Raimundo Soto - Catholic University of Chile
ERF Training on Advanced Panel Data Techniques Applied to Economic Modelling
29 -31 October, 2018
Cairo, Egypt
Raimundo Soto - Catholic University of Chile
ERF Training on Advanced Panel Data Techniques Applied to Economic Modelling
29 -31 October, 2018
Cairo, Egypt
Raimundo Soto - Catholic University of Chile
ERF Training on Advanced Panel Data Techniques Applied to Economic Modelling
29 -31 October, 2018
Cairo, Egypt
Raimundo Soto - Catholic University of Chile
ERF Training on Advanced Panel Data Techniques Applied to Economic Modelling
29 -31 October, 2018
Cairo, Egypt
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 5: Discrete Probability Distribution
5.1: Probability Distribution
Raimundo Soto - Catholic University of Chile
ERF Training on Advanced Panel Data Techniques Applied to Economic Modelling
29 -31 October, 2018
Cairo, Egypt
Raimundo Soto - Catholic University of Chile
ERF Training on Advanced Panel Data Techniques Applied to Economic Modelling
29 -31 October, 2018
Cairo, Egypt
Raimundo Soto - Catholic University of Chile
ERF Training on Advanced Panel Data Techniques Applied to Economic Modelling
29 -31 October, 2018
Cairo, Egypt
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 5: Discrete Probability Distribution
5.1: Probability Distribution
Please Subscribe to this Channel for more solutions and lectures
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Chapter 10: Correlation and Regression
10.2: Regression
To control variation in any process, it is absolutely essential that you understand which causes are generating which effects. By knowing which elements of your process are related and how they are related, you will know what to control or what to vary to affect a quality characteristic.
Regression Analysis is simplified in this presentation. Starting with simple linear to multiple regression analysis, it covers all the statistics and interpretation of various diagnostic plots. Besides, how to verify regression assumptions and some advance concepts of choosing best models makes the slides more useful SAS program codes of two examples are also included.
This Presentation course will help you in understanding the Machine Learning model i.e. Generalized Linear Models for classification and regression with an intuitive approach of presenting the core concepts
Paper Study: Melding the data decision pipelineChenYiHuang5
Melding the data decision pipeline: Decision-Focused Learning for Combinatorial Optimization from AAAI2019.
Derive the math equation from myself and match the same result as two mentioned CMU papers [Donti et. al. 2017, Amos et. al. 2017] while applying the same derivation procedure.
Many Decision Problems in business and social systems can be modeled using mathematical optimization, which seeks to maximize or minimize some objective which is a function of the decisions.
Stochastic Optimization Problems are mathematical programs where some of the data incorporated into the objective or constraints are Uncertain.
whereas, Deterministic Optimization Problems are formulated with known parameters.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 10: Correlation and Regression
10.2: Regression
To control variation in any process, it is absolutely essential that you understand which causes are generating which effects. By knowing which elements of your process are related and how they are related, you will know what to control or what to vary to affect a quality characteristic.
Regression Analysis is simplified in this presentation. Starting with simple linear to multiple regression analysis, it covers all the statistics and interpretation of various diagnostic plots. Besides, how to verify regression assumptions and some advance concepts of choosing best models makes the slides more useful SAS program codes of two examples are also included.
This Presentation course will help you in understanding the Machine Learning model i.e. Generalized Linear Models for classification and regression with an intuitive approach of presenting the core concepts
Paper Study: Melding the data decision pipelineChenYiHuang5
Melding the data decision pipeline: Decision-Focused Learning for Combinatorial Optimization from AAAI2019.
Derive the math equation from myself and match the same result as two mentioned CMU papers [Donti et. al. 2017, Amos et. al. 2017] while applying the same derivation procedure.
Many Decision Problems in business and social systems can be modeled using mathematical optimization, which seeks to maximize or minimize some objective which is a function of the decisions.
Stochastic Optimization Problems are mathematical programs where some of the data incorporated into the objective or constraints are Uncertain.
whereas, Deterministic Optimization Problems are formulated with known parameters.
