Assignment Bf

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Assignment

Subject: Business Forecasting

Program: MBA Semester: 3

Faculty Name: Dr. K V Narrasimham

Type: Individual Marks: 20

E Mail ID: [email protected]

Assignment Issue Date: 25/5/12 Date of Submission: 16/6/12

TITLE

Q1. What is Business forecasting? What is the need of Forecasting? What are the

various types of Forecasting? Describe the forecasting process.

Q2. What do we study and analyse Time series Analysis? Describe the least square

method for determining the trend in a time serious.

OBJECTIVES:-1. To understand the impact of various Forecasting in Business.2. To know about forecasting and its usefulness in Business environment

SUBMITTED BY:-

SOMESH NAUTIYAL


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Business Forecasting:-Business forecasting is a process used to estimate or predict future patterns using business data.

Some examples of business forecasting include estimating quarterly sales, product demand,customer lifetime value and churn potential, inventory and supply-chain reorder timing,

workforce attrition, website traffic, and predicting exposure to fraud and risk. Several powerful

estimation functions are commonly used to perform business forecasting: time series analysis,causal models, and regression analysis. Business forecasting supports executives, analysts and

end users in decision-making using decision support systems such as business intelligence.

Business Forecasting and Data Mining:-Business forecasting can be completed using a process called data mining. The data miningprocess uses predictive models based on existing and historical data to project potential

outcomes of business activities and transactions.

The ultimate goal of data mining is to find hidden predictive information in large amounts ofdata. The data mining process involves using existing information to gain new insights into

business activities by applying predictive models, using analysis techniques such as regression,

classification, clustering, and association. Data mining helps organizations leverage datawarehouses to forecast future business outcomes.

Business forecasting has always been one component of running an enterprise. However,forecasting traditionally was based less on concrete and comprehensive data than on face-to-face

meetings and common sense. In recent years, business forecasting has developed into a much

more scientific endeavor, with a host of theories, methods, and techniques designed for


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forecasting certain types of data. The development of information technologies and the Internet

propelled this development into overdrive, as companies not only adopted such technologies intotheir business practices, but into forecasting schemes as well. In the 2000s, projecting the

optimal levels of goods to buy or products to produce involved sophisticated software and

electronic networks that incorporate mounds of data and advanced mathematical algorithms

tailored to a company's particular market conditions and line of business.

Business forecasting involves a wide range of tools, including simple electronic spreadsheets,enterprise resource planning (ERP) and electronic data interchange (EDI) networks, advanced

supply chain management systems, and other Web-enabled technologies. The practice attempts

to pinpoint key factors in business production and extrapolate from given data sets to produceaccurate projections for future costs, revenues, and opportunities. This normally is done with an

eye toward adjusting current and near-future business practices to take maximum advantage of

expectations.

In the Internet age, the field of business forecasting was propelled by three interrelated

phenomena. First, the Internet provided a new series of tools to aid the science of businessforecasting. Second, business forecasting had to take the Internet itself into account in trying toconstruct viable models and make predictions. Finally, the Internet fostered vastly accelerated

transformations in all areas of business that made the job of business forecasters that much more

exacting. By the 2000s, as the Internet and its myriad functions highlighted the centralimportance of information in economic activity, more and more companies came to recognize

the value, and often the necessity, of business forecasting techniques and systems.

Business forecasting is indeed big business, with companies investing tremendous resources in

systems, time, and employees aimed at bringing useful projections into the planning process.

According to a survey by the Hudson, Ohio-based AnswerThink Consulting Group, which

specializes in studies of business planning, the average U.S. company spends more than 25,000person-days on business forecasting and related activities for every billion dollars of revenue.

Companies have a vast array of business forecasting systems and software from which to choose,but choosing the correct one for their particular needs requires a good deal of investigation.

According to theJournal of Business Forecasting Methods & Systems, any forecasting system

needs to be able to facilitate data-sharing partnerships between businesses, accept input fromseveral different data sources and platforms, operate on an open architecture, and feature an array

of analysis techniques and approaches.

