Population Growth, Fitness, and Selection - Revision history

Gorm at 11:54, 5 February 2025

2025-02-05T11:54:36Z

← Older revision		Revision as of 13:54, 5 February 2025
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	xpred = seq(0, 14.5, 0.1)		xpred = seq(0, 14.5, 0.1)
	yexp = predict(fit1, list(t = xpred))		yexp = predict(fit1, list(t = xpred))
	ypoly = predict(fit2, list(t = xpred))		ypoly = predict(fit2, list(t = xpred)) %>% as.numeric()
	dfpred = tibble(x=xpred, exponential=yexp, polynomium=ypoly) %>%		dfpred = tibble(x=xpred, exponential=yexp, polynomium=ypoly) %>%
	pivot_longer(cols=c(exponential, polynomium), names_to="model") %>%		pivot_longer(cols=c(exponential, polynomium), names_to="model") %>%

Gorm at 11:51, 5 February 2025

2025-02-05T11:51:49Z

← Older revision		Revision as of 13:51, 5 February 2025
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	formula = N ~ b0 + I(b1t) + I(b2t^2) + I(b3t^3) + I(b4t^4) + I(b5t^5) + I(b6t^6),		formula = N ~ b0 + I(b1t) + I(b2t^2) + I(b3t^3) + I(b4t^4) + I(b5t^5) + I(b6t^6),
	data = train,		data = train,
	start=list(b0=1, b1=1,b2=1,b3=1,b4=1,b5=1,b6=1)		start=list(b0=1, b1=1,b2=1,b3=1,b4=1,b5=1,b6=1),
	trace=TRUE,		trace=TRUE,
	control=list(warnOnly = TRUE)		control=list(warnOnly = TRUE)

Gorm at 11:50, 5 February 2025

2025-02-05T11:50:22Z

← Older revision		Revision as of 13:50, 5 February 2025
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	The free parameters of this model are: <math>b_0, b_1, b_2, b_3, b_4, b_5, b_6</math>. Note that we now have as many free parameters as data points in our training set		The free parameters of this model are: <math>b_0, b_1, b_2, b_3, b_4, b_5, b_6</math>. Note that we now have as many free parameters as data points in our training set

	Using the same approach as we did above for an exponential model, we now fit this polynomial model to the data~~. In this case it turns out we do not need to include guesses at starting values~~:		Using the same approach as we did above for an exponential model, we now fit this polynomial model to the data:
	fit2 = nls(		fit2 = nls(
	formula = N ~ b0 + I(b1t) + I(b2t^2) + I(b3t^3) + I(b4t^4) + I(b5t^5) + I(b6t^6),		formula = N ~ b0 + I(b1t) + I(b2t^2) + I(b3t^3) + I(b4t^4) + I(b5t^5) + I(b6t^6),
	data = train,		data = train,
			start=list(b0=1, b1=1,b2=1,b3=1,b4=1,b5=1,b6=1)
	trace=TRUE,		trace=TRUE,
	control=list(warnOnly = TRUE)		control=list(warnOnly = TRUE)
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	Plot the fitted model compared to the training data:		Plot the fitted model compared to the training data:
	xpred = seq(0, 13, 0.1)		xpred = seq(0, 13, 0.1)
	ypred = predict(fit2, list(t = xpred))		ypred = predict(fit2, list(t = xpred)) %>% as.numeric()
	dfpred = tibble(x=xpred, y=ypred)		dfpred = tibble(x=xpred, y=ypred)
	ggplot(train, aes(x=t, y=N)) +		ggplot(train, aes(x=t, y=N)) +

Gorm at 11:15, 5 February 2025

2025-02-05T11:15:39Z

← Older revision		Revision as of 13:15, 5 February 2025
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	)		)

