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b/zajecia10/.Rhistory new file mode 100644 index 0000000..adc9267 --- /dev/null +++ b/zajecia10/.Rhistory @@ -0,0 +1,99 @@ +load(url("http://ls.home.amu.edu.pl/data_sets/liver_data.RData")) +head(liver_data) +liver_data$condition <- ifelse(liver_data$condition == "Yes", 1, 0) +model_1 <- glm(condition ~ bilirubin + ldh, data = liver_data, family = 'binomial') +model_1 +summary(model_1) +step(model_1) +exp(coef(model_1)[2]) +exp(coef(model_1)[3]) +View(liver_data) +View(liver_data) +library(ROCR) +install.packages("ROCR") +library(ROCR) +pred_1 <- prediction(model_1$fitted, liver_data$condition) +plot(performance(pred_1, 'tpr', 'fpr'), main = "Model 1") +performance(pred_1, 'auc')@y.values +liver_data_new <- data.frame(bilirubin = c(0.9, 2.1, 3.4), ldh = c(100, 200, 300)) +(predict_glm <- stats::predict(model_1, +liver_data_new, +type = 'response')) +model_1_hat <- coef(model_1)[1] + +coef(model_1)[2] * liver_data$bilirubin + +coef(model_1)[3] * liver_data$ldh +model_1_temp <- seq(min(model_1_hat) - 1, max(model_1_hat) + 2.5, length.out = 100) +condition_temp <- exp(model_1_temp) / (1 + exp(model_1_temp)) +plot(model_1_temp, condition_temp, type = "l", xlab = "X beta", ylab = "condition", +xlim = c(-6, 9), ylim = c(-0.1, 1.1)) +points(model_1_hat, liver_data$condition, pch = 16) +points(coef(model_1)[1] + +coef(model_1)[2] * liver_data_new$bilirubin + +coef(model_1)[3] * liver_data_new$ldh, +predict_glm, pch = 16, col = "red") +mode_1_hat_c1 <- model_1_hat[liver_data$condition == 1] +which(model_1_hat == min(mode_1_hat_c1) & liver_data$condition == 1) +which(model_1_hat == min(mode_1_hat_c1[mode_1_hat_c1 > min(mode_1_hat_c1)]) +& liver_data$condition == 1) +mode_1_hat_c0 <- model_1_hat[liver_data$condition == 0] +which(model_1_hat == max(mode_1_hat_c0) & liver_data$condition == 0) +liver_data_wo <- liver_data[-c(18, 26, 29), ] +model_1_wo <- glm(condition ~ bilirubin + ldh, data = liver_data_wo, family = 'binomial') +model_1_wo +summary(model_1_wo) +step(model_1_wo) +exp(coef(model_1_wo)[2]) +exp(coef(model_1_wo)[3]) +pred_1_wo <- prediction(model_1_wo$fitted, liver_data_wo$condition) +plot(performance(pred_1_wo, 'tpr', 'fpr'), main = "Model 1 (wo)") +performance(pred_1_wo, 'auc')@y.values +liver_data_wo_new <- data.frame(bilirubin = c(0.9, 2.1, 3.4), ldh = c(100, 200, 300)) +(predict_glm_wo <- stats::predict(model_1_wo, +liver_data_wo_new, +type = 'response')) +model_1_wo_hat <- coef(model_1_wo)[1] + +coef(model_1_wo)[2] * liver_data_wo$bilirubin + +coef(model_1_wo)[3] * liver_data_wo$ldh +model_1_wo_temp <- seq(min(model_1_wo_hat) - 10, max(model_1_wo_hat) + 20, length.out = 100) +condition_wo_temp <- exp(model_1_wo_temp) / (1 + exp(model_1_wo_temp)) +plot(model_1_wo_temp, condition_wo_temp, type = "l", xlab = "X beta", ylab = "condition", +xlim = c(-40, 89), ylim = c(-0.1, 1.1)) +points(model_1_wo_hat, liver_data_wo$condition, pch = 16) +points(coef(model_1_wo)[1] + +coef(model_1_wo)[2] * liver_data_wo_new$bilirubin + +coef(model_1_wo)[3] * liver_data_wo_new$ldh, +predict_glm_wo, pch = 16, col = "red") +library(DAAG) +install.packages("DAAG") +library(DAAG) +head(moths) +model_1 <- glm(A ~ log(meters), data = moths, family = 'poisson') +model_1 +summary(model_1) +step(model_1) +data_new <- data.frame(meters = c(3, 20, 100)) +(pred_1 <- stats::predict(model_1, data_new, type = "response")) +moths$A_hat <- stats::predict(model_1, type = "response") +moths <- moths[with(moths, order(meters)), ] +plot(log(moths$meters), moths$A_hat, +type = "l", col = "red", lwd = 2, +xlab = "log(moths$meters)", ylab = "A", +ylim = c(0, 40), main = "Model 1") +points(log(moths$meters), moths$A, pch = 16) +points(log(data_new$meters), pred_1, pch = 16, col = "blue", lwd = 4) +cat("1.", "\n") +rm(moths) +model_2 <- glm(P ~ log(meters), data = moths, family = 'poisson') +model_2 +cat("2.", "\n") +summary(model_2) +step(model_2) +(pred_2 <- stats::predict(model_2, data_new, type = "response")) +moths$P_hat <- stats::predict(model_2, type = "response") +moths <- moths[with(moths, order(meters)), ] +plot(log(moths$meters), moths$P_hat, +type = "l", col = "red", lwd = 2, +xlab = "log(moths$meters)", ylab = "P", +ylim = c(0, 20), main = "Model 2") +points(log(moths$meters), moths$P, pch = 16) +points(log(data_new$meters), pred_2, pch = 16, col = "blue", lwd = 4) diff --git a/zajecia10/README.md b/zajecia10/README.md new file mode 100644 index 0000000..271dc2f --- /dev/null +++ b/zajecia10/README.md @@ -0,0 +1,35 @@ +# Zajęcia 10 + + +## Regresja logistyczna +Regresja logistyczna – jedna z metod regresji używanych w statystyce w przypadku, gdy zmienna zależna jest na skali dychotomicznej (przyjmuje tylko dwie wartości). Zmienne niezależne w analizie regresji logistycznej mogą przyjmować charakter nominalny, porządkowy, przedziałowy lub ilorazowy. W przypadku zmiennych nominalnych oraz porządkowych następuje ich przekodowanie w liczbę zmiennych zero-jedynkowych taką samą lub o 1 mniejszą niż liczba kategorii w jej definicji. + +Zwykle wartości zmiennej objaśnianej wskazują na wystąpienie, lub brak wystąpienia pewnego zdarzenia, które chcemy prognozować. Regresja logistyczna pozwala wówczas na obliczanie prawdopodobieństwa tego zdarzenia (tzw. prawdopodobieństwo sukcesu). + + +## Regresja Poissona +Rozkład Poissona – dyskretny rozkład prawdopodobieństwa, wyrażający prawdopodobieństwo szeregu wydarzeń mających miejsce w określonym czasie, gdy te wydarzenia występują ze znaną średnią częstotliwością i w sposób niezależny od czasu jaki upłynął od ostatniego zajścia takiego zdarzenia. + +Regresja to pewnie dla danych w rozkładzie Poissona. + + +## Krzywa ROC +W statystyce matematycznej krzywa ROC jest graficzną reprezentacją efektywności modelu predykcyjnego poprzez wykreślenie charakterystyki jakościowej klasyfikatorów binarnych powstałych z modelu przy zastosowaniu wielu różnych punktów odcięcia. Mówiąc inaczej – każdy punkt krzywej ROC odpowiada innej macierzy błędu uzyskanej przez modyfikowanie „cut-off point”. +

+![ROC](roc.png) + +Pojęcia: +- TPR (True Positive Rate) – określa zdolność klasyfikatora do wykrywania klasy pozytywnej + +- Lepiej jest powyżej przekątnej, gorzej poniżej. Przekątna to klasyfikator losowy. Przekątna idzie od początku układu współrzędnych. + +- False positive rate = 1-specyficzność + +- Punkt równowagi to klasyfikator gdzie czułość = specyficzność. Jest to na przecięciu drugiej przekątnej z wykresem. + +- AUC - Interpretacja AUROC (Area Under the ROC) to prawdopodobieństwo, że badany model predykcyjny oceni wyżej (wartość score) losowy element klasy pozytywnej od losowego elementu klasy negatywnej. + + +## Notatki +- Liczba stopni swobody – liczba niezależnych wyników obserwacji pomniejszona o liczbę związków, które łączą te wyniki ze sobą. +Liczbę stopni swobody można utożsamiać z **liczbą niezależnych zmiennych losowych**, które wpływają na wynik. 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