ISYE6501x - WEEK 3 HW

Question 7.1 Describe a situation or problem from your job, everyday life, current events, etc., for which exponential smoothing would be appropriate. What data would you need? Would you expect the value of α (the first smoothing parameter) to be closer to 0 or 1, and why?

Walmart can use exponential smoothing to help forecast their future sales of a variety of products for a store. The data needed would be the daily sales of the product in the store in the past years and the past months. I would expect the value of α to be closer to 1 since the recent demand for a product is likely to have more impact in forecasting the future sales than the randomness in the system.

Question 7.2 Using the 20 years of daily high temperature data for Atlanta (July through October) from Question 6.2 (file temps.txt), build and use an exponential smoothing model to help make a judgment of whether the unofficial end of summer has gotten later over the 20 years. (Part of the point of this assignment is for you to think about how you might use exponential smoothing to answer this question. Feel free to combine it with other models if you’d like to. There’s certainly more than one reasonable approach.)

Note: in R, you can use either HoltWinters (simpler to use) or the smooth package’s es function (harder to use,but more general).

If you use “es”,the Holt-Winters model uses model=”AAM” in the function call (the first and second constants are used “A”dditively, and the third (seasonality) is used “M”ultiplicatively; the documentation doesn’t make that clear).

Data_temps= read.csv("temps.txt",sep = "",check.names=FALSE)
str(Data_temps)

'data.frame':   123 obs. of  21 variables:
 $ DAY : Factor w/ 123 levels "1-Aug","1-Jul",..: 2 46 90 101 105 109 113 117 121 6 ...
 $ 1996: int  98 97 97 90 89 93 93 91 93 93 ...
 $ 1997: int  86 90 93 91 84 84 75 87 84 87 ...
 $ 1998: int  91 88 91 91 91 89 93 95 95 91 ...
 $ 1999: int  84 82 87 88 90 91 82 86 87 87 ...
 $ 2000: int  89 91 93 95 96 96 96 91 96 99 ...
 $ 2001: int  84 87 87 84 86 87 87 89 91 87 ...
 $ 2002: int  90 90 87 89 93 93 89 89 90 91 ...
 $ 2003: int  73 81 87 86 80 84 87 90 89 84 ...
 $ 2004: int  82 81 86 88 90 90 89 87 88 89 ...
 $ 2005: int  91 89 86 86 89 82 76 88 89 78 ...
 $ 2006: int  93 93 93 91 90 81 80 82 84 84 ...
 $ 2007: int  95 85 82 86 88 87 82 82 89 86 ...
 $ 2008: int  85 87 91 90 88 82 88 90 89 87 ...
 $ 2009: int  95 90 89 91 80 87 86 82 84 84 ...
 $ 2010: int  87 84 83 85 88 89 94 97 96 90 ...
 $ 2011: int  92 94 95 92 90 90 94 94 91 92 ...
 $ 2012: int  105 93 99 98 100 98 93 95 97 95 ...
 $ 2013: int  82 85 76 77 83 83 79 88 88 87 ...
 $ 2014: int  90 93 87 84 86 87 89 90 90 87 ...
 $ 2015: int  85 87 79 85 84 84 90 90 91 93 ...

print(summary(Data_temps))

      DAY           1996            1997            1998            1999      
 1-Aug  :  1   Min.   :60.00   Min.   :55.00   Min.   :63.00   Min.   :57.00  
 1-Jul  :  1   1st Qu.:79.00   1st Qu.:78.50   1st Qu.:79.50   1st Qu.:75.00  
 1-Oct  :  1   Median :84.00   Median :84.00   Median :86.00   Median :86.00  
 1-Sep  :  1   Mean   :83.72   Mean   :81.67   Mean   :84.26   Mean   :83.36  
 10-Aug :  1   3rd Qu.:90.00   3rd Qu.:88.50   3rd Qu.:89.00   3rd Qu.:91.00  
 10-Jul :  1   Max.   :99.00   Max.   :95.00   Max.   :95.00   Max.   :99.00  
 (Other):117                                                                  
      2000             2001            2002            2003      
 Min.   : 55.00   Min.   :51.00   Min.   :57.00   Min.   :57.00  
 1st Qu.: 77.00   1st Qu.:78.00   1st Qu.:78.00   1st Qu.:78.00  
 Median : 86.00   Median :84.00   Median :87.00   Median :84.00  
 Mean   : 84.03   Mean   :81.55   Mean   :83.59   Mean   :81.48  
 3rd Qu.: 91.00   3rd Qu.:87.00   3rd Qu.:91.00   3rd Qu.:87.00  
 Max.   :101.00   Max.   :93.00   Max.   :97.00   Max.   :91.00  
                                                                 
      2004            2005            2006            2007            2008      
 Min.   :62.00   Min.   :54.00   Min.   :53.00   Min.   : 59.0   Min.   :50.00  
 1st Qu.:78.00   1st Qu.:81.50   1st Qu.:79.00   1st Qu.: 81.0   1st Qu.:79.50  
 Median :82.00   Median :85.00   Median :85.00   Median : 86.0   Median :85.00  
 Mean   :81.76   Mean   :83.36   Mean   :83.05   Mean   : 85.4   Mean   :82.51  
 3rd Qu.:87.00   3rd Qu.:88.00   3rd Qu.:91.00   3rd Qu.: 89.5   3rd Qu.:88.50  
 Max.   :95.00   Max.   :94.00   Max.   :98.00   Max.   :104.0   Max.   :95.00  
                                                                                
      2009            2010            2011            2012       
 Min.   :51.00   Min.   :67.00   Min.   :59.00   Min.   : 56.00  
 1st Qu.:75.00   1st Qu.:82.00   1st Qu.:79.00   1st Qu.: 79.50  
 Median :83.00   Median :90.00   Median :89.00   Median : 85.00  
 Mean   :80.99   Mean   :87.21   Mean   :85.28   Mean   : 84.65  
 3rd Qu.:88.00   3rd Qu.:93.00   3rd Qu.:94.00   3rd Qu.: 90.50  
 Max.   :95.00   Max.   :97.00   Max.   :99.00   Max.   :105.00  
                                                                 
      2013            2014            2015     
 Min.   :56.00   Min.   :63.00   Min.   :56.0  
 1st Qu.:77.00   1st Qu.:81.50   1st Qu.:77.0  
 Median :84.00   Median :86.00   Median :85.0  
 Mean   :81.67   Mean   :83.94   Mean   :83.3  
 3rd Qu.:88.00   3rd Qu.:89.00   3rd Qu.:90.0  
 Max.   :92.00   Max.   :95.00   Max.   :97.0  
                                               

