test 1296
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#TOC>
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#TOC>
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#TOC> ==========================================================================
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#TOC> ==========================================================================
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@ -384,7 +385,9 @@ legend("topright",
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# misunderstanding about the requirements for the particular theory to apply. R
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# misunderstanding about the requirements for the particular theory to apply. R
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# has a general way to examine such differences: qqplot() plots the deviations
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# has a general way to examine such differences: qqplot() plots the deviations
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# between theory and observation ordered as ranks, i.e. not affected by absolute
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# between theory and observation ordered as ranks, i.e. not affected by absolute
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# scale.
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# scale. (See here for an accessible introduction to qqplot() and the
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# "quantiles" that it uses:
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# https://data.library.virginia.edu/understanding-q-q-plots/)
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oPar <- par(mfrow = c(1, 2))
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oPar <- par(mfrow = c(1, 2))
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qqplot( dpois(0:nMax, (250 * 256) / 6091),
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qqplot( dpois(0:nMax, (250 * 256) / 6091),
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@ -503,10 +506,12 @@ legend("topright",
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# ==== 2.2.2.1 Fit a normal distribution (using nls() )
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# ==== 2.2.2.1 Fit a normal distribution (using nls() )
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x <- 0:28
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x <- 0:28
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plot(0:28, tu, type="s")
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plot(x, tu, type="s")
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fit <- nls(tu ~ ( a / (sig*sqrt(2*pi)) ) * exp( (-1/2)*((x-mu)/sig)^2 ), start = c(a = 1, mu = 10, sig = 3)) # starting values
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fit <- nls(tu ~ ( a / (sig*sqrt(2*pi)) ) * exp( (-1/2)*((x-mu)/sig)^2 ),
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data = data.frame(x = 0:28, tu = tu),
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start = c(a = 1, mu = 10, sig = 3)) # starting values
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points(x, predict(fit, list(x = x)), col = "#CC00CC55", lwd = 2,
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points(x, predict(fit), col = "#CC00CC55", lwd = 2,
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type = "s")
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type = "s")
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coef(fit)
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coef(fit)
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@ -526,15 +531,14 @@ pkg <- "robustbase"
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if (! requireNamespace(pkg, quietly = TRUE)) { install.packages(pkg) }
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if (! requireNamespace(pkg, quietly = TRUE)) { install.packages(pkg) }
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x <- 0:28
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x <- 0:28
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plot(0:28, tu, type="s")
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plot(x, tu, type="s")
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fit <- robustbase::nlrob(tu ~ ( a / (sig*sqrt(6.2831853072)) ) *
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fit <- robustbase::nlrob(tu ~ ( a / (sig*sqrt(6.2831853072)) ) *
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exp( (-1/2)*((x-mu)/sig)^2 ),
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exp( (-1/2)*((x-mu)/sig)^2 ),
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data = data.frame(tu = tu,
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data = data.frame(x = 0:28,
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x = 0:28),
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tu = tu),
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start = c(a = 1, mu = 10, sig = 3)) # starting values
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start = c(a = 1, mu = 10, sig = 3)) # starting values
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points(x, predict(fit, list(x = x)), col = "#CC00CC55", lwd = 2,
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points(x, predict(fit), col = "#CC00CC55", lwd = 2, type = "s")
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type = "s")
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coef(fit)
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coef(fit)
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# a mu sig
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# a mu sig
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@ -546,18 +550,20 @@ sum(resid(fit)^2)
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# ==== 2.2.2.3 Extreme Value distributions: Gumbel
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# ==== 2.2.2.3 Extreme Value distributions: Gumbel
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# Many processes that involve best-of choices are better modelled with so called extreme-value distributions: here is the Gumbel distribution from the evd package.
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# Many processes that involve "best-of" choices are better modelled with
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# so-called extreme-value distributions: here is the Gumbel distribution
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# from the evd package.