Deep learning paper review ppt sourece -Direct clr taeseon ryu
딥러닝 이미지 분류 테스크에서는 Self-Supervision 학습 방법이 있습니다. 레이블이 없는 상태에서 context prediction 이나 jigsaw puzzle과 같은 방법으로 학습시키는 방법이지만 이러한 self-supervision 테스크에는 모든 차원에 분포하지 않고 특정 부분 차원으로만 학습이 되는 Dimensional Collapse 라는 고질적인 문제를 일으킵니다. Self-supervision 중 positive pair는 가깝게, 그리고 negative pair는 서로 멀어지게 학습을 시키는 Contrastive Learning 이 있습니다. 이로인해 Dimensional Collapse에 강인할 것 이라고 직관적으로 생각이 들지만, 그렇지 않았습니다. 이러한 문제를 해결하기 위해 등장한 Direct CLR이라는 방법론을 소개드립니다.
논문의 배경부터 Direct CLR논문에 대한 디테일한 설명까지,
펀디멘탈팀의 이재윤님이 자세한 리뷰 도와주셨습니다.
오늘도 많은 관심 미리 감사드립니다 !
In this presentation we describe the formulation of the HMM model as consisting of states that are hidden that generate the observables. We introduce the 3 basic problems: Finding the probability of a sequence of observation given the model, the decoding problem of finding the hidden states given the observations and the model and the training problem of determining the model parameters that generate the given observations. We discuss the Forward, Backward, Viterbi and Forward-Backward algorithms.
Aly Rashed - Economic Research Forum
ERF 25th Annual Conference
Knowledge, Research Networks & Development Policy
10-12 March, 2019
Kuwait City, Kuwait
The Future of Jobs is Facing the Biggest Policy Induced Price Distortion in H...Economic Research Forum
Lant Pritchett - University of Oxford
ERF 25th Annual Conference
Knowledge, Research Networks & Development Policy
10-12 March, 2019
Kuwait City, Kuwait
Massoud Karshenas - University of London
ERF 25th Annual Conference
Knowledge, Research Networks & Development Policy
10-12 March, 2019
Kuwait City, Kuwait
Rediscovering Industrial Policy for the 21st Century: Where to Start?Economic Research Forum
Rohinton P. Medhora - Centre for International Governance & Innovation
ERF 25th Annual Conference
Knowledge, Research Networks & Development Policy
10-12 March, 2019
Kuwait City, Kuwait
Rana Hendy - Doha Institute
Mahmoud Mohieldin - World Bank
ERF 25th Annual Conference
Knowledge, Research Networks & Development Policy
10-12 March, 2019
Kuwait City, Kuwait
Ibrahim Elbadawi - Economic Research Forum
ERF 25th Annual Conference
Knowledge, Research Networks & Development Policy
10-12 March, 2019
KuwaitCity, Kuwait
ZGB - The Role of Generative AI in Government transformation.pdfSaeed Al Dhaheri
This keynote was presented during the the 7th edition of the UAE Hackathon 2024. It highlights the role of AI and Generative AI in addressing government transformation to achieve zero government bureaucracy
A process server is a authorized person for delivering legal documents, such as summons, complaints, subpoenas, and other court papers, to peoples involved in legal proceedings.
Jennifer Schaus and Associates hosts a complimentary webinar series on The FAR in 2024. Join the webinars on Wednesdays and Fridays at noon, eastern.
Recordings are on YouTube and the company website.
https://www.youtube.com/@jenniferschaus/videos
Presentation by Jared Jageler, David Adler, Noelia Duchovny, and Evan Herrnstadt, analysts in CBO’s Microeconomic Studies and Health Analysis Divisions, at the Association of Environmental and Resource Economists Summer Conference.
Jennifer Schaus and Associates hosts a complimentary webinar series on The FAR in 2024. Join the webinars on Wednesdays and Fridays at noon, eastern.
