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Least Median of Squares Models. What Is Sufficiency. A sufficient statistic t for a parameter q is a function of the sample data x1. xn, which contains all information in the sample about the parameter q. More formally, sufficiency is defined in terms of the likelihood function for q. For a sufficient statistic t, the Likelihood L x1. xn q can be written as. Since the second term does not depend on qt is said to be a sufficient statistic for q. Another way of stating this for the usual problems is that one could construct a random process starting from the sufficient statistic, which will have exactly the same distribution as the full sample for all states of nature.

To illustrate, let the observations be independent Bernoulli trials with the same probability of success. Suppose that there are n trials, and that person A observes which observations are successes, and person B only finds out the number of successes. Then if B places these successes at random points without replication, the probability that B will now get any given set of successes is exactly the same as the probability that A will see that set, no matter what the true probability of success happens to be.

You Must Look at Your Scattergrams. All three sets have the same correlation and regression line. The important moral is look at your scattergrams. How to produce a numerical example where the two scatterplots show clearly different relationships strengths but yield the same covariance. Produce two sets of X, Y values that have different correlation s; 2. Perform the following steps.

Calculate the two covariances, say C1 and C2; 3. Suppose you want to make C2 equal to C1. Then you want to multiply C2 by C1 C2 ; 4. S yyou want two numbers one of them might be 1a and b such that a. Multiply all values of X in set 2 by a, and all values of Y by b for the new variables, C r. An interesting numerical example showing two identical scatterplots but with differing covariance is the following Consider a data set of X, Y values, with covariance C1. Now let V 2X, and W 3Y.

The covariance of V and W will be 2 3 6 times C1, but the correlation between V and W is the same as the correlation between X and Y. Power of a Test. Power of a test is the probability of correctly rejecting a false null hypothesis. This probability is one minus the probability of making a Type II error b. Recall also that we choose the probability of making a Type I error when we set a and that if we decrease the probability of making a Type I error we increase the probability of making a Type II error.

Power and Alpha. Power and the True Difference between Population Means Anytime we test whether a sample differs from a population or whether two sample come from 2 separate populations, there is the assumption that each of the populations we are comparing has it s own mean and standard deviation even if we do not know it.

The distance between the two iq option grupo whatsapp means will affect the power of our test. Power as a Function of Sample Size and Variance You should notice that what really made the difference in the size of b is how much overlap there is in the two distributions. When the means are close together the two distributions overlap a great deal compared to when the means are farther apart.

Thus, anything that effects the extent the two distributions share common values will increase b the likelihood of making a Type II error. Sample size has an indirect effect on power because it affects the measure of variance we use to calculate the t-test statistic. Thus, sample size is of interest because it modifies our estimate of the standard deviation. Since we are calculating the power of a test that involves the comparison of sample means, we will be more interested in the standard error the average difference in sample values than standard deviation or variance by itself.

When n is large we will have a lower standard error than when n is small. In turn, when N is large well have a smaller b region than when n is small. Pilot Studies When the needed estimates for sample size calculation is not available from existing database, a pilot study is needed for adequate estimation with a given precision. Further Readings Cohen J.Statistical Power Analysis for the Behavioral SciencesL. Erlbaum Associates, 1988.

Thiemann, How Many Subjects. Provides basic sample size tablesexplanations, and power analysis. Myors, Statistical Power AnalysisL. Erlbaum Associates, 1998. Provides a simple and general sample size determination for hypothesis tests. ANOVA Analysis of Variance. Thus, when the variability that we predict between the two groups is much greater than the variability we don t predict within each group then we will conclude that our treatments produce different results.

Levene s Test Suppose that the sample data does not support the homogeneity of variance assumption, however, there is a good reason that the variations in the population are almost the same, then in such a situation you may like to use iq option grupo whatsapp Levene s modified test In each group first compute the absolute deviation of the individual values from the median in that group. Apply the usual one way ANOVA on the set of deviation values and then interpret the results.

The Procedure for Two Populations Independent Means Test Click on the image to enlarge it and THEN print it. You may use the following JavaScript to Test of Hypothesis for Two Populations. The Procedure for Two Dependent Means Test Click on the image to enlarge it and THEN print it. You may use the following JavaScript to Two Dependent Populations Testing. The Procedure for More Than Two Independent Means Test Click on the image to enlarge it and THEN print it.