Data_temps

temps_mat <- as.vector(unlist(Data_temps[,2:21]))
str(temps_mat)

 int [1:2460] 98 97 97 90 89 93 93 91 93 93 ...

temps_mat

   [1]  98  97  97  90  89  93  93  91  93  93  90  91  93  93  82  91  96  95
  [19]  96  99  91  95  91  93  84  84  82  79  90  91  87  86  90  84  91  93
  [37]  88  91  84  90  89  88  86  84  86  89  90  91  91  90  89  90  91  91
  [55]  91  84  88  84  86  88  84  82  80  73  87  84  87  89  89  89  91  84
  [73]  86  88  78  79  86  82  82  78  79  79  78  81  84  84  87  84  79  75
  [91]  72  64  66  72  84  70  66  64  60  78  70  72  69  69  73  79  81  80
 [109]  82  66  63  68  79  81  69  73  73  75  75  81  82  82  81  86  90  93
 [127]  91  84  84  75  87  84  87  84  88  86  90  91  91  89  89  89  90  89
 [145]  84  87  88  89  89  91  91  89  88  72  80  84  88  89  88  84  84  80
 [163]  73  80  86  88  88  87  88  91  91  89  89  88  82  79  81  82  84  87
 [181]  90  90  91  91  88  88  91  93  81  81  82  86  88  84  80  82  86  87
 [199]  87  88  88  90  88  91  95  89  70  80  82  66  70  64  68  77  86  75
 [217]  73  75  78  81  82  82  82  80  82  82  79  80  68  63  57  66  64  69
 [235]  70  70  62  63  62  75  71  57  55  64  66  60  91  88  91  91  91  89
 [253]  93  95  95  91  91  86  88  87  91  87  90  91  95  91  91  89  91  91
 [271]  86  88  80  88  89  90  86  86  82  84  86  90  89  89  86  82  87  88
 [289]  84  86  80  82  86  84  87  90  79  84  87  87  88  90  91  89  90  93
 [307]  93  91  87  84  77  90  91  89  90  89  79  78  81  84  89  87  87  88
 [325]  87  82  80  82  82  88  84  81  82  84  87  80  75  75  86  78  77  82
 [343]  82  73  82  69  72  73  78  78  78  75  79  78  77  78  82  75  73  63
 [361]  63  72  75  79  79  79  78  82  79  84  82  87  88  90  91  82  86  87
 [379]  87  82  77  73  81  81  86  82  87  88  90  90  91  93  93  91  93  93
 [397]  93  93  97  99  96  93  88  89  91  93  93  93  91  90  96  98  97  98
 [415]  93  93  96  98  98  89  91  91  90  80  82  89  88  90  91  91  84  88
 [433]  91  84  93  96  96  91  91  77  87  87  87  86  87  89  81  81  82  79
 [451]  68  79  72  75  78  81  82  78  80  77  71  73  75  84  71  73  71  73
 [469]  73  72  72  73  70  64  75  73  77  80  71  66  60  64  73  57  59  64
 [487]  69  75  73  72  75  75  89  91  93  95  96  96  96  91  96  99  96  93
 [505]  91  93  93  93  91  97 100  99  93  96  87  82  75  82  88  91  89  87
 [523]  86  86  81  84  88  91  91  91  91  96  95  89  89  89  89  94  97  99
 [541] 101 101  97  87  86  88  92  92  90  90  92  92  88  87  79  81  82  87
 [559]  81  66  66  75  80  82  84  86  87  86  80  75  73  73  84  87  77  73
 [577]  81  84  82  68  71  75  73  75  77  79  82  81  82  73  66  55  55  64
 [595]  71  73  75  75  77  80  80  80  73  73  75  79  75  75  78  75  78  80
 [613]  75  77  78  84  87  87  84  86  87  87  89  91  87  90  90  86  82  82
 [631]  84  87  88  90  87  84  87  90  84  82  88  90  84  89  89  87  84  84
 [649]  84  86  88  84  86  88  87  88  86  86  81  87  84  90  91  91  87  86
 [667]  88  90  88  93  90  91  91  81  86  81  82  80  75  73  81  90  88  87
 [685]  86  86  89  87  84  84  86  77  77  81  81  82  84  86  87  88  69  66
 [703]  72  75  78  71  71  75  80  81  80  79  70  68  79  66  73  75  78  78
 [721]  75  75  62  60  64  71  75  79  80  81  79  73  64  51  55  63  72  71
 [739]  90  90  87  89  93  93  89  89  90  91  84  77  82  88  91  93  93  93
 [757]  93  91  95  91  89  87  84  86  89  91  91  88  90  93  91  91  91  93
 [775]  97  87  87  86  88  89  91  91  89  88  90  91  93  91  93  93  91  95
 [793]  93  91  88  84  82  82  78  77  84  84  89  95  93  91  88  87  91  95
 [811]  95  90  75  78  91  88  86  81  80  86  84  77  82  73  69  75  75  79
 [829]  73  79  82  84  84  82  87  86  80  71  66  70  78  84  79  68  57  66
 [847]  64  68  71  73  71  64  59  68  60  68  69  75  75  68  60  73  81  87
 [865]  86  80  84  87  90  89  84  84  86  87  84  86  88  88  88  88  88  89
 [883]  86  81  82  84  87  87  89  88  84  88  84  84  84  82  84  82  84  84
 [901]  86  87  84  81  87  89  90  86  89  90  90  87  88  88  90  89  88  89
 [919]  90  91  89  88  89  88  86  87  87  84  73  75  81  82  79  80  81  84
 [937]  82  82  81  81  81  84  87  82  75  81  80  82  82  82  73  66  71  72
 [955]  68  66  77  78  75  73  73  73  73  66  78  78  78  69  72  68  70  75
 [973]  78  84  78  78  73  73  68  64  57  70  77  75  82  81  86  88  90  90
 [991]  89  87  88  89  90  89  91  91  84  84
 [ reached getOption("max.print") -- omitted 1460 entries ]