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pkg <- "evd"
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pkg <- "evd"
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if (! requireNamespace(pkg, quietly = TRUE)) { install.packages(pkg) }
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if (! requireNamespace(pkg, quietly = TRUE)) { install.packages(pkg) }
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x <- 0:28
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x <- 0:28
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plot(0:28, tu, type="s")
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plot(x, tu, type="s")
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fit <- robustbase::nlrob(tu ~ evd::dgumbel(x, loc = L, scale = S),
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fit <- robustbase::nlrob(tu ~ evd::dgumbel(x, loc = L, scale = S),
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data = data.frame(tu = tu, x = 0:28),
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data = data.frame(tu = tu, x = 0:28),
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start = c(L = 7.3, S = 2.82))
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start = c(L = 7.3, S = 2.82))
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points(x, predict(fit, list(x = x)), type = "s", col = "#55DD88")
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points(x, predict(fit), type = "s", col = "#55DD88")
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coef(fit)
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coef(fit)
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# L S
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# L S
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@ -575,9 +581,9 @@ sum(resid(fit)^2)
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# NOTE: the parameter TH is > X
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# NOTE: the parameter TH is > X
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idx <- 4:28
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idx <- 4:28
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x <- idx
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myX <- idx
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y <- as.numeric(tu)[idx]
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myY <- as.numeric(tu)[idx]
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plot(x, y, type="s")
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plot(myX, myY, type="s")
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dWei <- function(x, th, l, k) {
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dWei <- function(x, th, l, k) {
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a <- k/l
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a <- k/l
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@ -589,8 +595,8 @@ dWei <- function(x, th, l, k) {
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set.seed(112358)
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set.seed(112358)
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fit <- robustbase::nlrob(y ~ dWei(x, th = TH, l = L, k = K),
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fit <- robustbase::nlrob(y ~ dWei(x, th = TH, l = L, k = K),
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data = data.frame(y = y,
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data = data.frame(y = myY,
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x = x),
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x = myX),
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lower = c(TH = 3.5,
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lower = c(TH = 3.5,
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L = 8,
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L = 8,
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K = 2.5),
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K = 2.5),
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@ -599,8 +605,7 @@ fit <- robustbase::nlrob(y ~ dWei(x, th = TH, l = L, k = K),
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K = 3),
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K = 3),
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method = "mtl")
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method = "mtl")
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points(x, predict(fit, list(x = x)),
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points(x, predict(fit), col = "#CC00CC55", lwd = 2, type = "s")
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col = "#CC00CC55", lwd = 2, type = "s")
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coef(fit)
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coef(fit)
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# TH L K
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# TH L K
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@ -615,9 +620,9 @@ sum(resid(fit)^2)
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# Similar to normal distribution, but with heavier tails
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# Similar to normal distribution, but with heavier tails
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# https://en.wikipedia.org/wiki/Logistic_distribution
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# https://en.wikipedia.org/wiki/Logistic_distribution
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x <- 0:28
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myX <- 0:28
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y <- as.numeric(tu)
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myY <- as.numeric(tu)
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plot(0:28, y, type="s")
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plot(myX, myY, type="s")
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dLogi <- function(x, mu, s) {
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dLogi <- function(x, mu, s) {
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y <- (exp(-(x-mu)/s)) / (s * (1+exp(-(x-mu)/s))^2)
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y <- (exp(-(x-mu)/s)) / (s * (1+exp(-(x-mu)/s))^2)
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@ -625,12 +630,11 @@ dLogi <- function(x, mu, s) {
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}
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}
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fit <- robustbase::nlrob(y ~ dLogi(x, mu = M, s = S),
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fit <- robustbase::nlrob(y ~ dLogi(x, mu = M, s = S),
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data = data.frame(y = y,
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data = data.frame(y = myY,
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x = 0:28),
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x = myX),
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start = c(M = 10, S = 3))
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start = c(M = 10, S = 3))
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points(x, predict(fit, list(x = x)),
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points(x, predict(fit), col = "#CC00CC55", lwd = 2, type = "s")
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col = "#CC00CC55", lwd = 2, type = "s")
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coef(fit)
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coef(fit)
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# M S
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# M S
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@ -642,12 +646,12 @@ sum(resid(fit)^2)
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# ==== 2.2.2.6 Log-Logistic distribution
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# ==== 2.2.2.6 Log-Logistic distribution
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# Spercial case of logistic, often used in survival analysis
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# Special case of logistic, often used in survival analysis
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# https://en.wikipedia.org/wiki/Log-logistic_distribution
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# https://en.wikipedia.org/wiki/Log-logistic_distribution
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x <- 0:28
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myX <- 0:28
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y <- as.numeric(tu)
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myY <- as.numeric(tu)
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plot(0:28, y, type="s")