The Procedure for More Than Two Dependent Populations Test Click on the image to enlarge it and THEN print it. You may use the following JavaScript iq option grupo whatsapp Three Dependent Means Comparison. Orthogonal Contrasts of Means in ANOVA. Further Readings Kachigan S.Multivariate Statistical Analysis A Conceptual IntroductionRadius Press, 1991. The Six-Sigma Quality. Sigma is a Greek symbol, which is used in statistics to represent standard deviation of a population.Statistical Analysis An Interdisciplinary Introduction to Univariate Multivariate MethodsRadius Press, 1986.

When a large enough random sample data are close to their mean i.the averagethen the population has a small deviation. If the data varies significantly from the mean, the data has a large deviation. In quality control measurement terms, you want to see that the sample is as close as possible to the mean and that the mean meets or exceed specifications. A large sigma means that there is a large amount of variation within the data. A lower sigma value corresponds to a small variation, and therefore a controlled process with a good quality.

The Six-Sigma means a measure of quality that strives for near perfection. Six-Sigma is a data-driven approach and methodology for eliminating defects to achieve six sigmas between lower and upper specification limits. 4 defects per million opportunities. Accordingly, to achieve Six-Sigma, e. Therefore, a Six-Sigma defect is defined for not meeting the customer s specifications.

A Six-Sigma opportunity is then the total quantity of chances for a defect. One sigma means only 68 of products are acceptable; three sigma means 99. 7 are acceptable. Six-Sigma is 99. Six-Sigma is a statistical measure expressing how close a product comes to its quality goal. 4 defects per million parts or opportunities. 9997 perfect or 3. The natural spread is 6 times the sample standard deviation.

The natural spread is centered on the sample mean, and all weights in the sample fall within the natural spread, meaning the process will produce relatively few out-of-specification products. Six-Sigma does not necessarily imply 3 defective units per million made; it also signifies 3 defects per million opportunities when used to describe a process. Some products may have tens of thousands of opportunities for defects per finished item, so the proportion of defective opportunities may actually be quite large.

Six-Sigma Quality is a fundamental approach to delivering very high levels of customer satisfaction through disciplined use of data and statistical analysis for maximizing and sustaining business success. What that means is that all business decisions are made based on statistical analysis, not instinct or past history.

Using the Six-Sigma approach will result in a significant, quantifiable improvement.in a manufacturing process it must not produce more than 3. 6 sigma defect-free good enough. Here are some examples of what life would be like if 99. 9 were good enough 1 hour of unsafe drinking water every month 2 long or short landings at every American cities airport each day 400 letters per hour which never arrive at their destination 3,000 newborns accidentally falling from the hands of nurses or doctors each year 4,000 incorrect drug prescriptions per year 22,000 checks deducted from the wrong bank account each hour As you can see, sometimes 99.

9 good just isn t good enough. Is it truly necessary to go for zero defects. Here are some examples of what life would be still like at Six-Sigma, 99. 9997 defect-free 13 wrong drug prescriptions per year 10 newborns accidentally falling from the hands of nurses or doctors each year 1 lost article of mail per hour. Now we see why the quest for Six-Sigma quality is necessary. Six-Sigma is the application of statistical methods to business processes to improve operating efficiencies.

It provides companies with a series of interventions and statistical tools that can lead to breakthrough profitability and quantum gains in quality. Six-Sigma allows us to take a real world problem with many potential answers, and translate it to a math problem, which will have only one answer. We then convert that one mathematical solution back to a real world solution. Six-Sigma goes beyond defect reduction to emphasize business process improvement in general, which includes total cost reduction, cycle-time improvement, increased customer satisfaction, and any other metric important to the customer and the company.

An objective of Six-Sigma is to eliminate any waste in the organization s processes by creating a road map for changing data into knowledge, reducing the amount of stress companies experience when they are overwhelmed with day-to-day activities and proactively uncovering opportunities that impact the customer and the company itself. The key to the Six-Sigma process is in eliminating defects. Organizations often waste time creating metrics that are not appropriate for the outputs being measured.

Executives can get deceptive results if they force all projects to determine a one size fits all metric in order to compare the quality of products and services from various departments. From a managerial standpoint, having one universal tool seems beneficial; however, it is not always feasible. In the airline industry, the US Air Traffic Control System Command Center measures companies on their rate of on time departure.

