# Harjoitukset R:llä
# H1, 16.1.2003
#########
# 1.1
#########
Y<-c(-1,3,-2,2,4,2,5,2,3)
Y<-matrix(data = Y, nrow = 3, ncol = 3, byrow = T)
Yf<-as.data.frame(Y)
 
attach(Yf)
mean(V1);mean(V2);mean(V3)
[1] 2
[1] 3
[1] 1
var(V1);var(V2);var(V3)
[1] 9
[1] 1
[1] 7

YK<-transform(Yf,v1k=V1-mean(V1),v2k=V2-mean(V2),v3k=V3-mean(V3))
# a)
YK
  V1 V2 V3 v1k v2k v3k
1 -1  3 -2  -3   0  -3
2  2  4  2   0   1   1
3  5  2  3   3  -1   2
s1<-sqrt(var(V1));s2<-sqrt(var(V2));s3<-sqrt(var(V3))
s1;s2;s3
[1] 3
[1] 1
[1] 2.645751
YKS<-transform(YK,v1s=v1k/s1,v2s=v2k/s2,v3s=v3k/s3)
attach(YKS)
# b)
YKS
  V1 V2 V3 v1k v2k v3k v1s v1s        v3s
1 -1  3 -2  -3   0  -3  -1   0 -1.1338934
2  2  4  2   0   1   1   0   1  0.3779645
3  5  2  3   3  -1   2   1  -1  0.7559289
> cov(Yf)
     V1   V2   V3
V1  9.0 -1.5  7.5
V2 -1.5  1.0 -0.5
V3  7.5 -0.5  7.0
# c)
det(cov(Yf))
[1] 0

###################
# 1.2
###################
cor(Yf)
           V1         V2         V3
V1  1.0000000 -0.5000000  0.9449112
V2 -0.5000000  1.0000000 -0.1889822
V3  0.9449112 -0.1889822  1.0000000
det(cor(Yf))
[1] 0
diag(cov(Yf))
V1 V2 V3 
 9  1  7 
prod(diag(cov(Yf)))
[1] 63

det(cov(Yf))
[1] 0

##################
# 1.3
###################
a<-c(1/2,1/4,1/4)
a<-as.matrix(a)
ym<-apply(Yf,2,mean)
ym<-as.matrix(ym)
ym
   [,1]
V1    2
V2    3
V3    1
a
     [,1]
[1,] 0.50
[2,] 0.25
[3,] 0.25
t(a)%*%ym
      
       [,1]
  [1,]    2
t(a)%*%cov(Yf)%*%a
      
         [,1]
  [1,] 4.1875
###################
# 1.4
##############
YKS
  V1 V2 V3 v1k v2k v3k v1s v1s        v3s
1 -1  3 -2  -3   0  -3  -1   0 -1.1338934
2  2  4  2   0   1   1   0   1  0.3779645
3  5  2  3   3  -1   2   1  -1  0.7559289

# a)
par(mfrow=c(2,2))
plot(V1,V2);plot(v1k,v2k);plot(v1s,v2s)
par(mfrow=c(2,2))

# Kuvioihin sama skaala

plot(V1,V2,xlim=c(-3,5),ylim=c(-1,4));plot(v1k,v2k,xlim=c(-3,5),ylim=c(-1,4))
plot(v1s,v2s,xlim=c(-3,5),ylim=c(-1,4))

############
# 1.7
############
# a)
pojat<-read.table("C:\\Kurssit\\Mmm\\mmm03\\Datat\\pojat.dat", header = T, skip=7)
m<-apply(pojat,2,mean)
m
 ppit1  plev1  ppit2  plev2 
185.72 151.12 183.84 149.24 

xv<-cbind(m[1:2])
yv<-cbind(m[3:4])
xv
        [,1]
ppit1 185.72
plev1 151.12
yv
        [,1]
ppit2 183.84
plev2 149.24
m<-rbind(xv,yv)
m
        [,1]
ppit1 185.72
plev1 151.12
ppit2 183.84
plev2 149.24

# b)
cov(pojat)
        ppit1    plev1     ppit2    plev2
ppit1 95.29333 52.86833  69.66167 46.11167
plev1 52.86833 54.36000  51.31167 35.05333
ppit2 69.66167 51.31167 100.80667 56.54000
plev2 46.11167 35.05333  56.54000 45.02333

cov(pojat[,1:2],pojat[,3:4])    # S_yx
         ppit2    plev2
ppit1 69.66167 46.11167
plev1 51.31167 35.05333

###########
# 1.8
###########
y1<-c(-1,2,5)
y2<-c(3,4,2)
y1<-y1-mean(y1) # Keskistys
y2<-y2-mean(y2)
y1
[1] -3  0  3
y2
[1]  0  1 -1
y1*y1
[1] 9 0 9
sum(y1*y1)  # y1:n pituuden neliö
[1] 18
sum(y2*y2)
[1] 2
sum(y1*y2)/sqrt(sum(y1*y1)*sum(y2*y2))  # korrelaatio = cos(alpha)
[1] -0.5
# sin(alpha)^2=1-cos(alpha)^2=0.75
sum(y1*y1)*sum(y2*y2)*0.75/4       # Suunnikkaan ala
[1] 6.75

Y<-as.data.frame(cbind(y1,y2))
Y
  y1 y2
1 -3  0
2  0  1
3  3 -1
det(cov(Y))   # Yleistetty varianssi
[1] 6.75

# det(cov(Y))=sum(y1*y1)*sum(y2*y2)*0.75/4

acos(-0.5)
[1] 2.094395
(2.094395/pi)*180
[1] 120 # Vektoreiden välinen kulma 120 astetta

#################
# 1.9
#################
# a)
P<-rbind(c(cos(kulma),sin(kulma)),c(-sin(kulma),cos(kulma)))
P
           [,1]      [,2]
[1,]  0.9396926 0.3420201
[2,] -0.3420201 0.9396926
a<-c(3,-2);a<-cbind(a)
a
      a
[1,]  3
[2,] -2
P%*%a
      
               a
  [1,]  2.135038
  [2,] -2.905446
b<-c(5,1)
b<-cbind(b)
b
     b
[1,] 5
[2,] 1

 P%*%b
      
                b
  [1,]  5.0404832
  [2,] -0.7704081

# b)

asin(-12.27/29)
[1] -0.4368677
(-0.4368677/pi)*180
[1] -25.03068