This Presentation describes, in short, Introduction to Time Series and the overall procedure required for Time Series Modelling including general terminologies and algorithms. However the detailed Mathematics is excluded in the slides, this ppt means to give a start to understanding the Time Series Modelling before going into detailed Statistics.
Generally it has been noticed that differential equation is solved typically. The Laplace transformation makes it easy to solve. The Laplace transformation is applied in different areas of science, engineering and technology. The Laplace transformation is applicable in so many fields. Laplace transformation is used in solving the time domain function by converting it into frequency domain. Laplace transformation makes it easier to solve the problems in engineering applications and makes differential equations simple to solve. In this paper we will discuss how to follow convolution theorem holds the Commutative property, Associative Property and Distributive Property. Dr. Dinesh Verma"Application of Convolution Theorem" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-4 , June 2018, URL: http://www.ijtsrd.com/papers/ijtsrd14172.pdf http://www.ijtsrd.com/mathemetics/applied-mathamatics/14172/application-of-convolution-theorem/dr-dinesh-verma
ARCH/GARCH model.ARCH/GARCH is a method to measure the volatility of the series, to model the noise term of ARIMA model. ARCH/GARCH incorporates new information and analyze the series based on the conditional variance where users can forecast future values with updated information. Here we used ARIMA-ARCH model to forecast moments. And forecast error 0.9%
Generally it has been noticed that differential equation is solved typically. The Laplace transformation makes it easy to solve. The Laplace transformation is applied in different areas of science, engineering and technology. The Laplace transformation is applicable in so many fields. Laplace transformation is used in solving the time domain function by converting it into frequency domain. Laplace transformation makes it easier to solve the problems in engineering applications and makes differential equations simple to solve. In this paper we will discuss how to follow convolution theorem holds the Commutative property, Associative Property and Distributive Property. Dr. Dinesh Verma"Application of Convolution Theorem" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-4 , June 2018, URL: http://www.ijtsrd.com/papers/ijtsrd14172.pdf http://www.ijtsrd.com/mathemetics/applied-mathamatics/14172/application-of-convolution-theorem/dr-dinesh-verma
ARCH/GARCH model.ARCH/GARCH is a method to measure the volatility of the series, to model the noise term of ARIMA model. ARCH/GARCH incorporates new information and analyze the series based on the conditional variance where users can forecast future values with updated information. Here we used ARIMA-ARCH model to forecast moments. And forecast error 0.9%
z-Transform is for the analysis and synthesis of discrete-time control systems.The z transform in discrete-time systems play a similar role as the Laplace transform in continuous-time systems
PROGRAMMA ATTIVITA’ DIDATTICA A.A. 2016/17
DOTTORATO DI RICERCA IN INGEGNERIA STRUTTURALE E GEOTECNICA
____________________________________________________________
STOCHASTIC DYNAMICS AND MONTE CARLO SIMULATION IN EARTHQUAKE ENGINEERING APPLICATIONS
Lecture Series by
Agathoklis Giaralis, Ph.D., M.ASCE., P.E. City, University of London
Visiting Professor Sapienza University of Rome
Avionics 738 Adaptive Filtering at Air University PAC Campus by Dr. Bilal A. Siddiqui in Spring 2018. This lecture covers background material for the course.
z-Transform is for the analysis and synthesis of discrete-time control systems.The z transform in discrete-time systems play a similar role as the Laplace transform in continuous-time systems
PROGRAMMA ATTIVITA’ DIDATTICA A.A. 2016/17
DOTTORATO DI RICERCA IN INGEGNERIA STRUTTURALE E GEOTECNICA
____________________________________________________________
STOCHASTIC DYNAMICS AND MONTE CARLO SIMULATION IN EARTHQUAKE ENGINEERING APPLICATIONS
Lecture Series by
Agathoklis Giaralis, Ph.D., M.ASCE., P.E. City, University of London
Visiting Professor Sapienza University of Rome
Avionics 738 Adaptive Filtering at Air University PAC Campus by Dr. Bilal A. Siddiqui in Spring 2018. This lecture covers background material for the course.
Kalman filter is a algorithm of predicting the future state of a system based on the previous ones.