Below is an example of the deceptiveness of metrics. This would obviously be a critical measurement to customers the flying public. Whenever an airplane departs 15 minutes or more later than scheduled, that event is considered as a defect. Unfortunately, the government measures the airlines on whether the plane pulls away from the airport gate within 15 minutes of scheduled departure, not when it actually takes off. Airlines know this, so they pull away from the gate on time but let the plane sit on the runway as long as necessary before take off.

The result to the customer is still a late departure. This defect metric is therefore not an accurate representation of the desires of the customers who are impacted by the process. If this were a good descriptive metric, airlines would be measured by the actual delay experienced by passengers. This example shows the importance of having the right metrics for each process. The method above creates no incentive to reduce actual delays, so the customer and ultimately the industry still suffers.

With a Six-Sigma business strategy, we want to see a picture that describes the true output of a process over time, along with additional metrics, to give an insight as to where the management has to focus its improvement efforts for the customer. The Six Steps of Six-Sigma Loop Process The process is identified by the following five major activities for each project Identify the product or service you provide What do you do.

Identify your customer base, and determine what they care about Who uses your products and services. What is really important to them. Identify your needs What do you need to do your work. Define the process for doing your work How do you do your work. Ensure continuous improvement by measuring, analyzing, and controlling the improved process How perfectly are you doing your customer-focused work.

Often each step can create dozens of individual improvement projects and can last for several months. It is important to go back to each step from time to time in order to determine actual data maybe with improved measurement systems. Eliminate wasted efforts How can you do your work better. Once we know the answers to the above questions, we can begin to improve the process. The following case study will further explain the steps applied in Six-Sigma to Measure, Analyze, Improve, and Control a process to ensure customer satisfaction.

The Six Sigma General Process and Its Implementation The Six-Sigma means a measure of quality that strives for near perfection. Six-Sigma is a data-driven approach and methodology for eliminating defects to achieve six-sigma s between lower and upper specification limits. The implementation of the Six Sigma system starts normally with a few days workshop of the top level management of the organization. Only if the advantages of Six Sigma can be clearly stated and supported of the entire Management, then it makes sense to determine together the first project surrounding field and the pilot project team.

The pilot project team members participate is a few days Six Sigma workshop to learn the system principals, the process, the tools and the methodology. The project team meets to compiles main decisions and identifying key stakeholders in the pilot surrounding field. Within the next days the requirements of the stakeholders are collected for the main decision processes by face-to-face interviews.

By now, the workshop of the top management must be ready for the next step. The next step for the project team is to decide which and how the achievements should be measured and then begin with the data collection and analysis. Whenever the results are understood well then suggestions for improvement will be collected, analyzed, and prioritized based on the urgency and inter-dependencies.

As the main outcome, the project team members will determine which improvements should be realized first. The activities must be carried out in parallel whenever possible by a network activity chart. The activity chart will become more and more realistic by a loop-process while spread the improvement throughout the organization. In this phase it is important that rapid successes are obtained, in order to even the soil for other Six Sigma projects in the organization.

The main objective of the Six-Sigma approach is the implementation of a measurement-based strategy that focuses on process improvement. The aim is variation reduction, which can be accomplished by Six-Sigma methodology. More and more processes will be included and employees are trained including Black Belts who are the six sigma masters, and the dependency of external advisors will be reduced.

The Six-Sigma is a business strategy aimed at the near-elimination of defects from every manufacturing, service and transactional process. The concept of Six-Sigma was introduced and popularized for reducing defect rate of manufactured electronic boards. Although the original goal of Six-Sigma was to focus on manufacturing process, iq option grupo whatsapp the marketing, purchasing, customer order, financial and health care processing functions also embarked on Six Sigma programs.

Motorola Inc. Case Motorola is a role model for modern manufacturers. There is a reason for this reputation. The maker of wireless communications products, semiconductors, and electronic equipment enjoys a stellar reputation for high-tech, high-quality products. A participative-management process emphasizing employee involvement is a key factor in Motorola s quality push.

In 1987, Motorola invested 44 million in employee training and education in a new quality program called Six-Sigma. Motorola measures its internal quality based on the number of defects in its products and processes. Motorola conceptualized Six-Sigma as a quality goal in the mid-1980. Their target was Six-Sigma quality, or 99. 9997 defect free products which is equivalent to 3.

