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A regression analysis overview
1. The relationship between the variables
Deterministic phenomenon (function), for example, a rectangular perimeter
Non-deterministic phenomena (statistical correlation), for example, height and weight
2. correlation and regression analysis
Correlation: Correlation and relevance of two (or more) variables (expressed using the correlation coefficient)
Regression analysis: already have relationships with, solving its causality, variable unequal status (a result of a State), according to changes in the independent variables can predict the law of motion.
For chestnut:
(1) playing basketball taller.
No, the reality is that taller people chose to play basketball, belongs to reverse causation.
(2) high social status of people living longer.
No, high social status by better health care, leading to better health and long life of some.
Tips : causality premise: chronological.
3, linear or nonlinear correlation analysis (Note: if the correlation forming correlation will be 0)
Linear correlation:
Two variables: covariance, correlation coefficient
A plurality of variables: computing partial correlation coefficients, multiple correlation coefficient
Second, the population regression function (PRF)
Under the conditions given the explanatory variables X, Y is the dependent variable desired trajectory generally called regression curve, the corresponding function
E(Y|X)=f(X)
F simplest form is a linear function. Which intercept and slope of the linear regression coefficients, as shown in the following expression, where beta] 0 representative of the spontaneous consumption, beta] . 1 Representative marginal consumption trend.
E (Y | X) = b 0 + b 1 X
Identifying: the dependent variable Y is the dependent variable, the variable is predicted, sub regression, response variable, independent variable X is the explanatory variables predictor variables, regressors, the control variable.
Third, the random error term
Y will be a real minus its mean, for the deviation:
μ = YE (Y | X)
then
Y = E (Y | X) + μ
Interprets variables consists of two parts, after determining the portion of a given X, since another portion of the random factors involved in, and the fluctuation reaction itself.
Random error term meaning:
1, unknown factors
2, incomplete data
3, a number of small factors
4, the observation data error
5, the model setting error
6, the inherent randomness
Fourth, the sample regression function
We must first understand a basic fact: in general is always unknown . Population parameter values, variance, etc. are not informed of the purpose of metrology is: the overall sample inference .
Scatter from the sample points fit a straight line, the regression line used to approximate the overall replaced.
For example, a linear regression:
Note symbol estimates (remember to wear a hat)
Key distinction:
** The overall regression function PRF:
E (Y) = b 0 + b 1 X i (i = 1,2 ,,,, n)
PRF random form:
E (Y) = b 0 + b 1 X i + m i
** Regression Function SRF:
SRF random form:
The aggregate data is a regression, a post-normalization based on the sample data, the regression results plus their random error, the true value can be obtained.
Fifth, the model assumes
Suppose a: Errors in the model no setting (correct variable, the correct functional form).
Hypothesis 2: X is a variable deterministic (non-random variables).
Suppose three: X takes at least two or more different values, the sample variance and converged (If X is a constant value, the significance of the problem would change) (as sample size increases, the convergence of the variance, which is a time series in order to avoid the spurious regression problem)
Hypothesis 4 : error term "zero mean with variance and zero covariance." (On the whole, we expect the overall mean error is 0) (same population variance, guarantee the same degree of variation on each point) (for two different sample points, covariance is zero, the correlation coefficient is 0, sent two sample point information related to possible)
Five assumptions: a random error term associated explanatory variable not (can be introduced from the front)
Assume Six: random error term normally distributed (with mean 0 and variance [sigma] 2 ) (optional, not this assumption, it is still automatically and then set up)
Sixth, parameter estimation
1. The ordinary least squares parameter estimation (the OLS) (commonly used)
Principle: The overall error is minimized → least squares idea that the minimum residual sum of squares, this problem will become optimized asked.
result
or
Classification: estimator: The function of the random variable
Estimated value: specific values
2. The maximum likelihood (ML) (ML when using unsuitable OLS)
Likelihood: refers to the probability or a possibility
Principle: there is reasonable (when you observe a set of samples, must be the mechanism behind the decision that it can appear in front of you). ---- maximum probability of maximum likelihood method; total least squares method is the smallest of errors.
3. Parameter Estimation moment method (MM)
This method uses less temporarily explain.
