Introduction to Time Series Analysis in Python

Time Series Analysis in Python

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Course Description...

From stock prices to climate data, time series data are found in a wide variety of domains, and being able to effectively work with such data is an increasingly important skill for data scientists. This course will introduce you to time series analysis in Python. After learning about what a time series is, you'll learn about several time series models ranging from autoregressive and moving average models to cointegration models. Along the way, you'll learn how to estimate, forecast, and simulate these models using statistical libraries in Python. You'll see numerous examples of how these models are used, with a particular emphasis on applications in finance.

TOPICS

.Correlation and Autocorrelation
.Some Simple Time Series
.Autoregressive (AR) Models
.Moving Average (MA) and ARMA Models
.Putting It All Together

And lot MORE...

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PREREQUISITES:

>>>Introduction to Python<<<

About the Author...

Ed Neil O. Maratas an instructor of Jose Rizal Memorial State University, Dapitan Campus, Philippines as regular status. He earned his Bachelor of Science in Statistics at Mindanao State University-Tawi-Tawi College of Technology and Oceanography in the year 2003 and finished Master of Arts in Mathematics at Jose Rizal Memorial State University year 2009. He Became a researcher, a data analyst, and engaged to several projects linked to the university as data processor.

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Numerical Methods of Summarizing Data

Percentages & Proportions

These are the most commonly used numerical measures for summarizing data. The formulas are given below.

Where;

f – the frequency or the number of cases in any category

N- the total number of cases in all categories.

Ratio

Used to compare categories in terms of relative frequency. The formula is given below;

Where;

- the number of cases in the first category

- the number of cases in the second category

Rates

Rates are defined as the number of actual occurrences of some phenomenon divided by the number of possible occurrences per some unit of time. The formula is given below;

Rate of change or percentage change

Is useful for comparing the actual change between time periods.

Where;

– frequency at the new/current time period

– frequency at the old/previous time period

Collection, Summarization and Presentation of Data

Introduction

Data can be collected in different ways. It can be obtained from original data or from previous students.

Methods of Collecting Data

The methods of collecting data are:

1. Direct or interview method. This is a personal communication with the individual you want to interview.

2. Indirect or questionnaires method. This is done by sending questionnaires to the person from whom you would like to get the information.

3. Registration method. Utilizing existing records from various agencies.

4. Observation method. This can be done directly or indirectly.

5. Experiment method. This is done with the participation of a certain researcher. In other words, there is a human intervention occur during the process of data collection.

Two Documented Sources of Data

1. Primary Data. Data documented by the primary source.

2. Secondary Data. Data documented by a secondary source.

Sampling Techniques

The study of the entire population of interest in some situations is impractical or even impossible to include the entire population. Thus, take a part of a population, the so-called sample.

Sampling

Sampling is the process of selecting the sample or the study units from a previously defined population.

Sampling Error

Sampling error is the difference or deviation of the sample from the population with respect to the characteristics of interest in the study.

Sampling Frame

The list of units from which the sample were drawn in any sampling procedure.

Sampling Procedure

Sampling procedure refers to the manner in which the members of the population are selected as part of the sample. These are classified into probability or random sampling and non-probability sampling procedures.

Sample Size Determination

An important aspect of the sampling design is the sample size. The number of members that you include in the study must not be too small in order to come up with reliable estimates. According to some researchers and statisticians suggest, Slovin’s formula is an alternative approach to computing the sample size. The formula is given below;

Where;

n – the sample size N – the population size e – the desired margin of error

Probability Sampling Procedures

These comprise all sampling methods done when there is a sampling frame which ensures that all the probable sampled have an equal chance or probability of being selected for the study.

Types of Probability Sampling Procedures

Simple Random Sampling

This is the basic method on which all other methods of probability sampling are built. Each member of the population has an equal and independent chance of being selected.

Systematic Sampling

This is done by selecting a sampling interval k and using the sampling frame, the researcher selects every kth member of the population beginning at some random point and cycling through the list.

Stratified Sampling

The members of the population are classified into non-overlapping groups or strata on the basis of characteristics to be properly represented in the sample.

Cluster Sampling

This is usually used in studies of huge populations where the sampling frame may be too large to study or too time-consuming that is better to divide them first into clusters or groups and randomly select a sample cluster of choice.