In the presentation, I introduce to basic Kalman filtering step by step, with providing examples for better understanding.
This short report briefly illustrates the main ingredients required to perform Nonlinear Time History Analyses (NLTHAs) of a Single Degree of Freedom (SDF) system having rate-independent hysteretic behavior.
The Vaiana Rosati Model - Analytical Formulation (VRM AF) is adopted to simulate the behavior of the rate-independent hysteretic element.
The second-order Ordinary Differential Equation (ODE) of motion is numerically solved by using the Chang's Family of Explicit structure-dependent time integration Methods (CFEMs).
This Chapter is part of previous published ch.1 and ch.3 and its use for undergraduate students in physics department. also, you can use it for mathematical and Statistical courses and for those experimental courses of data fitting.
Avionics 738 Adaptive Filtering at Air University PAC Campus by Dr. Bilal A. Siddiqui in Spring 2018. This lecture deals with introduction to Kalman Filtering. Based n Optimal State Estimation by Dan Simon.
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Empowering the Data Analytics Ecosystem: A Laser Focus on Value
The data analytics ecosystem thrives when every component functions at its peak, unlocking the true potential of data. Here's a laser focus on key areas for an empowered ecosystem:
1. Democratize Access, Not Data:
Granular Access Controls: Provide users with self-service tools tailored to their specific needs, preventing data overload and misuse.
Data Catalogs: Implement robust data catalogs for easy discovery and understanding of available data sources.
2. Foster Collaboration with Clear Roles:
Data Mesh Architecture: Break down data silos by creating a distributed data ownership model with clear ownership and responsibilities.
Collaborative Workspaces: Utilize interactive platforms where data scientists, analysts, and domain experts can work seamlessly together.
3. Leverage Advanced Analytics Strategically:
AI-powered Automation: Automate repetitive tasks like data cleaning and feature engineering, freeing up data talent for higher-level analysis.
Right-Tool Selection: Strategically choose the most effective advanced analytics techniques (e.g., AI, ML) based on specific business problems.
4. Prioritize Data Quality with Automation:
Automated Data Validation: Implement automated data quality checks to identify and rectify errors at the source, minimizing downstream issues.
Data Lineage Tracking: Track the flow of data throughout the ecosystem, ensuring transparency and facilitating root cause analysis for errors.
5. Cultivate a Data-Driven Mindset:
Metrics-Driven Performance Management: Align KPIs and performance metrics with data-driven insights to ensure actionable decision making.
Data Storytelling Workshops: Equip stakeholders with the skills to translate complex data findings into compelling narratives that drive action.
Benefits of a Precise Ecosystem:
Sharpened Focus: Precise access and clear roles ensure everyone works with the most relevant data, maximizing efficiency.
Actionable Insights: Strategic analytics and automated quality checks lead to more reliable and actionable data insights.
Continuous Improvement: Data-driven performance management fosters a culture of learning and continuous improvement.
Sustainable Growth: Empowered by data, organizations can make informed decisions to drive sustainable growth and innovation.
By focusing on these precise actions, organizations can create an empowered data analytics ecosystem that delivers real value by driving data-driven decisions and maximizing the return on their data investment.
As Europe's leading economic powerhouse and the fourth-largest hashtag#economy globally, Germany stands at the forefront of innovation and industrial might. Renowned for its precision engineering and high-tech sectors, Germany's economic structure is heavily supported by a robust service industry, accounting for approximately 68% of its GDP. This economic clout and strategic geopolitical stance position Germany as a focal point in the global cyber threat landscape.
In the face of escalating global tensions, particularly those emanating from geopolitical disputes with nations like hashtag#Russia and hashtag#China, hashtag#Germany has witnessed a significant uptick in targeted cyber operations. Our analysis indicates a marked increase in hashtag#cyberattack sophistication aimed at critical infrastructure and key industrial sectors. These attacks range from ransomware campaigns to hashtag#AdvancedPersistentThreats (hashtag#APTs), threatening national security and business integrity.
🔑 Key findings include:
🔍 Increased frequency and complexity of cyber threats.
🔍 Escalation of state-sponsored and criminally motivated cyber operations.