4 defects or less per 1 million parts. Quality is a competitive advantage because Motorola s reputation opens markets. When Motorola Inc. won the Malcolm Baldridge National Quality Award in 1988; it was in the early stages of a plan that, by 1992, would achieve Six-Sigma Quality. It is estimated that of 9. Shortly thereafter, many US firms were following Motorola s lead.

Control Charts, and the CUSUM. 2 billion in 1989 sales, 480 million was saved as a result of Motorola s Six-Sigma program. Developing quality control charts for variables X-Chart The following steps are required for developing quality control charts for variables. Decide what should be measured. Determine the sample size. Collect random sample and record the measurements counts. Calculate the average for each sample. Calculate the overall average. This is the average of all the sample averages X-double bar.

Calculate the average range R-bar. Determine the upper control limit UCL and lower control limit LCL for the average and for the range. Determine the range for each sample. Plot the chart. Determine if the average and range values are in statistical control. Take necessary action based on your interpretation of the charts.

Developing control charts for attributes P-Chart Control charts for attributes are called P-charts. The following steps are required to set up P-charts. Determine what should be measured. Collect sample data and record the data. Determine the required sample size. Calculate the average percent defective for the process p. Determine the control limits by determining the upper control limit UCL and the lower control limit LCL values for the chart.

Plot the data. Determine if the percent defectives are within control. Control charts are also used in industry to monitor processes that are far from Zero-Defect. However, among the powerful techniques is the counting of the cumulative conforming items between two nonconforming and its combined techniques based on cumulative sum and exponentially weighted moving average smoothing methods.

The general CUSUM is a statistical process control when measurements are multivariate. It is an effective tool in detecting a shift in the mean vector of the measurements, which is based on the cross-sectional antiranks of the measurements At each time point, the measurements, after being appropriately transformed, are ordered and their antiranks are recorded. When the process is in-control under some mild regularity conditions the antirank vector at each time point has a given distribution, which changes to some other distribution when the process is out-of-control and the components of the mean vector of the process are not all the same.

This latter shift, however, can be easily detected by a univariate CUSUM. Further Readings Breyfogle F. Therefore it detects shifts in all directions except the one that the components of the mean vector are all the same but not zero.Implementing Six Sigma Smarter Solutions Using Statistical Methods, Wiley, 1999. del Castillo E.Statistical Process and Adjustment Methods for Quality ControlWiley, 2002. Juran J, and A. Godfreym, Juran s Quality Handbook, McGraw-Hill, 1999.

Kuralmani, Statistical Models and Control Charts for High Quality ProcessesKluwer, 2002. Repeatability and Reproducibility.Concepts for R R StudiesASQ Quality Press, 1991. Lyday, Evaluating the Measurement ProcessStatistical Process Control Press, 1990. Further Readings Barrentine L. Statistical Instrument, Grab Sampling, and Passive Sampling Techniques. What is a statistical instrument. A statistical instrument is any process that aim at describing a phenomena by using any instrument or device, however the results may be used as a control tool.

Examples of statistical instruments are questionnaire and surveys sampling. What is grab sampling technique. The grab sampling technique is to take a relatively small sample over a very short period of time, the result obtained are usually instantaneous. However, the Passive Sampling is a technique where a sampling device is used for an extended time under similar conditions.

Depending on the desirable statistical investigation, the Passive Sampling may be a useful alternative or even more appropriate than grab sampling. However, a passive sampling technique needs to be developed and tested in the field. line transect sampling, in which the distances sampled are distances of detected objects usually animals from the line along which the observer travels. Distance Sampling.

point transect sampling, in which the distances sampled are distances of detected objects usually birds from the point at which the observer stands. cue counting, in which the distances sampled are distances from a moving observer to each detected cue given by the objects of interest usually whales. trapping webs, in which the distances sampled are from the web center to trapped objects usually invertebrates or small terrestrial vertebrates.

migration counts, in which the distances sampled are actually times of detection during the migration of objects usually whales past a watch point. Many mark-recapture models have been developed over the past 40 years. Monitoring of biological populations is receiving increasing emphasis in many countries. Data from marked populations can be used for the estimation of survival probabilities, how these vary by age, sex and time, and how they correlate with external variables.

Estimation of the finite rate of population change and fitness are still more difficult to address in a rigorous manner. Estimation of immigration and emigration rates, population size and the proportion of age classes that enter the breeding population are often important and difficult to estimate with precision for free-ranging populations. Further Readings Buckland S. Burnham, and J. Laake, Distance Sampling Estimating Abundance of Biological PopulationsChapman and Hall, London, 1993.