temps_ts <- ts(temps_mat, start=1996, end = 2015, frequency=123)
temps_ts

Time Series:
Start = c(1996, 1) 
End = c(2015, 1) 
Frequency = 123 
   [1]  98  97  97  90  89  93  93  91  93  93  90  91  93  93  82  91  96  95
  [19]  96  99  91  95  91  93  84  84  82  79  90  91  87  86  90  84  91  93
  [37]  88  91  84  90  89  88  86  84  86  89  90  91  91  90  89  90  91  91
  [55]  91  84  88  84  86  88  84  82  80  73  87  84  87  89  89  89  91  84
  [73]  86  88  78  79  86  82  82  78  79  79  78  81  84  84  87  84  79  75
  [91]  72  64  66  72  84  70  66  64  60  78  70  72  69  69  73  79  81  80
 [109]  82  66  63  68  79  81  69  73  73  75  75  81  82  82  81  86  90  93
 [127]  91  84  84  75  87  84  87  84  88  86  90  91  91  89  89  89  90  89
 [145]  84  87  88  89  89  91  91  89  88  72  80  84  88  89  88  84  84  80
 [163]  73  80  86  88  88  87  88  91  91  89  89  88  82  79  81  82  84  87
 [181]  90  90  91  91  88  88  91  93  81  81  82  86  88  84  80  82  86  87
 [199]  87  88  88  90  88  91  95  89  70  80  82  66  70  64  68  77  86  75
 [217]  73  75  78  81  82  82  82  80  82  82  79  80  68  63  57  66  64  69
 [235]  70  70  62  63  62  75  71  57  55  64  66  60  91  88  91  91  91  89
 [253]  93  95  95  91  91  86  88  87  91  87  90  91  95  91  91  89  91  91
 [271]  86  88  80  88  89  90  86  86  82  84  86  90  89  89  86  82  87  88
 [289]  84  86  80  82  86  84  87  90  79  84  87  87  88  90  91  89  90  93
 [307]  93  91  87  84  77  90  91  89  90  89  79  78  81  84  89  87  87  88
 [325]  87  82  80  82  82  88  84  81  82  84  87  80  75  75  86  78  77  82
 [343]  82  73  82  69  72  73  78  78  78  75  79  78  77  78  82  75  73  63
 [361]  63  72  75  79  79  79  78  82  79  84  82  87  88  90  91  82  86  87
 [379]  87  82  77  73  81  81  86  82  87  88  90  90  91  93  93  91  93  93
 [397]  93  93  97  99  96  93  88  89  91  93  93  93  91  90  96  98  97  98
 [415]  93  93  96  98  98  89  91  91  90  80  82  89  88  90  91  91  84  88
 [433]  91  84  93  96  96  91  91  77  87  87  87  86  87  89  81  81  82  79
 [451]  68  79  72  75  78  81  82  78  80  77  71  73  75  84  71  73  71  73
 [469]  73  72  72  73  70  64  75  73  77  80  71  66  60  64  73  57  59  64
 [487]  69  75  73  72  75  75  89  91  93  95  96  96  96  91  96  99  96  93
 [505]  91  93  93  93  91  97 100  99  93  96  87  82  75  82  88  91  89  87
 [523]  86  86  81  84  88  91  91  91  91  96  95  89  89  89  89  94  97  99
 [541] 101 101  97  87  86  88  92  92  90  90  92  92  88  87  79  81  82  87
 [559]  81  66  66  75  80  82  84  86  87  86  80  75  73  73  84  87  77  73
 [577]  81  84  82  68  71  75  73  75  77  79  82  81  82  73  66  55  55  64
 [595]  71  73  75  75  77  80  80  80  73  73  75  79  75  75  78  75  78  80
 [613]  75  77  78  84  87  87  84  86  87  87  89  91  87  90  90  86  82  82
 [631]  84  87  88  90  87  84  87  90  84  82  88  90  84  89  89  87  84  84
 [649]  84  86  88  84  86  88  87  88  86  86  81  87  84  90  91  91  87  86
 [667]  88  90  88  93  90  91  91  81  86  81  82  80  75  73  81  90  88  87
 [685]  86  86  89  87  84  84  86  77  77  81  81  82  84  86  87  88  69  66
 [703]  72  75  78  71  71  75  80  81  80  79  70  68  79  66  73  75  78  78
 [721]  75  75  62  60  64  71  75  79  80  81  79  73  64  51  55  63  72  71
 [739]  90  90  87  89  93  93  89  89  90  91  84  77  82  88  91  93  93  93
 [757]  93  91  95  91  89  87  84  86  89  91  91  88  90  93  91  91  91  93
 [775]  97  87  87  86  88  89  91  91  89  88  90  91  93  91  93  93  91  95
 [793]  93  91  88  84  82  82  78  77  84  84  89  95  93  91  88  87  91  95
 [811]  95  90  75  78  91  88  86  81  80  86  84  77  82  73  69  75  75  79
 [829]  73  79  82  84  84  82  87  86  80  71  66  70  78  84  79  68  57  66
 [847]  64  68  71  73  71  64  59  68  60  68  69  75  75  68  60  73  81  87
 [865]  86  80  84  87  90  89  84  84  86  87  84  86  88  88  88  88  88  89
 [883]  86  81  82  84  87  87  89  88  84  88  84  84  84  82  84  82  84  84
 [901]  86  87  84  81  87  89  90  86  89  90  90  87  88  88  90  89  88  89
 [919]  90  91  89  88  89  88  86  87  87  84  73  75  81  82  79  80  81  84
 [937]  82  82  81  81  81  84  87  82  75  81  80  82  82  82  73  66  71  72
 [955]  68  66  77  78  75  73  73  73  73  66  78  78  78  69  72  68  70  75
 [973]  78  84  78  78  73  73  68  64  57  70  77  75  82  81  86  88  90  90
 [991]  89  87  88  89  90  89  91  91  84  84
 [ reached getOption("max.print") -- omitted 1338 entries ]

class(temps_ts)

[1] "ts"

plot(temps_ts)

#Exponential Smoothing 
#Simple Exponential 
temps_single <- HoltWinters(temps_ts,beta=FALSE, gamma=FALSE)
#Double Exponential - model trend 
temps_double <- HoltWinters(temps_ts,gamma=FALSE)
#Triple Exponential - model trend and seasonality
temps_triple_additive <- HoltWinters(temps_ts, seasonal = "additive")
#Look at 3 kinds of ES
temps_single

Holt-Winters exponential smoothing without trend and without seasonal component.

Call:
HoltWinters(x = temps_ts, beta = FALSE, gamma = FALSE)

Smoothing parameters:
 alpha: 0.8396301
 beta : FALSE
 gamma: FALSE

Coefficients:
      [,1]
a 81.62444

temps_single$SSE

[1] 53704.15

# Single ES parameters:
# alpha: 0.8396301
# SSE(sum of squared error):53704.15
temps_double

Holt-Winters exponential smoothing with trend and without seasonal component.

