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QBUS2820 Predictive Analytics

创建日期：2024-04-13 17:43:28

所属分类：数学mathematics

标签： QBUS2820

Predictive Analytics

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QBUS2820

Predictive Analytics

Question 1 ()
What is overfitting and why is it a fundamental concern in supervised learning?
Solution:
We say that there is overfitting when an estimated model is excessively flexible,
incorporating minor variations in the training data that are likely to be noise
rather than predictive patterns.
Main issues with overfitting:
• Low training error, high generalization error
• Poor predictive performance
• Overreacts to minor fluctuations in training data
Question 2 ()
What is classification and how is it different to regression?
Solution:
In classification, the response variable Y is qualitative or categorical that takes
values in a finite unordered set Y = {1, . . . , C}, where C is the number of classes.
Our task is to predict which class a subject belongs to based on input variables.
However, in regression the response variable Y is numerical.
Question 3 ()
1. What is a confusion matrix?
Solution: A confusion matrix counts the number of true negatives, false posi-
tives, false negatives, and true positives for the test data.
2. Present how the confusion matrix look like for a binary classification problem.
Classification (Prediction)
Ŷ = 0 Ŷ = 1
A
ct
ua
l
Y = 0 True negatives (TN) False positives (FP)
Y = 1 False negatives (FN) True positives (TP)
Question 4 ()
Given the below table which contains actual Y and predicted Y from two models,
calculate Sensitivity, Specificity and Precision respectively for the two models.
Solution:
Note: as mentioned in the question: “Fill in your answers to all ques-
tions 1 to 3 in the answer box. You only need to present the final
solution.” Therefore, you only need to type your answers as below:
The Sensitivity for model 1 is: 1. The Sensitivity for model 2 is: 0.333.
The Specificity for model 1 is: 0.143. The Specificity for model 2 is: 1.
The Precision for model 1 is: 0.333. The Precision for model 2 is: 1.
Solution:
The details of the solutions are shown below, while you do NOT have
to type such solving details into the answer box.
For Model 1: we have: TP= 3; TN= 1; FP= 6; FN= 0;
Page 2
Therefore, we have Sensitivity:
P (Ŷ = 1|Y = 1) = TPTP + FN =
3
3 + 0 = 1
Specificity is :
P (Ŷ = 0|Y = 0) = TNTN + FP =
1
1 + 6 = 0.143
Precision is:
P (Y = 1|Ŷ = 1) = TPTP + FP =
3
3 + 6 = 0.333
For Model 2: we have: TP= 1; TN= 7; FP= 0; FN= 2;
Therefore, we have Sensitivity:
P (Ŷ = 1|Y = 1) = TPTP + FN =
1
1 + 2 = 0.333
Specificity is :
P (Ŷ = 0|Y = 0) = TNTN + FP =
7
7 + 0 = 1
Precision is:
P (Y = 1|Ŷ = 1) = TPTP + FP =
1
1 + 0 = 1
Question 5 ()
What are the disadvantages of KNN method? Present three of them.
Solution:
The KNN method has the following disadvantageous (any three are fine):
(a) The estimate of the regression function can be very unstable, since it is the
average of only a few points. This is the price that we pay for flexibility.
(b) Curse of dimensionality.
(c) The predictive performance is sensitive to noisy or irrelevant predictors.
(d) Generating predictions is computationally expensive.
Page 3
Question 6 ()
Consider the additive error model
Y = f(x) + ε,
where ε is i.i.d. with E(ε) = 0 and Var(ε) = σ2.
We can write the expected prediction error at a new input point X = x0 as:
EPE(x0) = ED
[(
Y − f̂(x0)
)2 |X = x0]
1. Decompose the expected prediction error into two parts: irreducible error and
reducible error.
2. Further, we can show that the reducible error can be decomposed into the following
two parts: squared bias and variance. First, identify which part is the squared bias
and which part is the variance. Second, explain the meaning of each part.
E([f̂(x0)− E(f̂(x0))]2) +
[
E(f̂(x0))− f(x0)
]2
Solution:
Note: as mentioned in the question: “Fill in your answers to both
questions 1 to 2 in the answer box. You only need to present the final
solution.” Therefore, you only need to type your answers as below:
1. The irreducible error is: σ2.
The reducible error is: E
[(
f(x0)− f̂(x0)
)2 |X = x0].
2.
[
E(f̂(x0))− f(x0)
]2
is the squared bias. Bias reflects the error between ex-
pectation of f̂(x0) differs from the true function.
E([f̂(x0) − E(f̂(x0))]2) is the variance. Variance reflects the expected squared
deviation of f̂(x0) around its mean.
Solution:
Page 4
The details of question 1 solutions are shown below, while you do NOT
have to type such solving details into the answer box.
Err(x0) = E
[(
Y0 − f̂(x0)
)2 |X = x0]
= E
[(
f(x0) + ε− f̂(x0)
)2 |X = x0]
= σ2 + E
[(
f(x0)− f̂(x0)
)2 |X = x0]
= Irreducible error + Reducible error
Page 5
Question 7 ()
Given the below Gaussian linear regression:
Y = β0 +
p∑
j=1
βjXj + εi, εi ∼ N(0, σ2).
1. Show the full form of the conditional distribution of Y |X.
2. Derive the complete form log-likelihood L(β, σ2) which is the log-density of the N
observed samples.
Solution:
Note: as mentioned in the question: “Fill in your answers to both
questions 1 to 2 in the answer box. You only need to present the final
solution.” Therefore, you only need to type your answers as below:
1. The full form of the conditional distribution of Y |X is:
Y |X = x ∼ N
β0 + p∑
j=1
βjxj, σ
2