Recordings are on YouTube and the company website.
https://www.youtube.com/@jenniferschaus/videos
What is the point of small housing associations.pptxPaul Smith
Given the small scale of housing associations and their relative high cost per home what is the point of them and how do we justify their continued existance
Many ways to support street children.pptxSERUDS INDIA
By raising awareness, providing support, advocating for change, and offering assistance to children in need, individuals can play a crucial role in improving the lives of street children and helping them realize their full potential
Donate Us
https://serudsindia.org/how-individuals-can-support-street-children-in-india/
#donatefororphan, #donateforhomelesschildren, #childeducation, #ngochildeducation, #donateforeducation, #donationforchildeducation, #sponsorforpoorchild, #sponsororphanage #sponsororphanchild, #donation, #education, #charity, #educationforchild, #seruds, #kurnool, #joyhome
Russian anarchist and anti-war movement in the third year of full-scale warAntti Rautiainen
Anarchist group ANA Regensburg hosted my online-presentation on 16th of May 2024, in which I discussed tactics of anti-war activism in Russia, and reasons why the anti-war movement has not been able to make an impact to change the course of events yet. Cases of anarchists repressed for anti-war activities are presented, as well as strategies of support for political prisoners, and modest successes in supporting their struggles.
Thumbnail picture is by MediaZona, you may read their report on anti-war arson attacks in Russia here: https://en.zona.media/article/2022/10/13/burn-map
Links:
Autonomous Action
http://Avtonom.org
Anarchist Black Cross Moscow
http://Avtonom.org/abc
Solidarity Zone
https://t.me/solidarity_zone
Memorial
https://memopzk.org/, https://t.me/pzk_memorial
OVD-Info
https://en.ovdinfo.org/antiwar-ovd-info-guide
RosUznik
https://rosuznik.org/
Uznik Online
http://uznikonline.tilda.ws/
Russian Reader
https://therussianreader.com/
ABC Irkutsk
https://abc38.noblogs.org/
Send mail to prisoners from abroad:
http://Prisonmail.online
YouTube: https://youtu.be/c5nSOdU48O8
Spotify: https://podcasters.spotify.com/pod/show/libertarianlifecoach/episodes/Russian-anarchist-and-anti-war-movement-in-the-third-year-of-full-scale-war-e2k8ai4
2. DISCRETE VARIABLE MODELS
• A discrete variable is usually represented by 𝑦𝑖𝑡 = 0,1
• For example,
0 = 𝑑𝑖𝑑 𝑛𝑜𝑡 𝑡𝑎𝑘𝑒 𝑡ℎ𝑒 𝑡𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡
1 = 𝑑𝑖𝑑 𝑡𝑎𝑘𝑒 𝑡ℎ𝑒 𝑡𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡
• Assume: 𝑃𝑟𝑜𝑏 𝑦𝑖𝑡 = 1 = 𝑝
• Hence: 𝑃𝑟𝑜𝑏 𝑦𝑖𝑡 = 0 = 1 − 𝑝
2
3. DISCRETE VARIABLE MODELS
• The expected value of 𝑦𝑖𝑡 is:
𝐸 𝑦 = 𝑃𝑟𝑜𝑏 𝑦𝑖𝑡 = 1 ∗ 1 + 𝑃𝑟𝑜𝑏 𝑦𝑖𝑡 = 0 ∗ 0
= 𝑃𝑟𝑜𝑏 𝑦𝑖𝑡 = 1 = 𝑝
• Let us assume now that 𝑝 = 𝐹 𝑥𝑖𝑡, 𝛽
• Then: 𝑃𝑟𝑜𝑏 𝑦𝑖𝑡 = 1 = 𝐹 𝑥𝑖𝑡, 𝛽 = 𝐸 𝑦|𝑥
3
4. LINEAR PROBABILITY MODEL
• Let us estimate a linear model of 𝐹 𝑥𝑖𝑡, 𝛽
𝑦𝑖𝑡 = 𝛼𝑖 + 𝛽𝑥𝑖𝑡 + 𝜀𝑖𝑡
• Naturally, it verifies that
𝐸 𝑌|𝑋 = 𝛼𝑖 + 𝛽𝑥𝑖𝑡
• Problem: forecast 𝑦𝑖𝑡 = 𝛼𝑖 + 𝛽𝑥𝑖𝑡 could lie outside 0,1
4
5. LINEAR PROBABILITY MODEL
• Furthermore, the model is heteroskedastic and rather
awkward because heteroskedasticity now depends on 𝛽.