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plot(myX, myY, type="s")
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dLogLogi <- function(x, a, b) {
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dLogLogi <- function(x, a, b) {
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y <- ((b/a)*(x/a)^(b-1)) / (1+((x/a)^b))^2
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y <- ((b/a)*(x/a)^(b-1)) / (1+((x/a)^b))^2
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@ -655,12 +659,11 @@ dLogLogi <- function(x, a, b) {
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}
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}
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fit <- robustbase::nlrob(y ~ dLogLogi(x, a = A, b = B),
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fit <- robustbase::nlrob(y ~ dLogLogi(x, a = A, b = B),
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data = data.frame(y = y,
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data = data.frame(y = myY,
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x = 0:28),
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x = myX),
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start = c(A = 9.5, B = 4.8))
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start = c(A = 9.5, B = 4.8))
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points(x, predict(fit, list(x = x)),
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points(x, predict(fit), col = "#CC00CC55", lwd = 2, type = "s")
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col = "#CC00CC55", lwd = 2, type = "s")
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coef(fit)
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coef(fit)
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# A B
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# A B
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@ -696,8 +699,8 @@ fit <- robustbase::nlrob(tu ~ dNB(x, r = RR, p = P),
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upper = c(P = 0.99),
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upper = c(P = 0.99),
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method = "mtl")
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method = "mtl")
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points(x, predict(fit, list(x = x)),
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points(x, predict(fit), col = "#CC00CC55", lwd = 2, type = "s")
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col = "#CC00CC55", lwd = 2, type = "s")
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coef(fit)
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coef(fit)
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# P
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# P
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# 0.9100729
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# 0.9100729
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@ -728,9 +731,9 @@ which(R == min(R))
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# cf. https://en.wikipedia.org/wiki/Binomial_distribution
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# cf. https://en.wikipedia.org/wiki/Binomial_distribution
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x <- 0:28
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myX <- 0:28
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y <- as.numeric(tu)
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myY <- as.numeric(tu)
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plot(x, y, type="s")
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plot(myX, myY, type="s")
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dBinom <- function(x, s, p) {
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dBinom <- function(x, s, p) {
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y <- choose(s, x) * (p^x) * ((1-p)^(s-x))
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y <- choose(s, x) * (p^x) * ((1-p)^(s-x))
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@ -738,12 +741,11 @@ dBinom <- function(x, s, p) {
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}
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}
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fit <- robustbase::nlrob(y ~ dBinom(x, s = S, p = P),
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fit <- robustbase::nlrob(y ~ dBinom(x, s = S, p = P),
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data = data.frame(y = y,
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data = data.frame(y = myY,
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x = 0:28),
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x = myX),
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start = c(S = 240, P = 256/6091))
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start = c(S = 240, P = 256/6091))
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points(x, predict(fit, list(x = x)),
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points(x, predict(fit), col = "#CC00CC55", lwd = 2, type = "s")
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col = "#CC00CC55", lwd = 2, type = "s")
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coef(fit)
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coef(fit)
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# S P
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# S P
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@ -752,7 +754,72 @@ coef(fit)
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sum(resid(fit)^2)
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sum(resid(fit)^2)
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# [1] 1.114272e-06
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# [1] 1.114272e-06
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# Here we go: near perfect fit. But note the parameters! Can you identify the relationship of S and P to our model of choosing unique transcription factors in a random sampling process. Because, honestly: I can't.
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# Here we go: near perfect fit. But note the parameters! Can you identify the
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# relationship of S and P to our model of choosing unique transcription factors
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# in a random sampling process?
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# Remember what we started out with:
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points(x, dBinom(x, 240, 256/6091),
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col = "#00AAFF55", lwd = 2, type = "s")
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# The model has improved a lot from our starting-value guess. But how does an
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# effective sample size of 122.78 relate to the actual value of 250 samples? And
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# how does a probability for choosing a transcription factor of 0.0838 relate to
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# having 256 transcription factors among 6091 genes (a fraction of 0.0420)? Well
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# - if you stare at the numbers for a bit you might notice that the fitted
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# sample size is about half of what we had, while the fitted probability is
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# about twice that. Let's correct for the factor of two, and look at the fit ...
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points(x, dBinom(x, coef(fit)["S"] * 2 , coef(fit)["P"] / 2),
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col = "#00FF8899", lwd = 2, type = "s")
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# interesting ... that's not so bad ... Essentially what this tells us is: our
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# model is very similar to one in which our effective sample size would have
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# been about 246 genes, not 250, and the number of our genes would have been
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# close to 0.04188461 * 6091 ~= 255 transcription factors, not 256. Close, to -
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# but not almost perfect.
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# Incidentally: would you have thought that we can
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# detect the number of transcription factors in our sample +- 1 if we just
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# sample frequently enough?
