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STK2100 –– Machine Learning and Statistical Methods
创建日期：2024-04-15 17:41:22
所属分类：统计学statistics
标签： STK2100
Machine Learning and Statistical Methods

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STK2100 –– Machine Learning and Statistical Methods

for Prediction and Classification - Home exam
Day of examination: June 15 -2021
Examination hours: 09.00 – 13.00.
This problem set consists of 9 pages.
Appendices: None
Permitted aids: Anything available
Please make sure that your copy of the problem set is
complete before you attempt to answer anything.
All subquestions are counted equally!
Problem 1
An important measure to keep low during the Covid-19 pandemic has been
the number of people ending up at hospital. The figure below shows the
number of new arrivals to hospitals among Oslo citizens for each day during
the pandemic. The additional plots show the number of positive tests and
the total number of tests in the same period.
0
5
10
15
20
Apr 2020 Jul 2020 Oct 2020 Jan 2021 Apr 2021
dates
va
lu
e variable
mat
Hospitalization Oslo
(Continued on page 2.)
Exam in STK2100, June 15 -2021 Page 2
0
100
200
300
400
Apr 2020 Jul 2020 Oct 2020 Jan 2021 Apr 2021
dates
va
lu
e variable
mat
Positive tests Oslo
0
1000
2000
3000
Apr 2020 Jul 2020 Oct 2020 Jan 2021 Apr 2021
dates
va
lu
e variable
mat
Total tests Oslo
Our aim in this exercise will be to see if the test data can be used for
prediction of the number of hospitalizations.
We will introduce the following variables (where each variable correspond to
citizens with residence in Oslo):
yt The number of new arrivals at hospital on day t
vt The number of positive tests at day t
zt The number of tests performed at day t
(a) Consider first a general setting where
yt ～Binom(N, pt);
logit(pt) =β0 +
p∑
j=1
βjxt,j.
where xt = (xt,1, ..., xt,p) is the collection of all covariates involved in
the modelling of pt. Here pt can be interpreted as the probability of a
random individual being hospitalized due to the Covid-19 virus at day
t. Further, define y?t = Np?t where
logit(p?t) =β?0 +
p∑
j=1
β?jxt,j.
(Continued on page 3.)
Exam in STK2100, June 15 -2021 Page 3
Show that
E[(yt ? y?t)2|xt]
=Npt(1? pt) + E[(y?t ?Npt)2|xt]? 2E[(yt ?Npt)(y?t ?Npt)|xt]
Give an interpretation of each term on the right hand side.
Can we neglect the last term on the right hand side in this case?
For which value of pt is the term Npt(1? pt) maximized?
Due to that we want to make predictions one-week ahead, we will consider the
following model, with N being the population size in Oslo (here for simplicity
assumed to be constant equal to 681 071 over the whole period):
yt ～Binom(N, pt)
logit(pt) =β0 +
3∑
j=1
βjvt?7?j +
3∑
j=1
β3+jzt?7?j
where we also assume all observations are independent.
Note that using test-data for some days earlier makes sense in this case due
to that it typically takes 10-12 days from infection until one (potentially)
becomes so sick that one needs to go to hospital. The delay from people get
infected until they take a test typically varies between 2 and 5 days.
Note: The test data we talk about here is something different from the test
data we have talked about during the course.
When fitting the model above, we obtained many non-significant coefficient,
so two model selection procedures were considered, giving the following
regression tables (where v8 corresponds to vt?8 and so on):
Model 1
Co e f f i c i e n t s :
Estimate Std . Error z va lue Pr(>| z | )
( I n t e r c ep t ) ?1.377 e+01 1 .049 e?01 ?131.303 < 2e?16 ???
v8 4 .353 e?03 6 .106 e?04 7 .130 1 .00 e?12 ???
v10 3 .858 e?03 6 .102 e?04 6 .322 2 .59 e?10 ???
z10 3 .953 e?04 6 .477 e?05 6 .102 1 .05 e?09 ???
Model 2
Co e f f i c i e n t s :
Estimate Std . Error z va lue Pr(>| z | )
( I n t e r c ep t ) ?1.370 e+01 1 .101 e?01 ?124.415 < 2e?16 ???
v8 3 .669 e?03 7 .469 e?04 4 .913 8 .98 e?07 ???
v9 1 .786 e?03 8 .549 e?04 2 .089 0 .0367 ?
z9 ?1.752e?04 9 .867 e?05 ?1.775 0 .0759 .
v10 2 .860 e?03 7 .344 e?04 3 .894 9 .85 e?05 ???
z10 5 .064 e?04 9 .487 e?05 5 .337 9 .45 e?08 ???
(Continued on page 4.)

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MOEC0021 STA 106 ENGG 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 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FN1024 MATH0094 MANG 6554 ECOS3003 ELEC6252 C2511C PHY328 EC1060 ECON32202 COMP4002-E1 MATH3063 EFIMM0124 QBUS2310 ECON4411 FUNDAMENTALS ACCT20001 18YX RPM COMP 2711 QBUS6600 BUSS5221 ST3189 FIT1047 MAT344 EGB375 MATH60013 MATH96011 COMP3557 MAST20029 ECM159 EAE1011 ENSCEN 105 SECTION A STA 141C SOSE 2022 PAM006 COMP0048 M40007 ISOM2500 CMPE 142 A06472 MCC150 ACST 8083 GEOG6087 ECON920 FINM7008 ENVENG 4110 COMPSCI 361 ENGRCEE 21 PHYS 1112 ACF2100 ECMM108 COMPS267F MAT00050M MAST30025 CSIT970 MSCI 224 INT202 PUBL0054 MSDM5004 BMAN 21020 BMAN21020 MGMT1002 MATH11150 LAWS7023 ACTL30002 CIVL3340 MATH1002 ECON 8069 COMPSCI 351 IBUS7302 STAT4528 CHEM1101 STAT3925 BUSN 7008 Comp 3120 ECOS2040 SWEN90010 ENGN6528 ECM158 ECON20022 COMP6452 Calculation CITS1401 S1889112 BUS 351 ACTL30008 FIT2091 ECON6002 IBUS6020 PHI 001 ECON30290 MATH7501 ECON10004 MFIN7017 ISYS90043 LAWS7012 ECON30025 COMP30023 COMM1190 ECOM20001 GSBS6009 STAT 337 EMET8002 ECOM90001 ECOM30001 FINC 2012 BUSINESS ENGCV512 ECO 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