Call:
HoltWinters(x = temps_ts, gamma = FALSE)

Smoothing parameters:
 alpha: 0.8455303
 beta : 0.003777803
 gamma: FALSE

Coefficients:
          [,1]
a 81.729657393
b -0.004838906

temps_double$SSE

[1] 54071.22

# Double ES parameters:
# alpha: 0.8455303
# beta : 0.003777803
# SSE: 54071.22
temps_triple

Holt-Winters exponential smoothing with trend and additive seasonal component.

Call:
HoltWinters(x = temps_ts)

Smoothing parameters:
 alpha: 0.6677614
 beta : 0
 gamma: 0.6297674

Coefficients:
              [,1]
a     66.739214602
b     -0.004362918
s1    17.167113056
s2    12.692593452
s3    11.926233267
s4    12.862822489
s5    11.026083880
s6     8.860499089
s7     9.547553333
s8     7.755384526
s9     4.419013466
s10    2.272689626
s11    4.628251667
s12    2.396834852
s13    3.512957136
s14    1.734948091
s15    3.035023890
s16    6.257944053
s17    5.086362292
s18    8.599153274
s19    5.507486014
s20   10.404819396
s21   10.115801978
s22    9.628840064
s23    7.658623118
s24    7.150473636
s25    6.306599371
s26    5.850691115
s27    5.770487458
s28    4.280481134
s29    7.229771199
s30    4.632381095
s31    6.006248308
s32    6.443645890
s33    5.701166527
s34    3.546887269
s35    3.879569716
s36    3.517339384
s37    2.828550977
s38    2.122971410
s39    2.627923984
s40    1.658896597
s41    0.165866282
s42   -0.001574460
s43   -1.557500303
s44   -2.159601227
s45   -2.260609558
s46    0.474052766
s47    2.501631056
s48    6.552191593
s49    7.240238719
s50    8.395899120
s51    8.633263084
s52    7.504540260
s53    4.804135812
s54    0.449902809
s55   -1.045831475
s56    1.562077049
s57    1.632745190
s58    0.857309158
s59    2.909614779
s60    0.626594899
s61    4.491805650
s62    4.567058619
s63    3.065433531
s64    3.787652805
s65   -2.147135463
s66    1.759895146
s67    1.541155061
s68    1.278521842
s69    0.895959617
s70    2.009912430
s71    3.695537344
s72    4.675235988
s73    4.535880359
s74    1.710420810
s75    0.822675780
s76    2.363162195
s77    1.925012161
s78   -1.656914701
s79   -1.809929506
s80   -0.427021203
s81    0.056812125
s82   -1.137248149
s83   -1.037423821
s84   -2.817503990
s85   -4.578240308
s86   -3.080091372
s87   -2.710719111
s88   -2.255335538
s89   -4.518502545
s90   -5.159556421
s91   -4.440834373
s92   -5.790113744
s93   -7.461163074
s94   -8.882612687
s95   -8.619859733
s96   -6.200719796
s97   -6.055889182
s98  -11.167287691
s99  -13.489975101
s100 -13.615536188
s101 -14.373453486
s102 -15.142110213
s103 -14.419874185
s104 -14.023613348
s105 -16.187082843
s106 -15.999259045
s107 -12.074075053
s108  -9.199729415
s109 -10.403127076
s110 -12.075113349
s111  -9.722863134
s112  -5.846856763
s113  -8.047801338
s114  -9.636669876
s115 -10.510269852
s116 -12.876648138
s117  -8.657362442
s118  -9.828539578
s119 -14.522204766
s120 -11.852457644
s121  -8.714763993
s122  -4.711332904
s123  18.737998957

temps_triple$SSE

[1] 63025.97

# Triple ES parameters:
# alpha: 0.6677614
# beta : 0
# gamma: 0.6297674
# SSE: 63025.97
#Single ES gives the smallest SSE.Its alpha is closer to 1 which means there is less randomness in the system. The recent temperature observations have more weight in predicting the current temperature.
#Seasonality can appear in two forms: 
#1. additive: amplitude of the seasonal variation is independent of the level,
#2. multiplicative: amplitude of the seasonal variation is connected. 
#Triple Exponential - use multiplicative decomposition
temps_triple_mul <- HoltWinters(temps_ts, seasonal = "multiplicative")
temps_triple_mul$SSE

[1] 65648.65

#SSE:65648.65
#Triple Exponential - use additive decomposition
temps_triple_additive <- HoltWinters(temps_ts, seasonal = "additive")
temps_triple_additive$SSE