Forecasting systems draw on several sources for their forecasting input, including databases, e-

mails, documents, and Web sites. After processing data from various sources, sophisticated

forecasting systems integrate all the necessary data into a single spreadsheet, which the companycan then manipulate by entering in various projectionssuch as different estimates of future

salesthat the system will incorporate into a new readout.

A flexible and sound architecture is crucial, particularly in the fast-paced, rapidly developing

Internet economy. If a system's base is rigid or inadequate, it can be impossible to reconfigure to

adjust to changing market conditions. Along the same lines, according to theJournal of Business


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Forecasting Methods & Systems, it's important to invest in systems that will remain useful over

the long term, weathering alterations in the business climate.

One of the distinguishing characteristics of forecasting systems is the mathematical algorithmsthey use to take various factors into account. For example, most forecasting systems arrange

relevant data into hierarchies, such as a consumer hierarchy, a supply hierarchy, a geographyhierarchy, and so on. To return a useful forecast, the system can't simply allocate down each

hierarchy separately, but must account for the ways in which those dimensions interact with each

other. Moreover, the degree of this interaction varies according to the type of business in which acompany is engaged. Thus, businesses need to fine-tune their allocation algorithms in order to

receive useful forecasts.

According to theJournal of Business Forecasting Methods & Systems, there are three models ofbusiness forecasting systems. In the time-series model, data simply is projected forward based on

an established methodof which there are several, including the moving average, the simpleaverage, exponential smoothing, decomposition, and Box-Jenkins. Each of these methods appliesvarious formulas to the same basic premise: data patterns from the recent past will continue more

or less unabated into the future. To conduct a forecast using the time-series model, one need only

plug available historical data into the formulas established by one or more of the above methods.Obviously, the time-series model is the most useful means for forecasting when the relevant

historical data reveals smooth and stable patterns. Where jumps and anomalies do occur, the

time-series model may still be useful, providing those jumps can be accounted for.

The second forecasting model is cause-and-effect. In this model, one assumes a cause, or driver

of activity, that determines an outcome. For instance, a company may assume that, for a

particular data set, the cause is an investment in information technology, and the effect is sales.This model requires the historical data not only of the factor with which one is concerned (in this

case, sales), but also of that factor's determined cause (here, information technology

expenditures). It is assumed, of course, that the cause-and-effect relationship is relatively stableand easily quantifiable.

The third primary forecasting model is known as the judgmental model. In this case, oneattempts to produce a forecast where there is no useful historical data. A company might choose

to use the judgmental model when it attempts to project sales for a brand new product, or when

market conditions have qualitatively changed, rendering previous data obsolete. In addition,

according to theJournal of Business Forecasting Methods & Systems, this model is useful whenthe bulk of sales derives only from a relative handful of customers. To proceed in the absence of

historical data, alternative data is collected by way of experts in the field, prospective customers,

trade groups, business partners, or any other relevant source of information.

Business forecasting systems often work hand-in-hand with supply chain management systems.

In such systems, all partners in the supply chain can electronically oversee all movement ofcomponents within that supply chain and gear the chain toward maximum efficiency. The

Internet has proven to be a panacea in this field, and business forecasting systems allow partners


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to project the optimal flow of components into the future so that companies can try to meet

optimal levels rather than continually catch up to them.

In integrated supply chain networks, for instance, a single company in the supply chain can enter

slight changes in their own production or purchasing schedules for all parties to see, and the

forecasting system immediately processes the effects of those changes through the entire supplychain, allowing each company to adjust their own schedules accordingly. With business

relationships and supply chains growing increasingly complexparticularly in the world of e-commerce, with heavy reliance on logistics outsourcing and just-in-time deliverysuch

forecasting systems become crucial for companies and networks to remain efficient.

Financial Forecasting Methods:-

There are a number of different methods by which a business forecast can be made. All

the methods fall into one of two overarching approaches: qualitativeandquantitative.

Qualitative Models:-Qualitative models have generally been successful with short-term predictions, where the

scope of the forecast is limited. Qualitative forecasts can be thought of as expert-driven,

in that they depend on market mavens or the market as a whole to weigh in with an

informed consensus. Qualitative models can be useful in predicting the short-term

success of companies, products and services, but meets limitations due to its reliance on

opinion over measurable data. Qualitative models include:-

Market Research Polling a large number of people on a specific product or service topredict how many people will buy or use it once launched.

