Statistical Modelling and Variable Selection in Climate Science
367
Here level = 0.95 indicates the value of 1 − α = 1 − 0.05 = 0.95 and
can be changed for the desired value of α. This outcome provides the lower and
upper confidence limits of parameters as follows: 6.7535814 ≤ β 0 ≤ 37.8097335,
0.9879101 ≤ β 1 ≤ 2.0495022, 0.2439594 ≤ β 2 ≤ 0.8567413 and 0.9500208 ≤
β 3 ≤ 1.2426673. For example, 0.9879101 ≤ β 1 ≤ 2.0495022 tells that there are
95% chances that β 1 is expected to lie between 0.9879101 and 2.0495022. Recall
that the point estimate of β 1 is 1.5187, which lies within this interval. A smaller
length of interval is preferable over a large length of interval.
Prediction:
Next we consider the prediction. Suppose the predictions on rain are to be made
for following two sets of observations on wind speed, precipitation, and relative
humidity.
Observation number (i)
Wind speed (x 1i )
Precipitation (x 2i )
Relative humidity (x 3i )
1
20
30
55
2
17
25
48
The R command to obtain the predicted values and prediction interval of rain is
predict(), which gives the following outcome:
> predict(lm(rain~ windspeed + precipitation + relhum), newdata =
data.frame(windspeed=c(20,17),precipitation=c(30,25),relhum=c(55,48)),
interval = "prediction")
fit
lwr
upr
1 129.4652 125.9713 132.9591
2 114.4829 111.0423 117.9236
The screenshot of this outcome, when executed over the R software, is as follows:
The first column under fit gives the predicted values of rain fall (y) as a point,
which are obtained by p f 1 = ˆ
β 0 + x 11 ˆ
β 1 + x 12 ˆ
β 2 + x 13 ˆ
β 3 for given values of x’s as
p f 1 = 22.2816575+1.5187061 × 20+0.5503503 × 30+1.0963440 × 55= 129.4652
and similarly,
p f 2 = 22.2816575+1.5187061 × 17+0.5503503 × 25+1.0963440 × 48=114.4829.
367
Here level = 0.95 indicates the value of 1 − α = 1 − 0.05 = 0.95 and
can be changed for the desired value of α. This outcome provides the lower and
upper confidence limits of parameters as follows: 6.7535814 ≤ β 0 ≤ 37.8097335,
0.9879101 ≤ β 1 ≤ 2.0495022, 0.2439594 ≤ β 2 ≤ 0.8567413 and 0.9500208 ≤
β 3 ≤ 1.2426673. For example, 0.9879101 ≤ β 1 ≤ 2.0495022 tells that there are
95% chances that β 1 is expected to lie between 0.9879101 and 2.0495022. Recall
that the point estimate of β 1 is 1.5187, which lies within this interval. A smaller
length of interval is preferable over a large length of interval.
Prediction:
Next we consider the prediction. Suppose the predictions on rain are to be made
for following two sets of observations on wind speed, precipitation, and relative
humidity.
Observation number (i)
Wind speed (x 1i )
Precipitation (x 2i )
Relative humidity (x 3i )
1
20
30
55
2
17
25
48
The R command to obtain the predicted values and prediction interval of rain is
predict(), which gives the following outcome:
> predict(lm(rain~ windspeed + precipitation + relhum), newdata =
data.frame(windspeed=c(20,17),precipitation=c(30,25),relhum=c(55,48)),
interval = "prediction")
fit
lwr
upr
1 129.4652 125.9713 132.9591
2 114.4829 111.0423 117.9236
The screenshot of this outcome, when executed over the R software, is as follows:
The first column under fit gives the predicted values of rain fall (y) as a point,
which are obtained by p f 1 = ˆ
β 0 + x 11 ˆ
β 1 + x 12 ˆ
β 2 + x 13 ˆ
β 3 for given values of x’s as
p f 1 = 22.2816575+1.5187061 × 20+0.5503503 × 30+1.0963440 × 55= 129.4652
and similarly,
p f 2 = 22.2816575+1.5187061 × 17+0.5503503 × 25+1.0963440 × 48=114.4829.