[1] 63025.97

temps_triple_additive$fitted

Time Series:
Start = c(1997, 1) 
End = c(2015, 1) 
Frequency = 123 
              xhat     level        trend        season
1997.000  87.17619  82.87739 -0.004362918   4.303159495
1997.008  90.32137  82.08762 -0.004362918   8.238118845
1997.016  92.95607  81.86865 -0.004362918  11.091777381
1997.024  90.93226  81.89363 -0.004362918   9.042996893
1997.033  83.99752  81.93450 -0.004362918   2.067387137
1997.041  84.04359  81.93179 -0.004362918   2.116167625
1997.049  75.06703  81.89832 -0.004362918  -6.826921806
1997.057  87.04230  81.84919 -0.004362918   5.197468438
1997.065  84.01782  81.81658 -0.004362918   2.205598519
1997.073  87.05847  81.80032 -0.004362918   5.262509089
1997.081  84.04758  81.75692 -0.004362918   2.295029414
1997.089  88.04397  81.72078 -0.004362918   6.327549739
1997.098  86.02650  81.68706 -0.004362918   4.343809902
1997.106  89.93127  81.66500 -0.004362918   8.270639170
1997.114  90.90776  81.70653 -0.004362918   9.205598519
1997.122  90.94873  81.76376 -0.004362918   9.189338357
1997.130  88.92982  81.79363 -0.004362918   7.140557869
1997.138  88.90728  81.83613 -0.004362918   7.075517219
1997.146  88.88353  81.89368 -0.004362918   6.994216406
1997.154  89.85938  81.96709 -0.004362918   7.896655430
1997.163  88.81884  82.05663 -0.004362918   6.766574129
1997.171  83.84602  82.17324 -0.004362918   1.677143235
1997.179  87.03391  82.27170 -0.004362918   4.766574129
1997.187  88.03942  82.24469 -0.004362918   5.799094454
1997.195  89.02500  82.21400 -0.004362918   6.815354617
1997.203  89.17467  82.19295 -0.004362918   6.986086324
1997.211  91.16749  82.07195 -0.004362918   9.099907462
1997.220  91.17324  81.95574 -0.004362918   9.221858682
1997.228  89.11010  81.83570 -0.004362918   7.278769251
1997.236  87.99157  81.75781 -0.004362918   6.238118845
1997.244  71.81397  81.75908 -0.004362918  -9.940742944
1997.252  79.86066  81.87894 -0.004362918  -2.013913676
1997.260  83.94121  81.96762 -0.004362918   1.977956243
1997.268  88.04928  82.00251 -0.004362918   6.051126975
1997.276  88.94697  81.96524 -0.004362918   6.986086324
1997.285  87.85607  81.99629 -0.004362918   5.864135105
1997.293  83.80148  82.08804 -0.004362918   1.717793641
1997.301  83.75082  82.21625 -0.004362918   1.538931853
1997.309  79.88033  82.37828 -0.004362918  -2.493588472
1997.317  72.87458  82.45383 -0.004362918  -9.574889285
1997.325  79.87267  82.53322 -0.004362918  -2.656190098
1997.333  85.84764  82.61388 -0.004362918   3.238118845
1997.341  87.86372  82.71126 -0.004362918   5.156818032
1997.350  87.89345  82.79790 -0.004362918   5.099907462
1997.358  87.04967  82.86469 -0.004362918   4.189338357
1997.366  88.15848  82.82716 -0.004362918   5.335679820
1997.374  91.23528  82.71697 -0.004362918   8.522671690
1997.382  91.20389  82.55550 -0.004362918   8.652752991
1997.390  89.07964  82.41499 -0.004362918   6.669013154
1997.398  88.97332  82.35745 -0.004362918   6.620232666
1997.407  87.97051  82.37090 -0.004362918   5.603972503
1997.415  82.05901  82.38623 -0.004362918  -0.322856765
1997.423  79.16971  82.34246 -0.004362918  -3.168385220
1997.431  81.10080  82.22477 -0.004362918  -1.119604733
1997.439  82.11856  82.15310 -0.004362918  -0.030173838
1997.447  84.01877  82.06956 -0.004362918   1.953565999
1997.455  87.03439  82.05267 -0.004362918   4.986086324
1997.463  90.15341  82.02534 -0.004362918   8.132427788
1997.472  90.25799  81.91854 -0.004362918   8.343809902
1997.480  91.22769  81.74190 -0.004362918   9.490151365
1997.488  91.20137  81.58550 -0.004362918   9.620232666
1997.496  84.21295  81.44667 -0.004362918   2.770639170
1997.504  80.81467  83.97116 -0.004362918  -3.152125058
1997.512  78.66530  88.76488 -0.004362918 -10.095214489
1997.520 100.93009  96.99715 -0.004362918   3.937305836
1997.528  92.62219  91.69738 -0.004362918   0.929175755
1997.537  87.89763  83.93216 -0.004362918   3.969826162
1997.545  85.36047  79.32183 -0.004362918   6.042996893
1997.553  83.25846  77.07348 -0.004362918   6.189338357
1997.561  85.11731  78.89981 -0.004362918   6.221858682
1997.569  89.11107  80.82040 -0.004362918   8.295029414
1997.577  78.74251  77.40306 -0.004362918   1.343809902
1997.585  81.62663  78.23840 -0.004362918   3.392590389
1997.593  83.89598  78.48336 -0.004362918   5.416980633
1997.602  75.35351  79.88398 -0.004362918  -4.526108798
1997.610  84.15061  87.65670 -0.004362918  -3.501718554
1997.618  92.97579  89.55504 -0.004362918   3.425110715
1997.626  85.64879  86.22804 -0.004362918  -0.574889285
1997.634  87.27138  87.79373 -0.004362918  -0.517978716
1997.642  85.13787  89.61143 -0.004362918  -4.469198229
1997.650  88.10164  91.51829 -0.004362918  -3.412287659
1997.659  90.10586  93.44934 -0.004362918  -3.339116928
1997.667  92.38587  96.71309 -0.004362918  -4.322856765
1997.675  93.20999  94.44778 -0.004362918  -1.233425871
1997.683  80.73941  78.94468 -0.004362918   1.799094454
1997.691  80.28195  78.44657 -0.004362918   1.839744861
1997.699  84.38418  79.58945 -0.004362918   4.799094454
1997.707  69.06292  67.30884 -0.004362918   1.758444048
1997.715  64.61113  67.93022 -0.004362918  -3.314726684
1997.724  60.10112  67.51777 -0.004362918  -7.412287659
1997.732  62.37945  72.78797 -0.004362918 -10.404157578
1997.740  64.16252  82.54665 -0.004362918 -18.379767334
1997.748  80.86233  97.12451 -0.004362918 -16.257816115
1997.756  82.99211  93.20551 -0.004362918 -10.209035627
1997.764  88.36418  86.52880 -0.004362918   1.839744861
1997.772  65.40321  77.60035 -0.004362918 -12.192775464
1997.780  69.82676  86.00764 -0.004362918 -16.176515302
1997.789  75.32411  93.46433 -0.004362918 -18.135864895
1997.797  75.81017  97.91788 -0.004362918 -22.103344570
1997.805  97.99605 102.04684 -0.004362918  -4.046434001
1997.813  79.34266  91.36094 -0.004362918 -12.013913676
1997.821  81.91546  91.79552 -0.004362918  -9.875702294
1997.829  79.04072  91.84761 -0.004362918 -12.802531562
1997.837  81.02871  93.81934 -0.004362918 -12.786271399
1997.846  83.65339  92.46029 -0.004362918  -8.802531562
1997.854  87.17691  90.01633 -0.004362918  -2.835051887
1997.862  76.35882  77.20636 -0.004362918  -0.843181968
1997.870  66.44208  68.28150 -0.004362918  -1.835051887
1997.878  62.12453  61.97208 -0.004362918   0.156818032
1997.886  48.70806  64.55560 -0.004362918 -15.843181968
1997.894  55.93132  74.76261 -0.004362918 -18.826921806
1997.902  69.66185  83.48500 -0.004362918 -13.818791725
1997.911  80.89142  83.70644 -0.004362918  -2.810661643
1997.919  75.67923  76.42921 -0.004362918  -0.745620993
1997.927  54.63797  67.29039 -0.004362918 -12.648060017
1997.935  64.29875  72.86987 -0.004362918  -8.566759204
1997.943  62.83254  71.33049 -0.004362918  -8.493588472
1997.951  72.95314  79.45109 -0.004362918  -6.493588472
1997.959  71.65267  78.14250 -0.004362918  -6.485458391
1997.967  67.81504  68.35364 -0.004362918  -0.534238879
1997.976  60.22077  59.79189 -0.004362918   0.433240796
1997.984  62.71564  62.31115 -0.004362918   0.408850552
1997.992  63.84754  64.49996 -0.004362918  -0.648060017
1998.000  65.97906  61.92636 -0.004362918   4.057062406