• Since 𝛼𝑖 + 𝛽𝑥𝑖𝑡 + 𝜀𝑖𝑡 must be either 1 o 0, then
𝜀𝑖𝑡 = −𝛼𝑖 − 𝛽𝑥𝑖𝑡 with probability 1 − 𝐹( 𝑥𝑖𝑡, 𝛼, 𝛽)
𝜀𝑖𝑡 = 1 − 𝛼𝑖 − 𝛽𝑥𝑖𝑡 with probability 𝐹( 𝑥𝑖𝑡, 𝛼, 𝛽)
• Thus, the variance of 𝜀𝑖𝑡 is:
𝑉𝑎𝑟 𝜀𝑖𝑡 = 𝛼𝑖 + 𝛽𝑥𝑖𝑡 1 − 𝛼𝑖 − 𝛽𝑥𝑖𝑡
5
6. LINEAR PROBABILITY MODEL
• Let’s keep the notion of studying the probability that 𝑦𝑖𝑡 = 1 but
change the specification so that the probability is properly defined
(between 0 and 1)
• The latter obtains if 𝐹 𝑥𝑖𝑡, 𝛽 is a cumulative distribution function
(CDF)
• There are two main families of models:
– Probabilistic models using the logistic CDF, called Logit
– Probabilistic models using the normal CDF, called Probit
• Both types of models are highly non-linear
– Cannot be estimated using Least Squares
– Must be estimated using likelihood functions when such estimator exists
6
7. NON-LINEAR PROBABILITY MODEL
• Logit Model
Λ 𝑤 =
𝑒 𝑤
1 + 𝑒 𝑤
– Note: variance is 𝜋2
3
• Probit Model
𝜙 𝑤 =
−∞
𝑤
1
2𝜋
𝑒
−𝑤2
2 𝑑𝑤
– Note: 𝜎2 = 1 , it cannot be identified in a two-state model
7
14. PARAMETERS AND MARGINAL EFFECTS
• In standard models (not panel) there is a certain
correspondence among estimators of linear models (L),
probit (𝜙) and logit (Λ):
𝛽Λ ≅ 𝛽 𝜙 ∗ 1.6
𝛽𝐿 ≅ 𝛽 𝜙 ∗ 0.4
Except the constant
𝛽𝐿 ≅ 𝛽 𝜙 ∗ 0.4 + 0.5
14
15. MULTI-STATE MODELS
• So far, we have specified the discrete variable as 𝑦𝑖𝑡 = 0,1
• We could have three (o more) states
• For example,
0 = 𝑡𝑟𝑎𝑣𝑒𝑙 𝑡𝑜 𝐶𝑎𝑖𝑟𝑜 𝑏𝑦 𝑏𝑢𝑠
1 = 𝑡𝑟𝑎𝑣𝑒𝑙 𝑡𝑜 𝐶𝑎𝑖𝑟𝑜 𝑏𝑦 𝑡𝑟𝑎𝑖𝑛
2 = 𝑡𝑟𝑎𝑣𝑒𝑙 𝑡𝑜 𝐶𝑎𝑖𝑟𝑜 𝑏𝑦 𝑝𝑙𝑎𝑛𝑒
• Problems are similar (slightly more complicated)
15
16. GENERAL PLAN
• It seems we have 4 cases:
16
Fixed
Effects
Random
Effects
Logit
Probit
17. GENERAL PLAN
• It seems we have 4 cases:
17
Fixed
Effects
Random
Effects
Logit Almost OK Bias
Probit Bias Ok
18. LOGIT MODEL
• Assume T=2 and consider the (log) likelihood function of a
sample of size N:
log 𝐿 = −
𝑖=1
𝑁
𝑡=1
2
log 1 + 𝑒𝑥𝑝 𝛼𝑖 + 𝛽𝑥𝑖𝑡 +
𝑖=1
𝑁
𝑡=1
2
𝑦𝑖𝑡 𝑒𝑥𝑝 𝛼𝑖 + 𝛽𝑥𝑖𝑡
• Suppose that 𝑥𝑖𝑡 = 0 if 𝑡 = 0 and 𝑥𝑖𝑡 = 1 if 𝑡 = 1. Then the
first derivatives are:
18
20. LOGIT MODEL
• The estimator of 𝛼𝑖 does not exist if 𝑡=1
𝑇
𝑦𝑖𝑡 = 0 o
𝑡=1
𝑇
𝑦𝑖𝑡 = 𝑇. Obviously!