2. The complete log-likelihood is the log-density of the N observed samples is:
L(β, σ2) = −N2 log(2pi)−
N
2 log(σ
2)− 12σ2
N∑
i=1
yi − β0 − p∑
j=1
βjxij
2
Solution:
The details of the solutions are shown below, while you do NOT have
to type such solving details into the answer box.
1. Show the full form of the conditional distribution of Y |X.
The expectation of Y |X:
E(Y |X) = E(β0 +
p∑
j=1
βjXj + εi|X) = β0 +
p∑
j=1
βjXj +E(εi|X) = β0 +
p∑
j=1
βjXj + 0
Page 6
The variance of Y |X:
V ar(Y |X) = V ar(β0 +
p∑
j=1
βjXj + εi|X)
= V ar(β0|X) + V ar(
p∑
j=1
βjXj|X) + V ar(εi|X)
= 0 + 0 + σ2
There is no covariance terms in the above second step as independence assump-
tion.
Last, linear transformations of the Gaussian variables will be still following Gaus-
sian distribution.
Therefore, the full conditional distribution of Y |X is:
Y |X = x ∼ N
β0 + p∑
j=1
βjxj, σ
2

2. Derive the complete form log-likelihood L(β, σ2) which is the log-density of
the N observed samples.
For one specific sample, we haev
Yi|Xi = xi ∼ N
β0 + p∑
j=1
βjxij, σ
2
 ,
the density for an observed value yi is
p(yi|xi;β, σ2) = 1√2piσ2 exp
−
(yi−β0−
∑p
j=1 βjxij)
2
2σ2 .
The likelihood function is the joint PDF of the data evaluated at the sample
values. In our Gaussian linear regression model, independence assumption implies
that we can multiply the PDFs for each observation:
p(y;β, σ2) =
N∏
i=1
1√
2piσ2
exp−
(
yi−β0−
∑p
j=1 βjxij
)2
2σ2
Page 7
Therefore, the complete log-likelihood is the log-density of the N observed sam-
ples,
L(β, σ2) = log
N∏
i=1
p(yi;β, σ2) =
N∑
i=1
log p(yi;β, σ2)
=− N2 log(2pi)−
N
2 log(σ
2)− 12σ2
N∑
i=1
yi − β0 − p∑
j=1
βjxij
2
Question 8 ()
Given the below time series model:
Yt = `t−1 + εt,
`t = αyt + (1− α)`t−1,
where εt i.i.d∼ N(0, σ2) , 0 ≤ α ≤ 1.
1. Derive the error correction formulation of the model.
2. Derive the h-step ahead point forecasts ŷt+h.
3. Derive the h-step ahead variance forecasts Var(Yt+h|y1, . . . , yt).
Solution:
Note: as mentioned in the question: “Fill in your answers to all ques-
tions 1 to 3 in the answer box. You only need to present the final
solution.” Therefore, you only need to type your answers as below:
1. We have model in error correction form as:
Yt+1 = `t + εt+1,
`t = `t−1 + αεt.
2. The point forecast for any horizon h is:
ŷt+h = `t
3. The variance forecast for any horizon h is:
Page 8
Var(Yt+h|y1:t) = σ2(1 + (h− 1)α2)
Solution:
The details of the solutions are shown below, while you do NOT have
to type such solving details into the answer box.
1. Derive the error correction formulation of the model.
First, we transform the `t equation as:
`t = αYt + (1− α)`t−1
= `t−1 + α(Yt − `t−1)
= `t−1 + αεt.
Therefore, we have model in error correction form as:
Yt+1 = `t + εt+1,
`t = `t−1 + αεt.
2. Derive the h-step ahead point forecasts ŷt+h.
Using Yt = `t−1 + εt and `t = `t−1 + αεt, we have:
Yt+1 = `t + εt+1
Yt+2 = `t+1 + εt+2
= `t + αεt+1 + εt+2
Yt+3 = `t+2 + εt+3
= `t + αεt+1 + αεt+2 + εt+3
...
Yt+h = `t+h−1 + εt+h
= `t +
h−1∑
i=1
αεt+i + εt+h
Page 9
Thus the point forecast for any horizon h is
ŷt+h = E(Yt+h|y1:t)
= E
(
`t +
h−1∑
i=1
αεt+i + εt+h
∣∣∣∣∣ y1:t
)
= `t
3. Derive the h-step ahead variance forecasts Var(Yt+h|y1, . . . , yt).
Based on the derivations of the last question, we have:
Var(Yt+h|y1:t) = Var
(
`t +
h−1∑
i=1
αεt+h−i + εt+h
∣∣∣∣∣ y1:t
)
= σ2(1 + (h− 1)α2)
Question 9 ()
Given the linear regression problem, the original centered input matrix X has N
samples and p features. Suppose we argument the centered input matrix X with
p additional rows
√
λI and augment y with p zeros correspondingly. I is a p × p
identity matrix.
Now we are implementing OLS on such augmented data set. Present the parameter
estimates.
You only need to present the final solution.
Hint: you might need to use
β̂ = (XTX)−1XTy.
Solution: Note: as mentioned in the question: “You only need to
present the final solution.” Therefore, you only need to type your
answers as below:
The parameter estimates are:
(XTX + λI)−1XTy,
Page 10
Solution:
The details of the solutions are shown below, while you do NOT have
to type such solving details into the answer box.
Denote X˜ and y˜ as the augmented data sets, e.g.,
X˜(N+p)×p =
 XN×p√
λIp×p