🔍 Active dark web exchanges of malicious tools and tactics.
Our comprehensive report delves into these challenges, using a blend of open-source and proprietary data collection techniques. By monitoring activity on critical networks and analyzing attack patterns, our team provides a detailed overview of the threats facing German entities.
This report aims to equip stakeholders across public and private sectors with the knowledge to enhance their defensive strategies, reduce exposure to cyber risks, and reinforce Germany's resilience against cyber threats.
Adjusting primitives for graph : SHORT REPORT / NOTESSubhajit Sahu
Graph algorithms, like PageRank Compressed Sparse Row (CSR) is an adjacency-list based graph representation that is
Multiply with different modes (map)
1. Performance of sequential execution based vs OpenMP based vector multiply.
2. Comparing various launch configs for CUDA based vector multiply.
Sum with different storage types (reduce)
1. Performance of vector element sum using float vs bfloat16 as the storage type.
Sum with different modes (reduce)
1. Performance of sequential execution based vs OpenMP based vector element sum.
2. Performance of memcpy vs in-place based CUDA based vector element sum.
3. Comparing various launch configs for CUDA based vector element sum (memcpy).
4. Comparing various launch configs for CUDA based vector element sum (in-place).
Sum with in-place strategies of CUDA mode (reduce)
1. Comparing various launch configs for CUDA based vector element sum (in-place).
Explore our comprehensive data analysis project presentation on predicting product ad campaign performance. Learn how data-driven insights can optimize your marketing strategies and enhance campaign effectiveness. Perfect for professionals and students looking to understand the power of data analysis in advertising. for more details visit: https://bostoninstituteofanalytics.org/data-science-and-artificial-intelligence/
Chatty Kathy - UNC Bootcamp Final Project Presentation - Final Version - 5.23...John Andrews
SlideShare Description for "Chatty Kathy - UNC Bootcamp Final Project Presentation"
Title: Chatty Kathy: Enhancing Physical Activity Among Older Adults
Description:
Discover how Chatty Kathy, an innovative project developed at the UNC Bootcamp, aims to tackle the challenge of low physical activity among older adults. Our AI-driven solution uses peer interaction to boost and sustain exercise levels, significantly improving health outcomes. This presentation covers our problem statement, the rationale behind Chatty Kathy, synthetic data and persona creation, model performance metrics, a visual demonstration of the project, and potential future developments. Join us for an insightful Q&A session to explore the potential of this groundbreaking project.
Project Team: Jay Requarth, Jana Avery, John Andrews, Dr. Dick Davis II, Nee Buntoum, Nam Yeongjin & Mat Nicholas
Levelwise PageRank with Loop-Based Dead End Handling Strategy : SHORT REPORT ...Subhajit Sahu
Abstract — Levelwise PageRank is an alternative method of PageRank computation which decomposes the input graph into a directed acyclic block-graph of strongly connected components, and processes them in topological order, one level at a time. This enables calculation for ranks in a distributed fashion without per-iteration communication, unlike the standard method where all vertices are processed in each iteration. It however comes with a precondition of the absence of dead ends in the input graph. Here, the native non-distributed performance of Levelwise PageRank was compared against Monolithic PageRank on a CPU as well as a GPU. To ensure a fair comparison, Monolithic PageRank was also performed on a graph where vertices were split by components. Results indicate that Levelwise PageRank is about as fast as Monolithic PageRank on the CPU, but quite a bit slower on the GPU. Slowdown on the GPU is likely caused by a large submission of small workloads, and expected to be non-issue when the computation is performed on massive graphs.
2. • Continuous Time Series
• Discrete Time Series
o E.g. Adjustment of a price P in response to non-zero excess
demand for a product can be modeled in continuous time as:
𝑑𝑃
𝑑𝑡
= λ
o A discrete signal or discrete-time signal is a time series
consisting of a sequence of quantities e.g. Weather data etc.
Time Series
4. General Approach
Plot the series and examine the main features of the graph, checking in particular whether there is:
o A trend,
o A seasonal component,
o Any apparent sharp changes in behavior,
o Any outlying observations.
Remove the trend and seasonal components to get stationary residuals.
Choose a model to fit the residuals, making use of various sample statistics including the sample
autocorrelation function.