Delphi Method: Asking field experts for general opinions and then compiling them intoa forecast. (For more on qualitative modeling, read Qualitative Analysis: What Makes ACompany Great?)

In a Delphi study the experts are all consulted separately to avoid some of the bias that might

result were they all brought together, e.g. domination by a strong willed individual, divergent

(but valid) views not being expressed for fear of humiliation.


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A typical question might be "In what year (if ever) do you expect automated rapid transit to have

become common in major cities in Europe?". The answers are assembled in the form of adistribution of years, with comments attached, and recirculated to provide revised estimates. This

process is repeated until a consensus view emerges. Plainly such a method has many deficiencies

but on the other hand is there a better way of getting a view of the future if we lack the relevant

data (numbers) which would be needed if we were to apply some of the more quantitativetechniques?

As an example of this there was a Delphi study published in Science Journal in October 1967

which tried to look forward into the future (now, of course, we are many years past 1967 so we

can see how well they forecast). Many questions were asked as to when something might happenand a selection of these questions are given below. For each question we give the upper quartile

answer, the time by which 75% of the experts believed something would have happened.

Automated rapid transit, upper quartile answer 1985, i.e. 75% of the experts asked in1967 thought that by 1985 there would be widespread automated rapid transit in most

urban areas, tell that to anyone who lives in London! Widespread use of sophisticated teaching machines, upper quartile answer 1990, i.e. 75%

of the experts asked in 1967 thought that by 1990 there would be widespread use of

sophisticated teaching machines, tell that to anyone who works in a UK school/university

Widespread use of robot services, upper quartile answer 1995, i.e. 75% of the expertsasked in 1967 thought that by 1995 there would be widespread use of robot services

It is clear that these forecasts, at least, were very inaccurate. Indeed looking over the full set offorecasts many of the 25 forecasts made (about all aspects of life/society in the future after 1967)

were wildly inaccurate.

This brings us to our first key point, we are interested in the difference between theoriginal forecast and the final outcome, i.e. in forecast error.

However, back in 1967 when this Delphi study was done, what other alternative approach did we

have if we wished to answer these questions?

In many respects the issue we need address with regard to forecasting is not whether a particular

method gives good (accurate) forecasts but whether it is the best available method - if it is then

what choice do we have about using it?

This brings us to our second key point, we need to use the most appropriate (best)

forecasting method, even if we know that (historically) it does not give accurate forecasts.


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Quantitative Models:-Quantitative models discount the expert factor and try to take the human element out ofthe analysis. These approaches are concerned solely with data and avoid the fickleness of

the people underlying the numbers. They also try to predict where variables like sales,

gross domestic product, housing prices and so on, will be in the long-term, measured inmonths or years. Quantitative models include:

The Indicator Approach: The indicator approach depends on the relationship betweencertain indicators, for example GDP and unemployment rates, remaining relatively

unchanged over time. By following the relationships and then following indicators that

are leading, you can estimate the performance of the lagging indicators, by using theleading indicator data.

Econometric Modeling: This is a more mathematically rigorous version of the indicatorapproach. Instead of assuming that relationships stay the same, econometric modelingtests the internal consistency of data sets over time and the significance or strength of the

relationship between data sets. Econometric modeling is sometimes used to create custom

indicators that can be used for a more accurate indicator approach. However, theeconometric models are more often used in academic fields to evaluate economic

policies. (For a basic explanation on applying econometric models, read Regression

Basics For Business Analysis.)

Regression methods:-You have probably already met linear regression where a straight line of the form Y = a +

bX is fitted to data. It is possible to extend the method to deal with more than one

independent variable X. Suppose we have k independent variables X1, X2, ..., Xkthen we

can fit the regression line

Y = a + b1X1 + b2X2 + ... + bkXk

This extension to the basic linear regression technique is known as multiple regression.

Plainly knowing the regression line enables us to forecast Y given values for the

Xi i=1,2,...,k.