1998.008  86.79653  78.63002 -0.004362918   8.170877414
1998.016  90.52589  79.42928 -0.004362918  11.100969520
1998.024  88.79432  79.74151 -0.004362918   9.057170233
1998.033  83.27356  81.21002 -0.004362918   2.067905689
1998.041  88.46776  86.36507 -0.004362918   2.107046284
1998.049  79.87081  86.71612 -0.004362918  -6.840946973
1998.057 100.66318  95.47892 -0.004362918   5.188618358
1998.065  93.89042  91.69291 -0.004362918   2.201869966
1998.073  97.67540  92.42948 -0.004362918   5.250275614
1998.081  90.24826  87.96755 -0.004362918   2.285073521
1998.089  94.77916  88.46517 -0.004362918   6.318350547
1998.098  86.93233  82.59842 -0.004362918   4.338264278
1998.106  91.58767  83.30701 -0.004362918   8.285019302
1998.114  89.45972  80.23918 -0.004362918   9.224897674
1998.122  90.45906  81.26336 -0.004362918   9.200065311
1998.130  86.10005  78.94917 -0.004362918   7.155241134
1998.138  88.63960  81.54904 -0.004362918   7.094917086
1998.146  90.13509  83.12087 -0.004362918   7.018585474
1998.154  94.28682  86.36510 -0.004362918   7.926077622
1998.163  90.96605  84.16593 -0.004362918   6.804479491
1998.171  85.88924  84.18424 -0.004362918   1.709361624
1998.179  91.01224  86.25712 -0.004362918   4.759479289
1998.187  92.03107  86.24459 -0.004362918   5.790845815
1998.195  92.35748  85.55172 -0.004362918   6.810124804
1998.203  88.24725  81.30207 -0.004362918   6.949538929
1998.211  90.19311  81.13261 -0.004362918   9.064862730
1998.220  83.50293  74.32168 -0.004362918   9.185612110
1998.228  84.57166  77.32029 -0.004362918   7.255732036
1998.236  86.50852  80.27300 -0.004362918   6.239883268
1998.244  72.69393  82.60011 -0.004362918  -9.901820071
1998.252  89.49191  91.48103 -0.004362918  -1.984759393
1998.260  91.13080  89.14491 -0.004362918   1.990256369
1998.268  89.07980  83.04335 -0.004362918   6.040816683
1998.276  86.63971  79.64689 -0.004362918   6.997182367
1998.285  85.10524  79.21535 -0.004362918   5.894250731
1998.293  84.23449  82.47952 -0.004362918   1.759331552
1998.301  87.24409  85.65738 -0.004362918   1.591069014
1998.309  84.35264  86.82555 -0.004362918  -2.468549302
1998.317  78.36822  87.92123 -0.004362918  -9.548646623
1998.325  87.70812  90.34203 -0.004362918  -2.629547590
1998.333  93.13045  89.86481 -0.004362918   3.269997443
1998.341  91.61551  86.43453 -0.004362918   5.185333016
1998.350  86.46267  81.34483 -0.004362918   5.122201679
1998.358  85.20610  81.03152 -0.004362918   4.178946324
1998.366  82.84888  77.55072 -0.004362918   5.302520582
1998.374  85.44859  76.97951 -0.004362918   8.473442899
1998.382  85.94909  77.34336 -0.004362918   8.610092846
1998.390  82.68546  76.03747 -0.004362918   6.652350510
1998.398  85.53564  78.91419 -0.004362918   6.625816049
1998.407  87.49673  81.89095 -0.004362918   5.610142549
1998.415  75.87323  76.21280 -0.004362918  -0.335203707
1998.423  78.42692  81.63518 -0.004362918  -3.203895037
1998.431  86.21053  87.35559 -0.004362918  -1.140696088
1998.439  87.81906  87.87840 -0.004362918  -0.054980233
1998.447  89.94014  87.99486 -0.004362918   1.949639387
1998.455  93.00500  88.03047 -0.004362918   4.978890292
1998.463  94.78322  86.68725 -0.004362918   8.100330384
1998.472  91.10654  82.82108 -0.004362918   8.289830648
1998.480  91.51595  82.07781 -0.004362918   9.442510769
1998.488  92.63817  83.06443 -0.004362918   9.578100181
1998.496  86.86034  83.30169 -0.004362918   3.563016184
1998.504  84.40855  86.06163 -0.004362918  -1.648715946
1998.512  80.26899  87.78774 -0.004362918  -7.514388171
1998.520  92.54851  90.27480 -0.004362918   2.278067894
1998.528  78.38081  79.88775 -0.004362918  -1.502570622
1998.537  90.16448  87.64223 -0.004362918   2.526614473
1998.545  93.53131  88.19579 -0.004362918   5.339876352
1998.553  91.92420  85.16560 -0.004362918   6.762960169
1998.561  90.69698  83.87633 -0.004362918   6.825012625
1998.569  89.96005  82.73879 -0.004362918   7.225625268
1998.577  77.01828  75.41573 -0.004362918   1.606918095
1998.585  79.53327  76.06692 -0.004362918   3.470711462
1998.593  82.89483  77.04198 -0.004362918   5.857211189
1998.602  75.68197  77.77561 -0.004362918  -2.089278380
1998.610  83.75462  86.66451 -0.004362918  -2.905532868
1998.618  91.20694  88.82729 -0.004362918   2.384010915
1998.626  85.92640  86.01370 -0.004362918  -0.082937756
1998.634  87.44258  87.39401 -0.004362918   0.052937961
1998.642  83.21940  87.09410 -0.004362918  -3.870345199
1998.650  83.46526  86.27548 -0.004362918  -2.805854177
1998.659  81.63768  83.95715 -0.004362918  -2.315100992
1998.667  79.15907  84.19473 -0.004362918  -5.031293693
1998.675  79.99334  86.08743 -0.004362918  -6.089722075
1998.683  93.06962  91.42960 -0.004362918   1.644385312
1998.691  87.56375  85.36889 -0.004362918   2.199218069
1998.699  81.92967  80.98151 -0.004362918   0.952515668
1998.707  82.97427  81.02412 -0.004362918   1.954511737
1998.715  78.25774  81.70470 -0.004362918  -3.442595878
1998.724  81.77414  87.53808 -0.004362918  -5.759580500
1998.732  78.99960  86.34902 -0.004362918  -7.345056098
1998.740  69.85887  83.67388 -0.004362918 -13.810644299
1998.748  69.61379  87.10256 -0.004362918 -17.484410097
1998.756  85.73620  98.04028 -0.004362918 -12.299715505
1998.764  91.90913  92.86998 -0.004362918  -0.956483800
1998.772  73.34840  82.90987 -0.004362918  -9.557111515
1998.780  74.83965  88.68272 -0.004362918 -13.838703967
1998.789  76.71635  93.45976 -0.004362918 -16.739047137
1998.797  70.16117  90.97376 -0.004362918 -20.808227064
1998.805  91.47721  98.87491 -0.004362918  -7.393335572
1998.813  71.98039  83.86113 -0.004362918 -11.876376715
1998.821  74.00749  83.86986 -0.004362918  -9.858012730
1998.829  71.00502  83.19274 -0.004362918 -12.183352127
1998.837  74.64424  87.85935 -0.004362918 -13.210744539
1998.846  80.52453  90.09583 -0.004362918  -9.566942039
1998.854  81.55383  88.40569 -0.004362918  -6.847496242
1998.862  80.38228  84.02493 -0.004362918  -3.638288239
1998.870  79.28252  83.09753 -0.004362918  -3.810647528
1998.878  83.20008  82.23675 -0.004362918   0.967694205
1998.886  65.44425  78.09221 -0.004362918 -12.643602052
1998.894  70.37521  86.47210 -0.004362918 -16.092522774
1998.902  80.47792  94.23032 -0.004362918 -13.748039245
1998.911  85.47415  90.56801 -0.004362918  -5.089506695
1998.919  78.62177  82.23390 -0.004362918  -3.607767550
1998.927  60.89511  71.79792 -0.004362918 -10.898446041
1998.935  64.14703  73.19912 -0.004362918  -9.047733112
1998.943  72.48656  78.43867 -0.004362918  -5.947753797
1998.951  73.20608  80.11269 -0.004362918  -6.902249366
1998.959  74.42164  83.97728 -0.004362918  -9.551282217
1998.967  83.81024  87.03017 -0.004362918  -3.215568915
1998.976  85.03334  83.81372 -0.004362918   1.223981060
1998.984  80.20445  79.11276 -0.004362918   1.096047196
1998.992  78.84995  80.30740 -0.004362918  -1.453091577
1999.000  89.69115  80.40324 -0.004362918   9.292270132
1999.008  85.01687  76.59855 -0.004362918   8.422683133
1999.016  85.77544  74.57964 -0.004362918  11.200169006
1999.024  84.90729  75.39299 -0.004362918   9.518671355
 [ reached getOption("max.print") -- omitted 1965 rows ]