Multistage Sampling

This is usually done in big community-based studies in which selection of the sampling unit is done by stages.

Non-probability Sampling Procedures

These are methods that do not include random sampling at some stage in the process. Further, these are applicable when there is no sampling frame available.

Types of Non-Probability Sampling Procedures

Convenience Sampling

In this method, the sample consists of elements that are most accessible or easiest to contact.

Judgement or Purposive Sampling

In this method, the researcher chooses a sample that agrees with his/her subjective judgment of a representative sample.

Quota Sampling

Is the non-probability sampling wherein the researcher just sets a quota or number of sampling units to be included in each grouping but uses convenience sampling to select the units within each grouping.

Snowball Sampling

Also called chain referral and referential sampling. This is used to find members of a group not otherwise visibly identified.

Estimation of Parameters

Learning Objectives:

Given the learning materials and activities of this chapter, they will be able to:

Ø  Distinguish classical methods from Bayesian methods of estimates.

Ø  Calculate the standard error of a sample.

Ø  Calculate the margin of error for interval estimate.

Ø  Construct interval estimates of the population mean given a specified level of confidence.

Ø  Construct interval estimates of the population proportion with the specified confidence level.

Introduction

         There are two major areas in statistical inference, the estimation of parameters and hypothesis testing. Estimation is the process of estimating the value of a parameter from information obtained in a sample. An important aspect of estimation is the size of the sample. An estimator is a formula, the function or procedure used in estimating a population parameter. There are two methods of estimating population parameters, such as:

a.                       Classical method is based strictly on information obtained from a random sample selected from the population.

b.                 Bayesian method utilizes prior subjective knowledge about the probability distribution of the unknown parameters in conjunction with the information provided from the sample data.

In this text, we shall utilize the classical method to estimate unknown population parameters such as the mean, proportion and the variance by computing statistics from a random sample and applying the theory of sampling distributions. There are two ways in the classical method of estimation, namely: point estimate and interval estimate.

Point estimate

            Point estimate consists of a single value used to estimate a population parameter. For most parts, the point estimate will be different from the population mean due to sampling error. There is now way of knowing how close the point estimate is to the population parameter. For this reason, statisticians prefer another type of estimate.

Interval estimate

            Is an interval or a range of values used to estimate the parameter. In an interval estimate, the parameter is specified as being between two values. A degree of confidence can be assigned before an interval estimate is made. The confidence level is the probability that the interval estimate will contain the true population mean or population proportion.

        Three common confidence level are 90%, 95% and 99% confidence intervals. The table below summarizes the values of the standard deviates and the margin of error for the most commonly used confidence level.

       A term level of significance is defined as the probability of erroneously concluding that a confidence interval generated will contain the parameter. As observed in the table, the greater the level of confidence, the larger the z values, the larger the margin of error and of course the wider the confidence interval.

        An interval estimate is constructed by subtracting and adding the margin of error to a point estimator. The length of the confidence interval is determined by the sample size, the standard deviation, and the desired confidence level.

Estimating Means

         Estimation (or estimating) is the process of finding an estimate, or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is derived from the best information available. Typically, estimation involves "using the value of a statistic derived from a sample to estimate the value of a corresponding population parameter". The sample provides information that can be projected, through various formal or informal processes, to determine a range most likely to describe the missing information. An estimate that turns out to be incorrect will be an overestimate if the estimate exceeded the actual result, and an underestimate if the estimate fell short of the actual result. 

Estimating Means Large Sample and the Standard Deviation is known

The Central Limit Theorem says that, for large samples (samples of size n ≥ 30), when viewed as a random variable the sample mean is normally distributed with mean and standard deviation. The Empirical Rule says that we must go about two standard deviations from the mean to capture 95% of the values of sample mean generated by sample after sample.

Confidence interval for means >=30 and the standard deviation is known the formula is

Example 1: A study of 40 bowlers showed that their average score was 186. The standard deviation of the population is 6.

a.       Find the 95% confidence interval of the mean score for all bowlers.

b.      Find the 99% confidence interval of the mean score of a sample of 100 bowlers instead of a sample of 40.