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Multiple equation methods:-Methods of this type are frequently used in economic modelling (econometrics) where there are

many dependent variables that interact with each other via a series of equations, theform of

which is given by economic theory. This is an important point. Economic theory gives us some

insight into the basic structural relationships between variables. The precise numeric relationshipbetween variables must often be deduced by examining data.

As an example consider the following simple model, let:

X = personal income Y = personal spending I = personal investment r = interest rate

From economic theory suppose that we have

Y = a1 + b1(X-a1) (spending a linear function of disposable income)

I = a2 + b2r (investment linearly related to the interest rate)

and the balancing equation

X = Y + I (income = spending + investment)

where a1,a2,b1,b2 are constants.

Here we have 3 equations in 4 variables (X,Y,I,r) and so to solve these equations one of the

variables must be given a value. The variable so chosen is known as an exogenous variablebecause its value is determined outside the system of equations whilst the remaining variables

are called endogenous variables as their values are determined within the system of equations,

e.g. in our model we might regard the interest rate r as the exogenous variable and be interestedin how X, Y and I change as we alter r.

Usually the constants a1,a2,b1,b2 are not known exactly and must be estimated from data (a

complex procedure). Note too that these constants will probably be different for different groups

of people, e.g. urban/rural, men/women, single/married, etc.

An example of an econometric model of this type is the UK Treasury model of the economy

which contains many variables (each with a time subscript), complicated equations, and is usedto look at the effect of interest rate changes, tax changes, oil price movements, etc.

For example the UK Treasury equation [New Scientist, 31st October 1993] to predict consumer

spending looks like:

DlogeCt = -0.018 + 0.0623DDlogeUt - 0.00448logeCt-1 + 0.004256logeYt-1 +0.0014336loge[(NFWt-1 + GPWt-1)/(Pt-1Yt-1)] + etc


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The purpose of the analysis is to discern some relationship between the Y t values observed so farin order to enable us to forecast future Yt values. We shall deal with two techniques for time

series analysis in detail and briefly mention a more sophisticated method.

1. A problem or data point is chosen. This can be something like "will people buy a high-end coffee maker?" or "what will our sales be in March next year?"

2. Theoretical variables and an ideal data set are chosen. This is where the forecasteridentifies the relevant variables that need to be considered and decides how to collect thedata.

3. Assumption time. To cut down the time and data needed to make a forecast, the forecastermakes some explicit assumptions to simplify the process.


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4. A model is chosen. The forecaster picks the model that fits the data set, selected variables

and assumptions.

5. Analysis. Using the model, the data is analyzed and a forecast made from the analysis.

6. Verification. The forecaster compares the forecast to what actually happens to tweak theprocess, identify problems or in the rare case of an absolutely accurate forecast, pat himself

on the back.

Moving average:-One, very simple, method for time series forecasting is to take a moving average (also known as

weighted moving average).

The moving average (mt) over the last L periods ending in period t is calculated by taking the

average of the values for the periods t-L+1, t-L+2, t-L+3, ..., t-1, t so that

mt = [Yt-L+1 + Yt-L+2 + Yt-L+3 + ... + Yt-1 + Yt]/L

To forecast using the moving average we say that the forecast for all periods beyond t is just

mt (although we usually only forecast for one period ahead, updating the moving average as the

actual observation for that period becomes available).

Consider the following example: the demand for a product for 6 months is shown below -calculate the three month moving average for each month and forecast the demand for month 7.

Month 1 2 3 4 5 6

Demand (100's) 42 41 43 38 35 37

Now we cannot calculate a three month moving average until we have at least 3 observations -

i.e. it is only possible to calculate such an average from month 3 onward. The moving averagefor month 3 is given by:

m3 = (42 + 41 + 43)/3 = 42

and the moving average for the other months is given by:


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m4 = (41 + 43 + 38)/3 = 40.7

m5 = (43 + 38 + 35)/3 = 38.7m6 = (38 + 35 + 37)/3 = 36.7

We use m6 as the forecast for month 7. Hence the demand forecast for month 7 is 3670 units.

