Tokyo.R#77 BeginneRSession-Data analysis

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77th Tokyo.R @kilometer BeginneR Session 5 -- Data analysis -- 2019.04.13 at SONY Co.

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Who！？

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Who！？名前：三村 @kilometer 職業：ポスドク (こうがくはくし) 専⾨：⾏動神経科学(霊⻑類) 脳イメージング医療システム⼯学 R歴： ~ 10年ぐらい流⾏: ベントー

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BeginneR Session

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BeginneR

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BeginneR

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Before After BeginneR Session BeginneR BeginneR

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BeginneR Advanced Hoxo_m If I have seen further it is by standing on the sholders of Giants. -- Sir Isaac Newton, 1676

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BeginneR Session 5 -- Data analysis --

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What is Data?

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What is Data? ℎ f X ℎℎ Truth Knowledge

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What is Data? ℎ f X ℎℎ Truth Knowledge Modeling Modeling

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“Strong” Hypothesis “Weaken” Hypothesis Data Data What is Data? Hypothesis Driven Data Driven

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What is Data? f X ℎℎ . f X ℎℎ . ℎ ℎ Hypothesis Driven Data Driven

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What is Data? Data is observed (partial) information about phenotype of the world. We can hypothesize a part of principle via statistical modeling with data.

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What is the most frequently used standard popular simple modeling?

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Regression Linear = + + Input Output coefficient intercept residual

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Regression Linear = + + ~(0, ) Input Output coefficient intercept residual Normal (Gaussian) distribution mean standard deviation

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Regression Linear = + + ~(0, ) Input Output coefficient intercept residual Normal (Gaussian) distribution mean standard deviation parameters

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Before we begin...

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Tools ①: pipe %>% ②: mutate

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Tool①: pipe %>% X %>% f X %>% f(y) X %>% f %>% g X %>% f(y, .) f(X) f(X, y) g(f(X)) f(y, X)

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Tool②: mutate

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Regression Linear = + + ~(0, ) Input Output coefficient intercept residual Normal (Gaussian) distribution mean standard deviation parameters

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[A , A ]

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# set environment library(tidyverse) set.seed(123) # set parameters N <- 7 a <- 4 b <- 3 s <- 15 # make data sample dat <- data.frame(x = runif(N, 0, 10), mutate(y = a * x + b + e) # attach {package} # set random seed # data No. # coefficient # intercept # standard deviation Random number generator e = rnorm(N, 0, s)) %>%

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# make data sample dat <- data.frame(x = runif(N, 0, 10), mutate(y = a * x + b + e) e = rnorm(N, 0, s)) %>%

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# visualization ggplot(dat, aes(x, y))+ geom_point()

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[A , A ] = +

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[A , A ] = + E E E observed predicted

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[A , A ] = + E E E E observed predicted

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[A , A ] = + E E E observed predicted F predicted F unobserved E

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# liner model fitting fit_lm <- lm(formula = y ~ x, data = dat) fit_lm <- lm(y ~ x, dat) abbreviated form (same meaning)

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# liner model fitting fit_lm <- lm(y ~ x, dat) # view help ?lm

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# liner model fitting fit_lm <- lm(y ~ x, dat) # view help ?lm Usage lm(formula, data, subset, weights, na.action, method = "qr", model = TRUE, ...)

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# liner model fitting fit_lm <- lm(y ~ x, dat)

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# liner model fitting fit_lm <- lm(y ~ x, dat)

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[A , A ] = + E E E E observed predicted

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[A , A ] = + E E E E observed predicted

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[A , A ] = + E E E observed predicted F predicted F unobserved E unobserved F

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dat <- data.frame(x = runif(N, 0, 10), e = rnorm(N, 0, s)) %>% mutate(y = a * x + b + e) dat_lm <- dat %>% mutate(predict = lm(y ~ x) %>% predict) E

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# visualization ggplot(dat_lm)+ geom_point(aes(x, y))+ geom_point(aes(x, predict), color = "blue")

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# visualization ggplot(dat_lm)+ geom_line(aes(x, predict), linetype = 2) geom_point(aes(x, y))+ geom_point(aes(x, predict), color = "blue")

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ggplot(dat_lm)+ geom_path(aes(x, predict), size = 3, color = "darkgrey")+ geom_point(aes(x, y), size = 3)+ geom_segment(aes(x, y, xend = x, yend = predict), color = "blue", linetype = 1, size = 0.7)+ geom_point(aes(x, predict), size = 3, color = "blue")+ theme_classic()+ theme(text = element_text(size = 21))+ ylab("y") ggsave("plot.png", width = 5, height = 5)

