##############
### 2a. model kvadraticky - parabola
x = c(1, 1.5, 2, 2, 2.5)
y = c(3, 5,   6, 6, 7)
vys2 <- lm(y ~ 1 + x + I(x^2))
## <> lm( y ~ poly(x,2) ) # - orthogonal
plot(x,y)
lines(x, vys2$fit)
# <=>:
par = function(x, c) {
  c[1] + c[2]*x + c[3]*x^2
}
xx = seq(min(x), max(x), .5);  ## FromToBy
lines(xx, par(xx, vys2$coef), col = "blue" )
xx = seq(min(x), max(x), .2);  ## FromToBy
lines(xx, par(xx, vys2$coef), col = "blue" )
xx = seq(min(x), max(x), .1);  ## FromToBy
lines(xx, par(xx, vys2$coef), col = "magenta" )

##############
### 2b. model maticovo - Imre
x = c(1, 1.5, 2, 2, 2.5)
y = c(3, 5,   6, 6, 7)
#X = cbind( rep(1,length(x)), x, x^2)
X = cbind( rep(1,length(x)), x)


### a) pomocou inverznej matice - pre teoriu
solve( t(X) %*% X) %*% t(X) %*% y   # - coeff

### b) ako riesenie n.sustavy - pre aplikacie
solve( t(X) %*% X, t(X) %*% y )  # - coeff

# <=> :
r2 = lm(y ~ 1 + x + I(x^2));
r2$coeff
r2$coeff[3]

summary(r2)

#=============
# Linearizovatelne nelinearne modely
x = c(  45.3,    96.5,   164.8,   260.6,  429.1,   596.6,   775.1)
y = c(8886.09, 8643.75, 8340.46, 8126.6, 7486.71, 7370.64, 7258.11)

plot(y~x)
z1=exp(-x);v1 = lm(y~z1);v1
z2=1/x;    v2 = lm(y~z2)
z3=log(x); v3 = lm(y~z3);

summary(v1)
summary(v2)
summary(v3)
anova(v1)$'Pr(>F)'[1]
anova(v2)$'Pr(>F)'[1]
anova(v3)$'Pr(>F)'[1]

lines(x, v1$coef[1]+v1$coef[2]*exp(-x),col="green")
lines(x, v2$coef[1]+v2$coef[2]/x,col="yellow")
lines(x, v3$coef[1]+v3$coef[2]*log(x),col="green")# <=>
lines(x, v3$fit, col="magenta")

### Kvadraticky model:
v4 = lm(y~x+I(x^2))
lines(x, v4$fit, col="blue")
anova(v3)$'Pr(>F)'[1]
anova(v4)$'Pr(>F)'[1]

### Rezudualny sucet stvorcov
### Pouzivaju to aj numerici:
sum((v1$resid)^2)
sum((v2$resid)^2)
sum((v3$resid)^2)
sum((v4$resid)^2)

####

# vys_ = lm(y ~ 1 + x + I(x^2) + I(x^3)) # NONO
# summary(vys_)

# !!!!!!!!!!!!!!!!!! MEGOLDAS
vys3_1 = lm(y ~ 1 + I(x^2) + I(x^3))
summary(vys3_1)

# alebo este lepsie:
vys3_2 = lm(y ~ 1 + x  + I(x^3))
summary(vys3_2)

lines(x, vys3_2$fit, col='gold')
vys3_2$fit


### =============
### Teploty v case t
### !!! http://pastebin.com
t = c(  7,	9,	11,	13,	15,	17,	19,	21,	23)
o = c(	8.2,	10.7,	18.9,	26.7,	24.6,	13,	11.4,	10.6,	10.2)
r = c(	38.4,	38.5,	35.8,	35.8,	37,	38.4,	38.2,	37.6,	38.1)
s = c(	13.5,	13.6,	15.3,	17.2,	18.1,	17.9,	16.7,	15.8,	15.4)
i = c(	15.6,	16,	17.7,	19.7,	20.4,	19.7,	18.6,	17.9,	17.4)
reg = lm(i ~ o + r + s)
summary(reg)

df = data.frame(t=t,o=o,r=r,s=s,i=i);
options(digits=4)
cor(df)
options(digits=7)

########
### MS
x1 = c(2310,2333,2356,2379,2402,2425,2448,2471,2494,2517,2540)
x2 = c(2,2,3,3,2,4,2,2,3,4,2)
x3 = c(2,2,1.5,2,3,2,1.5,2,3,4,3)
x4 = c(20,12,33,43,53,23,99,34,23,55,22)
y = c(142000,144000,151000,150000,139000,169000,126000,142900,163000,169000,149000)
r2 = lm(y ~ 1 + x1 + x2 + x3 + x4); r2$coeff
summary(r2)

##############
### 3. model sinus:
### 3. model sinus:
x = c(1, 1.5, 2, 2, 2.5)
y = c(3, 5, 6, 6, 5)
x1=pi/2*(x-1)
y3=y-3
plot(y3~x1)
r = lm(y3 ~ sin(x1))
summary(r)
lines(x1, r$fit)
plot(y~x)
lines((2*x1)/pi + 1, r$fit + 3)