• The estimator of 𝛽 is inconsistent with fixed T if 𝑛 → ∞
because of the incidental parameter problem (Neyman and
Scott, 1948): as N grows, so thus the number of parameters
to be estimated (one 𝛼𝑖 for each i)
• In fact, the estimator of 𝛽 is inconsistent, 𝑝𝑙𝑖𝑚 𝛽 = 2𝛽
(Hsiao, pp. 160-161)
20
21. LOGIT MODEL
• However, there is a Logit estimator that is consistent, called
conditional logit
• It uses only the information from units that switch states,
i.e., when a unit from 0 → 1 or from 1 → 0
• Therefore, the estimator is conditional on observing a
change in state, which allows identifying first 𝛽 and later
𝛼𝑖
• Note that this estimator eliminates:
– All units that do not change state (always 0 or 1)
– All variables that do not change in time.
21
22. LOGIT MODEL
• The conditional Logit model allows estimating 𝛽 using
Newton-Raphson algorithm iterative (o similar techniques,
e.g. BHHH). The same algorithm produces the variance of
the estimators, 𝑣𝑎𝑟 𝛽 .
• Scores (1st derivatives) 𝑠 𝛽 =
𝜕ℒ 𝑐 𝛽
𝜕𝛽
= 𝑖 1(0 < 𝑦𝑖+ <
22
23. PROBIT MODEL
23
• Following the linear probability model, we assume that
individual effects are random and distribute with Normal
distribution
𝑦𝑖𝑡 ≠ 0 ↔ 𝛽𝑥𝑖𝑡 + 𝛼𝑖 + 𝜀𝑖𝑡 > 0
• Let 𝜈𝑖𝑡 = 𝛼𝑖 + 𝜀𝑖𝑡
• Then, 𝜈𝑖𝑡 ∼ 𝑁 0, 𝜎𝜈
2
• And: 𝐹 𝑦, 𝑧 =
𝜙 𝑧 𝑠𝑖 𝑦 ≠ 0
1 − 𝜙 𝑧 𝑦 = 0
24. PROBIT MODEL
24
• The likelihood of each panel is
𝑃𝑟 𝑦𝑖1, 𝑦𝑖2, … , 𝑦𝑖𝑛|𝑥𝑖1, 𝑥𝑖2, … , 𝑥𝑖𝑛 =
−∞
∞
𝑒
− 𝜈𝑖𝑡
2
2𝜎2
2𝜋𝜎2
𝑡=1
𝑛
𝐹 𝑦𝑖𝑡, 𝛼𝑖 + 𝛽𝑥𝑖𝑡 𝑑𝜈𝑖
• This integral can be represented by
=
−∞
∞
𝑔 𝑦𝑖𝑡, 𝑥𝑖𝑡, 𝜈𝑖 𝑑𝜈𝑖
• And approximated using Gauss-Hermite quadrature
methods (e.g., using 𝑒−𝑥2
):
25. PROBIT MODEL
25
• Quadrature’s notion:
−∞
∞
𝑒−𝑥2
ℎ 𝑥 𝑑𝑥 ≈
𝑚=1
𝑀
𝜔 𝑚
∗ℎ 𝑎 𝑚
∗
where 𝜔 𝑚 are weights, ℎ 𝑎 𝑚
∗
are quadrature ordinates and M are
quadrature points
– The idea is properly approximate that ℎ(𝑥) using a polynomial of
order M (this is the key issue)
• The above integral de can be written as :
−∞
∞
𝑓 𝑥 𝑑𝑥 ≈
𝑚=1
𝑀
𝜔 𝑚
∗ 𝑒 𝑎 𝑚
∗ 2
ℎ 𝑎 𝑚
∗
26. PROBIT MODEL
26
• The likelihood function of each panel (unit) is
approximated using:
𝑙𝑖 = 2 𝜎𝑖
𝑚=1
𝑀
𝑤 𝑚
∗ 𝑒 𝑎 𝑚
∗ 2
𝑔 𝑦𝑖𝑡, 𝑥𝑖𝑡, 2 𝜎𝑖 𝑎 𝑚
∗ + 𝜀𝑖
• The likelihood function of the complete sample (all units)
is approximated using:
𝐿 ≈
𝑖=1
𝑛
𝑤𝑖 log 2 𝜎𝑖
𝑚=1
𝑀
𝑤 𝑚
∗
𝑒 𝑎 𝑚
∗ 2 𝑒− 2 𝜎 𝑖 𝑎 𝑚
∗ + 𝜀 𝑖
2𝜋𝜎2
𝑡=1
𝑛
𝐹 𝑦𝑖𝑡, 𝛼𝑖 + 𝛽𝑥𝑖𝑡 + 2 𝜎𝑖 𝑎 𝑚
∗
+ 𝜀𝑖
27. Likelihood Estimation
• The likelihood function is “the probability that a sample of
observations of size n is a realization of a particular
distribution 𝑓 𝑦𝑖𝑡, 𝑥𝑖𝑡|𝜃 "
• Let’s call it ℒ 𝑛 𝜃|𝑦𝑖𝑡, 𝑥𝑖𝑡
27
28. Example of Likelihood Estimation