The package input for this problem is shown below.

The output from the package for a three month moving average is shown below.


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NEED OF FORECASTING :-

Forecasting is the process by which companies ponder and prepare for the future. It involves

predicting the future outcome of various business decisions. This includes the future of thebusiness as a whole, the future of an existing or proposed product or product line, and the future

of the industry in which the business operates, to name a few. Forecasting is used to answer

important questions, such as:

How much profit will the business make?

How much demand will there be for a product or service?

How much will it cost to produce the product or offer the service?

How much money will the company need to borrow?

When and how will borrowed funds be repaid?

Businesses must understand and use forecasting in order to answer these important questions.

This helps the company prepare for the future. It also helps the organization make plans that will

lead to becoming a financially successful business. This article will discuss the reasons financial

forecasting is important to an organization.

Why is Forecasting Important to an Organization?Financial forecasting is important for several reasons. First, it enables management to change

operations at the right time in order to reap the greatest benefit. It also helps the company prevent

losses by making the proper decisions based on relevant information. Organizations that can

create high quality and accurate forecasts are able to "see what interventions are required to meet

their business performance targets" (Vadasz).

Forecasting is also important when it comes to developing new products or new product lines. It

helps management decide whether the product or product line will be successful. Forecasting

prevents the company from spending time and money developing, manufacturing, and marketing

a product that will fail.


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Stockholder expectations highlight another reason behind the importance of forecasting. Public

companies experience scrutiny and pressure for short-term performance from investors.

Operational results will be examined by investors and investment analysts, and actual results that

differ from forecasts will be bad for the company and its stock price. This is because both

meeting predictions and exceeding predictions will reduce investor confidence. This will causeinvestors to believe that the company does not understand its own business model.

Conclusion

The elements of doing business are constantly changing. Interest rates rise and fall, customer

preferences change, suppliers go out of business, and the list goes on and on. As the factors that

affect business change, the company forecasts must also change. In this time of rapid change,

accurate and timely forecasts have become even more important. Recognizing this is the first

step in becoming a successful organization.

Why do we forecast, anyway?The answer, is that nearly every function within a company needs an estimate of what the futurewill look like as a foundation to build their current plans. For example:

Marketing people need forecasts to estimate the expected demand that will be generated fromtheir activities; to anticipate which customer segments will be attracted, where products are in

their life cycle, and actions needed to be taken and their expected impact.

Salespeople use forecasts to understand what volumes are expected in each period and to

evaluate the sales pipeline activity to make sure that it supports those expectations.

Managers in the supply chain use forecasts to make timely purchasing requests, developproduction plans, evaluate capacity needs, and develop logistical plans.

Finance professionals use forecasts to prepare and understand financial plans. They also use

them to report to Wall Street about their earnings expectations.

It is difficult to quantify the benefits of forecasting; however, in the supply chain, there are

certain components that can be quantified, showing the benefits of forecasting, the columncontinues. One way to measure the benefits of forecasting is to see how much would have been


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lost if the forecast was not accurate. Another way to measure them is how much would have

been gained (or saved) with improved forecasts.

Yes, these are basic, fundamental points. They are also valuable points to cover as you rethink

your companys forecasting approach. As you do, I encourage you to hold up fundamentalaspects of your forecasting approach and ask some very basic questions:

Why do we do this?

What value does it provide?

Can we do this more effectively?

THE FORECASTING PROCESS:-

When beginning the forecasting process, there are typical steps that must be followed. These

steps follow an acceptable decision-making process that includes the following elements:

1. Identification of the problem. Forecasters must identify what is going to be forecasted, orwhat is of primary concern. There must be a timeline attached to the forecasting period. This

will help the forecasters to determine the methods to be used later.

2. Theoretical considerations. It is necessary to determine what forecasting has been done inthe past using the same variables and how relevant these data are to the problem that iscurrently under consideration. It must also be determined what economic theory has to say

about the variables that might influence the forecast.

3. Data concerns. How easy will it be to collect the data needed to be able to make theforecasts is a significant issue.

4. Determination of the assumption set. The forecaster must identify the assumptions that willbe made about the data and the process.