Forecasting will be achieved by forecasting the residuals and then inverting the transformations
described above to arrive at forecasts of the original series {𝑋𝑡}.
1
5. Stationarity
• Strict Stationarity
• Weak Stationarity
{𝑋𝑡} is (weakly) stationary if
The mean function of { 𝑋𝑡} = µ 𝑋 (t) = E(𝑋𝑡) is independent of t,
The covariance function of {𝑋𝑡} = γ 𝑋(r, s) = Cov(𝑋𝑟, 𝑋𝑠) = E[(𝑋𝑟 − µ 𝑋 (r))( 𝑋𝑠 − µ 𝑋 (s))] for all integers r and s
and γ 𝑋 (h) := γ 𝑋 (h, 0 ) = γ 𝑋(t + h, t) = γ 𝑋 (t + h, t) is independent of t for each h.
{𝑋𝑡} is a strictly stationary time series if
• (𝑋1, … … . , 𝑋 𝑛) ′ ≜ (𝑋1+ℎ, … … . , 𝑋 𝑛+ℎ) ′, for all integers h and n ≥ 1.
1
6. 𝑋𝑡 = 𝑚 𝑡 + 𝑠𝑡 + 𝑌𝑡, t = 1,…..,n, where E 𝑌𝑡 = 0, 𝑠𝑡+𝑑 = 𝑠𝑡, and
𝑗=1
𝑑
𝑠𝑗 = 0
Removal of Trend
• By estimation of 𝑚 𝑡 and 𝑠𝑡
• By differencing the series {𝑋𝑡 }
• Smoothing with a finite
moving average filter
• Exponential smoothing
• Smoothing by elimination
of high-frequency
components
• Polynomial fitting
lag-1 difference operator ∇
1
7. Removal of Trend
• By estimation of 𝑚 𝑡 and 𝑠𝑡
• Smoothing with a finite
moving average filter
• Let q be a nonnegative integer and consider the two-sided moving average: 𝑊𝑡 =
1
(2𝑞+1) 𝑗=−q
𝑞
𝑋𝑡−𝑗.
• Assuming that 𝑚 𝑡 is approximately linear over the interval [t − q, t + q] and that the average of the error terms
over this interval is close to zero.
• The moving average thus provides us with the estimates 𝑚 𝑡 =
1
(2𝑞+1) 𝑗=−q
𝑞
𝑋𝑡−𝑗 .
• For large q it will attenuate noise as 𝑚 𝑡 ≈ 0. So, to overcome this we can use the Spencer 15-point moving
average as a filter that passes polynomials of degree 3 without distortions
Linear Filter
{𝒙 𝒕} { 𝒎 𝒕 = 𝒂𝒋 𝒙 𝒕−𝒋}
1
8. Removal of Trend
• By estimation of 𝑚 𝑡 and 𝑠𝑡
• Exponential smoothing
• For any fixed α ∈ [0 , 1], the one-sided moving averages 𝑚 𝑡, t = 1 ,...,n , defined by the recursions:
𝑚 𝑡 = α𝑋𝑡 + ( 1 − α) 𝑚 𝑡−1 , t = 2 ,...,n, and 𝑚1 = 𝑋1
• Weighted moving average with weights decreasing exponentially (except for the last one).
1
9. Removal of Trend
• By differencing the series {𝑋𝑡 }
lag-1 difference operator ∇
• We define the lag-1 difference operator ∇ by:
∇𝑋𝑡 = 𝑋𝑡 − 𝑋𝑡−1= (1 − B) 𝑋𝑡, where B is the backward shift operator.
• B𝑋𝑡 = 𝑋𝑡−1.
• Powers of the operators B and ∇ are defined as 𝐵 𝑗
(𝑋𝑡) = 𝑋𝑡−𝑗 and 𝛻 𝑗
(𝑋𝑡) = ∇(𝛻 𝑗−1
(𝑋𝑡)), j ≥ 1, with 𝛻0
(𝑋𝑡) = 𝑋𝑡.
