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CROSS-VALIDATION AND
MODEL SELECTION
Many �ݺ�ߣs are from: Dr. Thomas Jensen -Expedia.com and
Prof. Olga Veksler - CS9840 - Learning and Computer Vision

How to check if a model fit is good?
• The R2
statistic has become the almost universally standard measure
for model fit in linear models.
• What is R2
?
• It is the ratio of error in a model over the total variance in the
dependent variable.
• Hence the lower the error, the higher the R2
value.

How to check if a model fit is good?

OVERFITTING
• Modeling techniques tend to overfit the data.
• Multiple regression:
Every time you add a variable to the regression, the model’s R2
goes
up.
Naïve interpretation: every additional predictive variable helps to
explain yet more of the target’s variance. But that can’t be true!
Left to its own devices, Multiple Regression will fit too many patterns.
A reason why modeling requires subject-matter expertise.

OVERFITTING
• Error on the dataset used to fit
the model can be misleading
› Doesn’t predict future
performance.
• Too much complexity can
diminish model’s accuracy on
future data.
› Sometimes called the Bias-
Variance Tradeoff.

OVERFITTING
• What are the consequences of overfitting?
›“Overfitted models will have high R2
values, but will perform poorly in
predicting out-of-sample cases”

WHY WE NEED CROSS-VALIDATION?
• R2
, also known as coefficient of determination, is a popular measure
of quality of fit in regression. However, it does not offer any
significant insights into how well our regression model can predict
future values.
• When an MLR equation is to be used for prediction purposes it is
useful to obtain empirical evidence as to its generalizability, or its
capacity to make accurate predictions for new samples of data. This
process is sometimes referred to as “validating” the regression
equation.

• One way to address this issue is to literally obtain a new sample of
observations. That is, after the MLR equation is developed from the
original sample, the investigator conducts a new study, replicating the
original one as closely as possible, and uses the new data to assess
the predictive validity of the MLR equation.
• This procedure is usually viewed as impractical because of the
requirement to conduct a new study to obtain validation data, as well
as the difficulty in truly replicating the original study.
• An alternative, more practical procedure is cross-validation.

CROSS-VALIDATION
• In cross-validation the original sample is split into two parts. One part
is called the training (or derivation) sample, and the other part is
called the validation (or validation + testing) sample.
1)What portion of the sample should be in each part?
If sample size is very large, it is often best to split the sample in half. For
smaller samples, it is more conventional to split the sample such that
2/3 of the observations are in the derivation sample and 1/3 are in
the validation sample.

CROSS-VALIDATION
2) How should the sample be split?
The most common approach is to divide the sample randomly, thus
theoretically eliminating any systematic differences. One alternative is to
define matched pairs of subjects in the original sample and to assign one
member of each pair to the derivation sample and the other to the
validation sample.
•Modeling of the data uses one part only. The model selected for this part is
then used to predict the values in the other part of the data. A valid model
should show good predictive accuracy.
•One thing that R-squared offers no protection against is overfitting. On the
other hand, cross validation, by allowing us to have cases in our testing set
that are different from the cases in our training set, inherently offers
protection against overfitting.

CROSS VALIDATION – THE IDEAL PROCEDURE
1.Divide data into three sets, training, validation and test sets
2.Find the optimal model on the training set, and use the test set to
check its predictive capability
3.See how well the model can predict the test set
4.The validation error gives an unbiased estimate of the predictive
power of a model

TRAINING/TEST DATA SPLIT
Talked about splitting data in training/test sets
• training data is used to fit parameters
• test data is used to assess how classifier generalizes to new data
What if classifier has “non tunable” parameters?
‐
• a parameter is “non tunable” if tuning (or training) it on the training
‐
data leads to overfitting

TRAINING/TEST DATA SPLIT
What about test error? Seems appropriate
• degree 2 is the best model according to the test error
Except what do we report as the test error now?
• Test error should be computed on data that was not used for training
at all
• Here used “test” data for training, i.e. choosing model

VALIDATION DATA
Same question when choosing among several classifiers
• our polynomial degree example can be looked at as choosing among
3 classifiers (degree 1, 2, or 3)
• Solution: split the labeled data into three parts

Training/Validation/Test Data
•Training Data
•Validation Data
d = 2 is chosen
•Test Data
1.3 test error computed for d = 2

LOOCV (Leave‐one‐out Cross Validation)
• For k=1 to R
1. Let (xk
,yk
) be the k
example

LOOCV (Leave‐one‐out Cross Validation)

LOOCV for Quadratic Regression

Which kind of Cross Validation?

K-FOLD CROSS VALIDATION
›Since data are often scarce, there might not be enough to set aside for a
validation sample
›To work around this issue k-fold CV works as follows:
1. Split the sample into k subsets of equal size
2. For each fold estimate a model on all the subsets except one
3. Use the left out subset to test the model, by calculating a CV metric of
choice
4. Average the CV metric across subsets to get the CV error
›This has the advantage of using all data for estimating the model, however
finding a good value for k can be tricky

K-fold Cross Validation Example
1. Split the data into 5
samples
2. Fit a model to the training
samples and use the test
sample to calculate a CV
metric.
3. Repeat the process for the
next sample, until all
samples have been used to
either train or test the
model

Improve cross-validation
• Even better: repeated cross-validation
Example:
10-fold cross-validation is repeated 10 times and results are
averaged (reduce the variance)

Cross Validation - Metrics
• How do we determine if one model is predicting better than another
model?