	The option trace=TRUE causes nls to print the sum-of-squared residuals and the parameter values ~~are printed~~ at the conclusion of each iteration. Have a look at how well the fitted values match the training data:		The option trace=TRUE causes nls to print the sum-of-squared residuals and the parameter values at the conclusion of each iteration. Have a look at how well the fitted values match the training data:
	xpred = seq(0, 15, 0.1)		xpred = seq(0, 15, 0.1)
	ypred = predict(fit1, list(t = xpred))		ypred = predict(fit1, list(t = xpred))

Gorm at 11:13, 5 February 2025

2025-02-05T11:13:54Z

← Older revision		Revision as of 13:13, 5 February 2025
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	We will use the [https://stat.ethz.ch/R-manual/R-devel/library/stats/html/nls.html ''nls'' function] ("Non-linear Least Squares") in R to fit our models. This function takes as input a formula (the model) and some data, and returns estimates of the free model parameters, along with summaries of model fit. The function also requires a set of "start values" - guesses at approximate values for the free parameters in the model - to run. These do not need to be very close to the final result, they should just help by placing the numerical model fitting algorithm in a reasonable neighbourhood in parameter space.		We will use the [https://stat.ethz.ch/R-manual/R-devel/library/stats/html/nls.html ''nls'' function] ("Non-linear Least Squares") in R to fit our models. This function takes as input a formula (the model) and some data, and returns estimates of the free model parameters, along with summaries of model fit. The function also requires a set of "start values" - guesses at approximate values for the free parameters in the model - to run. These do not need to be very close to the final result, they should just help by placing the numerical model fitting algorithm in a reasonable neighbourhood in parameter space.

	In the R commands below, replace "???" by your guess about reasonable parameter values, and then check how well they fit by plotting the data points along with the line corresponding to your initial values. You will probably need to repeat this step a few times until you are in the vicinity of the correct values (but dont overdo it - the nls function should do most of the work). '''Note:''' It will make it simpler to repeat these commands if you place them in a small R script (click the plus icon in the upper left corner of RStudio and select "R Script"):		In the R commands below, replace "???" by your guess about reasonable parameter values, and then check how well they fit by plotting the data points along with the line corresponding to your initial values. You will probably need to repeat this step a few times until you are in the vicinity of the correct values (but dont overdo it - the nls function should do most of the work). '''Note:''' It will make it simpler to repeat these commands if you place them in a small R script (click the plus icon in the upper left corner of RStudio and select "R Script"). '''Note 2:''' <math>N_t = N_0 \exp(r \times t) = N_0 \exp(r)^t</math>, so <math>\exp(r)</math> is the number that will be multiplied by itself, and hence the value that will determine whether the curve is growing (<math>\exp(r) > 1</math>) or declining (<math>\exp(r) < 1</math>):
	N0_init = ???		N0_init = ???
	r_init = ???		r_init = ???

Gorm at 10:57, 5 February 2025

2025-02-05T10:57:47Z

← Older revision		Revision as of 12:57, 5 February 2025
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	df3 = df2 %>% pivot_longer(cols = -c("t"))		df3 = df2 %>% pivot_longer(cols = -c("t"))

	: The [https://thatdatatho.com/2019/03/13/tutorial-about-magrittrs-pipe-operator-and-placeholders/ "%>%" is the so-called "pipe" symbol], and works like pipe in UNIX, i.e, by sending the output from one command into a second command. This is extremely useful for making long, combined manipulations in R, while keeping them readable. The select command selects the columns we want to use for plotting (t, and all columns with population info: N, N_A, N_a). The pivot_longer command converts data from ~~[https://seananderson.ca/2013/10/19/reshape/~~ wide format to long format], which is useful when you want to automatically create legends for multiple variables (More generally, you want to understand what "tidy" data is, and why that will help with your data analyses: [https://r4ds.had.co.nz/tidy-data.html R For Data Science: Tidy Data]). You may want to inspect the intermediate data frames by simply typing their names in RStudio:		: The [https://thatdatatho.com/2019/03/13/tutorial-about-magrittrs-pipe-operator-and-placeholders/ "%>%" is the so-called "pipe" symbol], and works like pipe in UNIX, i.e, by sending the output from one command into a second command. This is extremely useful for making long, combined manipulations in R, while keeping them readable. The select command selects the columns we want to use for plotting (t, and all columns with population info: N, N_A, N_a). The pivot_longer command converts data from wide format to long format, which is useful when you want to automatically create legends for multiple variables (More generally, you want to understand what "tidy" data is, and why that will help with your data analyses: [https://r4ds.had.co.nz/tidy-data.html R For Data Science: Tidy Data]). You may want to inspect the intermediate data frames by simply typing their names in RStudio:
	df		df