1
10. Removal of Seasonality
𝑋𝑡 = 𝑚 𝑡 + 𝑠𝑡 + 𝑌𝑡, t = 1,…..,n, where E 𝑌𝑡 = 0, 𝑠𝑡+𝑑 = 𝑠𝑡, and
𝑗=1
𝑑
𝑠𝑗 = 0
Estimation of Trend and then
Seasonal Components
lag- d differencing operator 𝛻𝑑
lag-d difference operator 𝛻𝑑
1
11. Removal of Seasonality
Estimation of Trend and then
Seasonal Components
• The trend is first estimated by applying a moving average filter specially chosen to eliminate the seasonal
component and to dampen the noise.
• If the period d is even, say d = 2q , then we use, 𝑚 𝑡 = (0.5 𝑥𝑡−𝑞 + 𝑥𝑡−𝑞+1 + ……… + 𝑥𝑡+𝑞)/d, q <t ≤ n −q.
• If the period is odd, say d = 2 q + 1, then we use the simple moving average.
• The second step is to estimate the seasonal component. For each k = 1 ,...,d ,we compute the average 𝜔 𝑘 of the
deviations {( 𝑥 𝑘+𝑗𝑑 -m 𝑘+𝑗𝑑), q < k + jd ≤ n−q}.
• Since these average deviations do not necessarily sum to zero, we estimate the seasonal component 𝑠 𝑘 as, s 𝑘 =
𝜔 𝑘 -
1
𝑑 𝑖−1
𝑑
𝜔𝑖, k = 1, ….., d, and s 𝑘 = s 𝑘−𝑑, k > d.
• The deseasonalized data is then defined to be the original series with the estimated seasonal component
removed, i.e., 𝑑 𝑡 = 𝑥𝑡 - s𝑡, t = 1, …. ,n. Finally, we re-estimate the trend from the deseasonalized data { 𝑑 𝑡} using
one of the methods already described.
1
12. lag- d differencing operator 𝛻𝑑
Removal of Seasonality
• Lag- d differencing operator 𝛻𝑑 defined as: 𝛻𝑑 𝑋𝑡 = 𝑋𝑡 - 𝑋𝑡−𝑑 = (1 – 𝐵 𝑑) 𝑋𝑡.
• Applying the operator 𝛻𝑑 to the model 𝑋𝑡 = 𝑚 𝑡 + 𝑠𝑡 + 𝑌𝑡, where {𝑠𝑡} has period d, we obtain 𝛻𝑑 𝑋𝑡 = 𝑚 𝑡 - 𝑚 𝑡−𝑑
+ 𝑌𝑡 - 𝑌𝑡−𝑑, which gives a decomposition of the difference 𝛻𝑑 𝑋𝑡 into a trend component (𝑚 𝑡 - 𝑚 𝑡−𝑑) and a noise
term (𝑌𝑡 - 𝑌𝑡−𝑑).
• The trend, 𝑚 𝑡 - 𝑚 𝑡−𝑑, can then be eliminated using the methods already described, in particular by applying a
power of the operator ∇.
• This doubly differenced series can in fact be well represented by a stationary time series model.
lag-d difference operator 𝛻𝑑
1
13. Test of Randomness1
• The Portmanteau test
• The Turning point test
• The Difference-sign test
• The rank test
• Fitting an Auto-regressive model
• Ljung and Box test
• McLeod and Li Test
14. Stationary Processes
• Linear Processes
o The time series {𝑋𝑡} is a linear process if it has the representation 𝑋𝑡 = 𝑗= −∞
∞
𝜓𝑗 𝑍𝑡−𝑗 or 𝑋𝑡 =
𝜓(𝐵)𝑍𝑡, where 𝜓 𝐵 = 𝑗= −∞
∞
𝜓𝑗 B 𝑗 for all t, where {𝑍𝑡} ∼ WN(0, σ 2) and {𝜓𝑗} is a sequence of
constants with 𝑗= −∞
∞
𝜓𝑗 < ∞.
o The class of linear time series models includes the class of Auto-Regressive Moving-Average (ARMA)
models
o Every second-order stationary process is either a linear process or can be transformed to a linear
process by subtracting a deterministic component
2
15. • MA(q) Process
o {𝑋𝑡} is a moving-average process of order q if 𝑋𝑡 = 𝑍𝑡 + θ1 𝑍𝑡−1 +.…+ θq 𝑍𝑡−q, where {𝑍𝑡] ∼ WN(0 ,σ 2)
and θ 1 ,...,θ q are constants
o Every q-correlated process is an MA(q) process.