#SSE:63025.97
plot(fitted(temps_triple_additive))

plot(fitted(temps_triple_mul))

Triple ES with additive seasonal factor has better SSE.

From looking at the chart: the trend subchart shows a straight line which means there is no trend; this is also evidenced by alpha=0.

The season subchart shows that thee duration of each season has been pretty constant throughout these years. The next step I would do is to apply cumsum method to the level data for the daily temperature from 1997 to 2005 for each year and set the C and T values to see whether the last day of summer has become earlier or later.

Question 8.1 Describe a situation or problem from your job, everyday life, current events, etc., for which a linear regression model would be appropriate. List some (up to 5) predictors that you might use.

The credit card company will analyze cardholder information to predict the balance of a new cardholder for customer analysis purpose. Some predictors to be used are: 1. the past monthly balance 2. the cardholder’s income 3. the industry cardholder works in 4. spending on essential goods (food, medical expenses) 5. spending on nonessential goods (luxury brands, travelling, etc.)

Question 8.2 Using crime data from http://www.statsci.org/data/general/uscrime.txt (file uscrime.txt, description at http://www.statsci.org/data/general/uscrime.html ), use regression (a useful R function is lm or glm) to predict the observed crime rate in a city with the following data: M = 14.0 So = 0 Ed = 10.0 Po1 = 12.0 Po2 = 15.5 LF = 0.640 M.F = 94.0 Pop = 150 NW = 1.1 U1 = 0.120 U2 = 3.6 Wealth = 3200 Ineq = 20.1 Prob = 0.04 Time = 39.0

Show your model (factors used and their coefficients), the software output, and the quality of fit.

Note that because there are only 47 data points and 15 predictors, you’ll probably notice some overfitting. We’ll see ways of dealing with this sort of problem later in the course.

Data_crime = read.csv("uscrime.txt",sep = "")
str(Data_crime)

'data.frame':   47 obs. of  16 variables:
 $ M     : num  15.1 14.3 14.2 13.6 14.1 12.1 12.7 13.1 15.7 14 ...
 $ So    : int  1 0 1 0 0 0 1 1 1 0 ...
 $ Ed    : num  9.1 11.3 8.9 12.1 12.1 11 11.1 10.9 9 11.8 ...
 $ Po1   : num  5.8 10.3 4.5 14.9 10.9 11.8 8.2 11.5 6.5 7.1 ...
 $ Po2   : num  5.6 9.5 4.4 14.1 10.1 11.5 7.9 10.9 6.2 6.8 ...
 $ LF    : num  0.51 0.583 0.533 0.577 0.591 0.547 0.519 0.542 0.553 0.632 ...
 $ M.F   : num  95 101.2 96.9 99.4 98.5 ...
 $ Pop   : int  33 13 18 157 18 25 4 50 39 7 ...
 $ NW    : num  30.1 10.2 21.9 8 3 4.4 13.9 17.9 28.6 1.5 ...
 $ U1    : num  0.108 0.096 0.094 0.102 0.091 0.084 0.097 0.079 0.081 0.1 ...
 $ U2    : num  4.1 3.6 3.3 3.9 2 2.9 3.8 3.5 2.8 2.4 ...
 $ Wealth: int  3940 5570 3180 6730 5780 6890 6200 4720 4210 5260 ...
 $ Ineq  : num  26.1 19.4 25 16.7 17.4 12.6 16.8 20.6 23.9 17.4 ...
 $ Prob  : num  0.0846 0.0296 0.0834 0.0158 0.0414 ...
 $ Time  : num  26.2 25.3 24.3 29.9 21.3 ...
 $ Crime : int  791 1635 578 1969 1234 682 963 1555 856 705 ...

print(summary(Data_crime))

