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JAYKANTDIVYA &RAHUL

CORRELATION REGRESSION & TIME SERIES ANALYSIS

CORRELATION

• A Statistical technique that is used to analyse the strength and direction of the relationship between two quantitative variable is called Correlational analysis.

• Two variables are said to be in correlation if the change in one of the variable results in a change in other variable.

E g :- 1) Frequency of smoking and lungs damage , 2) Sales revenue and expenses incurred on advertising.

Importance of correlation

• If variables are linearly related to each other then it helps in estimation of one from the other.– Advertisement and sales– Prices and Demand

• We use Regression Analysis to find the value of one variable from the other

TYPES OF CORRELATION

• POSITIVE AND NEGATIVE

• LINEAR AND NON-LINEAR• SIMPLE ,PARTIAL AND MULTIPLE

POSITIVE CORRELATION ANDNEGATIVE CORRELATION

POSITIVE CORRELATION NEGATIVE CORRELATION

• If the variables vary in same direction, correlation is said to be POSITIVE.

– If one variable increases, the other also increases on the other hand, if one variable decreases, the other also decreases.

• If both variables vary in the opposite direction, correlation is said to be NEGATIVE.

– If one variable increases and the other decreases, or one decreases the other increases.

LINEAR CORRELATION NON-LINEAR CORRELATION

LINEAR CORRELATION

• If the extent of change in one variable tends to have a constant ratio in the extent of change in another variable, then the correlation is said to be LINEAR.

NON-LINEAR CORRELATION

• If the extent of change in one variable tends to have no consistent ratio in the extent of change in another variable, then the correlation is said to be NON-LINEAR.

SIMPLE,PARTIAL AND MULTIPLE CORRELATION

• When only two variables are involved, it is simple correlation

• When three or more than three variables are involved, we can compute either partial or multiple correlation

Methods of correlation

graphic

Scatter diagram

algebraic

1. Karl pearson2. Rank method

Scatter Diagram

Scatter diagram is a graph or chart which helps to determine whether there is a relationship between two variables by examining the graph of the observed data.

A scattered diagram can give us two types of information:

• Pattern that indicate that the variables are related.• If the variables are related,what kind of line or

estimating equation,describes this relationship.

KARL’S PEARSONSCOEFFIENT OF CORRELATION

• Karl Pearson’s Coefficient of Correlation denoted by- ‘r’ The coefficient of correlation ‘r’ measure the degree of linear relationship between two variables say x & y.

r = N Σdxdy - Σdx Σdy √N Σdx²-(Σdx)²√N Σdy²-(Σdy)²

The value of correlation coefficient ‘r’ ranges from -1 to +1 If r = +1, then the correlation between the two variables is said to be perfect and positive If r = -1, then the correlation between the two variables is said to be perfect and negative If r = 0, then there exists no correlation between the variables

Interpretation of Correlation Coefficient (r)

REGRESSION• The statistical technique that express the

relationship between two or more variables in the form of an equation to estimate the value of a variable, based on the given value of another variable is called regression analysis.

eg :- Profit after Sales of a firm.

Difference between dependent variable and independent variable

Independent Variable

1. The known variable is called the independent variable.

2. What we typically call “X”.3. Variable that is controlled or

manipulated.4. It is plotted on horizontal axis.

5. An input variable.

Dependent Variable

1. The variable we are trying to predict is the dependent variable.

2. What we typically call ”Y”.3. Variable that cannot be

controlled or manipulated.4. It is plotted on vertical axis.5. An output variable.

Difference between Regression and Correlation

Regression

• A statistical method used to describe the nature of relationship.

• In linear regression analysis one variable is considered as dependent variable and other as independent variable

Correlation

• A statistical method used to determine whether a relationship between two or more variables exist.

• In correlation analysis we examine the degree of association between two variables

Advantages of Regression Analysis

• It helps in developing a regression equation by which the value of a dependent variable can be estimated given a value of an independent variable.

• It helps to determine standard error of estimate to measure the variability or spread of values of a dependent variable with respect to the regression line.