Borchers, and L. Thomas, Introduction to Distance SamplingOxford University Press, 2001. Data Mining and Knowledge Discovery. The continuing rapid growth of on-line data and the widespread use of databases necessitate the development of techniques for extracting useful knowledge and for facilitating database access. The challenge of extracting knowledge from data is of common interest to several fields, including statistics, databases, pattern recognition, machine learning, data visualization, optimization, and high-performance computing.

The data mining process involves identifying an appropriate data set to mine or sift through to discover data content relationships. Data mining sometimes resembles the traditional scientific method of identifying a hypothesis and then testing it using an appropriate data set. Data mining tools include techniques like case-based reasoning, cluster analysis, data visualization, fuzzy query and analysis, and neural networks.

Sometimes however data mining is reminiscent of what happens when data has been collected and no significant results were found and hence an ad hoc, exploratory analysis is conducted to find a significant relationship. The combination of fast computers, cheap storage, and better communication makes it easier by the day to tease useful information out of everything from supermarket buying patterns to credit histories.

For clever marketers, that knowledge can be worth as much as the stuff real miners dig from the ground. The process thus consists of three basic stages exploration, model building or pattern definition, and validation verification. Data mining as an analytic process designed to explore large amounts of typically business or market related data in search for consistent patterns and or systematic relationships between variables, and then to validate the findings by applying the detected patterns to new subsets of data.

What distinguishes data mining from conventional statistical data analysis is that data mining is usually done for the purpose of secondary analysis aimed at finding unsuspected relationships unrelated to the purposes for which the data were originally collected. Data warehousing as a process of organizing the storage of large, multivariate data sets in a way that facilitates the retrieval of information for analytic purposes. Data mining is now a rather vague term, but the element that is common to most definitions is predictive modeling with large data sets as used by big companies.

Therefore, data mining is the extraction of hidden predictive information from large databases. It is a powerful new technology with great potential, for example, to help marketing managers preemptively define the information market of tomorrow. Data mining tools predict future trends and behaviors, allowing businesses to make proactive, knowledge-driven decisions.

The automated, prospective analyses offered by data mining move beyond the analyses of past events provided by retrospective tools. Data mining answers business questions that traditionally were too time-consuming to resolve. Data mining tools scour databases for hidden patterns, finding predictive information that experts may miss because it lies outside their expectations.

Data mining techniques can be implemented rapidly on existing software and hardware platforms across the large companies to enhance the value of existing resources, and can be integrated with new products and systems as they are brought on-line. Knowledge discovery in databases aims at tearing down the last barrier in enterprises information flow, the data analysis step.

It is a label for an activity performed in a wide variety of application domains within the science and business communities, as well as for pleasure. When implemented on high performance client-server or parallel processing computers, data mining tools can analyze massive databases while a customer or analyst takes a coffee break, then deliver answers to questions such as, Which clients are most likely to respond to my next promotional mailing, and why.

The activity uses a large and heterogeneous data-set as a basis for synthesizing new and relevant knowledge. The knowledge is new because hidden relationships within the data are explicated, and or data is combined with prior knowledge to elucidate a given problem. The term relevant is used to emphasize that knowledge discovery is a goal-driven process in which knowledge is constructed to facilitate the solution to a problem. Knowledge discovery maybe viewed as a process containing many tasks.

Some of these tasks are well understood, while others depend on human judgment in an implicit matter. Further, the process is characterized by heavy iterations between the tasks. This is very similar to many creative engineering process, e.the development of dynamic models. In this reference mechanistic, or first principles based, models are emphasized, and the tasks involved in model development are defined by.

Initialize data collection and problem formulation. The initial data are collected, and some more or less precise formulation of the modeling problem is developed. Tools selection. The software tools to support modeling and allow simulation are selected. Conceptual modeling. The system to be modeled, e.a chemical reactor, a power generator, or a marine vessel, is abstracted at first. Model representation. The essential compartments and the dominant phenomena occurring are identified and documented for later reuse.

A representation of the system model is generated. Often, equations are used; however, a graphical block diagram or any other formalism may alternatively be used, depending on the modeling tools selected above. Computer implementation. The model representation is implemented using the means provided by the modeling system of the software employed. These may range from general programming languages to equation-based modeling languages or graphical block-oriented interfaces.