       M               So               Ed             Po1       
 Min.   :11.90   Min.   :0.0000   Min.   : 8.70   Min.   : 4.50  
 1st Qu.:13.00   1st Qu.:0.0000   1st Qu.: 9.75   1st Qu.: 6.25  
 Median :13.60   Median :0.0000   Median :10.80   Median : 7.80  
 Mean   :13.86   Mean   :0.3404   Mean   :10.56   Mean   : 8.50  
 3rd Qu.:14.60   3rd Qu.:1.0000   3rd Qu.:11.45   3rd Qu.:10.45  
 Max.   :17.70   Max.   :1.0000   Max.   :12.20   Max.   :16.60  
      Po2               LF              M.F              Pop        
 Min.   : 4.100   Min.   :0.4800   Min.   : 93.40   Min.   :  3.00  
 1st Qu.: 5.850   1st Qu.:0.5305   1st Qu.: 96.45   1st Qu.: 10.00  
 Median : 7.300   Median :0.5600   Median : 97.70   Median : 25.00  
 Mean   : 8.023   Mean   :0.5612   Mean   : 98.30   Mean   : 36.62  
 3rd Qu.: 9.700   3rd Qu.:0.5930   3rd Qu.: 99.20   3rd Qu.: 41.50  
 Max.   :15.700   Max.   :0.6410   Max.   :107.10   Max.   :168.00  
       NW              U1                U2            Wealth    
 Min.   : 0.20   Min.   :0.07000   Min.   :2.000   Min.   :2880  
 1st Qu.: 2.40   1st Qu.:0.08050   1st Qu.:2.750   1st Qu.:4595  
 Median : 7.60   Median :0.09200   Median :3.400   Median :5370  
 Mean   :10.11   Mean   :0.09547   Mean   :3.398   Mean   :5254  
 3rd Qu.:13.25   3rd Qu.:0.10400   3rd Qu.:3.850   3rd Qu.:5915  
 Max.   :42.30   Max.   :0.14200   Max.   :5.800   Max.   :6890  
      Ineq            Prob              Time           Crime       
 Min.   :12.60   Min.   :0.00690   Min.   :12.20   Min.   : 342.0  
 1st Qu.:16.55   1st Qu.:0.03270   1st Qu.:21.60   1st Qu.: 658.5  
 Median :17.60   Median :0.04210   Median :25.80   Median : 831.0  
 Mean   :19.40   Mean   :0.04709   Mean   :26.60   Mean   : 905.1  
 3rd Qu.:22.75   3rd Qu.:0.05445   3rd Qu.:30.45   3rd Qu.:1057.5  
 Max.   :27.60   Max.   :0.11980   Max.   :44.00   Max.   :1993.0  

model <- lm(Crime ~ .  ,Data_crime)
model

Call:
lm(formula = Crime ~ ., data = Data_crime)

Coefficients:
(Intercept)            M           So           Ed          Po1          Po2  
 -5.984e+03    8.783e+01   -3.803e+00    1.883e+02    1.928e+02   -1.094e+02  
         LF          M.F          Pop           NW           U1           U2  
 -6.638e+02    1.741e+01   -7.330e-01    4.204e+00   -5.827e+03    1.678e+02  
     Wealth         Ineq         Prob         Time  
  9.617e-02    7.067e+01   -4.855e+03   -3.479e+00  

print(summary(model))

Call:
lm(formula = Crime ~ ., data = Data_crime)

Residuals:
    Min      1Q  Median      3Q     Max 
-395.74  -98.09   -6.69  112.99  512.67 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -5.984e+03  1.628e+03  -3.675 0.000893 ***
M            8.783e+01  4.171e+01   2.106 0.043443 *  
So          -3.803e+00  1.488e+02  -0.026 0.979765    
Ed           1.883e+02  6.209e+01   3.033 0.004861 ** 
Po1          1.928e+02  1.061e+02   1.817 0.078892 .  
Po2         -1.094e+02  1.175e+02  -0.931 0.358830    
LF          -6.638e+02  1.470e+03  -0.452 0.654654    
M.F          1.741e+01  2.035e+01   0.855 0.398995    
Pop         -7.330e-01  1.290e+00  -0.568 0.573845    
NW           4.204e+00  6.481e+00   0.649 0.521279    
U1          -5.827e+03  4.210e+03  -1.384 0.176238    
U2           1.678e+02  8.234e+01   2.038 0.050161 .  
Wealth       9.617e-02  1.037e-01   0.928 0.360754    
Ineq         7.067e+01  2.272e+01   3.111 0.003983 ** 
Prob        -4.855e+03  2.272e+03  -2.137 0.040627 *  
Time        -3.479e+00  7.165e+00  -0.486 0.630708    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 209.1 on 31 degrees of freedom
Multiple R-squared:  0.8031,    Adjusted R-squared:  0.7078 
F-statistic: 8.429 on 15 and 31 DF,  p-value: 3.539e-07

I apply all the predictors as my independent variables in the linear regression model. The adjusted R-Square (quality of fit) of 0.7 is not bad. The p-value for each predictor tests the null hypothesis that the coefficient is equal to zero (no effect). A low p-value (< 0.05) indicates that you can reject the null hypothesis.

Given there are too many predictors for too few number of observations (probably overfitting issue), I tried to remove some predictors with high p-values which are: So, Po2, LF, M.F, Pop, NW, U1,Wealth,Time. I create this new lr model below:

model_2 <- lm(Crime ~ M+Ed+Po1+U2+Ineq+Prob ,Data_crime)
print(summary(model_2))

Call:
lm(formula = Crime ~ M + Ed + Po1 + U2 + Ineq + Prob, data = Data_crime)

Residuals:
    Min      1Q  Median      3Q     Max 
-470.68  -78.41  -19.68  133.12  556.23 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) -5040.50     899.84  -5.602 1.72e-06 ***
M             105.02      33.30   3.154  0.00305 ** 
Ed            196.47      44.75   4.390 8.07e-05 ***
Po1           115.02      13.75   8.363 2.56e-10 ***
U2             89.37      40.91   2.185  0.03483 *  
Ineq           67.65      13.94   4.855 1.88e-05 ***
Prob        -3801.84    1528.10  -2.488  0.01711 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 200.7 on 40 degrees of freedom
Multiple R-squared:  0.7659,    Adjusted R-squared:  0.7307 
F-statistic: 21.81 on 6 and 40 DF,  p-value: 3.418e-11

Interestingly, removing some predictors actually help improve the model’s adjusted R-squared.

# Use the first linear regression model to make the prediction
test<-data.frame(M = 14.0,So = 0,Ed = 10.0, Po1 = 12.0,Po2 = 15.5,
                 LF = 0.640, M.F = 94.0,Pop = 150,NW = 1.1,U1 = 0.120,
                 U2 = 3.6, Wealth = 3200,Ineq = 20.1,Prob = 0.04, Time = 39.0)
print(predict(model,test))

       1 
155.4349