          Thus, it can be 95% confident that the true mean score of bowlers is between 184.14 and 187.86. This means that 95% of the time, the population mean score of bowlers will be roughly between 184 and 188. 

          Thus, the 99% confidence interval for the population mean score is ranging from 184.542 to 187.548. This means that we can be 99% confident that the population mean score is roughly between 185 to 188.

Confidence interval for means < 30 and the standard deviation is unknown (small sample)

         When the population standard deviation is unknown and the sample size is less than 30, the standard deviation from the sample can be used in place of the population standard deviation. In this case, the t-distribution is used to determine the confidence interval and the random variable is approximately normally distributed. The formula is:

          To determine the value of t critical locates the critical value from the table in t distribution with the corresponding degrees of freedom. The degrees of freedom are the values that are free to vary after a sample statistic has been computed. The degrees of freedom for the confidence interval for the mean is n – 1. Also, note that the sample standard deviation is used instead of the population standard deviation. 

Example 2: A sample of 20 tuna showed that they swim an average of 8.6 miles per hour. The standard deviation for the sample was 1.6. Find the 95% confidence interval of the true mean.

Solution: Given n = 20, the sample mean of 8.6 miles per hour, and the sample standard deviation s = 1.6 miles per hour. The degrees of freedom is n – 1 = 20 – 1, using the t-distribution table yielded a critical value of t =2.093. Hence, 

          Thus, the 95% confidence interval for the population mean time is ranging from 7.851 to 9.349 miles per hour. This means that we can be 95% confident that the true mean time of tuna can swim roughly between 8 and 9 miles per hour. 

When to use the z and t distribution:

-          If the population standard deviation is known and sample size is large, use z-test.

-          If the population standard deviation is unknown and sample size is large, use z-test.

-          If the population standard deviation is unknown and sample size is small, use t-test.

Estimating Proportions

       When the variable of interest is qualitative and are summarized in terms of frequencies, confidence intervals for estimating proportion may be constructed. 

          To construct confidence interval for estimating a population proportion based on a proportion obtain from a random sample, similar procedure used to estimate population mean. 

          Example 3: A local polling organization reports that based on a local-wide survey of 500 respondents, 43% of the vote will be in favor of the administration governatorial candidate in the May 2016 elections. Construct the 95% confidence interval for the proportion indicating preference for the administration candidate.

         Thus, the 95% confidence interval is from 38.7% to 47.3%. This means that with a sample of 500, the poll has a margin of error of ±4.3% and the pollster can be 95% confident that the administration candidate will obtain roughly between 39% and 47% of the votes.

Powerpoint presentation: Estimation of Parameters

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Note: For Comments, Questions, and Suggestions feel free to contact at enomaratas@jrmsu.edu.ph or ednielmaratas@gmail.com. You can also post at the comment section below.

Population and Sample, Parameter and Statistic, Descriptive and Inferential

What is Population? Sample?

In statistics, researchers, and educators commonly use the terms population and sample (Alferez & Duro, 2006). The population is often too large for us to examine each of its members.

           The population is the entire collection of all elements/experimental units under consideration in a statistical inquiry or to be studied. Sample a part of the population. If the sample is to provide information about the entire population, it must be representative of that group in some way. In actuality, unless a sample is picked at random, it cannot be expected to be representative of a population. This is because any nonrandom rule for selecting a sample will almost always produce one that is skewed toward some data values over others.

        For example, if we wish to determine the average income of households in Zamboanga del Norte, then the population of interest is the collection of all households in Zamboanga del Norte. However due to some constraints, the budget, the time, the manpower for instance, and then we would have to redefine the interest. This time we can delimit the scope of the study we utilized sampling size to include only the collection of all households in Dapitan and Dipolog City, considering this is done thru sampling techniques.

What is Parameter? Statistic?
          Parameter refers to any numerical value describing a characteristic of a population and is usually denoted by some Greek letters such as population standard deviation, σ and population mean, μ. 

         While the term statistic is a numerical measurement describing some characteristics of a sample. The symbols x and s are statistics which is unbiased of the parameter μ and σ.

Two Major Areas of Statistics

Descriptive Statistics

Is concerned with the methods for collecting, organizing, and describing a set of data so as to yield meaningful information (Walpole, 2000). Construction of tables, charts, and graphs and the computation of descriptive statistical measures also fall in this area.