5. Modeling methodology. After careful examination of the problem, the types of models mostappropriate for the problem must be determined.

6. Preparation of the forecast. This is the analysis part of the process. After the model to beused is determined, the analysis can begin and the forecast can be prepared.

7. Forecast verification. Once the forecasts have been made, the analyst must determinewhether they are reasonable and how they can be compared against the actual behavior of

the data.


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ANSWER 2:-

WHAT IS A TIME SERIES?A time series is a collection of observations of well-defined data items obtained through

repeated measurements over time. For example, measuring the value of retail sales each

month of the year would comprise a time series. This is because sales revenue is well

defined, and consistently measured at equally spaced intervals. Data collected irregularlyor only once are not time series.

An observed time series can be decomposed into three components: the trend (long term

direction), the seasonal (systematic, calendar related movements) andthe irregular (unsystematic, short term fluctuations).

Many types of data are collected over time. Stock prices, sales volumes, interest rates, and

quality measurements are typical examples. Because of the sequential nature of the data, special

statistical techniques that account for the dynamic nature of the data are required.

STATGRAPHICS Centurion provides an extensive set of procedures designed for analyzingtime series data:

1.Descriptive Methods- time sequence plots, autocorrelation functions, partial autocorrelation

functions, periodograms, and cross-correlation functions are all important tools for characterizingtime series data.

2. Smoothing- a variety of smoothers are available to estimate the underlying trend in a timeseries.

3. Seasonal Decomposition- decomposes time series data into trend, cycle, seasonal, andirregular components, and returns seasonally adjusted data if desired.

4.Forecasting- creation of forecasts beyond the end of the data, using trend models, moving

averages, exponential smoothers, or ARIMA models.

5.Automatic Forecasting- selects the best forecasting method for a time series by optimizing a

specified information criterion.


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Descriptive Methods:-Characterizing a time series involves estimating not only a mean and standard deviation but also

the correlations between observations separated in time. Tools such as the autocorrelationfunction are important for displaying the manner in which the past continues to affect the future.

Other tools, such as the periodogram, are useful when the data contain oscillations at specificfrequencies.


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SmoothingWhen a time series contains a large amount of noise, it can be difficult to visualize any

underlying trend. Various linear and nonlinear smoothers are provided to separate the signalfrom the noise.


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Seasonal DecompositionWhen the data contain a strong seasonal effect, it is often important to separate the seasonality

from the other components in the time series. This enables one to estimate the seasonal patternsand to generate seasonally adjusted data.


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Forecasting:-A common goal of time series analysis is extrapolating past behavior into the future. The

STATGRAPHICS forecasting procedures include random walks, moving averages, trendmodels, simple, linear, quadratic, and seasonal exponential smoothing, and ARIMA parametric

time series models. Users may compare various models by withholding samples at the end of thetime series for validation purposes.


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Automatic ForecastingIf desired, users may elect to let STATGRAPHICS select a forecasting model for them by

comparing multiple models and automatically picking the model that maximizes a specifiedinformation criterion. The available criteria are based on the mean squared forecast error,

penalized for the number of model parameters that must be estimated from the data. A commonuse of this procedure in Six Sigma is to select an ARIMA model on which to base an ARIMAcontrol chart, which unlike most control charts does not assume independence between

successive measurements. In such cases, the analyst may elect to consider only models of the

ARMA(p,p-1) form, which theory suggests can characterize many dynamic processes.

LEAST SQUARE METHOD:-

A statistical technique to determine the line of best fit for a model. The least squaresmethod is specified by an equation with certain parameters to observed data. This method

is extensively used in regression analysis and estimation.

In the most common application - linear or ordinary least squares - a straight line issought to be fitted through a number of points to minimize the sum of the squares of the

distances (hence the name "least squares") from the points to this line of best fit.

In contrast to a linear problem, a non-linear least squares problem has no closed solution

and is generally solved by iteration. The earliest description of the least squares methodwas by Carl Freidrich Gauss in 1795.