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[A , A ] = + , H ≅ 0.65

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[A , A ] = + , H ≅ 0.65

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= + , H ≅ 0.84 = + , H ≅ 0.62 = + , H ≅ 0.37 summary(fit_lm)$r.squared

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= + E E E A − S S H = 1 − ∑ A H A ∑ A − S H A A Coefficient of determination = S

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= + , H ≅ 0.84 = + , H ≅ 0.62 = + , H ≅ 0.37

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= + , H ≅ 0.84 = + , H ≅ 0.62 = + , H ≅ 0.37 ~(0, = 5) ~(0, = 15) ~(0, = 25)

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Regression Linear = W + + Input Output coefficient intercept residual Multiple Linear Regression = W + X X + H H + Input Output intercept coefficient residual coefficient Input

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N <- 15; s <- 10; a0 <- 3; a1 <- 3; a2 <- 2 dat <- data.frame(x1 = runif(N, 0, 10), x2 = runif(N, 0, 10)) %>% mutate(y = a0 + a1 * x1 + a2 * x2 + rnorm(N, 0, s)) Multiple Linear Regression

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Multiple Linear Regression plot(dat)

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library(rgl) plot3d(dat) Multiple Linear Regression

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N <- 15; s <- 10; a0 <- 3; a1 <- 3; a2 <- 2 dat <- data.frame(x1 = runif(N, 0, 10), x2 = runif(N, 0, 10)) %>% mutate(y = a0 + a1 * x1 + a2 * x2 + rnorm(N, 0, s)) Multiple Linear Regression fit_lm <- lm(y ~ x1 + x2, dat)

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coeff <- fit_lm$coefficients plot3d(dat) planes3d(coeff[2], coeff[3], -1, coeff[1], alpha = 0.5) Multiple Linear Regression

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Multiple Linear Regression W , X , H H

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Agenda Introduction What is data? Liner Regression Multivariate Analysis 済済済

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plot(dat)

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X = XX X + XH [ + ⋯ + X] ] H = HX X + HH [ + ⋯ + H] ] [ = [X [ + [H [ + ⋯ + [] ] ⋮ X ⋮ ] = XX … X] ⋮ ⋱ ⋮ ]X … ]] X ⋮ ] Principal Component Analysis eigenvector matrix principal components

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Principal Component Analysis fit_pca <- prcomp(dat, scale = T)

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Principal Component Analysis fit_pca <- prcomp(dat, scale = T)

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Principal Component Analysis compression

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References

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Agenda Introduction What is data? Liner Regression Multivariate Analysis 済済済済

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Appendix…

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Nested data modeling

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dat1 <- data.frame(...) dat_lm1 <- ... dat2 <- data.frame(...) dat_lm2 <- ... dat3 <- data.frame(...) dat_lm3 <- ... Nested data modeling

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Nested data modeling

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Nested data modeling

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Nested data modeling

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Nested data modeling

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dat <- data.frame(x = runif(N, 0, 10)) %>% nest %>% list %>% rep(3) %>% bind_rows %>% rowid_to_column("id") %>% mutate(s = c(5, 10, 25)) Nested data modeling

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dat <- data.frame(x = runif(N, 0, 10)) %>% nest %>% list %>% rep(3) %>% bind_rows %>% rowid_to_column("id") %>% mutate(s = c(5, 10, 25)) %>% mutate(data = map2(data, s, ~mutate(.x, y = a * x + b + rnorm(N, 0, .y)))) %>% mutate(lm = map(data, ~lm(.$y ~ .$x)), predict = map(lm, predict)) Nested data modeling

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dat %>% select(id, data, predict) %>% unnest %>% ggplot(...)+ geom_...+ facet_wrap(~id) Nested data modeling

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Nested data modeling

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Nested data modeling

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Summary

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What is Data? ℎ f X ℎℎ Truth Knowledge Modeling Modeling

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Regression Linear = + + ~(0, ) Input Output coefficient intercept residual Normal (Gaussian) distribution mean standard deviation parameters

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# liner model fitting fit_lm <- lm(y ~ x, dat)

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fit_lm <- lm(y ~ x1 + x2, dat) coeff <- fit_lm$coefficients plot3d(dat) planes3d(coeff[2], coeff[3], -1, coeff[1], alpha = 0.5) Multiple Linear Regression

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X ⋮ ] = XX … X] ⋮ ⋱ ⋮ ]X … ]] X ⋮ ] Principal Component Analysis eigenvector matrix principal components fit_pca <- prcomp(dat, scale = T) biplot