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# Let's examine the effect that applying various factors to size and probability
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# have on our distribution:
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z <- c(1/8:2 , 1, 2:8, 10, 20, 50)
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myCols <- colorRampPalette(c("#00000022",
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"#00000088",
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"#DD0000DD",
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"#0088FF88",
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"#0088FF55",
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"#0088FF22"),
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|
alpha=TRUE)(length(z))
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||||||
|
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||||||
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myLwd <- rep(1.25, length(z))
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myLwd[z == 1] <- 2.5 # make the actual fitted value stand out
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||||||
|
|
||||||
|
plot(x, y, type="s", ylim =c(0, 0.22), lwd = 5, col="#FF990077")
|
||||||
|
|
||||||
|
for (i in seq_along(z)) {
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||||||
|
points(x, dBinom(x, coef(fit)["S"] * z[i] , coef(fit)["P"] / z[i]),
|
||||||
|
col = myCols[i], type = "s")
|
||||||
|
}
|
||||||
|
|
||||||
|
legend("topright",
|
||||||
|
legend = sprintf("%3.2f", z),
|
||||||
|
cex = 0.5,
|
||||||
|
col = myCols,
|
||||||
|
lwd = myLwd,
|
||||||
|
bty = "n")
|
||||||
|
|
||||||
|
# We see how factors less than 1 applied to our actual sample size and fraction
|
||||||
|
# of transcription factors increasingly narrow the distribution; factors larger
|
||||||
|
# than one increasingly broaden the distribution but in a much slower manner. A
|
||||||
|
# factor of one (i.e. half size, double probability) hits the sweet spot and
|
||||||
|
# recaptures the essence of our observation. Clearly, this is a case of
|
||||||
|
# fortuitously cancelling errors.
|
||||||
|
|
||||||
|
|
||||||
# What is the take-home message here?
|
# What is the take-home message here?
|
||||||
# - The standard Poisson and hypergeometric distributions apply to
|
# - The standard Poisson and hypergeometric distributions apply to
|
||||||
@ -927,7 +994,6 @@ qqplot(dl, dg) # clearly not equal
|
|||||||
|
|
||||||
# = 4 Quantifying the difference ==========================================
|
# = 4 Quantifying the difference ==========================================
|
||||||
|
|
||||||
|
|
||||||
# Quantile-quantile plots give us a visual estimate, but how do we quantify the
|
# Quantile-quantile plots give us a visual estimate, but how do we quantify the
|
||||||
# difference between distributions? Let's compare two types of extreme-value
|
# difference between distributions? Let's compare two types of extreme-value
|
||||||
# distributions, a lognormal distribution with the same distribution that is
|
# distributions, a lognormal distribution with the same distribution that is
|
||||||
@ -964,7 +1030,7 @@ legend("topright",
|
|||||||
|
|
||||||
# The chi2 test can be used to compare discrete distributions - even though
|
# The chi2 test can be used to compare discrete distributions - even though
|
||||||
# it is not ideal for this purpose.
|
# it is not ideal for this purpose.
|
||||||
# (https://stats.stackexchange.com/questions/113692/test-identicality-of-discrete-distributions)
|
# (https://stats.stackexchange.com/questions/113692)
|
||||||
#
|
#
|
||||||
# Let's draw 100 random variables according to our target distributions
|
# Let's draw 100 random variables according to our target distributions
|
||||||
N <- 100
|
N <- 100
|
||||||
@ -989,9 +1055,9 @@ hist(rG1.5, breaks = myBreaks, col = myCols[4])
|
|||||||
# package information - plotrix has _many_ useful utilities to enhance
|
# package information - plotrix has _many_ useful utilities to enhance
|
||||||
# plots or produce informative visualizations.
|
# plots or produce informative visualizations.
|
||||||
|
|
||||||
if (! requireNamespace("plotrix", quietly = TRUE)) {
|
pkg <- "plotrix"
|
||||||
install.packages("plotrix")
|
if (! requireNamespace(pkg, quietly = TRUE)) { install.packages(pkg) }
|
||||||
}
|
|
||||||
# Package information:
|
# Package information:
|
||||||
# library(help = plotrix) # basic information
|
# library(help = plotrix) # basic information
|
||||||
# browseVignettes("plotrix") # available vignettes
|
# browseVignettes("plotrix") # available vignettes
|
||||||
@ -1139,7 +1205,6 @@ pmfPC <- function(cts, nam) {
|
|||||||
}
|
}
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
# The definition of the Kullback-Leibler divergence itself is quite simple
|
# The definition of the Kullback-Leibler divergence itself is quite simple
|
||||||
# actually: for two distributions, p and q it is sum(p * log( p / q))
|
# actually: for two distributions, p and q it is sum(p * log( p / q))
|
||||||
|
|
||||||
|
|||||||
Loading…
Reference in New Issue
Block a user