A line of best fit is a straight line that is the best approximation of the given set of data. It is used to study the nature of the relation between two variables. A line of best fit can be roughly determined using an eyeball method by drawing a

straight line on a scatter plot so that the number of points above the line and below the

line is about equal (and the line passes through as many points as possible).

A more accurate way of finding the line of best fit is the least square method .
http://hotmath.com/hotmath_help/topics/scatter-plots.htmlhttp://hotmath.com/hotmath_help/topics/scatter-plots.html


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Use the following steps to find the equation of line of best fit for a set ofordered pairs.Step 1: Calculate the mean of thex-values and the mean of they-values.

Step 2: Compute the sum of the squares of thex-values.

Step 3: Compute the sum of eachx-value multiplied by its correspondingy-value.Step 4: Calculate the slope of the line using the formula:

where n is the total number of data points.

Step 5: Compute they-intercept of the line by using the formula:

where are the mean of thex- andy-coordinates of the data points respectively.

Step 6: Use the slope and they -intercept to form the equation of the line.
http://hotmath.com/hotmath_help/topics/ordered-pair.htmlhttp://hotmath.com/hotmath_help/topics/slope.htmlhttp://hotmath.com/hotmath_help/topics/y-intercepts.htmlhttp://hotmath.com/hotmath_help/topics/y-intercepts.htmlhttp://hotmath.com/hotmath_help/topics/y-intercepts.htmlhttp://hotmath.com/hotmath_help/topics/y-intercepts.htmlhttp://hotmath.com/hotmath_help/topics/slope.htmlhttp://hotmath.com/hotmath_help/topics/ordered-pair.html


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Example:Use the least square method to determine the equation of line of best fit for the data. Then plot

the line.

Solution:Plot the points on a coordinate plane.

Calculate the means of thex-values and they-values, the sum of squares of thex-values, and the

sum of eachx-value multiplied by its correspondingy-value.
http://hotmath.com/hotmath_help/topics/cartesian-plane.htmlhttp://hotmath.com/hotmath_help/topics/cartesian-plane.html


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Calculate the slope.

Calculate they-intercept.

First, calculate the mean of thex-values and that of they-values.

Use the formula to compute they-intercept.


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Use the slope andy-intercept to form the equation of the line of best fit.

The slope of the line is1.1 and they -intercept is 14.0.

Therefore, the equation isy =1.1x + 14.0.

Draw the line on the scatter plot.

Curve Fitting, Regression

Field data is often accompanied by noise. Even though all control parameters (independent

variables) remain constant, the resultant outcomes (dependent variables) vary. A process of

quantitatively estimating the trend of the outcomes, also known as regression or curve fitting,

therefore becomes necessary.

The curve fitting process fits equations of approximating curves to the raw field data.

Nevertheless, for a given set of data, the fitting curves of a given type are generallyNOT unique.Thus, a curve with a minimal deviation from all data points is desired. This best-fitting curve can

be obtained by the method of least squares.


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The Method of Least Squares

The method of least squares assumes that the best-fit curve of a given type is the curve that has

the minimal sum of the deviations squared (least square error) from a given set of data.

Suppose that the data points are , , ..., where is the independent

variable and is the dependent variable. The fitting curve has the deviation

(error) from each data point, i.e., , , ..., .

According to the method of least squares, the best fitting curve has the property that:

Polynomials Least-Squares Fitting

Polynomials are one of the most commonly used types of curves in regression. The applications

of the method of least squares curve fitting using polynomials are briefly discussed as follows.

To obtain further information on a particular curve fitting, please click on the link at the end of

each item. Or try the calculator on the right

The Least-Squares Line: The least-squares line method uses a straightline to approximate the given set of data, , , ..., ,

where .

The Least-Squares Parabola: The least-squares parabola method uses a second degreecurve to approximate the given set of data, , ,

..., , where .

The Least-Squaresmth Degree Polynomials: The least-squares mth degree Polynomialsmethod uses m

thdegree polynomials to approximate

the given set of data, , , ..., , where .

Multiple Regression Least-Squares: Multiple regression estimates the outcomes whichmay be affected by more than one control parameter or there may be more than one

control parameter being changed at the same time, e.g., .
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