• Consider “bicycle accidents in the campus” in a
given year. Suppose we record the following
sample
{2,0,3,4,1,3,0,2,3,4,3,5}
• What do you think is the model that generated this
sample?
• What do you think is the distribution that
generated this sample?
28
29. Example of Likelihood Estimation
• OK, let’s try Poisson (although I think it is Normal)
• Poisson distribution of each observation:
• When observations are independent, the joint probability
or likelihood function is the product of the marginals
29
30. Example of Likelihood Estimation
• We want to pick 𝜃 so as to make this probability
(likelihood) to be a maximum. There are two ways:
– Trying different values (most often used)
– Use calculus
• Our likelihood function is really ugly (non linear)
• But we can use logs to make it much nicer
30
31. Example of Likelihood Estimation
• Taking logs:
• To get the optimal 𝜃: derive twice, equalize to zero,
make sure second derivative is negative, and
obtain 𝜃 from first derivative:
• 1st derivative: −12 + 30 ∗
1
𝜃
• 2nd derivative: −30 ∗
1
𝜃2 which is negative
• Therefore 𝜃 = 2.5
31
32. Example of Likelihood Estimation
• Of all Poisson distributions, the one that best
describes the data is that with parameter 2.5
• What about my normal?
– Well I can fit the normal to the data and find 𝜇, 𝜎2
• Is you model better than mine?
– No way!!!
32
33. Likelihood Estimation
• The likelihood function is “the probability that a sample of
observations of size n is a realization of a particular
distribution 𝑓 𝑦𝑖𝑡, 𝑥𝑖𝑡|𝜃 "
ℒ 𝑛 𝜃|𝑦𝑖𝑡, 𝑥𝑖𝑡
• The joint distribution is the product of the conditional
density times the marginal density:
𝑓 𝑦𝑖𝑡, 𝑥𝑖𝑡|𝜃 = 𝑓 𝑦𝑖𝑡|𝑥𝑖𝑡, 𝜃 𝑓 𝑥𝑖𝑡|𝜃
33
34. Likelihood Estimation
• A statistic is sufficient with regards to a model and its
unknown parameters if "no other statistic can be calculated
using the same sample that could bring additional
information vis-à-vis the true value of the parameters”.
• Usually, a sufficient statistic is a simple function of the data,
e.g., the sum of the observations.
• In our case, a sufficient statistic of 𝑓 𝑥𝑖𝑡|𝜃 is the sum of
the observations (the units that change state, because in
the others 𝛼𝑖 is undefined)
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35. Likelihood Estimation
• Therefore, the conditional log likelihood function is:
ℒ 𝑐 𝛽 =
𝑖
1 0 < 𝑦𝑖+ < 𝑇
𝑡
𝑦𝑖𝑡 𝑥𝑖𝑡
′
𝛽 − log
𝑧 𝑦 𝑖+
𝑒 𝑡 𝑧 𝑡 𝑥 𝑖𝑡´𝛽
Where the 𝑧𝑡 represents all possible cases where there is a
change in state.
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