• AR(p) Process
o {𝑋𝑡} is an Auto-Regressive process of order p if 𝑋𝑡 = φ1 X 𝑡−1 +.…+ φp X 𝑡−p + 𝑍𝑡, where {𝑍𝑡] ∼ WN(0
,σ 2) and 𝑍𝑡 is uncorrelated with 𝑋𝑠 for each s < t.
• ARMA(p, q) Process
o {𝑋𝑡} is an ARMA(p, q) process if 𝑋𝑡 - φ1 X 𝑡−1 -.…- φp X 𝑡−p = 𝑍𝑡 + θ1 𝑍𝑡−1 +.…+ θq 𝑍𝑡−q, where {𝑍𝑡] ∼
WN(0 ,σ 2
) and the polynomials (1 - φ1z -…- φp 𝑧 𝑝
) and (1 + θ1z +…+ θq 𝑧 𝑞
) have no common factors.
AR(p), MA(q) and ARMA(p, q) Processes2
16. ARMA(p, q) Processes
• ARMA(p, q) Process
o {𝑋𝑡} is an ARMA(p, q) process if 𝑋𝑡 - φ1 X 𝑡−1 -.…- φp X 𝑡−p = 𝑍𝑡 + θ1 𝑍𝑡−1 +.…+ θq 𝑍𝑡−q, where {𝑍𝑡] ∼ WN(0 ,σ 2
)
and the polynomials (1 - φ1z -…- φp 𝑧 𝑝
) and (1 + θ1z +…+ θq 𝑧 𝑞
) have no common factors.
o 𝑋𝑡 in above definition must be Stationary.
o A stationary solution {𝑋𝑡} of above equation exists (and is also the unique stationary solution) if and only if φ(z)
= 1 − φ1z − ··· − φpzp ≠ 0 for all |z| = 1.
o An ARMA(p, q) process {𝑋𝑡} is causal, or a causal function of {𝑍𝑡} , if there exist constants {ψ 𝑗} such that 𝑋𝑡 =
𝑗=0
∞
|𝜓𝑗|𝑍𝑡−𝑗. Causality is equivalent to the condition φ(z) = 1 − φ1z − ··· − φpzp ≠ 0 for all |z| ≤ 1.
o An ARMA(p, q) process {𝑋𝑡} is Invertible if there exist constants {𝜋𝑗} such that 𝑗=0
∞
𝜋𝑗 < ∞ and 𝑍𝑡 = 𝑗=0
∞
𝜋𝑗 𝑋𝑡−𝑗
for all t. Invertibility is equivalent to the condition θ(z) = 1 + θ1z +…+ θq 𝑧 𝑞 ≠ 0 for all |z| ≤ 1.
o We will focus our attention principally on Causal and Invertible ARMA processes.
2
17. ACF and PACF of ARMA(p, q) process
• PACF
o The partial autocorrelation function (PACF) of an ARMA process {𝑋𝑡} is the function α(·) defined by the
equations: α(0) = 0, and α(h) = ∅ℎℎ, h ≥ 1, where ∅ℎℎ = Γℎ
−1
𝛾ℎ, Γℎ =[𝛾(I - j)]
ℎ
𝑖, 𝑗 = 1
and 𝛾ℎ =
[𝛾(1), 𝛾(2),…, 𝛾(h)]′
o PAC For a causal AR(p) process is zero for lags greater than p.