Best Practice for Reporting Model Fit
1.Use Cross Validation to find the best model
2.Report the RMSE and MAPE statistics from the cross validation
procedure
3.Report the R Squared from the model as you normally would.
The added cross-validation information will allow one to evaluate not
how much variance can be explained by the model, but also the
predictive accuracy of the model. Good models should have a high
predictive AND explanatory power!

EXAMPLE
• The following table gives the size of the floor area (ha) and the price ($000), for
15 houses sold in the Canberra (Australia) suburb of Aranda in 1999.
• For simplicity, we will use 3-fold cross validation
> library(DAAG)
Loading required package: lattice
> data(houseprices)
> summary(houseprices)
area bedrooms sale.price
Min. : 694.0 Min. :4.000 Min. :112.7
1st Qu.: 743.5 1st Qu.:4.000 1st Qu.:213.5
Median : 821.0 Median :4.000 Median :221.5
Mean : 889.3 Mean :4.333 Mean :237.7
3rd Qu.: 984.5 3rd Qu.:4.500 3rd Qu.:267.0
Max. :1366.0 Max. :6.000 Max. :375.0

> houseprices$bedrooms=as.factor(houseprices[,2])
> summary(houseprices)
area bedrooms sale.price
Min. : 694.0 4:11 Min. :112.7
1st Qu.: 743.5 5: 3 1st Qu.:213.5
Median : 821.0 6: 1 Median :221.5
Mean : 889.3 Mean :237.7
3rd Qu.: 984.5 3rd Qu.:267.0
Max. :1366.0 Max. :375.0
plot(sale.price ~ area, data = houseprices, log = "y",pch = 16, xlab = "Floor Area",
ylab = "Sale Price", main = "log(sale.price) vs area")

hist(log(houseprices$sale.price), xlab="Sale Price (logarithmic
scale)", main="Histogram of log(sale.price)")

> #Split row numbers randomly into 3 groups
> rand<- sample(1:15)%%3 + 1
> # a%%3 is a remainder of a modulo 3
> #Subtract from a the largest multiple of 3 that is <= a; take
remainder
> (1:15)[rand == 1] # Observation numbers from the first group
[1] 2 3 5 7 12
> (1:15)[rand == 2] # Observation numbers from the second group
[1] 4 8 9 11 14
> (1:15)[rand == 3] # Observation numbers from the third group
[1] 1 6 10 13 15

> houseprice.lm<- lm(sale.price ~ area, data= houseprices)
> CVlm(houseprices, houseprice.lm, plotit=TRUE)
Analysis of Variance Table
Response: sale.price
Df Sum Sq Mean Sq F value Pr(>F)
area 1 18566 18566 8 0.014 *
Residuals 13 30179 2321
fold 1
Observations in test set: 5
11 20 21 22 23
area 802 696 771.0 1006.0 1191
cvpred 204 188 199.3 234.7 262
sale.price 215 255 260.0 293.0 375
CV residual 11 67 60.7 58.3 113
Sum of squares = 24351 Mean square = 4870 n = 5

fold 2
10 13 14 17 18
area 905 716 963.0 1018.00 887.00
cvpred 255 224 264.4 273.38 252.06
sale.price 215 113 185.0 276.00 260.00
CV residual -40 -112 -79.4 2.62 7.94
fold 3
9 12 15 16 19
area 694.0 1366 821.00 714.0 790.00
cvpred 183.2 388 221.94 189.3 212.49
sale.price 192.0 274 212.00 220.0 221.50
CV residual 8.8 -114 -9.94 30.7 9.01
Overall (Sum over all 5 folds)
ms
3934

CROSS-VALIDATION AND MODEL SELECTION (1).ppt

houseprice.lm2<- lm(sale.price ~ area + bedrooms, data= houseprices)
CVlm(houseprices, houseprice.lm2, plotit=TRUE)
Analysis of Variance Table
Response: sale.price
Df Sum Sq Mean Sq F value Pr(>F)
area 1 18566 18566 17.0 0.0014 **
bedrooms 1 17065 17065 15.6 0.0019 **
Residuals 12 13114 1093
fold 1
11 20 21 22 23
Predicted 206 249 259.8 293.3 378
cvpred 204 188 199.3 234.7 262
sale.price 215 255 260.0 293.0 375
CV residual 11 67 60.7 58.3 113

fold 2
10 13 14 17 18
Predicted 220.5 193.6 228.8 236.6 218.0
cvpred 226.1 204.9 232.6 238.8 224.1
sale.price 215.0 112.7 185.0 276.0 260.0
CV residual -11.1 -92.2 -47.6 37.2 35.9
fold 3
9 12 15 16 19
Predicted 190.5 286.3 208.6 193.3 204
cvpred 174.8 312.5 200.8 178.9 194
sale.price 192.0 274.0 212.0 220.0 222
CV residual 17.2 -38.5 11.2 41.1 27
Overall (Sum over all 5 folds)
ms
2816

MEASURING THE MODEL ACCURACY
44

45

46

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CROSS-VALIDATION AND MODEL SELECTION (1).ppt

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