The model implementation is verified to really capture the intent of the modeler. No simulations for the actual problem to be solved are carried out for this purpose. Reasonable initial values are provided or computed, the numerical solution process is debugged. The results of the simulation are validated against some reference, ideally against experimental data.

The modeling process, the model, and the simulation results during validation and application of the model are documented. The model is used in some model-based process engineering problem solving task. For other model types, like neural network models where data-driven knowledge is utilized, the modeling process will be somewhat different.

Some of the tasks like the conceptual modeling phase, will vanish. Typical application areas for dynamic models are control, prediction, planning, and fault detection and diagnosis. A major deficiency of today s methods is the lack of ability to utilize a wide variety of knowledge. Model application. As an example, a black-box model structure has very limited abilities to utilize first principles knowledge on a problem. this has provided a basis for developing different hybrid schemes.

Two hybrid schemes will highlight the discussion. First, it will be shown how a mechanistic model can be combined with a black-box model to represent a pH neutralization system efficiently. Second, the combination of continuous and discrete control inputs is considered, utilizing a two-tank example as case. The hybrid approach may be viewed as a means to integrate different types of knowledge, i.being able to utilize a heterogeneous knowledge base to derive a model.

Standard practice today is that almost any methods and software can treat large homogeneous data-sets. A typical example of a homogeneous data-set is time-series data from some system, e.temperature, pressure, and compositions measurements over some time frame provided by the instrumentation and control system of a chemical reactor. If textual information of a qualitative nature is provided by plant personnel, the data becomes heterogeneous.

The above discussion will form the basis for analyzing the interaction between knowledge discovery, and modeling and identification of dynamic models. In particular, we will be interested in identifying how concepts from knowledge discovery can enrich state-of-the- art within control, prediction, planning, and fault detection and diagnosis of dynamic systems. Further Readings Marco D.Building and Managing the Meta Data Repository A Full Lifecycle GuideJohn Wiley, 2000.

Thuraisingham B.Iq option grupo whatsapp Mining Technologies, Techniques, Tools, and TrendsCRC Press, 1998. Westphal Ch. Different approaches to handle this heterogeneous case are considered. Blaxton, Data Mining Solutions Methods and Tools for Solving Real-World ProblemsJohn Wiley, 1998. Neural Networks Applications. The classical approaches are the feedforward neural networks, trained using back-propagation, which remain the most widespread and efficient technique to implement supervised learning.

Applications include data mining, and stock market predictions. Further Readings Schurmann J.Pattern Classification A Unified View of Statistical and Neural ApproachesJohn Wiley Sons, 1996. Bayes and Empirical Bayes Methods. Bayes and EB methods can be implemented using modern Markov chain Monte Carlo MCMC computational methods.

The main steps are preprocess the data, the appropriate selection of variables, postprocessing of the results, and a final validation of the global strategy. Properly structured Bayes and EB procedures typically have good frequentist and Bayesian performance, both in theory and in practice. This in turn motivates their use in advanced high-dimensional model settings e.longitudinal data or spatio-temporal mapping modelswhere a Bayesian model implemented via MCMC often provides the only feasible approach that incorporates all relevant model features.

Smith, Bayesian TheoryWiley, 2000. Louis, Bayes and Empirical Bayes Methods for Data AnalysisChapman and Hall, 1996.Bayesian Statistical ModellingWiley, 2001. Markovian Memory Theory. Memory Theory and time series share the additive property and inside a single term there can be multiplication, but like general regression methods this does not always mean that they are all using M Iq option grupo whatsapp.

One may use standard time series methods in the initial phase of modeling things, but instead proceed as follows using M Theory s Cross-Term Dimensional Analysis CTDA. Suppose that you postulate a model y af x - bg z ch u where f, g, h are some functions and x, z, u are what are usually referred to as independent variables. Notice the minus sign - to the left of b and the sign to the left of c and implicitly to the left of a, where a, b, c are positive constants. The variable y is usually referred to as a dependent variable.

According to M Theory, not only do f, g, and h influence cause y, but g influences causes f and h at least to some extent. In fact, M Theory can formulate this in terms of probable influence as well as deterministic influence. All this generalizes to the case where the functions f, g, h depend on two or more variables, e.f x, wg z, t, retc.

One can reverse this process.

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