• ACF
o If the sample ACF 𝜌(h) is significantly different from zero for 0 ≤ h ≤ q and negligible for h > q, then it
is MA(q) process
o In order to apply this criterion we need to take into account the random variation expected in the
sample autocorrelation function before we can classify ACF values as “negligible.” To resolve this
problem we can use Bartlett’s formula (Section 2.4), which implies that for a large sample of size n from
an MA( q ) process, the sample ACF values at lags greater than q are approximately normally distributed
with means 0 and variances 𝜔ℎℎ/n = (1 + 2𝜌2
(1) +…+ 2𝜌2
(q))/n
o This means that if the sample is from MA(q) process and if h > q, then 𝜌(h) should fall between the
bounds ±1.96 𝜔ℎℎ/𝑛 with probability approximately 0.95. In practice we frequently use the more
stringent values ±1.96
2
18. Forecasting ARMA Processes
• Innovations Algorithm
o It provides us with a recursive method for forecasting second-order zero-mean processes that are not
necessarily stationary.
o For the causal ARMA process φ(B) 𝑋𝑡 = θ(B) 𝑍𝑡, {𝑍𝑡} ∼ WN(0, 𝜎2
), it is possible to simplify the application of
the algorithm drastically.
2
19. 3 Modelling and Forecasting with ARMA Processes
• General
o Estimation of the parameters φ = (φ𝑖 ,…, φ 𝑝), θ = (θ𝑖 ,…, θ 𝑞), and 𝜎2
when p and q are assumed to be known
o Assumption that data have been “mean-corrected” by subtraction of the sample mean, so that it is appropriate
to fit a zero-mean ARMA model to the adjusted data x1 ,..., x 𝑛 . If the model fitted to the mean-corrected data is
φ(B)X 𝑡 = θ(B)Z 𝑡, {Z 𝑡} ∼ WN(0, 𝜎2
)
o When p and q are known, good estimators of φ and θ can be found by imagining the data to be observations of a
stationary Gaussian time series and maximizing the likelihood with respect to the p + q + 1 parameters φ1 ,..., φ 𝑝,
θ1 ,..., θ 𝑞 and 𝜎2
. The estimators obtained by this procedure are known as maximum likelihood (or maximum
Gaussian likelihood) estimators
• Preliminary Estimation of parameters
o Yule-Walker Estimation: The Yule–Walker and Burg procedures apply to the fitting of pure autoregressive models.
(Although the former can be adapted to models with q > 0, its performance is less efficient than when q = 0.).
Assumption is that the ACF of {X 𝑡} coincides with the sample ACF at lags 1,…,p.
o Burg’s Algorithm: Assumption is that the PACF of {X 𝑡} coincides with the sample ACF at lags 1,…,p.
o The Innovations Algorithm:
o The Hannan-Rissanen Algorithm:
• After getting Preliminary Estimates we apply Maximum Likelihood Estimation (MLE) (or maximum Gaussian likelihood)
to estimate the parameters.
20. Diagnostic Checking3
• Residuals are defined by: 𝑊𝑡 = (𝑋𝑡 - 𝑋𝑡(φ, 𝜃)) / (𝑟𝑡−1(φ, 𝜃))
1/2
, t = 1,…,n.
o E(𝑋 𝑛+1 − 𝑋 𝑛+1)
2
= 𝜎2
E(𝑊𝑛+1 − 𝑊𝑛+1)
2
= 𝜎2
𝑟𝑛
• Rescaled Residuals 𝑅𝑡, t = 1 ,…, n, are obtained by dividing the residuals 𝑊𝑡, t = 1 ,…, n, by the estimate 𝜎 =
( 𝑡=1
𝑛
𝑊𝑡
2
)/𝑛 of the white noise standard deviation. Thus, 𝑅𝑡 = 𝑊𝑡/ 𝜎
• If the fitted model is appropriate, the rescaled residuals should have properties similar to those of a
WN(0,1) sequence or of an iid(0,1) sequence if we make the stronger assumption that the white noise {𝑍𝑡}
driving the ARMA process is independent white noise.
21. Diagnostic Checking3
The Graph of { 𝑅𝑡, t = 1 ,…, n}
• If the fitted model is appropriate, then the graph of the rescaled residuals { 𝑅𝑡, t = 1 ,…, n} should resemble
that of a white noise sequence with variance one.
Rescaled residuals after fitting the ARMA(1,1) model to some data
25. Reference(s)
1) Introduction to Time Series and Forecasting, Brockwell, Peter J., Davis, Richard A.,
https://www.springer.com/us/book/9781475777505
2) Discrete time Series, https://www.wikiwand.com/en/Discrete-time_signal