Inferential Statistics

Inferential Statistics is also called Inductive Statistics or Statistical Inference. Comprises those procedures for drawing inferences or making generalizations about characteristics of a population-based on partial and incomplete information obtained sample data to infer to populations.

SEE YOU ON THE NEXT TOPIC:

Introduction to Statistics

Types of Variables

What are the Types of Variables?

Variable refers to a characteristics or phenomena that changes or varies over time for different individual or objects under consideration (Mendenhall, et.al., 2012).

Data

Data are the values that the variables can assume (Reston, 2004).

Experimental unit

Experimental unit is the individual or object on which a variable is measured.

Types of variable according to a functional relationship

Independent variable. This is sometimes termed as a predictor variable if the object is to predict the value of one variable on the basis of the other.

Dependent variable. This is sometimes called the criterion variable and whose value is predicted. For example, academic achievement is dependent on Intelligent Quotient, study habits, interests, attitudes and many more. Hence, IQ, study habits, interests, attitudes are independent variables. On the other hand, academic achievement is the dependent variable.

Types of variable according to the attribute of objects they classify

         Qualitative Variables. These are words or codes that represent a class or category. Further, produce data that can be categorized according to similarities or differences in kind. Also known as a categorical variable. 

Here are some examples: gender, taste ranking, religious affiliation, academic achievement, marital status, type of high school attended and many more.

Quantitative Variables. These are variables that classify objects or represent an amount or a count. This is a variable often represented by an arbitrary letter, let say x, produce numerical data. 

Here are some examples: Height, student enrolment, class size, family size, test scores, entrance test results, crime rate, salary, number of passengers, a volume of orange juice, etc.

Types of variable according to the continuity of values

Discrete variable. This refers to variables that can be obtained, can assume only a finite or through a countable number of values.

Examples are: Number of family members, number of new car sales, number of defective bulbs, faculty size, hospital staff size, number of students enrolled in Statistics course, number of bedrooms in a house, etc.

Continuous variable. Variables that can assume many values corresponding to the points on a line interval. 

Here are some examples: Crime rates, cell density, rainfall, temperature, air pressure, weight, height, study hours, time, salary, distance traveled, etc. 

Types of variable according to the Scale of Measurements

Nominal scale is often referred to as a categorical scale. This only satisfies the identity property of measurement. 

For example, gender, Religion, and political affiliation.

Ordinal scale has an ordered relationship to every other value on the scale. 

Example: Academic achievement, taste ranking, honors received, educational qualification etc.

           Interval scale has equal units of measurement, thus making it possible to interpret not only the order of scale scores but also the distance between them. It has the properties of identity, magnitude, and equal intervals. 

     

            Example: test scores, height (in cm), Intelligent Quotient (IQ) and many more.

Ratio scale has the property allows one to make statements of equality of intervals. This is the highest level of measurement which includes the inherent zero starting point. 

Examples: number of children in a family, student enrolment income and many more.

                                                                                                  

Why Study Statistics?

Study Statistics?

As early as in the Old Testament, statistical methods occur particularly censuses of population and wealth were taken by the Pharaohs and the ancient Hebrews long before Christ was born. With the turn of 20^th century, the term came to be applied to a study of scientific methods of dealing with quantitative data with applications to practically all fields of study.

There are reasons why the scope of statistics and the need to study statistics have grown enormously in the last few decades.

First, Knowledge of basic statistics is essential for people going into research in various fields of human endeavor.

Second, numerical information is everywhere. 

Third, statistical techniques are used to make decisions that affect our daily lives.

Fourth, a person with an understanding of statistics is between able to decide whether his or her professional colleagues use their statistics to illuminate or merely to support their personal biases; that is, it helps one to decide whether the claims are valid or not.

Fifth, knowledge of statistics is essential for persons who wish to keep their education up-to-date.

Finally, an understanding of statistics can help anyone discriminate between facts and fancy in everyday life – in reading newspapers and watching television, and in making daily comparison and evaluations.

Generally, no matter what your future line of work, you will make decisions that involve data. That is, an understanding of statistical methods will help you make decisions more effectively.