Estimation using the Regression Line

• The equation for a straight line where the dependent variable Y is determined by the independent variable X is:

Y = a + bxWhere,

a = y-intercept b = slope of the line Y = value of dependent variable X = value of independent variable

THE METHOD OF LEAST SQUARE

• It is a method of having a “good fit” of a line which minimizes the error between the estimated points on the line and actual points that were used to draw it.

• In this method Y represents the individual value of the observed points measured along the Y-axis and Y(y-hat) symbolize the individual values of the estimated points.

• The Estimated Line is: = a + bx

b = ∑XY - ∑X2 –n 2

a = -b

Where, a = Y-intercept b = slope of the best-fitting estimating line. X = value of independent variable Y = value of dependent variable = mean of the values of the independent variable = mean of the values of the dependent variable

x

xy

xn y

y x

COEFFICIENT OF DETERMINATION

• The convenient way of interpreting the value of correlation coefficient is to use of square of coefficient of correlation which is called Coefficient of Determination.

• The Coefficient of Determination is r2.

r2= 1- ∑(Y- )2

∑(Y-YH )2

STANDARD ERROR OF ESTIMATE

Standard error of estimate measures the variability of the scatter of the observed values around the regression line.

It is given by: Se= ∑(Y- )2

n-2If Se=0, the estimating equation is expected to be a “perfect” estimator of the dependent variable.

WHAT DOES TIME-SERIES MEAN?

• A time series is a sequence of data points, measured typically at successive points in time spaced at uniform time intervals.

• Time series is a set of measurements of a variable that are ordered through time

• Time series analysis comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data

DIFFERENCE WITH REGRESSION ANALYSIS

• Time –series Analysis • Regression Analysis

Time series forecasting is the use of a model to predict future values based on

previously observed values.

Regression analysis is often employed in such a way as to test theories that the current value of one time series affects the current value of another time series.

Regression analysis cannot explain seasonal and cyclical effects.

It shows or suggests

periodicity of a data like seasonal and cyclical effects.

COMPONENTS OF TIME SERIES

• SECULAR TREND• CYCLICAL VARIATIONS• SEASONAL VARIATIONS• IRREGULAR VARIATIONS

SECULAR TRENDA time-series which displays a steady tendency of either upward or downward movement in the average (or mean) value of the forecast variable (let us say ‘y’)over a long period of time is called “Trend”.

If we talk about commodities, Secular Trend is affected by prices, productions and sales of the commodity as well as the population of the area. Examples-1. We find that over the last few years the sales of Laptop in

Ranchi has increased. so, we can say that the sales of Laptop is showing an “ Upward Trend”.

2. Use of Landline Phone has decreased over the last few years. This shows the “Declining Trend” of using Landline Phone.

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Upward trend of sales of Laptops in Ranchi

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CYCLICAL VARIATION Cyclical variations are long-term movements that

represent consistently recurring rises and declines in activity.

Timing is the most important factor which affect the Cyclical Variations.

for example- Business Cycle, it consists of the recurrence ofthe up and down movements of business activity

depression

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recession

recession

Prosperity or boom

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Cyclical Variation(Business cycle)

SEASONAL VARIATION

Seasonal variations are those periodic movements in business activity which occur regularly every year. Since these variations repeat during a period of twelve months

so, they can be predicted fairly accurately.

Seasonal Variations are caused by climate and weather conditions, customs, festivals and habits.

for example-Sales of Cold-drinks goes up in summer season than any other season

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IRREGULAR VARIATION Irregular variations refer to such variations in businessactivity which do not repeat in a definite pattern.In these type of variations the pattern of the variable isunpredictable.Irregular Variations are caused by unpredictable factors like natural disasters (earthquakes, floods, wars etc.).These are unpredictable and no one has control over it.

For example-Production of cars tremendously went down after earthquake came in Japan in Nov 2011.

2005 2006 2007 2008 2009 2010 2011

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NEED OF TIME-SERIES ANALYSIS

Helpful in evaluating current accomplishments Actual performances can be compared with the expected performance and the cause of the variations analysed

Facilitates comparison. Different time-series can be compared and important conclusions can be drawn from this with the help of this we can take decisions

Thank you…