A regression analysis overview
1. The relationship between the variables
Deterministic phenomenon (function), for example, a rectangular perimeter
Non-deterministic phenomena (statistical correlation), for example, height and weight
2. correlation and regression analysis
Correlation: Correlation and relevance of two (or more) variables (expressed using the correlation coefficient)
Regression analysis: already have relationships with, solving its causality, variable unequal status (a result of a State), according to changes in the independent variables can predict the law of motion.
For chestnut:
(1) playing basketball taller.
No, the reality is that taller people chose to play basketball, belongs to reverse causation.
(2) high social status of people living longer.
No, high social status by better health care, leading to better health and long life of some.
Tips : causality premise: chronological.
3, linear or nonlinear correlation analysis (Note: if the correlation forming correlation will be 0)
Linear correlation:
Two variables: covariance, correlation coefficient
A plurality of variables: computing partial correlation coefficients, multiple correlation coefficient
Second, the population regression function (PRF)
Under the conditions given the explanatory variables X, Y is the dependent variable desired trajectory generally called regression curve, the corresponding function
E(Y|X)=f(X)
F simplest form is a linear function. Which intercept and slope of the linear regression coefficients, as shown in the following expression, where beta] 0 representative of the spontaneous consumption, beta] . 1 Representative marginal consumption trend.
E (Y | X) = b 0 + b 1 X
Identifying: the dependent variable Y is the dependent variable, the variable is predicted, sub regression, response variable, independent variable X is the explanatory variables predictor variables, regressors, the control variable.
Third, the random error term
Y will be a real minus its mean, for the deviation:
μ = YE (Y | X)
then
Y = E (Y | X) + μ
Interprets variables consists of two parts, after determining the portion of a given X, since another portion of the random factors involved in, and the fluctuation reaction itself.
Random error term meaning:
1, unknown factors
2, incomplete data
3, a number of small factors
4, the observation data error
5, the model setting error
6, the inherent randomness
Fourth, the sample regression function
We must first understand a basic fact: in general is always unknown . Population parameter values, variance, etc. are not informed of the purpose of metrology is: the overall sample inference .
Scatter from the sample points fit a straight line, the regression line used to approximate the overall replaced.
For example, a linear regression:
Note symbol estimates (remember to wear a hat)
Key distinction:
** The overall regression function PRF:
E (Y) = b 0 + b 1 X i (i = 1,2 ,,,, n)
PRF random form:
E (Y) = b 0 + b 1 X i + m i
** Regression Function SRF:
SRF random form:
The aggregate data is a regression, a post-normalization based on the sample data, the regression results plus their random error, the true value can be obtained.
Fifth, the model assumes
Suppose a: Errors in the model no setting (correct variable, the correct functional form).
Hypothesis 2: X is a variable deterministic (non-random variables).
Suppose three: X takes at least two or more different values, the sample variance and converged (If X is a constant value, the significance of the problem would change) (as sample size increases, the convergence of the variance, which is a time series in order to avoid the spurious regression problem)
Hypothesis 4 : error term "zero mean with variance and zero covariance." (On the whole, we expect the overall mean error is 0) (same population variance, guarantee the same degree of variation on each point) (for two different sample points, covariance is zero, the correlation coefficient is 0, sent two sample point information related to possible)
Five assumptions: a random error term associated explanatory variable not (can be introduced from the front)
Assume Six: random error term normally distributed (with mean 0 and variance [sigma] 2 ) (optional, not this assumption, it is still automatically and then set up)
Sixth, parameter estimation
1. The ordinary least squares parameter estimation (the OLS) (commonly used)
Principle: The overall error is minimized → least squares idea that the minimum residual sum of squares, this problem will become optimized asked.
result
or
Classification: estimator: The function of the random variable
Estimated value: specific values
2. The maximum likelihood (ML) (ML when using unsuitable OLS)
Likelihood: refers to the probability or a possibility
Principle: there is reasonable (when you observe a set of samples, must be the mechanism behind the decision that it can appear in front of you). ---- maximum probability of maximum likelihood method; total least squares method is the smallest of errors.
3. Parameter Estimation moment method (MM)
This method uses less temporarily explain.