Gorm at 10:54, 5 February 2025

2025-02-05T10:54:58Z

← Older revision		Revision as of 12:54, 5 February 2025
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	ggplot(df3, aes(x=t, y=value, color=name)) + geom_line()		ggplot(df3, aes(x=t, y=value, color=name)) + geom_line()

	: The ggplot command plots the population data as a function of time, with automatic legend and coloration based on variable (N, N_A, and N_a). Specifically, we here plot total population size ("N"), the number of organisms with allele 'A' ("NA"), and the number of organisms with allele 'a' ("Na") for each generation. Recall that in this run the two alleles had the same fitness (rate_A = rate_a = 1.2) but different initial population sizes (fA = 0.3, fa = 0.7).		: The ggplot command plots the population data as a function of time, with automatic legend and coloration based on variable (N, N_A, and N_a). Specifically, we here plot total population size ("N"), the number of organisms with allele 'A' ("N_A"), and the number of organisms with allele 'a' ("N_a") for each generation. Recall that in this run the two alleles had the same fitness (rate_A = rate_a = 1.2) but different initial population sizes (fA = 0.3, fa = 0.7).

	Now plot the allele frequencies: In RStudio type the following:		Now plot the allele frequencies: In RStudio type the following:

Gorm: /* Exploration of models of Population Growth, Fitness, and Selection */

2025-02-05T10:46:59Z

Exploration of models of Population Growth, Fitness, and Selection

← Older revision		Revision as of 12:46, 5 February 2025
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	library(tidyverse)		library(tidyverse)

	: The 'tidyverse' is "a set of packages that work in harmony because they share common data representations and user interface. This package is designed to make it easy to install and load multiple 'tidyverse' packages in a single step. Learn more about the 'tidyverse' at <https://tidyverse.org>."		: The 'tidyverse' is "a set of packages that work in harmony because they share common data representations and user interface. This package is designed to make it easy to install and load multiple 'tidyverse' packages in a single step. Learn more about the 'tidyverse' at [https://tidyverse.org https://tidyverse.org]."

	Now, read data from the file you just created, select variables, and reshape it for plotting: In RStudio type the following:		Now, read data from the file you just created, select variables, and reshape it for plotting: In RStudio type the following:

Gorm: /* Getting started */

2025-02-05T10:41:42Z

Getting started

← Older revision		Revision as of 12:41, 5 February 2025
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	'''3. Start extra Terminal window and cd to working directory:'''		'''3. Start extra Terminal window and cd to working directory:'''

	Start a second Terminal app ~~via the start menu~~. Also cd to today's working directory in that terminal window:		Start a second Terminal app. Also cd to today's working directory in that terminal window:

	cd /path/to/molevol/simulation		cd /path/to/molevol/simulation

Gorm: /* Overview */

2025-02-05T10:40:08Z

Overview

← Older revision		Revision as of 12:40, 5 February 2025
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	Finally, you will start learning how to work in the UNIX environment that we will use in the rest of the course (and that is alo used by bioinformaticians in the real world).		Finally, you will start learning how to work in the UNIX environment that we will use in the rest of the course (and that is alo used by bioinformaticians in the real world).

	~~If you are interested in learning more about UNIX you can browse [http://www.ee.surrey.ac.uk/Teaching/Unix/ this set of tutorials].~~

	=== Getting started ===		=== Getting started ===