y˜(N+p)×1 =
yN×1
0p×1

The OLS solution of regression parameters estimates with the augmented data
set is:
β̂new = (X˜TX˜)−1X˜T y˜.
Firstly, we have:
X˜TX˜ = (XTp×N ,
√
λIp×p)
 XN×p√
λIp×p
 = XTX + λI
Secondly (as y˜ has p additional 0 rows),
X˜T y˜ = XTy
Therefore, finally we have:
β̂new = (X˜TX˜)−1X˜T y˜ = (XTX + λI)−1XTy,
END OF EXAMINATION.

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1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 BCPM0008 ECN 620 QBUS6830 ECSE-415 ECON5102 MGMT20005 MAT 653 ECO 480 COMP 5/600 CSSE 5600 ME 571 7CCMMS61 CS 550 COMP2865 MATH3075 STATS 506 QTM 220 ISYS2120 FIT5106 STAT 153 MKTG3112 ECON2102 ELE2018 COMP3031 MSIN0010 SCC461 MATH237 CS246 CSC3065 CSE 101 ICTWEB411 COMP9313 ECOS 2025 FN620 MATH11111 MATH 3018 KE1068 FINA 6112 C++ MISM 6202 MSIN0105 CSC8631 MECO6935 COMP4121 ACCT2011 DPBS1150 EEL 3701C F19PL OPM 661 CSE 310 CS918 BIOL30001 ACCT3563 BANK3600 MET CS622 MATH367 ACCT5907 Continuing COMP5048 COMP61021 CS3334 GSOE9210 CSCI-570 COMP 5322 HW 2 ALY-6020 IEOR 4706 DATA 200 EBUS504 CGE12412 COMP4108 7SSMM800 ECMM171P MATE50002 CS225 RS 6003 BH2274 ELEC3111 MAT6669 ECON-UA 227 PHIL 1012 7SSMM603 TB028A P6 ECS708 CSC 427 CS6476 CSE 250A A1A2 ITS61504 EECS 376 model HW6 UV0010 ECON 6074 MG4F7 CMP-7030Y INFO3008 BFIP013 ECON 5068 MATH6002 CS3483 PSTAT 131 C31FM CSOR W4246 AAEC 5307 DATA ELEC362 COMSM0047 ECN6006 MA20224 BST351 MATH 323 MATH323 AA2021 COMP3811 COMP2013 IB9X60 INVP001 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